In this article, we give a necessary and sufficient condition of Kalman type for the indirect controllability of systems of groups of linear operators, under some " regularity and locality " conditions on the control operator that will be made precise later and fits very well the case of distributed controls. Moreover, in the case of first order in time systems, when the Kalman rank condition is not satisfied, we characterize exactly the initial conditions that can be controlled. Some applications to the control of systems of Schrödinger or wave equations are provided. The main tool used here is the fictitious control method coupled with the proof of an algebraic solvability property for some related underdetermined system and some regularity results.