A018216 - OEIS

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

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OFFSET

1,2

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

LINKS

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

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

EXAMPLE

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

PROG

(GAP) a:=function(n)

local gr, mx, t, g;

mx := 0;

gr := AllSmallGroups(n);