Let $G$ be a finitely generated group such that every non-trivial normal subgroup has finite index. Does it follow that every non-trivial subgroup of $G$ has finite index?
This question arose as a side novelty from another problem I was working on. I think it's interesting in its own right, but I cannot for the life of me prove it or find a counter-example. This would be cut-and-dry if I knew that every non-trivial subgroup contained a non-trivial normal subgroup, but I don't see how to get that. As far as I can tell, I don't have a guarantee the normal core isn't trivial.
Any help is appreciated.