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:


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*$Z(G)$ 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. 

This can be understood intuitively with group actions. Say $G$ acts on a set $X$:


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*A subset $Y\subseteq X$ is pointwise fixed if $gy=y$ for all $g\in G$.

*A subset $Y\subseteq X$ is setwise fixed if $gY:=\{gy:y\in Y\}=Y$ for all $g\in G$.


The group $G$ acts on itself by conjugation. Then:


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*A subset $H\subseteq G$ is central if $~[G,H]=1 ~~\Leftrightarrow~H\subseteq Z(G)~ \Leftrightarrow~  H$ is pointwise fixed.

*A subset $H\subseteq G$ is normal if it is setwise fixed.
For general group actions, pointwise fixed is much stronger than ($\Rightarrow$) setwise fixed. Therefore we may directly conclude that central ($H\subseteq Z(G)$) implies normal ($H\trianglelefteq G$) in this case.
