I have found some interesting results as follows:
- If $o(G)\not|\ i(H)!$ then $H$ contains a non-trivial normal subgroup of $G$, where $i(H)=[G:H]$.
2.If $o(G)=2m$, where m is an odd prime number then $G$ contains a non-trivial normal subgroup.
I am collecting this kind of property.Actually I want to know that if we can say a group is simple or not by observing its order only. So anyone who knows more generals result please share the result and give some hint to prove those results.