Cannonical Homomorphisms

Cannonical Homomorphisms

Let $G$ be a group and let $N$ be a normal subgroup. Let $\pi\colon G \to
G/N$ denote the canonical homomorphism. Prove that if $H$ is a subgroup of
$G$ then $\pi(H) = \pi(HN)$. Then prove that if $H$ and $K$ are subgroups
of $G$, then $\pi(H) = \pi(K)$ if and only if $HN = KN$.
I have been able to show that if $H$ is any subgroup of $G$ then $HN$ is
also a subgroup. But I am not really sure where to go to from there.