Abstract
Let lambda >= 2 and let H(lambda) be the Hecke group associated to lambda. Also let H(lambda)\U be the Riemann surface associated to the Hecke group H(lambda). In this article, we study the even subgroup H(e)(lambda) and the power subgroups H(m)(lambda) of the Hecke groups H(lambda). We also study some genus 0 normal subgroups of finite index of H(lambda). Finally, we discuss free normal subgroups of H(lambda).