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Photonic crystals (PhCs) have now been firmly established as an efficient means for light extraction from light emitting diodes (LEDs). We analyze the diffraction properties from thin GaN micro-cavity LEDs with hexagonal lattices that feature three guided TE modes only. In contrast to common design rules, we find that high order diffraction contributes significantly to the light extraction and increases the directionality of the emitted light. The implementation of the PhC leads to an enhancement in light extraction by a factor of up to 1.8 and the directionality of the light is greatly improved with a radiant intensity enhancement factor of 4.3, which can only be explained by the higher order diffraction that has been hitherto neglected. Furthermore, we show that higher order diffraction contributes significantly to the high azimuthal extraction uniformity we observe, suggesting that the use of quasi-crystal lattices is not necessary. We use a model including mode absorption where each in-plane angle of the guided modes is treated separately in order to explain the experimental results.