The paper provides conditions for a structured system to have a property that the supremal \(L_ p\)-almost invariant subspace is generically the entire state space and that the infimal \(L_ p\)-almost conditional invariant subspace is generally the zero subspace. The above mentioned conditions are used to assure the existence of decentralized stabilizing controllers for linear time-invariant systems. As a consequence of the independence on the system parameters, the presented conditions are only sufficient for stabilizability and normally they provide conservative results. Fortunately this undesirable feature (conservativeness) is compensated by its easy numerical testability involving only binary calculations.