Schizophrenia (SCZ) is highly heterogenous and no subtypes characterizing treatment response or longitudinal course well. Cognitive impairment is a core clinical feature of SCZ and a determinant of poorer outcome. Genetic overlap between SCZ and cognitive traits is complex, with limited studies of comprehensive epidemiological and genomic evidence. To examine the relation between SCZ and three cognitive traits, educational attainment (EDU), premorbid cognitive ability, and intellectual disability (ID), we used two Swedish samples: a national cohort (14,230 SCZ cases and 3,816,264 controls) and a subsample with comprehensive genetic data (4992 cases and 6009 controls). Population-based analyses confirmed worse cognition as a risk factor for SCZ, and the pedigree and SNP-based genetic correlations were comparable. In the genotyped cases, those with high EDU and premorbid cognitive ability tended to have higher polygenetic risk scores (PRS) of EDU and intelligence and fewer rare exonic variants. Finally, by applying an empirical clustering method, we dissected SCZ cases into four replicable subgroups characterized by EDU and ID. In particular, the subgroup with higher EDU in the national cohort had fewer adverse outcomes including long hospitalization and death. In the genotyped subsample, this subgroup had higher PRS of EDU and no excess of rare genetic burdens than controls. In conclusion, we found extensive evidence of a robust relation between cognitive traits and SCZ, underscoring the importance of cognition in dissecting the heterogeneity of SCZ.