Subgroups of some Fuchsian groups defined by two linear congruences

In this article we define a new family of subgroups of Fuchsian groups $\mathcal{H}(\sqrt{m})$, for a squarefree positive integer $m$, and calculate their index in $\mathcal{H}(\sqrt{m})$ and their parabolic class number. Moreover, we will show that the index of these subgroups is closely related to the solvability of a quadratic congruence $x^2\equiv m(\textrm{mod }n)$ and the number of inequivalent solutions of a quadratic congruence $x^2\equiv 1(\textrm{mod }n)$. Finally, we will show that the results obtained by Yilmaz and Keskin [Acta Math. Sin 25 (2005), 215-222] are immediate corollaries of one of the main theorems of this article.