Cosets and Lagrange's Theorem · Hard problem

$23$ is prime, so by Fermat $\operatorname{ord}(5)$ in $U(23)$ divides $22$. It is given that $5^{11}\equiv-1\pmod{23}$ (not $1$), and direct computation shows $5^2\equiv2\pmod{23}$ (not $1$). Using the divisors of $22$, find the exact order of $5$ mod $23$.