Consider the triangle group $\Delta(a, b, c)$ given by three generators $x$, $y$, and $z$, such that $x^2 = y^2 = z^2 = (xy)^a = (yz)^b = (zx)^c = 1$. [Here, we may presume $a, b, c > 1$]

Can anyone provide an example of a non-normal subgroup of some $\Delta(a, b, c)$ of finite index, containing only products of even numbers of generators (i.e., being a subgroup of the corresponding von Dyck group), and containing no conjugate of any nontrivial power of $xy$, $yz$, or $zx$.

Furthermore, I would particularly be interested in examples where $a = 2$.