A018216 -id:A018216 - cp's OEIS frontend

A368538 Integers k such that there exists a group of order k with exactly k subgroups.

1, 2, 6, 8, 28, 36, 40, 48, 54, 72, 96, 100, 104, 128, 132, 144, 160, 176, 180, 192, 216, 240, 252, 260, 288, 324, 336, 368, 384, 416, 456, 480, 496, 560, 576, 588, 624, 640, 672, 704, 720

Offset: 1

Robin Jones, Dec 29 2023

Powers of 4 cannot appear in this sequence. This is because for a group of order p^n, the number of subgroups of order p^k is congruent to 1 mod p, for 0 <= k <= n. It follows from p=2 and Lagrange's theorem that the number of subgroups of order 2^n for n even is congruent to 1 mod 2, i.e. not equal to 2^n. - Robin Jones, Feb 17 2024

a(34) >= 512. The smallest term strictly larger than 512 is 560. - Robin Jones, Feb 18 2024

			1 is a term since the trivial group (order 1) has exactly 1 subgroup.
2 is a term since the cyclic group C_2 has exactly 2 subgroups.
6 is a term since the symmetric group S_3 has exactly 6 subgroups.

Dave Benson, Congruence mod four of the number of subgroups of a finite 2-group, discussion in MathOverflow, Jun 11 2025.
Richard Stanley, What finite groups have as many elements as subgroups? Question in MathOverflow, answered by Dave Benson and others, Jun 07 2025.

Cf. A018216, A061034, A384727, A384800.

Missing term 36 added by Hugo Pfoertner, Jun 10 2025, following a suggestion by Dave Benson in the MathOverflow discussion.

a(34)-a(41) from Richard Stanley, Jun 11 2025, using results by Dave Benson in MathOverflow discussion of question 496010.

A061034 Maximal number of subgroups in an Abelian group with n elements.

Original entry on oeis.org

1, 2, 2, 5, 2, 4, 2, 16, 6, 4, 2, 10, 2, 4, 4, 67, 2, 12, 2, 10, 4, 4, 2, 32, 8, 4, 28, 10, 2, 8, 2, 374, 4, 4, 4, 30, 2, 4, 4, 32, 2, 8, 2, 10, 12, 4, 2, 134, 10, 16, 4, 10, 2, 56, 4, 32, 4, 4, 2, 20, 2, 4, 12, 2825, 4, 8, 2, 10, 4, 8, 2, 96, 2, 4, 16, 10, 4, 8, 2, 134, 212, 4, 2

Offset: 1

Ola Veshta (olaveshta(AT)my-deja.com), May 26 2001

a(n) is multiplicative: if m and n are relatively prime then a(m*n) = a(n) * a(m). For n >= 2, a(n)>=2 with equality iff n is prime.

			a(16) = 67: C16 has 5 subgroups, C2 X C8 has 11 subgroups, (C2)^2 X C4 has 27 subgroups, (C2)^4 has 67 subgroups, (C4)^2 has 15 subgroups.

Eric M. Schmidt, Table of n, a(n) for n = 1..10000
Max Alekseyev, PARI scripts for various problems
G. A. Miller, On the Subgroups of an Abelian Group, The Annals of Mathematics, 2nd Ser., Vol. 6, No. 1. (1904), pp. 1-6. doi:10.2307/2007151 [See paragraph 4 entitled "Total number of subgroups in a group of order p^m". - M. F. Hasler, Dec 03 2007]
G. A. Miller, Determination of the number of subgroups of an abelian group, Bull. Amer. Math. Soc. 33 (1927), 192-194.

Cf. A006116, A018216, A083573.

{ A061034(n) = my(f=factorint(n)); prod(i=1,#f~, vecmax( apply( x->numsubgrp(f[i,1],x), partitions(f[i,2]) ) ) ); } \\ See Alekseyev link for numsubgrp(), Max Alekseyev, 2008

(C_2)^m has A006116(m) subgroups, so this is a lower bound if n is a power of 2 (e.g. a(16) >= 67). - N. J. A. Sloane, Dec 01 2007

More terms from Victoria A Sapko (vsapko(AT)canes.gsw.edu), Jun 13 2003

A083573 Maximal number of subgroups in a non-Abelian group with n elements, or zero if there are no non-Abelian groups of order n.

