Sporadic group

The mentioned classification theorem states that the list of finite simple groups consists of 18 countably infinite families[a] plus 26 exceptions that do not follow such a systematic pattern.

[2] Five of the sporadic groups were discovered by Émile Mathieu in the 1860s and the other twenty-one were found between 1965 and 1975.

The full list is:[1][3][4] Various constructions for these groups were first compiled in Conway et al. (1985), including character tables, individual conjugacy classes and lists of maximal subgroup, as well as Schur multipliers and orders of their outer automorphisms.

These are also listed online at Wilson et al. (1999), updated with their group presentations and semi-presentations.

It does not show the numerous non-sporadic simple subquotients of the sporadic groups.

These twenty have been called the happy family by Robert Griess, and can be organized into three generations.