Groups of order 40

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

The list
There are 14 groups of order 40:

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.