We have applied ab initio electronic structure method to search minimum energetic structures of the supported and free bimetallic PdCo nanosized particles by using unbiased (surface modified) Birmingham Cluster Genetic Algorithm code, coupling with density functional theory including dispersion correction (Van der Waals). A detailed analysis of structural motifs and segregation effects of free and supported PdCo nanoparticles has been performed by investigating more than one thousand isomers. Alloying and oxide support effects on bimetallic PdCo in term of electronic, magnetic, structural, energetic and stability were also examined comprehensively by calculating pdos, lowden charge analyses, charge density (differences), second finite energies, mixing (formation) energies, dipole moments, ionization energies, electron affinities, chemical reactivity descriptors and HOMO-LUMO gaps. The optimized structures were characterized by vibrational, dipole moment, STM and XRD analyses for a comparison of further experimental studies. Furthermore, nitric oxide (NO) trapping capabilities of the free or supported nanoparticles and clean or strained (1%, 2%, 3%, 4% and 5%) MgO surfaces have been discussed.