# Groups of order 5.2^n

From Groupprops

This article discusses the groups of order , where varies over nonnegative integers. Note that any such group has a 5-Sylow subgroup which is cyclic group:Z5, and a 2-Sylow subgroup, which is of order . Further, because order has only two prime factors implies solvable, any such group is a solvable group.

## Number of groups of small orders

Exponent | Value | Value | Number of groups of order | Reason/explanation/list |
---|---|---|---|---|

0 | 1 | 5 | 1 | only cyclic group:Z5, see equivalence of definitions of group of prime order |

1 | 2 | 10 | 2 | cyclic group:Z10 and dihedral group:D10; see classification of groups of order a product of two distinct primes |

2 | 4 | 20 | 5 | See groups of order 20 |

3 | 8 | 40 | 14 | See groups of order 40 |

4 | 16 | 80 | 52 | See groups of order 80 |

5 | 32 | 160 | 238 | See groups of order 160 |

6 | 64 | 320 | 1640 | See groups of order 320 |

7 | 128 | 640 | 21541 | See groups of order 640 |

8 | 256 | 1280 | 1116461 | See groups of order 1280 |