Sylow Theorems are very powerful in finite group theory.
Two natural questions come to mind:
1) Given a finite group $G$ and a $p$-subgroup $H$ of $G$, how many Sylow $p$-subgroups of $G$ contain $H$?
2) Let $P$ be a Sylow $p$-subgroup of a finite group $G$ and $k$ be a fixed positive integer. How many subgroups of $G$ of order $p^k$ are contained in $P$?
Does anybody know of if there are some results which address the above two questions?