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A property of graphs is any isomorphism closed class of simple graphs. For given properties of graphs P 1 , P 2 ,. .. , P n a vertex (P 1 , P 2 ,. .. , P n)-partition of a graph G is a partition {V 1 , V 2 ,. .. , V n } of V (G) such that for each i = 1, 2,. .. , n the induced subgraph G[V i ] has property P i. The class of all graphs having a vertex (P 1 , P 2 ,. .. , P n)partition is denoted by P 1 •P 2 • • • • •P n. A property R is said to be reducible with respect to a lattice of properties of graphs L if there are n ≥ 2 properties P 1 , P 2 ,. .. , P n ∈ L such that R =P 1 •P 2 • • • • •P n ; otherwise R is irreducible in L. We study the structure of different lattices of properties of graphs and we prove that in these lattices every reducible property of graphs has a finite factorization into irreducible properties.