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Design smell detection is one of the essential activities for improving software quality. Mainly, detecting the co-occurrence of design smells contributes to reducing future maintainability costs. In this study, we empirically investigate the co-occurrence between God Class and 13 types of design smells in a set of 24 software systems. A group of five design smell detection tools and 42 human experts have been used for evaluation. The results indicate a strong co-occurrence between the pairs of (God Class, Feature Envy) and (God Class, Long Method).