There are better accounts that specifically say who was taken but here it is
One day when Pope Sixtus was giving a talk to the faithful, the police broke in, arrested Sixtus and his chief clerics, and carried them off to the prefect. On this occasion they do not seem to have bothered about the lay people. According to tradition, the touching scene between St. Sixtus and his chief deacon, St. Lawrence, occurred at this time. Lawrence was absent when the police made their swoop. On hearing the news, he hastened to meet the Pope and asked him, "Where are you going, father, without your son? Where are you going, O priest, without your deacon?" Pope Sixtus replied, "My son, you I am not abandoning. Greater strife awaits you. Stop weeping; you will follow me in three days" (Paul Allard, Les dernieres persecutions du troisieme siecle, p. 91). And so it happened. The police pounced on St. Lawrence and put pressure on him to deliver up the treasures of the Church. St. Lawrence agreed to lead the prefect to the treasures, and since the reserve money of the Church had been distributed to the poor, Lawrence, even as Cornelia pointed to her children as her jewels, pointed to the poor as the Church treasure. The prefect was disappointed. Lawrence met death like a hero.
How do you figure that this story of yours suggests that apostolic succession was broken? The Romans would have had to have killed every bishop in the world, before any of them could secretly consecrate a successor.