$H$ must contain every Sylow $p$-subgroup of $G$

Question

Let $p$ be a prime factor of the order of a finite group $G$. If $H$ is a normal subgroup $G$ whose index is not a multiple of $p$, show that $H$ must contain every Sylow $p$-subgroup of $G$.

Answer 1

Hints:

1. Show that $H$ must contain a Sylow $p$-subgroup $P$ of $G$.
2. Show that $H$ contains all the subgroups of $G$ that are conjugates of $P$.

Answer 2

Hint: If $N$ is normal in $G$, then $N$ contains every element of $G$ that has order coprime to $[G:N]$.