Conjugators of Fuchsian groups and quasiplatonic surfaces

Let G be a Fuchsian group containing two torsion free subgroups defining isomorphic Riemann surfaces. Then these surface subgroups K and αKα⁻¹ are conjugate in PSL(2,R), but in general the conjugating element α 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 α 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 of how many different regular dessins of the same type can exist on a given quasiplatonic Riemann surface.