## Order 1: 1 Group

Abelian | Cyclic | Simple | |

ℤ_{1} = S_{1} = A_{2} |
Y | Y | Y |

## Order 2: 1 Group

Abelian | Cyclic | Simple | |

ℤ_{2} = S_{2} |
Y | Y | Y |

## Order 3: 1 Group

Abelian | Cyclic | Simple | |

ℤ_{3} = A_{3} |
Y | Y | Y |

## Order 4: 2 Groups

Abelian | Cyclic | Simple | |

ℤ_{4} |
Y | Y | |

ℤ_{2}^{2} = K_{4} |
Y |

## Order 5: 1 Group

Abelian | Cyclic | Simple | |

ℤ_{5} |
Y | Y | Y |

## Order 6: 2 Groups

Abelian | Cyclic | Simple | |

ℤ_{6} = ℤ_{3} × ℤ_{2} |
Y | Y | |

Dih_{3} = S_{3} |

## Order 7: 1 Group

Abelian | Cyclic | Simple | |

ℤ_{7} |
Y | Y | Y |

## Order 8: 5 Groups

Abelian | Cyclic | Simple | |

ℤ_{8} |
Y | Y | |

ℤ_{4} × ℤ_{2} |
Y | ||

ℤ_{2}^{3} |
Y | ||

Dih_{4} |
|||

Dic_{2} = Q_{8} |

## Order 9: 2 Groups

Abelian | Cyclic | Simple | |

ℤ_{9} |
Y | Y | |

ℤ_{3} × ℤ_{3} |
Y |

## Order 10: 2 Groups

Abelian | Cyclic | Simple | |

ℤ_{10} = ℤ_{5} × ℤ_{2} |
Y | Y | |

Dih_{5} |

## Order 11: 1 Group

Abelian | Cyclic | Simple | |

ℤ_{11} |
Y | Y | Y |

## Order 12: 5 Groups

Abelian | Cyclic | Simple | |

ℤ_{12} = ℤ_{4} × ℤ_{3} |
Y | Y | |

ℤ_{6} × ℤ_{2} = ℤ_{3} × ℤ_{2}^{2} |
Y | ||

Dic_{3} |
|||

A_{4} |
|||

Dih_{6} = Dih_{3} × ℤ_{2} |

## Order 13: 1 Group

Abelian | Cyclic | Simple | |

ℤ_{13} |
Y | Y | Y |

## Order 14: 2 Groups

Abelian | Cyclic | Simple | |

ℤ_{14} = ℤ_{7} × ℤ_{2} |
Y | Y | |

Dih_{7} |

## Order 15: 1 Group

Abelian | Cyclic | Simple | |

ℤ_{15} = ℤ_{5} × ℤ_{3} |
Y | Y |

## Order 16: 14 Groups

Abelian | Cyclic | Simple | |

ℤ_{16} |
Y | Y | |

ℤ_{8} × ℤ_{2} |
Y | ||

ℤ_{4}^{2} |
Y | ||

ℤ_{4} × ℤ_{2}^{2} |
Y | ||

ℤ_{2}^{4} |
Y | ||

Dih_{8} |
|||

Dic_{4} |
|||

Dih_{4} × ℤ_{2} |
|||

Dic_{2} × ℤ_{2} |
|||

K_{4} ⋊ ℤ_{4} |
|||

ℤ_{4} ⋊ ℤ_{4} |
|||

ℤ_{8} ⋊ ℤ_{2} |
|||

(ℤ_{4} × ℤ_{2}) ⋊ ℤ_{2} |
|||

QD_{16} |

## Order 17: 1 Group

Abelian | Cyclic | Simple | |

ℤ_{17} |
Y | Y | Y |

## Order 18: 5 Groups

Abelian | Cyclic | Simple | |

ℤ_{18} = ℤ_{9} × ℤ_{2} |
Y | Y | |

ℤ_{6} × ℤ_{3} = ℤ_{3}^{2} × ℤ_{2} |
Y | ||

Dih_{9} |
|||

S_{3} × ℤ_{3} |
|||

(ℤ_{3} × ℤ_{3}) ⋊ ℤ_{2} |

## Order 19: 1 Group

Abelian | Cyclic | Simple | |

ℤ_{19} |
Y | Y | Y |

## Order 20: 5 Groups

Abelian | Cyclic | Simple | |

ℤ_{20} = ℤ_{5} × ℤ_{4} |
Y | Y | |

ℤ_{10} × ℤ_{2} = ℤ_{5} × ℤ_{3} × ℤ_{2} |
Y | ||

Dih_{10} = Dih_{5} × ℤ_{2} |
|||

