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SARS-CoV-2 is notable both for its rapid spread, and for the heterogeneity of its patterns of transmission, with multiple published incidences of superspreading behaviour. Here, we applied a novel network reconstruction algorithm to infer patterns of viral transmission occurring between patients and health care workers (HCWs) in the largest clusters of COVID-19 infection identified during the first wave of the epidemic at Cambridge University Hospitals NHS Foundation Trust, UK. Based upon dates of individuals reporting symptoms, recorded individual locations, and viral genome sequence data, we show an uneven pattern of transmission between individuals, with patients being much more likely to be infected by other patients than by HCWs. Further, the data were consistent with a pattern of superspreading, whereby 21% of individuals caused 80% of transmission events. Our study provides a detailed retrospective analysis of nosocomial SARS-CoV-2 transmission, and sheds light on the need for intensive and pervasive infection control procedures.