Recent efforts combining the optimization techniques of apodized pupil Lyot coronagraphs (APLC) and shaped pupils have demonstrated the viability of a binary-transmission mask architecture for extremely high contrast (10-10) exoplanet imaging. We are now building on those innovations to carry out a survey of Lyot coronagraph performance for large, segmented telescope apertures. These apertures are of the same kind under considera- tion for NASA's Large UV/Optical/IR (LUVOIR) observatory concept. To map the multi-dimensional design parameter space, we have developed a software toolkit to manage large sets of mask optimization programs and execute them on a computing cluster. Here we summarize a preliminary survey of 500 APLC solutions for 4 reference hexagonal telescope apertures. Several promising designs produce annular, 10-10 contrast dark zones down to inner working angle 4λ0=D over a 15% bandpass, while delivering a half-max PSF core throughput of 18%. We also report our progress on devising solutions to the challenges of Lyot stop alignment/fabrication tolerance that arise in this contrast regime.