To design a large aperture rectangular mirror with high stiffness, high strength and good thermal dimensional stability precisely, an analytical mathematical model of the mirror supporting system was established and its modal characteristics were investigated. Firstly, the support location of the mirror was determined according to the conclusion obtained by modal analytical solution that the dynamic and static stiffness is the highest when the three supports keep the mass distribution relatively homogenous. Then, the support structure was analyzed in detail by Finite Element Method(FEM) and a new type of flexible support was presented. By adjusting two parameters, the thinnest thickness and the arc radius of a flexible hinge, the surface figure accuracy of the mirror was modified. Finally, the modal analysis of mirror component was performed. Analytical and experimental results indicate that the surface figure accuracy of the mirror can reach RMS 12.3 nm under the load case of gravity and 4℃ uniform temperature rise, respectively, when the thickness of the flexible hinge is 4 mm and its circular radius is 2 mm. Moreover, its first order natural frequency is 146 Hz, which shows an error under 5% compared to FEA results. Research demonstrates that the mirror structure satisfies all design indexes.