$G$ Group, $H< J$, and $\forall J < G \Rightarrow H \subset Z(G)$ Let $G$ be a group and $H$ a subgroup such that $H$ nontrivial is a subgroup of J, for all $J$  nontrivial subgroup of $G$. Show that $H  \subset Z (G)$.
Thanks

Hint: $\,\,H\rlap{\,/}{\subset} Z(G)\,\Longrightarrow \exists\,x\in G\,\,s.t.\,\,hx\neq xh$ , for some $\,h\in H\,$ . Now take a closer look at $\,\langle x\rangle\,$
