  We have performed a systematic study of J/$\psi$ and $\psi$(2S) production in
p--p collisions at different Large Hadron Collider (LHC) energies and at
different rapidities using the leading order non-relativistic quantum
chromodynamics model of heavy quarkonium production. We have included the
contributions from hiher excited states decaying to J$\psi$ . The calculated
values have been compared with the available data from the four experiments at
LHC, namely ALICE, ATLAS, CMS and LHCb. In the case of ALICE, inclusive
J/$\psi$ and $\psi$(2S) cross-sections have been calculated by including the
feed-down from B mesons using a fixed-order next-to-leading logarithm
formalism. It is found that all the experimental cross-sections are well
described for $p_{\rm T} >$ 3 GeV within the theoretical uncertainties arising
due to the choices of the factorization and renormalization scales.
