If $H \leq G$ and $H \subset Z(G)$, the center of $G$, is $H \trianglelefteq G$? This is probably a very dumb question. Is it true that, in general, if $H$ is a subgroup of a group $G$, and $H \subset Z(G)$, the center of $G$, does it follow that $H$ is normal in $G$?
What I know so far that could potentially be useful:


*

*$Z(G)$ is normal in $G$.

*$H$ is normal in $G \Leftrightarrow gHg^{-1} \subseteq H$ for all $g \in G$.


This is part of an intermediate step for a homework problem. 

Since any $h\in H\le Z(G)$ then $hx=xh$ for all $x\in G$, so $xhx^{-1}=h\in H$ i.e. 
$$xHx^{-1}\subseteq H,$$
which is enough to warranty that $H$ is normal in $G$.  
