Quantum error correction protocols have been developed in an attempt to offset the high sensitivity to noise inherent in quantum systems. However, much is still unknown about the behaviour of noise in a quantum error correction code, making it difficult to correct errors effectively; this is largely due to the computational cost of simulating quantum systems large enough to perform nontrivial encodings. In this paper, we develop methods for reducing the computational complexity of calculating the set of effective logical noises conditioned on recovery operations by finding conditions under which symmetries in a code cause different recovery operations to produce equivalent logical noise. These symmetries can be used to reduce the size of a lookup table to speed up the error correction step in implementations of quantum error correcting codes. We give examples of such symmetries for the 3-qubit, 5-qubit, Steane, Shor, and toric codes.