e derive asymptotic expansions for the numbers $U(n)$ of isomorphism classes of sensed maps on orientable surfaces with given number of edges $n$, where we do not specify the genus and for the numbers $A(n)$ of reflexible maps with $n$ edges. As expected the ratio $A(n)/U(n)\to 0$ for $n\to \infty$. This shows that almost all maps are chiral. Moreover, we show $\log A(n)\sim\frac{1}{2}\log U(n)\sim (n/2)\log n$. Due to a correspondence between sensed maps with given number of edges and torsion-free subgroups of the group $\Gamma=\langle x,y|y^2=1\rangle$ of given index, the obtained results give an information on asymptotic expansions for the number of conjugacy classes of such subgroups of given index.