The criterion of achieving efficient passive Q-switching is analyzed to design an off-axis pumped Nd:YVO4/Cr4+:YAG laser with a degenerate cavity. Experimental results reveal that pure high-order HG0,m or HGm,0 eigenmodes with the order m between 0 and 14 can be generated, depending on the off-axis displacement along the y axis or the x axis. On the other hand, lasing modes naturally turn into planar geometric modes when the off-axis displacement is larger than the value for exciting the HG0,m or HGm,0 eigenmodes with m>14. The overall peak powers for high-order eigenmodes or geometric modes can exceed 140 W. Furthermore, the high-order eigenmodes and geometric modes are employed to generate vortex beams with large orbital angular momentum by using an external cylindrical mode converter. Theoretical analyses are performed to confirm experimental results and to manifest the phase structures of the generated vortex beams.