The space of subgroups of a countable group G is a compact Polish space equipped with a natural G-action. It is the crucible where certain properties of the non-free actions of G boil down, whether they are of a topological or measured nature. We will discuss several approaches to determining the Cantor-Bendixson decomposition of this space. In particular, we find the perfect kernel and the Cantor Bendixson rank of the subgroup space of many new groups, including infinitely ended groups, limit groups and hyperbolic 3manifold groups. We also give conditions under which the action on the perfect kernel is topologically transitive. This is joint work with Damien Gaboriau.