  We show all the possible structures of finite subgroups of the automorphism
groups of elliptic curves. Here the automorphism means the biholomorphic
transformation. Using the result, we determine every Galois group G at outer
Galois point for genus-one curve with singular points. In particular, if G is
abelian, then its order is at most nine. However if not so, the order is
unbounded. Furthermore, we give every defining equation of the curve with the
Galois point in the case where G is abelian.
