Let $G$ be a finite group, $N\mathrel{\lhd}G$ a normal subgroup of $G$, and $H\leq G$ a subgroup of $G$. Suppose that $|H|$ and $|N|$ are relatively prime (i.e., $\gcd(|H|,|N|)=1$). Show that the quotient group $G/N$ contains a subgroup isomorphic to $H$.
I am trying to show that there is an injective homomorphism $\varphi:H\rightarrow G/N$ , but I have no clue for the next step.
Hint: Show that $HN$ is a subgroup of $G$ containing $N$. Prove that, under the canonical projection $G\to (G/N)$, the subgroup $HN$ of $G$ is mapped onto a subgroup of $G/N$ isomorphic to $H$ (this subgroup is clearly, $HN/N$). More generally, we have $HN/N\cong H/(H\cap N)$.