We study hedonic games with heterogeneous player types that reflect her nationality, ethnic background, or skill type. Agents' preferences are dictated by status-seeking where status can be either local or global. The two dimensions of status define the two components of a generalized constant elasticity of substitution utility function. In this setting, we characterize the core as a function of the utility's parameter values and show that in all cases the corresponding cores are non-empty. We further discuss the core stable outcomes in terms of their segregating versus integrating properties.