Is there any element of order $420$ in the symmetric group $S_{19}$?

Question

Is there any element of order $420$ in the symmetric group $S_{19}$?

The first thing that I checked was Lagrange's theorem. But, $420$ indeed divides $19!$, so that's no good as we could only use it to eliminate the possibility. What other method can I use?

Answer 1

$420 = 4 \cdot 3 \cdot 5 \cdot 7$, and there's just enough room in $\{1,\ldots,19\}$ to fit in a $4$-cycle, a $3$-cycle, a $5$-cycle and a $7$-cycle and have them disjoint.