We search for vast planes of satellites (VPoS) in a high resolution simulation of the Local Group performed by the CLUES project, which improves significantly the resolution of former similar studies. We use a simple method for detecting planar configurations of satellites, and validate it on the known plane of M31. We implement a range of prescriptions for modelling the satellite populations, roughly reproducing the variety of recipes used in the literature, and investigate the occurence and properties of planar structures in these populations. The structure of the simulated satellite systems is strongly non-random and contains planes of satellites, predominantly co-rotating, with, in some cases, sizes comparable to the plane observed in M31 by Ibata et al.. However the latter is slightly richer in satellites, slightly thinner and has stronger co-rotation, which makes it stand out as overall more exceptional than the simulated planes, when compared to a random population. Although the simulated planes we find are generally dominated by one real structure, forming its backbone, they are also partly fortuitous and are thus not kinematically coherent structures as a whole. Provided that the simulated and observed planes of satellites are indeed of the same nature, our results suggest that the VPoS of M31 is not a coherent disc and that one third to one half of its satellites must have large proper motions perpendicular to the plane.
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