The Sylow Theorems · Superchallenge problem

In the non-abelian group of order $57=3\cdot19$ (which exists since $19-1=18$ is divisible by $3$), the Sylow $19$-subgroup is always normal, but the Sylow $3$-subgroup cannot also be normal – otherwise $G$ would be a direct product of the two and hence abelian. How many elements of order $3$ does this non-abelian group have?