Existence of Subgroup whose Index is Prime Power

Theorem
Let $G$ be a finite group.

Let $H$ be a normal subgroup of $G$ which has a finite index in $G$.

Let:
 * $p^k \divides \index G H$

where:
 * $p$ is a prime number
 * $k \in \Z_{>0}$ is a (strictly) positive integer
 * $\divides$ denotes divisibility.

Then $G$ contains a subgroup $K$ such that:
 * $\index K H = p^k$