Largest Normal Subgroup of a Group G

Question

Let $G$ be a group and let $Z(G)$ be the center of $G$.
We know that $Z(G) \unlhd G$, but does that mean that $Z(G)$ is the largest normal subgroup of $G$?

Answer 1

No. For $S_5$, centre is trivial. But is $S_5$ simple? NO. what non trivial proper normal subgroup is there? Think.

$\textbf{HINT-}$ Easiest ones are which are for every $S_n $, The younger brothers of symmetric groups!!

Answer 2

Hint. You can take D8 . here Z(G)={1,s} where s is a rotation by 180 but it has a normal subgroup of order 4 namely the set of aĺl rotation.