We study the semiclassical and quantum transport properties of the two-dimensional electron system in GaAs/AlGaAs semiconductor heterostructures under a perpendicular magnetic field. We studied two different samples, A and B under different experimental frameworks. For sample A, we mainly probe the applicable range of the Shubnikov-de Haas (SdH) theory when the thermal damping is reduced with decreasing temperature. We found the applicable range of the SdH theory can be extended, and the positive magneto-resistance and the disorder-induced damping play important roles. In the experiment of sample B, we removed the influence of thermal resistance by measuring forward and backward currents. We found the semiclassical Lifshitz-Kosevich formula in the SdH theory is more robust in describing the magneto-oscillations. Surprisingly, we found the existence of the mobility gap, which indicates the quantum Hall effect. The semiclassical behavior is attributed to the electrons away from the tails of Landau bands. Our study hints that we shall consider the quantum diffusion model to modify the Dingle term.