[Math] Example of not so simple group ?

Question

Help me with an example of a group having subgroups but it doesn't admit a normal subgroup.. ??

The alternate definition of simple groups using non trivial homomorphic image. Searching for a map that doesn't have a non trivial kernal ??

Answer 1

One classic example is $A_5$, which has plenty of non-trivial subgroups, none of which are normal. In fact, this is the smallest suitable example.