Maximal solvable subgroups of size 2 integer matrices

Studying the solvable subgroups of 2 Г— 2 matrix groups over Z, we find a maximal finite order primitive solvable subgroup of GL(2, Z) unique up to conjugacy in GL(2, Z). We describe the maximal primitive solvable subgroups whose maximal abelian normal divisor coincides with the group of units of a quadratic ring extension of Z. We prove that every real quadratic ring R determines h classes of conjugacy in GL(2, Z) of maximal primitive solvable subgroups of GL(2, Z), where h is the number of ideal classes in R.