The magnon energy band in a four-layer ferromagnetic superlattice is studied by using the linear spin-wave approach and Green’s function technique.It is found that three modulated energy gaps exist in the magnon energy band along Kx direction perpendicular to the superlattice plane.The spin quantum numbers and the interlayer exchange couplings all affect the three energy gaps.The magnon energy gaps of the four-layer ferromagnetic superlattice are different from those of the three-layer one.For the four-layer ferromagnetic superlattice,the disappearance of the magnon energy gaps △ω12,△ω23 and △ω34 all correlates with the symmetry of this system.The zero energy gap △ω23 correlates with the symmetry of interlayer exchange couplings,while the vanishing of the magnon energy gaps △ω12 and △ω34 corresponds to a translational symmetry of x-direction in the lattice.When the parameters of the system deviate from these symmetries,the three energy gaps will increase.