• Elliptic curves
• endomorphism rings
• finite fields
• isogenies

In this thesis, we give a brief survey on elliptic curves over finite fields, complex multiplication and isogeny volcanos. We then present a series of results that describe the possible orders and structures of the groups G_k=E(F_{q^k}), as k varies, assuming the curve E is defined over F_q. Several of these do not seem to appear in standard literature.