Learned image compression (LIC) has become more and more important in recent years. The hyperprior-module-based LIC models, which use hyperprior module to predict the distribution of image features and improve entropy coder performance, have achieved remarkable rate-distortion (RD) performance. However, the storage and memory costs of these LIC models are too high, resulting in higher difficulty to be applied to various devices, especially portable or edge devices. The storage and memory cost are directly linked to the parameter number. As a preliminary experiment, we manually assigned half channels for the hyperprior module in LIC models, reducing about 30% parameters in the model. The pruned models still kept similar RD performance to the original ones. This reveals that the hyperprior module in LIC models is highly redundant. In the meanwhile, LIC models with different reconstruction qualities require different amounts of parameters for the hyperprior module. Based on these phenomena, we propose a quality-aware hyperprior pruning method that efficiently reduces the storage and memory cost of the hyperprior module and various context models. It consists of two parts. The first part is the pruning method itself, called enhanced ResRep on hyper path (ERHP). The second part is a quality-aware threshold searching method, called pruning threshold searching (PTS), which prunes the hyperprior module based on the reconstruction qualities of LIC models. The experiments on various LIC models show that our methods reduce large volumes of storage cost (up to 74.6%) and memory cost (up to 41.5%), while keeping the performance the same before pruning.