It is well-known that all finitely-generated Fuchsian groups contain torsion-free normal subgroups of finite index and the particular case of the (2,3,7)-triangle group has been much studied as the corresponding quotient groups are maximal groups of automorphisms of compact Riemann surfaces. This work investigates the minimum index of such a subgroup in any (p,q,r)-triangle group where p, q, r are distinct primes so that the corresponding quotient groups are finite non-abelian simple groups. The author shows, in a well-defined number theoretic sense, that amongst all known finite simple groups, these quotients are almost always PSL\({}_ 2\)(\({\mathbb{F}}_ S)\) for some S. He also constructs exceptions from the families of alternating, sporadic simple and \(PSL_ 3\) groups. In addition, a constructive method based on branched coverings is given for obtaining nilpotent and solvable quotients of more general Fuchsian groups of signature \((0;n_ 1,n_ 2,...,n_ k;0)\) with torsion-free kernel.