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Intergenerational transmission of socio-economic standing, SES is widespread. This is often measured as education, occupation, and income. We analyze how these dimensions contribute to intergenerational mobility, both independently as well as in their overlap, and add wealth as a fourth dimension. We use Swedish administrative registers, which contain unusually rich measures of these SES dimensions over individual’s lives for both parents’ and children’s generations. We calculate sibling correlations to get the total effect of family background and then decompose by SES dimension and their overlaps. We find that: (1) the four dimensions of SES can explain up to 50 percent of the family background effect; (2) most of the SES effect for all children’s outcomes except wealth, belongs to the overlap of parent’s education, occupation and income; (3) over and above parent’s education, occupation and income, wealth adds another 5 percentage points of explanation for children’s education, occupation, and income, which is on a par with the largest of the net contributions of parent’s education, occupation, and income; (4) in a comparison of relative contributions, parents’ wealth underlies 15-20 percent of the total SES effect for children’s education, occupation and income, 35 percent for children’s income and 80 percent for children’s wealth; (5) wealth moderates intergenerational reproduction in other dimensions: The wealthiest have higher rates of reproduction in children’s occupation, income, and wealth. We conclude that wealth is an important but also a unique dimension of social stratification.