# Groups of order 2160

From Groupprops

## Contents |

This article gives information about, and links to more details on, groups of order 2160

See pages on algebraic structures of order 2160| See pages on groups of a particular order

## Statistics at a glance

The number 2160 has the prime factorization:

All groups of this order have not yet been classified. The information below is therefore partial.

Quantity | Value | Explanation |
---|---|---|

Total number of groups up to isomorphism | 3429 | |

Number of abelian groups (i.e., finite abelian groups) up to isomorphism | 15 | (Number of abelian groups of order ) times (Number of abelian groups of order ) times (Number of abelian groups of order ) = (number of unordered integer partitions of 4) times (number of unordered integer partitions of 3) times (number of unordered integer partitions of 1) = . See classification of finite abelian groups and structure theorem for finitely generated abelian groups. |

Number of nilpotent groups (i.e., finite nilpotent groups) up to isomorphism | 70 | (Number of groups of order 16) times (Number of groups of order 27) times (Number of groups of order 5) = . See number of nilpotent groups equals product of number of groups of order each maximal prime power divisor, which in turn follows from equivalence of definitions of finite nilpotent group. |

Number of solvable groups (i.e., finite solvable groups) up to isomorphism | unknown, but likely between 1000 and 10000 | |

Number of non-solvable groups up to isomorphism | PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE] |
There are two possibilities for the composition factors of non-solvable groups: First: alternating group:A5 (order 60) as the only simple non-abelian composition factor, cyclic group:Z2 (2 times), cyclic group:Z3 (2 times) Second: alternating group:A6 (order 360) as the only simple non-abelian composition factor, cyclic group:Z2 (1 time), cyclic group:Z3 (1 time) |

Number of simple groups up to isomorphism | 0 | |

Number of almost simple groups up to isomorphism | 0 | |

Number of quasisimple groups up to isomorphism | 1 | Schur cover of alternating group:A6 |