[ABRIDGED] We recently presented evidence of a connection between the brightness profiles of nearby early-type galaxies and the properties of the AGN they host. The radio loudness of the AGN appears to be univocally related to the hosts brightness profile: radio-loud nuclei are only hosted by ``core galaxies while radio-quiet AGN are only found in ``power-law galaxies. We extend our analysis here to a sample of 42 nearby (V < 7000 km/s) Seyfert galaxies hosted by early-type galaxies. We used the available HST images to study their brightness profiles. Having excluded complex and highly nucleated galaxies, in the remaining 16 objects the brightness profiles can be successfully modeled with a Nuker law with a steep nuclear cusp characteristic of ``power-law galaxies (with logarithmic slope 0.51 - 1.07). This result is what is expected for these radio-quiet AGN based on our previous findings, thus extending the validity of the connection between brightness profile and radio loudness to AGN of a far higher luminosity. We explored the robustness of this result against a different choice of the analytic form for the brightness profiles, using a Sersic law. In no object could we find evidence of a central light deficit with respect to a pure Sersic model, the defining feature of ``core galaxies in this modeling framework. We conclude that, regardless of the modeling strategy, the dichotomy of AGN radio loudness can be univocally related to the hosts brightness profile. Our general results can be re-phrased as ``radio-loud nuclei are hosted by core galaxies, while radio-quiet AGN are found in non-core galaxies.
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