---
tag: PZ49Z4CBFZ
kind: concept
name: Sporadic group
msc: [20D08, 20D05]
specializes: [0YMYMNQ0QZ]
sources:
  - url: https://en.wikipedia.org/wiki/Sporadic_group
same_as:
  wikidata: Q2682286
  wikipedia: Sporadic group
---

## Summary

A sporadic group is a finite simple group that belongs to none of the infinite families of the classification.
These families are the cyclic groups of prime order, the alternating groups, and the groups of Lie type.
By the classification of finite simple groups there are exactly $26$ of them.
The smallest is the Mathieu group $M_{11}$, of order $7920$, and the largest is the Monster group.

## Explanations

### L1 human

Almost all finite simple groups come in infinite families with a regular pattern.
The sporadic groups are the $26$ exceptions that fit no pattern.
They include the five Mathieu groups, symmetry groups of special codes and designs, the Conway groups, symmetry groups of the Leech lattice, and the Monster.
Twenty of them live inside the Monster as quotients of its subgroups, and the other six are called the pariahs.