We present an analytical study of the mode structure of a semiconductor laser subject to filtered optical feedback (FOF). First, we derive expressions for the solution sets that define the structure of the modes. We then employ continuation techniques to investigate how this structure changes as filter width, filter detuning and feedback phase are varied. This leads to a comprehensive picture of the FOF laser's mode structure. In particular, we find an equation for the boundary of the parameter region in which one finds two separate components of modes.