New lineages of SARS-CoV-2 are of potential concern due to higher transmissibility, risk of severe outcomes, and/or escape from neutralizing antibodies. Lineage B.1.1.7 (the Alpha variant) became dominant in early 2021, but the association between transmissibility and risk factors, such as age of primary case and viral load remains poorly understood. Here, we used comprehensive administrative data from Denmark, comprising the full population (January 11 to February 7, 2021), to estimate household transmissibility. This study included 5,241 households with primary cases; 808 were infected with lineage B.1.1.7 and 4,433 with other lineages. Here, we report an attack rate of 38% in households with a primary case infected with B.1.1.7 and 27% in households with other lineages. Primary cases infected with B.1.1.7 had an increased transmissibility of 1.5-1.7 times that of primary cases infected with other lineages. The increased transmissibility of B.1.1.7 was multiplicative across age and viral load.