Overgroups of regular unipotent elements in reductive groups

We study reductive subgroups \(\it H\) of a reductive linear algebraic group \(\it G\) – possibly nonconnected – such that \(\it H\) contains a regular unipotent element of \(\it G\). We show that under suitable hypotheses, such subgroups are \(\it G\)-irreducible in the sense of Serre. This generalises results of Malle, Testerman and Zalesski. We obtain analogous results for Lie algebras and for finite groups of Lie type. Our proofs are short, conceptual and uniform