We study the analytical features of the output beam diffracted from a phase-hologram grating when the incident vortex beam is misaligned with respect to the grating. The analytical representation describing the diffracted beam of the 1st order is derived theoretically. Based on the representation, the central of gravity and the central intensity of the diffracted beam are investigated in the cases of the alignment, lateral displacement, angular tilt and simultaneous lateral misalignment and angular tilt, separately. It is shown that the diffracted beam is described through confluent hypergeometrical function. The misalignment of the incident vortex beam can give rise to the displacement of the beam center of gravity, which is independent of the misalignment direction and azimuth angle. The displacement is more obvious for the lager misalignment. In the case of angular tilt, the direction of beam center of gravity is nearly identical to the misalignment direction, whatever the topological charge of the incident beam and the fork number of the grating are. Besides, if the sum of the topological charge of the incident beam and the fork number of the grating is zero, with radial displacement and deflection angule increasing, the central intensity of the diffracted beam decreases gradually, otherwise, the central intensity is non-zero value any more. That means the misalignment between the phase-hologram grating and the incident vortex beam can influence the measurement of the topological charge of the vortex beam.