Previous clustering analysis of low-power radio active galactic nuclei (AGNs) has indicated that they preferentially live in massive groups. The X-ray surveys of the Cosmic Evolution Survey (COSMOS) field have achieved a sensitivity at which these groups are directly detected out to z = 1.3. Making use of the surveys Chandra, XMM–Newton and VLA-COSMOS, we identify radio AGN members [1023.6≲ L1.4 GHz/(W Hz−1) ≲ 1025] of galaxy groups (1013.2≲ M200/M⊙≲ 1014.4; 0.1 < z < 1.3) and study (i) the radio-AGN–X-ray group occupation statistics as a function of group mass; and (ii) the distribution of radio AGNs within the groups. We find that radio AGNs are preferentially associated with galaxies close to the centre (<0.2r200). Compared to our control sample of group members matched in stellar mass and colour to the radio-AGN host galaxies, we find a significant enhancement of radio-AGN activity associated with 1013.6≲ M200/M⊙≲ 1014 haloes. We present the first direct measurement of the halo occupation distribution (HOD) for radio AGNs, based on the total mass function of galaxy groups hosting radio AGNs. Our results suggest a possible deviation from the usually assumed power-law HOD model. We also find an overall increase in the fraction of radio AGNs in galaxy groups (<1r200), relative to that in all environments.