Several sets (e.g. lists of integers, polynomials, etc.) which remain invariant under a group of transformations including static output feedback control of a linear dynamic system are identified. A nested feedback look decomposition of any given system is given which not only identifies the new invariants but also gives an algorithmic procedure of computing them. As is known, structural invariants such as the ones introduced here, play important roles in many theoretical studies of control theory. To show one such case, here an upper bound to the minimum order of a dynamic output feedback compensator required to stabilize a given system is calculated in terms of the newly identified invariants. This paper can be considered as an extension of the work of Morse who identified structural invariants under static-state feedback, where, as this paper identifies, there are structural invariants under static output feedback.