Let $\mathbf{Grp}$ be the category of groups and $\mathbf{Top}$ be the category of topological spaces. To each group $(G, \circ_G)$, we can associate a topological space $(G,\tau_G)$ the basis for this topology being given by the set of all subgroups of $G$. Call this topology on $G$ to be its **Subgroup Topology** Thus we get a functor $\mathscr{F}:\mathbf{Grp}\to\mathbf{Top}$ which associates a given group to its. Also note that any homomorphism $f:(G,\circ_G)\to (H,\circ_H)$ induces a continuous function between the corresponding **Subgroup Topological Spaces**.

This process looks like a sort of "inverse process" to what we do in Algebraic Topology especially when we try to associate the Fundamental Group to a given topological space. In Algebraic Topology in general, we are trying to find algebraic invariants of a given topological space whereas here I am trying to find topological invariants for a group.

However, it is clear that the functor which I have defined above is just an example of a functor from $\mathbf{Grp}$ to $\mathbf{Top}$ and (I believe) is not going to be much useful.

So my question is,

Does there exist any useful topological invariant of a group? More specifically, given any group can we associate a topological space to it (in the same way we did for Fundamental Groups)? If so can some literature be mentioned?