Coupled cell networks have been used extensively in modeling nonlinear dynamics that arise from complex networks. Among others, synchronization is one of the most important collective dynamics that arise from complex networks. To understand the emergence of synchronization in relation to the network topology, we study the synchrony-breaking bifurcations in coupled cell networks. Based on existing degree theories, with an incorporation of the network structure characterized by a lattice of invariant subspaces, we define a lattice degree, which can be used to describe topological structure of bifurcating solutions around a bifurcation point. As an example, we will analyze a synchrony-breaking Hopf-bifurcation around a fully synchronous equilibrium in a regular 10-dimensional 5-cell network and obtain a classification of the bifurcating branches of oscillating solutions according to their synchrony types.