We introduce a functorial construction C which takes unitary magmas M as input and produces operads. The obtained operads involve configurations of chords labeled by elements of M, called M-decorated cliques and generalizing usual configurations of chords. By considering combinatorial subfamilies of M-decorated cliques defined, for instance, by limiting the maximal number of crossing diagonals or the maximal degree of the vertices, we obtain suboperads and quotients of CM. This leads to a new hierarchy of operads containing, among others, operads on noncrossing configurations, Motzkin configurations, forests, dissections of polygons, and involutions. Moreover, the construction C leads to alternative definitions of the operads of simple and double multi-tildes, and of the gravity operad.