Group Theory problem.

Let $G$ be a group such that every subgroup of $G$ is subnormal. Suppose that there exists $N \lhd G$ such that $Z(N)$ is nontrivial and $G/N$ is cyclic. Prove that $Z(G)$ is nontrivial.

$H$ is a subnormal group of $G$ if there exist subgroups $H_1,H_2,\dots,H_n=G$ such that
$H\lhd H_1\lhd \dots \lhd H_n=G$.