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We investigate the properties of 1262 red giant stars with high photospheric abundances of lithium observed by the GALAH and \Ktwo-HERMES surveys, and discuss them in the context of proposed mechanisms for lithium enrichment and re-depletion in giant stars. We confirm that Li-rich giants are rare, making up only 1.2 per cent of our giant star sample. We use stellar parameters from the third public data release from the GALAH survey and a Bayesian isochrone analysis to divide the sample into first-ascent red giant branch and red clump stars, and confirm these classifications using asteroseismic data from \Ktwo. We find that red clump stars are 2.5 times as likely to be lithium-rich as red giant branch stars, in agreement with other recent work. The probability for a star to be lithium-rich is affected by a number of factors, though the causality in those correlations is not entirely clear. We show for the first time that primary and secondary red clump stars have distinctly different lithium enrichment patterns. The data set discussed here is large and heterogeneous in terms of evolutionary phase, metallicity, rotation rate and mass. We expect that if the various mechanisms that have been proposed for lithium enrichment in evolved stars are in fact active, they should all contribute to this sample of lithium-rich giants at some level.