Perfect Subgroups of $GL(2,p)$, where $p$ is an odd prime.

Question

It is known that $SL(2,p)$ is a perfect subgroup of $GL(2,p)$ if $p>3$. My question is:

Are they the only perfect subgroups of $GL(2,p)$?

If not, can $GL(2,p)$ have perfect subgroups whose order is not divisible by $p$?

Answer 1

By the classification of subgroups of $PSL(2,p)$ by Dickson, see Reference for the subgroup structure of $\rm{PSL}_2(q)$, the only perfect subgroups of $SL(2,p)$ can be

• The trivial subgroup
• $SL(2,p)$ itself
• $SL(2,5)\cong 2.A_5$. These arise for $p\equiv \pm1\pmod{10}$, thus $p=11$ is a counterexample.