A finitely generated group can be thought of as a metric space when equipped with the word metric with respect to a finite generating set. This metric space is well-defined up to quasi-isometry. A major program in geometric group theory, initiated by Gromov, is determining to what extent the coarse geometry of a group determines its algebra. In this talk, we investigate when normal and commensurated subgroups, and their associated quotient groups and spaces, are preserved by quasi-isometries.