The high-order Laguerre-Gaussian (LG) mode output from an end-pumped Nd:YVO₄ laser cavity with strong spherical aberration (SA) induced by short-focal-length lens is studied in this work. A long-focal-length lens L1 is used in the cavity to expand and collimate the beam, so that the beam incident on another short-focal-length lens L2 in the cavity undergoes a strong SA. Since the ring-shaped LG modes with different values of angular index m have different beam radii, the actual focal points of each order of beam are then spatially displaced. A flat output coupler (OC) is located near the focal point of L2, which is composed of a cat-eye retroreflector together with the lens. Such a retroreflector can provide only ideal retroreflection to the incident beam with a focal point exactly on the OC. Given the focal point displacements of the LG beams with different orders, such a mechanism can be used for implementing the transverse mode selection. The mode which has an actual focal point on the OC has a smaller loss than the other defocused modes. With an a-cut Nd:YVO₄ as laser crystal, scalar (linear-polarized) single-mode LG output with radical index p = 0 and angular index m>0 is obtained. The laser mode-order is selectable from LG_{0, ±10} to LG_{0, ±33} under 878.6-nm incident diode pump power of 1.03 W, by simply adjusting the distance between the OC and L2 in a range of 0.5 mm, when using lens L1 with f = 150 mm and lens L2 of f = 33.9 mm. It is found that sufficient SA which makes the optical paths of the neighboring modes well distinguishable is essential for single-mode operation of a wanted order of LG mode. However, too strong an SA can stop the high-order mode beam from oscillating, since the width and radius of the ring-shaped LG mode are an increasing function of indices p and m, which bring a stronger loss to the corresponding mode. Based on this analysis, we turn to a focal-length combination of f₁ = 100 mm and f₂ = 51.8 mm, to reduce the SA to a level suitable for further higher mode operation. A highest-order LG_{0, ±75} is obtained by such an SA mode-selecting technique under fixed pump power of 1.03 W.