The space of subgroups of a countable group is a compact topological space with respect to the pointwise topology. The group acts on it by conjugation. In joint work with Gaboriau, Le Maître, and Stalder, we conducted a study on the space of subgroups of Baumslag-Solitar groups. We determine the isolated points and the Cantor-Bendixson rank. We show that the action by conjugation is far from being topologically transitive: we find a countable invariant partition consisting of open and closed sets. Furthermore, we establish that the action by conjugation on each atom of the partition is topologically transitive.