Genome-wide association studies (GWAS) have revealed hundreds of genetic loci associated with the vulnerability to major psychiatric disorders, and post-GWAS analyses have shown substantial genetic correlations among these disorders. This evidence supports the existence of a higher-order structure of psychopathology at both the genetic and phenotypic levels. Despite recent efforts by collaborative consortia such as the Hierarchical Taxonomy of Psychopathology (HiTOP), this structure remains unclear. In this study, we tested multiple alternative structural models of psychopathology at the genomic level, using the genetic correlations among fourteen psychiatric disorders and related psychological traits estimated from GWAS summary statistics. The best-fitting model included four correlated higher-order factors - externalizing, internalizing, thought problems, and neurodevelopmental disorders - which showed distinct patterns of genetic correlations with external validity variables and accounted for substantial genetic variance in their constituent disorders. A bifactor model including a general factor of psychopathology as well as the four specific factors fit worse than the above model. Several model modifications were tested to explore the placement of some disorders - such as bipolar disorder, obsessive-compulsive disorder, and eating disorders - within the broader psychopathology structure. The best-fitting model indicated that eating disorders and obsessive-compulsive disorder, on the one hand, and bipolar disorder and schizophrenia, on the other, load together on the same thought problems factor. These findings provide support for several of the HiTOP higher-order dimensions and suggest a similar structure of psychopathology at the genomic and phenotypic levels.