We give a complete description of conjugacy classes of finite subgroups of the mapping class group of the sphere with r marked points. As a corollary we obtain a description of conjugacy classes of maximal finite subgroups of the hyperelliptic mapping class group. In particular, we prove that, for a fixed genus g, there are at most five such classes.
STUKOW, MICHAL (2004) "Conjugacy Classes of Finite Subgroups of Certain Mapping Class Groups," Turkish Journal of Mathematics: Vol. 28: No. 2, Article 1. Available at: https://journals.tubitak.gov.tr/math/vol28/iss2/1