Venn diagram of the 181 unique galaxies significantly detected in SPIRE which are host to a powerful AGN selected by the various methodologies presented in Sect. 3.5. While some galaxies are selected as hosting a powerful AGN in with multiple methods, only one galaxy is selected by all four methods, and a large fraction of the AGN selected are selected in only one of the methodologies, underscoring the importance of selecting AGN at multiple wavelengths using multiple selection techniques. Less powerful AGN are (generally) not separable or, in some cases, detectable in our dataset and are thus not represented in this diagram.