Classify the nonabelian groups of order $16p$, where $p$ is a prime number I need to classify the nonabelian groups of order $16p$, $p$ is a prime number. Is there any classification of groups of order $16p$?
 A: It's a (lengthy) exercise. Denote by $C_p$ the cyclic group of order $p$.  If $p>7$, then a simple verification (not based on the classification) shows that no 2-group of order $\le 16$ has an automorphism of order $p$. It follows that if $G$ has $C_p$ as a quotient, then $G$ is direct product of a group of order 16 and $C_p$. In particular, if $p>7$, then the intersection $G_2$ of subgroups of index 2 is a proper subgroup of $G$. This also applies to $G_2$ itself and it follows that the $p$-Sylow of $G$ is normal. Hence $G=C_p\rtimes D$ with $D$ a 2-Sylow. Since the case of direct products was already obtained, the remaining case is when $D$ comes with a homomorphism onto $C_p$, which provides the action on $C_p$. (Hence, for each $D$ we obtain as many groups as homomorphisms from $D$ to $C_2$ up to composition by an automorphism of $D$.)
For $p=3,5,7$, the above provides all examples in which the $p$-Sylow is normal, but there are also a few examples with a non-normal $p$-Sylow.
For $p=5$ the only 2-group of order $\le 16$ with an automorphism of order 5 is the elementary 2-abelian group $C_2^4$ of order 16, and this automorphism is unique up to conjugation. Hence for $p=5$ the only further example is the nontrivial semidirect product $C_2^4\rtimes C_5$.
For $p=2,3,7$ there are a few more examples; lists are available even if everything must be doable by hand with some patience.
