Propagation properties of Hermite-Gaussian beams through a misaligned first-order ABCD optical systems are studied using the generalized Huygens-Fresnel diffraction integral, augmented matrix and Wigner distribution function methods. It is shown that, as a Hermite-Gaussian beam passes through the misaligned first-order ABCD optical system, the closed property is not preserved. The action of the misaligned first-order ABCD optical system turns the output beam into a more general class of beam having the decentered property, which can be called the decentered Hermite-Gaussian beam. The usual Hermite-Gaussian beam and decentered Gaussian beam can be regarded as its special cases.