I've been refreshing on some group theory and I came to realize, that while I understand the idea of using finite simple groups as building blocks for finite groups, I'm not entirely sure what are some neat consequences of this method.
If I have a group $G$, I can tell/classify all sorts of things like automorphisms, homomorphisms out of this group etc., from the fact that $G=A \oplus B$. But I don't really know many consequences of knowing that a certain group $G$ is the extension of a simple group $\frac{G}{H}$ by $H$ - I'll have to work with a quotient group, and that seems quite limiting.
What are some interesting consequences of using simple groups as building blocks?
There's cardinality of the group itself. Deciding if the group is artinian, noetherian should be possible.
Can I tell something about the automorphisms? Homomorphisms out of the group?
All sorts of examples will be appreciated, but given my knowledge, I'd especially appreciate examples that won't require knowledge of advanced concepts in group theory.