If $\displaystyle X$ is a cyclic normal subgroup of $\displaystyle Y$ I need to prove that any subgroup of $\displaystyle X$ is also a normal subgroup of $\displaystyle Y$.
This seems trivial but I'm not sure that it is.
If X has generator a, and Z is a subgroup of X, generated by $\displaystyle a^k$, then Z is the set of all elements of the form $\displaystyle a^{nk}$.
If $\displaystyle g\in G$, then $\displaystyle g^{-1}Zg$ is generated by $\displaystyle g^{-1}a^kg = (g^{-1}ag)^k$, from which you can see that $\displaystyle g^{-1}ZG\subseteq Z$.