Lucifer is the king of Hell, he's been so for many, many years- no, lives. Almost all of his lives. The times where someone who hasn't been Lucifer has been on the throne, it's been in the few years between his reincarnations, so it's usually very brief.
In his first reincarnation, As Lucifer the 1st, he ended up falling from heaven and asked Satan for help, offering him his skills and smarts. Satan took him in, breathed unholy into his concept, and made him a devil. It wasn't like Satan had total trust of Lucifer, but Lucifer's ideas weren't bad. When presented with it, Satan picked out the things he saw fit and implimented them, leading to some small, but meaningful fixes in Hell's overall rule. Over time, Lucifer had proved himself useful enough, that he was allowed more range and with it, Lucifer presented his idea for his revised version of Heaven's Virtues.
Granted, Satan wasn't too excited to head about his ieda of making a bunch of representatives that symbolized sin and vices, and then connect it to Hell. He knew quite well the bad and evil image Unholy magic had. Lucifer was understanding and aware of his hesitation, but argued for it, for a couple of different reasons. First and foremost he had learnt that sins were inevitable, flexible and subjective. The words themselves, what they did, weren't so black and white, and showing that in a bigger scale, presenting shamelessly that sort of grey nuance, could be benefitial to the culture to come. On top of that, even if sins were inherently bad and gave Hell a bad imagine in turn, then it was a worthwhile sacrifice for the overall balance of the world. That sort of arguing: sacrificing a par tof one's image for the betterment of the world at large, really worked weel on Satan. With some hesitation, he let Lucifer set his project in motion, althought he had some specific rules he wanted kept.
Satan The Sins could be initially selected in the species that Lucifer created, much like Heaven, but he wanted the species to be quite small, forcing it to bleed out with other species. This would let it be tested out in a more controlled envirorment, before it was shared. Satan's own vision was that much like The Horsemen, these people would with their job earn specific powers, which were lend from him, that would aid them in keeping the overall balance of the world. As well as they could, of course. It wouldn't be a super active job, but more an assigment, to make sure it didn't get out of hand. After all, he couldn't really trust Obsolete God to consider that.
On top of that, he laid other ground rules, which Lucifer kept in mind and respected. Now, Lucifer and Satan did make a sub species of Devils, the King Devils, and it was in fact quite small. What Satan didn't really consider, is that he put a rather loose limit on how old they could get. Satan was quite used to a lot of beings often found their end, before their old age, so he hadn't quite predicted how The King Devils later would fight tooth and nail to stay alive for as long as possible, extending this little experiment quite a bit.
Lucifer's grip on Hell started to really show and Satan was very aware of the possibility of Lucifer attempting the same stunt he tried to pull in Heaven. It wasn't hard to realize, as criticism of Satan's decisions seemed to echo louder in Hell and it's leaders. Lucifer actually didn't have any active plan at the time to overthrow Satan, finding himself quite content in the way Satan ruled. He felt acknowledged and appreciated, even if he and Satan didn't always see eye to eye in te visions they had on how to rule. It was therefor a mixed reception that came, when it was announced that Lucifer now RULED Hell.