Every finitely generated group G has an associated topological space, called a Morse boundary, that captures the hyperbolic-like behavior of G at infinity. It was introduced by a combination of Charney--Sultan and Cordes, and later generalized to the sublinear Morse boundary by Qing-Rafi-Tiozzo. In this talk, we study Charney--Sultan's and Corde's variant of the Morse boundary. We introduce the notion of point-convergence and show that if the set of non-singleton connected components of the Morse boundary of a finitely generated group G is point-convergent, then every non-singleton connected component is the (relative) Morse boundary of its stabiliser. The above property only depends on the topology of the Morse boundary and hence is invariant under quasi-isometry. This shows that the Morse boundary can be used to detect certain subgroups which in some sense are invariant under quasi-isometry. This is joint work with Bobby Miraftab and Stefanie Zbinden.