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

Author

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

Keywords

• nonn
• nice

Comments

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

Examples

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

Links

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

Crossrefs

Cf. A061034.

Programs

GAP

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

Formula

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

Extensions

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

More terms from Eric M. Schmidt, Sep 07 2012