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Unique properties of graphene originate from its highly symmetric hexagonal structure. However, real-life graphene usually features various defects such as non-hexagonal rings. Curiously, those defects may even enhance the properties of graphene. For example, a stripe of 5 and 8- membered rings has been shown to improve transport characteristics of a graphene fragment [1].Graphene analogs with regularly distributed non-hexagonal cycles are known as haeckelites [2]. Previous studies have demonstrated that haeckelites can be stable enough compared to other carbon nanostructures. One can expect that inclusion of non-hexagonal cycles may offer more versatile possibilities of tuning the electronic properties, e.g. due to enhanced local reactivity and formation of electron- withdrawing substructures for charge-transfer binding. Unfortunately, the vast structural library of haeckelite networks has not yet been addressed systematically enough. In the present work, we provide a DFT survey of a series of haeckelite structures with hexagonal unit cells composed of 5, 6 and 7-membered rings. We report their geometric and electronic structure and assess their potential for noncovalent functionalization. In terms of per-atom energy, haeckelites without adjacency of pentagons fall into the gap between graphene and C60. The haeckelite family demonstrate lower Fermi energy than in graphene and a range of band structure types: metallic, semimetallic, and semiconducting. The present work was supported by the RFBR grants 16-33-00496 and 15-03-05083.