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Let $q$ be a power of a prime $p$ and let $G$ be a completely reduciblesubgroup of $\mathrm{GL}(d,q)$. We prove that the number of composition factorsof $G$ 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 showingthis bound is sharp infinitely often.

The number of composition factors of order $p$ in completely reducible groups of characteristic $p$