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Full sets of inequivalent elementary band co-representations (coEBRs) for spinless and spinful systems with grey line group (LG) symmetry are calculated and, together with recently reported elementary band representation (EBRs) for colorless single and double LGs by Milošević et al (2020 J. Phys. A: Math. Theor. 53 455204), the task of characterization of nonmagnetic quasi-one-dimensional (quasi-1D) topological crystals by means of the theory of topological quantum chemistry can be completed. Effects of additional time reversal symmetry on band topology and applicability of topological quantum chemistry is thoroughly analyzed. The main conclusions of the latter are illustrated on spinful Su-Schrieffer-Heeger model (topological mirror chain). Technique of induction and decomposition of coEBRs is fully developed and presented in detail. Complete sets of coEBRs for all thirteen families of single and double grey LGs are tabulated. Quasi-1D structures with symmetry enforced hourglass fermions are singled out. © 2022 IOP Publishing Ltd.