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Lexicase selection has been proven highly successful for finding effective solutions to problems in genetic programming, especially for test-based problems where there are many distinct test cases that must all be passed. However, lexicase (as with most selection schemes) requires all prospective solutions to be evaluated against most test cases each generation, which can be computationally expensive. Here, we propose reducing the number of per-generation evaluations required by applying random subsampling: using a subset of test cases each generation (down-sampling) or by assigning test cases to subgroups of the population (cohort assignment). Tests are randomly reassigned each generation, and candidate solutions are only ever evaluated on test cases that they are assigned to, radically reducing the total number of evaluations needed while ensuring that each lineage eventually encounters all test cases. We tested these lexicase variants on five different program synthesis problems, across a range of down-sampling levels and cohort sizes. We demonstrate that these simple techniques to reduce the number of per-generation evaluations in lexicase can substantially improve overall performance for equivalent computational effort.