We report a diversity of stable gap solitons in a spin-orbit coupled Bose-Einstein condensate subject to a spatially periodic Zeeman field. It is shown that the solitons, can be classified by the main physical symmetries they obey, i.e. symmetries with respect to parity (P), time (T), and internal degree of freedom, i.e. spin, (C) inversions. The conventional gap and gap-stripe solitons are obtained in lattices with different parameters. It is shown that solitons of the same type but obeying different symmetries can exist in the same lattice at different spatial locations. PT and CPT symmetric solitons have anti-ferromagnetic structure and are characterized respectively by nonzero and zero total magnetizations. The progressively growing interest in the physics of mixtures of spinor Bose-Einstein condensates (BECs) and in particular in spin-orbit (SO) coupled BECs (SO-BECs) [1, 2], is motivated on the one hand by their fundamental importance for the atomic physics, and on the other hand by the rich possibilities they offer for emulating synthetic electric and magnetic fields in solids [3]. The last property allows one using neutral atoms to simulate numerous condensed matter phenomena [4] in a tun-able way, what would be impossible in direct condensed matter experiments. In other words, a spinor BEC is a promising candidate for implementation of a quantum simulator, which due to the technologies available nowadays was already implemented using cold atoms [5], ions [6], or photonics [7]. In particular, we mention a recent experiment [8] with coupled (linear) arrays of optical waveguides simulating a discrete version of SO-coupling which in the present letter is considered for the matter waves. Meanwhile, the physics of BECs is characterized by two essential factors, which are not typical for the physics which is aimed to be simulated. First, a BEC is a nonlin-ear system, with the nonlinearity stemming from the two-body interactions [9]. Moreover, by changing the configuration of the system (i.e. of the laser beams) one can create effective nonlinear interactions of different types [10]. Respectively, such nonlinear objects as skyrmions [11], solitons [12, 13], anti-ferromagnetic structures and symmetry breaking [14], have been recently reported for SO-BECs. The second feature, is that a (quasi-) stationary state of the condensate requires the presence of the external potential. This, in particular, imposes constraints on the lower bound of the kinetic energy, what is particularly relevant for SO-BECs (see e.g. [14]). What concerns the external potentials, since the very first experiments with cold atoms loaded in optical lattices [15] it was widely recognized that lattices (i.e. periodic potentials) are particularly efficient in manipulating BECs [16]. Nowadays, the nonlinear properties of the atomic BECs held in optical lattices, described in the mean-field approximation are very well studied [17] (see also [18] for a brief review on BEC mixtures in optical lattices). It is therefore natural that studies of SO-BE...