A061034 -id:A061034 - cp's OEIS frontend

A018216 Maximal number of subgroups in a group with n elements.

1, 2, 2, 5, 2, 6, 2, 16, 6, 8, 2, 16, 2, 10, 4, 67, 2, 28, 2, 22, 10, 14, 2, 54, 8, 16, 28, 28, 2, 28, 2, 374, 4, 20, 4, 78, 2, 22, 16, 76, 2, 36, 2, 40, 12, 26, 2, 236, 10, 64, 4, 46, 2, 212, 14, 98, 22, 32, 2, 80, 2, 34, 36, 2825, 4, 52, 2, 58, 4, 52, 2, 272

Offset: 1

Ola Veshta (olaveshta(AT)my-deja.com), May 23 2001

For n >= 2 a(n)>=2 with equality iff n is prime.

The minimal number of subgroups is A000005, the number of divisors of n, attained by the cyclic group of order n. - Charles R Greathouse IV, Dec 27 2016

			a(6) = 6 because there are two groups with 6 elements: C_6 with 4 subgroups and S_3 with 6 subgroups.

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

Cf. A061034.

a:=function(n)
  local gr, mx, t, g;
  mx := 0;
  gr := AllSmallGroups(n);
  for g in gr do
    t := Sum(ConjugacyClassesSubgroups(g),Size);
    mx := Maximum(mx, t);
  od;
  return mx;
end; # Charles R Greathouse IV, Dec 27 2016

a(n)=Maximum of {A061034(n), A083573(n)}. - Lekraj Beedassy, Oct 22 2004

(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

More terms from Eric M. Schmidt, Sep 07 2012

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

Original entry on oeis.org

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.

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

cp's OEIS Frontend

A018216 Maximal number of subgroups in a group with n elements.

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

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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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