  Wilhelm (2021) has recently defended a criterion for comparing structure of
mathematical objects, which he calls Subgroup. He argues that Subgroup is
better than SYM * , another widely adopted criterion. We argue that this is
mistaken; Subgroup is strictly worse than SYM *. We then formulate a new
criterion that improves on both SYM * and Subgroup, answering Wilhelm's
criticisms of SYM * along the way. We conclude by arguing that no criterion
that looks only to the automorphisms of mathematical objects to compare their
structure can be fully satisfactory.
