Nonabelian group of order pq

Let $p>q$ be prime numbers. A nonabelian group of order $pq$ has a unique Sylow $p$-subgroup and exactly $p$ Sylow $q$-subgroups. Hence the Sylow count satisfies $p\equiv 1(\mathrm{mod}q)$, so $q\mid p-1$.

Up to isomorphism, the unique nonabelian group of order $21$ isIts nonidentity elements of $C_7$ form two conjugacy classes of size three, and the other two $C_7$ cosets form conjugacy classes of size seven.