We study the inclusive production of a Higgs boson in association with a high-pT photon at the LHC, detailing the leading-order features of the main processes contributing to the Hγ final state. Requiring an extra hard photon in Higgs production upsets the cross-section hierarchy for the dominant channels. The Hγ inclusive production comes mainly from photons radiated in vector-boson fusion (VBF), which accounts for about 2/3 of the total rate, for pT^(γ,jT)>30 GeV, at leading order. On the other hand, radiating a high-pT photon in the main top-loop Higgs channel implies an extra parton in the final state, which suppresses the production rate by a further αS power. As a result, the Hγ production via top loops at the LHC has rates comparable with the ones arising from either the Htt ̄ production or the HW(Z)γ associated production. Then, in order of decreasing cross section, comes the single-top-plus-Higgs channel, followed in turn by the heavy-flavor fusion processes bb ̄→Hγ and c barc→Hγ. The Hγ production via electroweak loops has just a minor role. At larger c.m. energies, the Httbar-γ channel surpasses the total contribution of top-loop processes. In particular, requiring pT(γ,j)>30 GeV at Sqrt(S)≃100 TeV, Htt ̄γ accounts for about 1/4 of the inclusive Hγ production at leading order, about half of the total being due to VBF production.
