We study shifts of the beam centroid of linearly polarized paraxial Laguerre-Gaussian beams reflected and refracted at an interface of isotropic dielectric in the cylindrical coordinate system. According to the Fresnel approximation and the Taylor series expansion, we derive analytical expressions for transverse and longitudinal shifts and their general relationship with topological charge in the cases of partial and total reflection, respectively. It is shown that in the partial reflection regime, transverse shifts of both reflected and refracted beams are in proportion to the topological charge of beams and their directions are determined by the sign of the topological charge, while only the value of longitudinal shift is related to the topological charge. In the regime of total reflection, the shift of reflected beam centroid is independent of topological charge. To verify our analytical results, we perform numerical simulations of the shifts of reflected and refracted beams directly and indicate the applicable condition of the analytical expressions. The shifts of the beam centroid can be applied to the measurement of the topological charge and radial node number of Laguerre-Gaussian beams, which is of significance for information encoding and communication.