Let $p$ be a prime number, $S = C_p$ a cyclic group of order $p$, $G = \mathbb{Z}_p$ the profinite additive group of $p$-adic integers. It is well known that all the closed nontrivial subgroups of $G$ are open (equivalently, of finite index).
Motivated by this example, we call an infinite profinite group $H$ thin if all its closed nontrivial subgroups are open.
What is known about thin profinite groups? Is there a complete list?
Is there an analogue of $G$ for a nonabelian finite simple group $T$? That is:
Is there a thin profinite group $M$ such that the composition factors of every nontrivial continuous finite image of $M$ are all isomorphic to $T$?