Cosets and Lagrange's Theorem · Medium problem

A group $G$ has order $36$. An element $g\in G$ satisfies $g^{20}=e$. By Lagrange, $\operatorname{ord}(g)$ divides $36$, and $g^{20}=e$ forces $\operatorname{ord}(g)$ to divide $20$ as well. What is the largest possible value of $\operatorname{ord}(g)$?