The so(2,1) analysis for the bound state sector of the hypergeometric Natanzon potentials (HNP) is extended to the scattering sector by considering the continuous series of the so(2, 1) algebra. As a result a complete algebraic treatment of the HNP by means of the so(2, 1) algebra is achieved. In the bound state sector we discuss a set of satellite potentials which arises from the action of the so(2, 1) generators. It is shown that the set of new potentials are not related to the one obtained by means of SUSYQM or of the potential algebra approach using the so(2, 2) algebra.