  Let $q$ be a power of a prime $p$ and let $G$ be a completely reducible
subgroup of $\mathrm{GL}(d,q)$. We prove that the number of composition factors
of $G$ that have prime order $p$ is at most $(\varepsilon_q d-1)/(p-1)$, where
$\varepsilon_q$ is a function of $q$ satisfying
$1\leqslant\varepsilon_q\leqslant 3/2$. For every $q$, we give examples showing
this bound is sharp infinitely often.
