Subgroup structure of projective general linear group of degree two over a finite field

Sylow subgroups
We consider the group $$PGL(2,q)$$ over the field $$\mathbb{F}_q$$ of $$q$$ elements. $$q$$ is a prime power of the form $$p^r$$ where $$p$$ is a prime number and $$r$$ is a positive integer. $$p$$ is hence also the characteristic of $$\mathbb{F}_q$$. We call $$p$$ the characteristic prime.

Sylow subgroups for other primes: cases and summary
For any prime $$\ell$$, the $$\ell$$-Sylow subgroup is nontrivial iff $$\ell \mid q^3 - q$$. If $$\ell \ne p$$, then it does not divide $$q$$, so we get that $$\ell \mid q^2 - 1$$ which means that either $$\ell \mid q - 1$$ or $$\ell \mid q + 1$$. Further, if $$\ell \ne 2$$, exactly one of these cases can occur. For $$\ell = 2$$, we make cases based on the residue of $$q$$ mod 4: