Conservative chaos systems have been investigated with some special advantages. Taking symmetry as a starting point, this study proposes a class of fifive-dimensional(5D) conservative hyperchaotic systems by constructing a generalized Hamiltonian conservative system. The proposed systems can have diffffe ent types of coordinate-transformation and time-reversal symmetries. Also, the constructed systems are conservative in both volume and energy. The constructed systems are analyzed, and their conservative and chaotic properties are verifified by relevant analysis methods, including the equilibrium points, phase diagram, Lyapunov exponent diagram, bifurcation diagram, and two-parameter Lyapunov exponent diagram. An interesting phenomenon, namely, that the proposed systems have multistable features when the initial values are changed, is observed. Furthermore, a detailed multistable characteristic analysis of two systems is performed, and it is found that the two systems have difffferent numbers of coexisting orbits under the same energy. In addition, this type of system can also exhibit the coexistence of infifinite orbits of difffferent energies. Finally, the National Institute of Standards and Technology tests confifirmed that the proposed systems can produce sequences with strong pseudo-randomness.