  We say that a Hopf algebra has the Chevalley property if the tensor product
of any two simple modules over this Hopf algebra is semisimple. In this paper
we classify finite dimensional triangular Hopf algebras with the Chevalley
property, over the field of complex numbers. Namely, we show that all of them
are twists of triangular Hopf algebras with R-matrix having rank <=2, and
explain that the latter ones are obtained from group algebras of finite
supergroups by a simple modification procedure. We note that all examples of
finite dimensional triangular Hopf algebras which are known to the authors, do
have the Chevalley property, so one might expect that our classification
potentially covers all finite dimensional triangular Hopf algebras.
