Epimorphic subgroups of simple algebraic groups

A morphism of linear algebraic groups $\phi :K\to G$ is called an epimorphism if it admits right cancellation. A subgroup $H\le G$ is epimorphic if the inclusion map is an epimorphism. For $G$ a simple algebraic group over an algebraically closed field of arbitrary characteristic we construct epimorphic subgroups of bounded dimension (at most five).