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We study the Cantor--Bendixson rank of the space of subgroups for members ofa general class of finitely generated self-replicating branch groups. Inparticular, we show for $G$ either the Grigorchuk group or the Gupta--Sidki $3$group, the Cantor--Bendixson rank of $\mathrm{Sub}(G)$ is $\omega$. For eachnatural number $n$, we additionally characterize the subgroups of rank $n$ andgive a description of subgroups in the perfect kernel.

On the Cantor-Bendixson rank of the Grigorchuk group and the Gupta-Sidki $3$ group