Isomorphisms between lattices of hyperinvariant subspaces

Given two nilpotent endomorphisms, we determine when their lattices of hyperinvariant subspaces are isomorphic. The study of the lattice of hyperinvariant subspaces can be reduced to the nilpotent case when the endomorphism has a Jordan-Chevalley decomposition; for example, it occurs if the underlying field is the field of complex numbers.