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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 for an explicit list of exceptions and that 
$S$
is always ‘large’ in the sense that 
$|T|^{1/3}<|S|\leq |T|^{1/2}$
. 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 
$T$
is at most 
$|T|^{2\lfloor \log _{r}(4(\ell +1)r)\rfloor /\ell }=|T|^{O(\log _{r}(\ell )/\ell )}$
, where 
$\ell$
is the Lie rank of 
$T$
.
