Let $G$ be a finite group of Lie type. Let $H$ be a subgroup of $G$ which contains unipotent elements. I want to find a 'nice' subgroup of $G$ that contains $H$, for example a Levi subgroup of $G$ which is minimal with this property. Do you have an idea how we can do this?
My motivation comes from the following theorem:
Let $P$ be a $p$-subgroup of $G$. Then there exists a parabolic subgroup $Q$ of $G$ such that $P$ lies in the unipotent radical of $Q$.