Dic_{5} |
|||

ℤ_{5} ⋊ ℤ_{4} |

## Order 21: 2 Groups

Abelian | Cyclic | Simple | |

ℤ_{20} = ℤ_{7} × ℤ_{3} |
Y | Y | |

ℤ_{7} ⋊ ℤ_{3} |

## Order 22: 2 Groups

Abelian | Cyclic | Simple | |

ℤ_{22} = ℤ_{11} × ℤ_{2} |
Y | Y | |

Dih_{11} |

## Order 23: 1 Group

Abelian | Cyclic | Simple | |

ℤ_{23} |
Y | Y | Y |

## Order 24: 14 Groups

Abelian | Cyclic | Simple | |

ℤ_{24} = ℤ_{8} × ℤ_{3} |
Y | Y | |

ℤ_{12} × ℤ_{2} = ℤ_{6} × ℤ_{4} |
Y | ||

ℤ_{6} × ℤ_{2}^{2} = ℤ_{3} × ℤ_{2}^{3} |
Y | ||

Dih_{12} |
|||

Dih_{6} × ℤ_{2} |
|||

Dih_{4} × ℤ_{3} |
|||

S_{4} |
|||

S_{3} × ℤ_{4} |
|||

A_{4} × ℤ_{2} |
|||

Dic_{6} |
|||

Dic_{3} × ℤ_{2} |
|||

Dic_{2} × ℤ_{3} |
|||

ℤ_{3} ⋊ ℤ_{8} |
|||

ℤ_{3} ⋊ Dih_{4} |
|||

SL(2,3) = 2T |

## Order 25: 2 Groups

Abelian | Cyclic | Simple | |

ℤ_{25} |
Y | Y | |

ℤ_{5}^{2} |
Y |

## Order 26: 2 Groups

Abelian | Cyclic | Simple | |

ℤ_{26} = ℤ_{13} × ℤ_{2} |
Y | Y | |

Dih_{13} |

## Order 27: 5 Groups

Abelian | Cyclic | Simple | |

ℤ_{27} |
Y | Y | |

ℤ_{9} × ℤ_{3} |
Y | ||

ℤ_{3}^{3} |
Y | ||

ℤ_{3}^{2} ⋊ ℤ_{3} |
|||

ℤ_{9} ⋊ ℤ_{3} |

## Order 28: 3 Groups

Abelian | Cyclic | Simple | |

ℤ_{28} = ℤ_{7} × ℤ_{4} |
Y | Y | |

ℤ_{14} × ℤ_{2} = ℤ_{7} × ℤ_{2}^{2} |
Y | ||

Dih_{14} |
|||

Z_{7} ⋊ Z_{4} |

## Order 29: 1 Group

Abelian | Cyclic | Simple | |

ℤ_{29} |
Y | Y | Y |

## Order 30: 4 Groups

Abelian | Cyclic | Simple | |

ℤ_{30} = ℤ_{15} × ℤ_{2} = ℤ_{10} × ℤ_{3} = ℤ_{6} × ℤ_{5} |
Y | Y | |

Dih_{15} |
|||

Dih_{5} × ℤ_{3} |
|||

S_{3} × ℤ_{5} |

## Order 31: 1 Group

(Order p)

Abelian | Cyclic | Simple | |

ℤ_{31} |
Y | Y | Y |

## Order 32: 51 Groups

I’m not going to list these all here, but seven of them are abelian.

(All cycle graphs stolen from Wikipedia)

# Notable Groups

A_{5}: Smallest simple group that isn’t cyclic. Order 60.

**ℤ**_{2}^{2}: Smallest non-cyclic group.

Dih_{3}: Smallest non-abelian group.

A_{4}: First example where n divides |G| but there is no element of order n. (n = 6)

Z_{7} ⋊ Z_{3}: Smallest non-abelian group of odd size.

Dih_{3}: Smallest group with a normal subgroup that isn’t isomorphic to one of its subgroups.

# General Patterns

1 group of order p

2 groups of order p^{2}

5 groups of order p^{3}

15 groups of order p^{4}, for p > 2

2 groups of order pq for q-1 divisible by p

1 group of order pq for q-1 not divisible by p