This question is part of a larger question, but I don't understand the solution I have been given here.

Show that there is no element of order 171 in the symmetric group $\displaystyle S_{20}$.

Solution: The smallest n such that $\displaystyle S_n$ contains an element of order $\displaystyle 171 = 3^2 \cdot 19 $ is n = 9 + 19 = 28.

I don't understand at all why that statement is true. The question comes at the end of a section about the Sylow theorems. Any help with this would be appreciated