Sporadic group

Remarks

By the classification of finite simple groups, there are exactly $26$ sporadic groups up to isomorphism. The Tits group ${}^2F_4(2)'$ is usually counted with the groups of Lie type in clause 3; some authors count it as a $27$th sporadic group. Example: the Mathieu group $M_{11}$ is sporadic. It is simple and not cyclic, its order $7920$ is not of the form $n!/2$, and no simple group of Lie type has order $7920$. Non-example: $A_5$ is simple but not sporadic, since it is an alternating group, so clause 2 fails. The sporadic groups include the five Mathieu groups, the four Janko groups, the three Conway groups and the three Fischer groups. The others are the Higman–Sims, McLaughlin, Held, Rudvalis, Suzuki, O'Nan, Harada–Norton, Lyons and Thompson groups, the Baby Monster and the Monster.