Let H be a subgroup of a group G of prime order |H|=p. Describe up to isomorphism all possible quotient groups F/N where N is a normal subgroup of G and N $\subset$ F $\subset$ $\langle H, N \rangle$.
I know that $H/(H \cap N) \cong HN/N$ and $H \cong$ $\mathbb{Z}_p$, but I don't know how to use this to get the desired result.
The subgroup $\langle H, N \rangle$ generated by $H$ and $N$ is equal to $HN$ since $N$ is normal. So by the second isomorphism theorem, we have $HN/N \cong H / (H \cap N)$. Now, the subgroups between $HN$ and $N$ is the same as the subgroups of $HN/N$, which by the isomorphism is the subgroups of $H / (H \cap N)$. $H$ is cyclic of order $p$, and so $H / (H \cap N)$ either has two subgroups if $H \cap N = \{e\}$ and one subgroup if $H \cap N = H$.