  Let $T$ be a finite simple group of Lie type in characteristic $p$, and let
$S$ be a Sylow subgroup of $T$ with maximal order. It is well known that $S$ is
a Sylow $p$-subgroup except in an explicit list of exceptions, and that $S$ is
always `large' in the sense that $|T|^{1/3} < |S| \leqslant |T|^{1/2}$. One
might anticipate that, moreover, the Sylow $r$-subgroups of $T$ with $r \neq p$
are usually significantly smaller than $S$. We verify this hypothesis by
proving that for every $T$ and every prime divisor $r$ of $|T|$ with $r \neq
p$, the order of the Sylow $r$-subgroup of $T$ at most
$|T|^{2\lfloor\log_r(4(\ell+1) r)\rfloor/\ell}=|T|^{{\rm
O}(\log_r(\ell)/\ell)}$, where $\ell$ is the Lie rank of $T$.