Original entry on oeis.org

0, 0, 0, 0, 0, 6, 0, 10, 0, 8, 0, 16, 0, 10, 0, 35, 0, 28, 0, 22, 10, 14, 0, 54, 0, 16, 19, 28, 0, 28, 0, 158, 0, 20, 0, 78, 0, 22, 16, 76, 0, 36, 0, 40, 0, 26, 0, 236, 0, 64, 0, 46, 0, 212, 14, 98, 22, 32, 0, 80, 0, 34, 36, 937, 0, 52, 0, 58, 0, 52, 0, 272

Offset: 1

Victoria A. Sapko (vsapko(AT)canes.gsw.edu), Jun 13 2003

nonn

A group G is non-Abelian iff there are two elements x,y such that xy != yx. Then and are nontrivial subgroups whose order divides the order of G which therefore cannot be prime (neither the square of a prime: there are only two nonisomorphic groups of that order which are both abelian; see A051532 for more). This also implies that a(n) >= 2+2+2 = 6 for all nonzero elements of this sequence and for even n=2m>4 there is the non-Abelian dihedral group D_m with A007503(m)=sigma(m)+tau(m)=A000005(m)+A000203(m), providing a lower bound. - M. F. Hasler, Dec 03 2007

			a(6)=6 because the only non-Abelian group with 6 elements is S_3 with 6 subgroups.

Eric M. Schmidt, Table of n, a(n) for n = 1..511

Cf. A018216, A061034.

Cf. A051532, A060689, A007503.

A083573 := function(n) local max, grp, i; max := 0; for i in [1..NumberSmallGroups(n)] do grp := SmallGroup(n, i); if (not IsAbelian(grp)) then max := Maximum(max, Sum(ConjugacyClassesSubgroups(grp), Size)); fi; od; return max; end; # Eric M. Schmidt, Sep 07 2012

a(n) = 0 <=> A060689(n)=0 <=> n is in A051532 ; otherwise a(n) >= 6 and a(2n) >= A007503(n). - M. F. Hasler, Dec 03 2007

More terms from Eric M. Schmidt, Sep 07 2012

A274847 a(n) = number of similarity classes of groups with exactly n subgroups (see reference for precise definition of similarity classes).

Original entry on oeis.org

1, 1, 1, 2, 2, 5, 1, 7, 2, 12, 4, 11, 1, 17, 8, 22, 3, 22, 5

Offset: 1

Michael C Slattery, Jul 08 2016

See Slatterly references for precise definition of similarity classes and a proof of the first 12 terms.

See Betz and Nash for correction of a(10) and proof of terms 13-19.

			For n = 6 the a(6) = 5 similarity classes of groups with 6 subgroups are Z_{p^5}, Z_p X Z_{q^2}, Z_3 X Z_3, S_3, Q_8.

Alexander Betz and David A. Nash, Classifying groups with a small number of subgroups, arXiv:2006.11315 [math.GR], 2020.
Angsuman Das and Arnab Mandal, Solvability of a group based on its number of subgroups, arXiv:2403.01262 [math.GR], 2024.
George A. Miller, Groups having a small number of subgroups, Proc. Natl. Acad. Sci. U S A, vol. 25 (1939) 367-371.
David A. Nash and Alexander Betz, Classifying groups with a small number of subgroups, arXiv:2006.11315 [math.GR], 2020.
Michael C. Slattery, On a property motivated by groups with a specified number of subgroups, Amer. Math. Monthly, 123 (2016), 78-81.
Michael C. Slattery, Groups with at most twelve subgroups, arXiv:1607.01834 [math.GR], 2016.

Cf. A018216, A289445.

Correction of a(10) and extension to 19 terms by David A. Nash, Jun 29 2020

A335887 Maximal sum of subgroup orders for a finite group of order n.

Original entry on oeis.org

1, 3, 4, 11, 6, 16, 8, 51, 22, 26, 12, 60, 14, 36, 24, 307, 18, 130, 20, 98, 50, 56, 24, 284, 56, 66, 184, 136, 30, 144, 32, 2451, 48, 86, 48, 498, 38, 96, 92, 466, 42, 200, 44, 212, 132, 116, 48, 1740, 106, 456, 72, 250, 54, 1696, 122, 648, 134, 146, 60, 552, 62

Offset: 1

Sébastien Palcoux, Jun 28 2020

nonn

The GAP Group, GAP - Groups, Algorithms, and Programming, Version 4.9.3, 2018. gap-system.org.

Sébastien Palcoux, On the sum the subgroup orders of a finite group (version: 2020-06-29), MathOverflow.

Cf. A000005, A000203, A018216, A335888.

L:=[];;
for n in [1..100] do
  Mn:=0;
  r:=NrSmallGroups(n);
  for d in [1..r] do
    g:=SmallGroup(n,d);
    lat:=AllSubgroups(g);
    sg:=Sum(List(lat,Order));
    if sg>Mn then
      Mn:=sg;
    fi;
  od;
  Add(L,Mn);
od;
Print(L);

A335888 Maximal sum of subgroup indices for a finite group of order n.

Original entry on oeis.org

1, 3, 4, 11, 6, 18, 8, 51, 22, 38, 12, 74, 14, 66, 24, 307, 18, 162, 20, 166, 74, 146, 24, 378, 56, 198, 184, 298, 30, 308, 32, 2451, 48, 326, 48, 722, 38, 402, 212, 886, 42, 564, 44, 682, 132, 578, 48, 2458, 106, 888, 72, 934, 54, 2268, 182, 1626, 422, 902, 60, 1444

Offset: 1

Sébastien Palcoux, Jun 28 2020

nonn

The GAP Group, GAP - Groups, Algorithms, and Programming, Version 4.9.3, 2018. gap-system.org.

Cf. A000005, A000203, A018216, A335887.

L:=[];;
for n in [1..100] do
  Mn:=0;
  r:=NrSmallGroups(n);
  for d in [1..r] do
    g:=SmallGroup(n,d);
    lat:=AllSubgroups(g);
    sg:=Sum(List(lat, h->Order(g)/Order(h)));
    if sg>Mn then
      Mn:=sg;
    fi;
  od;
  Add(L,Mn);
od;
Print(L);

A335917 a(n) is the number of similarity classes of abelian groups with exactly n subgroups (see reference for precise definition of similarity classes).

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 1, 5, 2, 5, 2, 5, 1, 6, 4, 9, 2, 7, 1, 11, 2, 6, 3, 11, 3, 8, 4, 9, 3, 14, 1, 16, 3, 6, 4, 15, 2, 8, 2, 21, 2, 13, 2, 13, 8, 6, 2, 23, 4

Offset: 1

David A. Nash, Jun 29 2020

See Slattery references for a precise definition of similarity.
See Betz and Nash first reference for proof of the first 22 terms.
See Betz and Nash second reference for proof of terms 23--49.

			For n = 6, a(6) = 3 and the three similarity classes of abelian groups with exactly six subgroups are Z_{p^5}, Z_{p^2q}, and Z_3 X Z_3.

Alexander Betz and David A. Nash, Classifying groups with a small number of subgroups, arXiv:2006.11315 [math.GR], (2020).
Alexander Betz and David A. Nash, A note on abelian groups with fewer than 50 subgroups, preprint, (2020).
G. A. Miller, Groups having a small number of subgroups, Proc. Natl. Acad. Sci. U S A, vol. 25 (1939) 367-371.
M. C. Slattery, On a property motivated by groups with a specified number of subgroups, Amer. Math. Monthly, 123 (2016), 78-81.
M. C. Slattery, Groups with at most twelve subgroups, arXiv:1607.01834 [math.GR], 2016-2020.

Cf. A274847, A018216, A289445.

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A368538 Integers k such that there exists a group of order k with exactly k subgroups.

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A061034 Maximal number of subgroups in an Abelian group with n elements.

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A083573 Maximal number of subgroups in a non-Abelian group with n elements, or zero if there are no non-Abelian groups of order n.

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A274847 a(n) = number of similarity classes of groups with exactly n subgroups (see reference for precise definition of similarity classes).

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A335887 Maximal sum of subgroup orders for a finite group of order n.

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GAP

A335888 Maximal sum of subgroup indices for a finite group of order n.

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GAP

A335917 a(n) is the number of similarity classes of abelian groups with exactly n subgroups (see reference for precise definition of similarity classes).

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