Is there a general method of finding the number of Sylow p-subgroups, given only the order of the group?

Question

Given that one is only told the order of the group, one will often reach conclusions as "its either $1$ or $x$ Sylow $p$-subgroups." Is there a general method to find out if one can rule one of the alternatives out, given only the order of the group? (That is, I am aware of the fact that for some orders we have to content with "$1$ or $x$, but I have also seen--seen, not understood--cases where one can rule one of the alternatives out.")

Answer 1

To answer you question quickly, yes. There are more parts to Sylow's Theorem than more people are usually aware. Two of them are

Thm: If $G$ is a finite group, $p$ a prime and $p^n \mid o(G)$ but $p^{n+1} \nmid o(G)$, then any two subgroups of $G$ of order $p^n$ are conjugate.

Corollary: The number of $p$-Sylow subgroups in $G$ equal $o(G) / o(N(P))$, where $P$ is any $p$-Sylow subgroup of $G$.

Does this help?