  Let $G$ be a reductive group over a field $k$ which is algebraically closed
of characteristic $p \neq 0$. We prove a structure theorem for a class of
subgroup schemes of $G$, for $p$ bounded below by the Coxeter number of $G$. As
applications, we derive semi-simplicity results, generalizing earlier results
of Serre proven in 1998, and also obtain an analogue of Luna's \'etale slice
theorem for suitable bounds on $p$.
