Cluster synchronization is a very common phenomenon occurring in single-layer complex networks, and it can also be observed in many multilayer networks in real life. In this paper, we study cluster synchronization of an isolated network and then focus on that of the network when it is influenced by an external network. We mainly explore how the influence layer impacts the cluster synchronization of the interest layer in a multilayer network. Considering that the clusters are changeable, we introduce a term called "cluster synchronizability" to measure the ability of a network to reach cluster synchronization. Since cluster synchronizability is intimately associated with the structure of the coupled external layer, we consider community networks and networks with different densities as the coupled layer. Besides the topology structure, the connection between two layers may also have an influence on the cluster synchronization of the interest layer. We study three different patterns of connection, including typical positive correlation, negative correlation, and random correlation and find that they all have a certain influence. However, the general theoretical analysis of cluster synchronization on multilayer networks is still a challenging topic. In this paper, we mainly use numerical simulations to discuss cluster synchronization.