%I #20 Dec 27 2016 12:01:29

%S 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,

%T 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,

%U 212,14,98,22,32,2,80,2,34,36,2825,4,52,2,58,4,52,2,272

%N Maximal number of subgroups in a group with n elements.

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

%C 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

%H Eric M. Schmidt, <a href="/A018216/b018216.txt">Table of n, a(n) for n = 1..511</a>

%F a(n)=Maximum of {A061034(n), A083573(n)}. - _Lekraj Beedassy_, Oct 22 2004

%F (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

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

%o (GAP) a:=function(n)

%o local gr, mx, t, g;

%o mx := 0;

%o gr := AllSmallGroups(n);

%o for g in gr do

%o t := Sum(ConjugacyClassesSubgroups(g),Size);

%o mx := Maximum(mx, t);

%o od;

%o return mx;

%o end; # _Charles R Greathouse IV_, Dec 27 2016

%Y Cf. A061034.

%K nonn,nice

%O 1,2

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

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

%E More terms from _Eric M. Schmidt_, Sep 07 2012