This paper studies the Shannon capacity of lexicographic products of finite simple graphs, alongside the Lovász theta function and the fractional Haemers number. Shannon capacity is proved to be supermultiplicative under lexicographic products in both orders, and these products are compared with the strong product. We explicitly construct three countably infinite lexicographic-power families based on the Schläfli graph, the McLaughlin graph, and its second subconstituent, in which strict supermultiplicativity is obtained by pairing members with their complements, while arbitrarily large multiplicative gaps are achieved within each family. Bounds and exact-capacity criteria for lexicographic products are derived, and the resulting upper bounds are shown to be incomparable. The capacities of lexicographic products involving Kneser graphs, their complements, and $q$-analogues are determined, and a complete outer factor is shown to preserve the inner factor's capacity. Capacities of iterated lexicographic powers are determined, including powers of self-complementary graphs that are vertex-transitive or strongly regular. Elementary, self-contained proofs are also given for three known results: multiplicativity of the Lovász theta function and the fractional Haemers number under lexicographic products, and equality of the fractional and ordinary Lovász theta functions. Finally, an open problem is posed concerning the Shannon capacity of lexicographic and strong products.

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