Number of subgroups of a group G

dumbQuestion

I was wondering if there are any theorems that specify an exact number of subgroups that a group G has, maybe given certain conditions.


The closest thing I know is a theorem that says if G is finite and cyclic of order n it has exactly one subgroup of order d for each divisor d or n. I am not sure what the formal name of this theorem is.


I also know Lagrange's theorem (if H a subgroup of G, order of H divides order of G), Sylow's theorem (if G a finite group of order n, then if you take the prime factorization of n, n=p₁^kp₂^j...p_m^z then for each p_m^k in that factorization G has at least one subgroup of order p_mⁱ for 0<=i<=k) I also know another theorem which says if G is finite and Abelian, it has at least one subgroup of order d for every divisor d or n.


The thing that gets me is the "at least one subgroup" in these theorems. Are there theorems other than the first one I posted up there which specify exactly how many subgroups of a certain size there are? Like if I have a group of order 500 (or any finite number), say there's no knowledge if it's cyclic or not, is there a way to say exactly how many subgroups it has? What if it's gauranteed to be Abelian? I know if it's Abelian I can say it's isomorphic to direct sums Z_m + Z_n + ... + Z_z for the different combinations of its prime factorization (what I mean by that is say I have an Abelian group of order 24 so its prime factorization is 2*2*2*3, then its isomorphic to Z₂ + Z₂ + Z₂ + Z₃, to Z₄ + Z₂ + Z₃, to Z₈ + Z₃, and to Z₂₄) so do I just then look at the number of subgroups of say Z₂₄? Is there a theorem which would tell me exactly how many subgroups Z₂₄ has?

Vargo

A google search yielded this paper.

http://math.ubbcluj.ro/~calu/nrsubgroup.pdf

It doesn't look simple.