In this paper the effects of external links on the synchronization performance of community networks, especially on the competition between individual community and the whole network, are studied in detail. The study is organized from two aspects: the number or portion of external links and the connecting strategy of external links between different communities. It is found that increasing the number of external links will enhance the global synchronizability but degrade the ynchronization performance of individual community before some critical point. After that the individual community will synchronize better and better as part of the whole network because the community structure is not so prominent. Among various connection strategies, connecting nodes belonging to different communities randomly rather than connecting nodes with larger degrees is the most efficient way to enhance global synchronization of the network. However, a preferential connection scheme linking most of the hubs from the communities will allow rather efficient global synchronization while maintaining strong dynamical clustering of the communities. Interestingly, the observations are found to be relevant in a realistic network of cat cortex. The synchronization state is just at the critical point, which shows a reasonable combination of segregated function in individual communities and coordination among them. Our work sheds light on principles underlying the emergence of modular architectures in real network systems and provides guidance for the manipulation of synchronization in community networks.