# Groups of order 40

## Contents

See pages on algebraic structures of order 40| See pages on groups of a particular order

This article gives basic information comparing and contrasting groups of order 40. The prime factorization of 40 is $40 = 2^3 \cdot 5$.

## Statistics at a glance

Quantity Value
Total number of groups 14
Number of abelian groups 3
Number of nilpotent groups 5
Number of solvable groups 14
Number of simple groups 0

## The list

There are 14 groups of order 40:

Group Second part of GAP ID (GAP ID is (40,second part)) Abelian? Nilpotent?
semidirect product of Z5 and Z8 via inverse map 1 No No
cyclic group:Z40 2 Yes Yes
semidirect product of Z5 and Z8 via square map 3 No No
nontrivial semidirect product of Z5 and Q8 4 No No
direct product of D10 and Z4 5 No No
dihedral group:D40 6 No No
? 7 No No
? 8 No No
direct product of Z20 and Z2 (also direct product of Z10 and Z4) 9 Yes Yes
direct product of D8 and Z5 10 No Yes
direct product of Q8 and Z5 11 No Yes
direct product of GA(1,5) and Z2 12 No No
direct product of D10 and V4 13 No No
direct product of E8 and Z5 14 Yes Yes

## Sylow subgroups

### 5-Sylow subgroups

Combining the congruence condition on Sylow numbers and the divisibility condition on Sylow numbers, we see that the number of 5-Sylow subgroups must be congruent to 1 modulo 5 and also must be a divisor of 8. The only possibility for both of these to hold simultaneously is that there is exactly one 5-Sylow subgroup, and hence it is a normal Sylow subgroup and the 2-Sylow subgroups are its permutable complements. In particular, this means that the whole group is a semidirect product with normal subgroup equal to the 5-Sylow subgroup and quotient/complement of order 8.