Subgroups of the simple groups

Question

True of false? Every subgroup of a simple group is itself simple. May i also get some examples on this as well to verify my answer.

Answer 1

NO. $A_5$ is simple but the subgroup $V$= {$e,(12)(34),(13),(24),(14)(23)$} is not simple as it klein-$4$ group and of order $4$, and all group of order $4$ are abelian, hence non simple, as every subgroup of an abelian subgroup is normal, and our $V$ here has a (normal) subgroup of order $2$, guess which one?