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Clustering a group of vertices in networks facilitates applications across different domains, such as social computing and Internet of Things. However, challenges arises for clustering networks with increased scale. This paper proposes a solution which consists of two motif clustering techniques: standard acceleration CHIEF-ST and approximate acceleration CHIEF-AP. Both algorithms first find the maximal k-edge-connected subgraphs within the target networks to lower the network scale, then employ higher-order motifs in clustering. In the first procedure, we propose to lower the network scale by optimizing the network structure with maximal k-edge-connected subgraphs. For CHIEF-ST, we illustrate that all target motifs will be kept after this procedure when the minimum node degree of the target motif is equal or greater than k. For CHIEF-AP, we prove that the eigenvalues of the adjacency matrix and the Laplacian matrix are relatively stable after this step. That is, CHIEF-ST has no influence on motif clustering, whereas CHIEF-AP introduces limited yet acceptable impact. In the second procedure, we employ higher-order motifs, i.e., heterogeneous four-node motifs clustering in higher-order dense networks. The contributions of CHIEF are two-fold: (1) improved efficiency of motif clustering for big networks; (2) verification of higher-order motif significance. The proposed solutions are found to outperform baseline approaches according to experiments on real and synthetic networks, which demonstrates CHIEF's strength in large network analysis. Meanwhile, higher-order motifs are proved to perform better than traditional triangle motifs in clustering.