Let G be a Fuchsian group containing two torsion free subgroups defining isomorphic Riemann surfaces. Then these surface subgroups K and alpha-Kalpha exp(-1) are conjugate in PSl(2,R), but in general the conjugating element alpha cannot be taken in G or a finite index Fuchsian extension of G. We will show that in the case of a normal inclusion in a triangle group G these alpha can be chosen in some triangle group extending G. It turns out that the method leading to this result allows also to answer the question how many different regular dessins of the same type can exist on a given quasiplatonic Riemann surface
