Order 1: 1 Group
Abelian | Cyclic | Simple | |
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Y | Y | Y |
Order 2: 1 Group
Abelian | Cyclic | Simple | |
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Y | Y | Y |
Order 3: 1 Group
Abelian | Cyclic | Simple | |
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Y | Y | Y |
Order 4: 2 Groups
Abelian | Cyclic | Simple | |
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Y | Y | |
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Y |
Order 5: 1 Group
Abelian | Cyclic | Simple | |
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Y | Y | Y |
Order 6: 2 Groups
Abelian | Cyclic | Simple | |
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Y | Y | |
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Order 7: 1 Group
Abelian | Cyclic | Simple | |
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Y | Y | Y |
Order 8: 5 Groups
Abelian | Cyclic | Simple | |
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Y | Y | |
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Y | ||
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Y | ||
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Order 9: 2 Groups
Abelian | Cyclic | Simple | |
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Y | Y | |
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Y |
Order 10: 2 Groups
Abelian | Cyclic | Simple | |
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Y | Y | |
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Order 11: 1 Group
Abelian | Cyclic | Simple | |
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Y | Y | Y |
Order 12: 5 Groups
Abelian | Cyclic | Simple | |
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Y | Y | |
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Y | ||
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Order 13: 1 Group
Abelian | Cyclic | Simple | |
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Y | Y | Y |
Order 14: 2 Groups
Abelian | Cyclic | Simple | |
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Y | Y | |
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Order 15: 1 Group
Abelian | Cyclic | Simple | |
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Y | Y |
Order 16: 14 Groups
Abelian | Cyclic | Simple | |
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Y | Y | |
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Y | ||
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Y | ||
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Y | ||
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Y | ||
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Order 17: 1 Group
Abelian | Cyclic | Simple | |
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Y | Y | Y |
Order 18: 5 Groups
Abelian | Cyclic | Simple | |
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Y | Y | |
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Y | ||
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Order 19: 1 Group
Abelian | Cyclic | Simple | |
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Y | Y | Y |
Order 20: 5 Groups
Abelian | Cyclic | Simple | |
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Y | Y | |
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Y | ||
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Order 21: 2 Groups
Abelian | Cyclic | Simple | |
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Y | Y | |
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Order 22: 2 Groups
Abelian | Cyclic | Simple | |
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Y | Y | |
Dih11 |
Order 23: 1 Group
Abelian | Cyclic | Simple | |
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Y | Y | Y |
Order 24: 14 Groups
Abelian | Cyclic | Simple | |
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Y | Y | |
ℤ12 × ℤ2 = ℤ6 × ℤ4 | Y | ||
ℤ6 × ℤ22 = ℤ3 × ℤ23 | Y | ||
Dih12 | |||
Dih6 × ℤ2 | |||
Dih4 × ℤ3 | |||
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S3 × ℤ4 | |||
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Dic3 × ℤ2 | |||
Dic2 × ℤ3 | |||
ℤ3 ⋊ ℤ8 | |||
ℤ3 ⋊ Dih4 | |||
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Order 25: 2 Groups
Abelian | Cyclic | Simple | |
ℤ25 | Y | Y | |
ℤ52 | Y |
Order 26: 2 Groups
Abelian | Cyclic | Simple | |
ℤ26 = ℤ13 × ℤ2 | Y | Y | |
Dih13 |
Order 27: 5 Groups
Abelian | Cyclic | Simple | |
ℤ27 | Y | Y | |
ℤ9 × ℤ3 | Y | ||
ℤ33 | Y | ||
ℤ32 ⋊ ℤ3 | |||
ℤ9 ⋊ ℤ3 |
Order 28: 3 Groups
Abelian | Cyclic | Simple | |
ℤ28 = ℤ7 × ℤ4 | Y | Y | |
ℤ14 × ℤ2 = ℤ7 × ℤ22 | Y | ||
Dih14 | |||
Z7 ⋊ Z4 |
Order 29: 1 Group
Abelian | Cyclic | Simple | |
ℤ29 | Y | Y | Y |
Order 30: 4 Groups
Abelian | Cyclic | Simple | |
ℤ30 = ℤ15 × ℤ2 = ℤ10 × ℤ3 = ℤ6 × ℤ5 | Y | Y | |
Dih15 | |||
Dih5 × ℤ3 | |||
S3 × ℤ5 |
Order 31: 1 Group
(Order p)
Abelian | Cyclic | Simple | |
ℤ31 | 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
A5: Smallest simple group that isn’t cyclic. Order 60.
ℤ22: Smallest non-cyclic group.
Dih3: Smallest non-abelian group.
A4: First example where n divides |G| but there is no element of order n. (n = 6)
Z7 ⋊ Z3: Smallest non-abelian group of odd size.
Dih3: 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 p2
5 groups of order p3
15 groups of order p4, 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