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This paper reports analyses of subsequences of linear congruential pseudorandom numbers by means of the spectral test. Such subsequences occur in particular simulation setups or as methods to obtain parallel streams of pseudorandom numbers for parallel and distributed simulation. Especially in the latter case, two kinds of substreams are of special interest: lagged random numbers with step sizes k, and consecutive streams of random numbers of length l. We show how to analyze correlations within and between lagged subsequences with arbitrary step sizes k. Analyzing consecutive streams with the spectral test is related to the well-known long-range correlation analysis of linear congruential generators. Whereas the latter was carried out to show correlations between pairs of processors only, the spectral test provides a convenient method to study correlations between larger numbers of parallel streams as well.