Algorithmic fairness and privacy are essential pillars of trustworthy machine learning. Fair machine learning aims at minimizing discrimination against protected groups by, for example, imposing a constraint on models to equalize their behavior across different groups. This can subsequently change the influence of training data points on the fair model, in a disproportionate way. We study how this can change the information leakage of the model about its training data. We analyze the privacy risks of group fairness (e.g., equalized odds) through the lens of membership inference attacks: inferring whether a data point is used for training a model. We show that fairness comes at the cost of privacy, and this cost is not distributed equally: the information leakage of fair models increases significantly on the unprivileged subgroups, which are the ones for whom we need fair learning. We show that the more biased the training data is, the higher the privacy cost of achieving fairness for the unprivileged subgroups will be. We provide comprehensive empirical analysis for general machine learning algorithms.