In this paper, multistability control of a 5D autonomous hyperjerk oscillator through linear augmentation scheme is investigated. The space magnetization is characterized by the coexistence of five different stable states including an asymmetric pair of chaotic attractors, an asymmetric pair of period-3 cycle, and a symmetric chaotic attractor for a given/fixed set of parameters. The linear augmentation method is applied here to control, for the first time, five coexisting attractors. Standard Lyapunov exponents, bifurcation diagrams, basins of attraction, and 3D phase portraits are presented as methods to conduct the efficaciousness of the control scheme. The results of the applied methods reveal that the monostable chaotic attractor is obtained through three important crises when varying the coupling strength. In particular, below the first critical value of the coupling strength, five distinct attractors are coexisting. Above that critical value, three and then two chaotic attractors are now coexisting, respectively. While for higher values of the coupling strength, only the symmetric chaotic attractor is viewed in the controlled system. The process of annihilation of coexisting multiple attractors to monostable one is confirmed experimentally. The important results of the controlled hyperjerk system with its unique survived chaotic attractor are suited in applications like secure communications.