Product of Subgroups of Prime Power Order

Theorem
Let $p$ be a prime number.

Let $G$ be a group of order $p^a k$, where:
 * $a \in \Z_{>0}$ is a (strictly) positive integer
 * $p$ is not a divisor of $k$.

Let $P \le G$ be a subgroup of $G$ of order $p^a$.

Let $Q \le G$ be a subgroup of $G$ of order $p^b$, where $0 < b \le a$.

Let it be the case that $Q$ is not a subgroup of $P$.

Then $P Q$ is not a subgroup of $G$.