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In this paper, we propose an alternative methodology to capture the impact of the inequality factors on poverty by decomposing the traditional Foster-Greer-Thorbecke index. We focus on the incidence and the intensity of poverty and on the inequality of the distribution of the poor. In particular, our proposal allows to evaluate the effect of each factor on the inequality part, which is further analyzed into the within-, between-, and the overlapping- components through Dagum's Gini index decomposition. We also introduce a subgroup decomposition able to detect the contribution of each subgroup to the poverty index. A case study on Italian income distribution highlights the usefulness of our proposal evaluating the effects of inequality factors as gender, education, and the area of residence on the Foster-Greer-Thorbecke index. [ABSTRACT FROM AUTHOR]