To support a wide range of vertical services in the fifth-generation (5G), a concept of private 5G has been introduced and has gained a great attention. Even though several deployment models for private 5G have been reported in the literature, they have not fully utilized the benefit of modularized control plane (CP) network functions (NFs) for better performance and cost efficiency. However, to leverage this benefit, a sophisticated CP placement design is essential. In this paper, we propose a novel CP framework for private 5G dubbed divide and cache (D&C). In D&C, the dependency and frequency of CP NFs are first analyzed based on the 3rd generation partnership project (3GPP) specification. Based on the analysis results, NFs for private 5G are split into on-premise and edge/public clouds to balance the tradeoff between performance and deployment costs. Also, a new NF called NF profile cache function (NFPCF) is built in on-premise to mitigate the signaling overhead. We implemented D&C over a private 5G testbed using an open-source software (i.e., free5GC) running on multiple virtual machines (VMs). Evaluation results demonstrate that D&C can achieve 19% lower deployment cost at the expense of slightly increased latency compared to the existing model.