Dynamical systems are ubiquitous in science and engineering. Inferring the mathematical laws that govern dynamical systems typically requires a 'scientist-in-the-loop' to guide the discovery process, via their expert knowledge and intuition about the system. Getting computers to perform this task automatically, without the guidance of a domain expert in the loop, is a grand challenge in the field of data science. A number of algorithms have been developed to infer such laws. One leading algorithm is Sparse Identification of Nonlinear Dynamics (SINDy), which applies simple linear regression coupled with sparsification to optimize a model over a large library of candidate functions. This algorithm is purely data-driven and makes no use of information that may be known previously about the dynamical system, such as symmetries and conservation laws. In this work, we develop a framework for incorporating and enforcing symmetries and conservation laws in SINDy so that the inferred models are consistent with prior domain knowledge. We analytically show how to propagate symmetries and conservation laws through the SINDy function library, and from this analytically derive linear constraints on the resultant linear regression. These constraints can be incorporated into the regression problem using options available in standard quadratic optimization packages. We implement this method and show that it provides improved accuracy and robustness on the task of inferring several canonical dynamical systems.