Mental illnesses extract a high personal and societal cost, and thus explorations of the links between mental illness and functional connectivity in the brain are critical. Investigating major mental illnesses, believed to arise from disruptions in sophisticated neural connections, allows us to comprehend how these neural network disruptions may be linked to altered cognition, emotional regulation, and social interactions. Although neuroimaging has opened new avenues to explore neural alterations linked to mental illnesses, the field still requires precise and sensitive methodologies to inspect these neural substrates of various psychological disorders. In this study, we employ a hierarchical methodology to derive double functionally independent primitives (dFIPs) from resting state functional magnetic resonance neuroimaging data (rs-fMRI). These dFIPs encapsulate canonical overlapping patterns of functional network connectivity (FNC) within the brain. Our investigation focuses on the examination of how combinations of these dFIPs relate to different mental disorder diagnoses. The central aim is to unravel the complex patterns of FNC that correspond to the diverse manifestations of mental illnesses. To achieve this objective, we used a large brain imaging dataset from multiple sites, comprising 5805 total individuals diagnosed with schizophrenia (SCZ), autism spectrum disorder (ASD), bipolar disorder (BPD), major depressive disorder (MDD), and controls. The key revelations of our study unveil distinct patterns associated with each mental disorder through the combination of dFIPs. Notably, certain individual dFIPs exhibit disorder-specific characteristics, while others demonstrate commonalities across disorders. This approach offers a novel, data-driven synthesis of intricate neuroimaging data, thereby illuminating the functional changes intertwined with various mental illnesses. Our results show distinct signatures associated with psychiatric disorders, revealing unique connectivity patterns such as heightened cerebellar connectivity in SCZ and sensory domain hyperconnectivity in ASD, both contrasted with reduced cerebellar-subcortical connectivity. Utilizing the dFIP concept, we pinpoint specific functional connections that differentiate healthy controls from individuals with mental illness, underscoring its utility in identifying neurobiological markers. In summary, our findings delineate how dFIPs serve as unique fingerprints for different mental disorders.