We present a complete evaluation for the prompt η_{c} production at the LHC at next-to-leading order in α_{s} in nonrelativistic QCD. By assuming heavy-quark spin symmetry, the recently observed η_{c} production data by LHCb results in a very strong constraint on the upper bound of the color-octet long-distance matrix element ⟨O^{J/ψ}(^{1}S_{0}^{[8]})⟩ of J/ψ. We find this upper bound is consistent with our previous study of the J/ψ yield and polarization and can give good descriptions for the measurements, but the upper bound is inconsistent with some other theoretical estimates. This may provide important information for understanding the nonrelativistic QCD factorization formalism.