Sato-Tate subgroups and supergroups (reviewed)

For a fixed weight $w$ and degree $d$, the set of (conjugacy classes of) Sato-Tate groups $G$ with the same identity component $G^0$ form a partially ordered set under inclusion. This partially ordered set is finite if we restrict to rational Sato-Tate groups, but it need not contain a unique maximal element.

The set of maximal subgroups of a Sato-Tate group is always finite, since component groups are finite, but the list of minimal supergroups is not finite in general.

For rational Sato-Tate groups the set of minimal supergroups that are also rational is finite, and we restrict to these when listing minimal supergroups of rational Sato-Tate groups; for irrational Sato-Tate groups we list just the first few minimal supergroups.