Abstract Severe geomagnetic storms appear to be ordered by the solar cycle in a number of ways. They occur more frequently close to solar maximum and the declining phase, are more common in larger solar cycles, and show different patterns of occurrence in odd- and even-numbered solar cycles. Our knowledge of the most extreme space-weather events, however, comes from spikes in cosmogenic-isotope (14C, 10Be, and 36Cl) records that are attributed to significantly larger solar energetic-particle (SEP) events than have been observed during the space age. Despite both storms and SEPs being driven by solar-eruptive phenomena, the event-by-event correspondence between extreme storms and extreme SEPs is low. Thus, it should not be assumed a priori that the solar-cycle patterns found for storms also hold for SEPs and the cosmogenic-isotope events. In this study, we investigate the solar-cycle trends in the timing and magnitude of the 67 SEP ground-level enhancements (GLEs) recorded by neutron monitors since the mid-1950s. Using a number of models of GLE-occurrence probability, we show that GLEs are around a factor of four more likely around solar maximum than around solar minimum, and that they preferentially occur earlier in even-numbered solar cycles than in odd-numbered cycles. There are insufficient data to conclusively determine whether larger solar cycles produce more GLEs. Implications for putative space-weather events in the cosmogenic-isotope records are discussed. We find that GLEs tend to cluster within a few tens of days, likely due to particularly productive individual active regions, and with approximately 11-year separations, owing to the solar-cycle ordering. However, these timescales would not explain any cosmogenic-isotope spikes requiring multiple extreme SEP events over consecutive years.