Inheritance and the titles one hold is one thing, actual power is another.

Anyhow, Augustus is the most senior male member of the Lieu clan who is a direct male descendant of Edward and Rose - his father, Heracles, is the eldest son of William, Edward and Rose's eldest son. Anna, meanwhile, is the eldest daughter of Altus, who is the eldest son of Alice, Edward and Rose's eldest daughter. Their marriage was in fact a way to reconcile Altus and Heracles, who essentially fought over who deserved to lead the family after William and Alice - who led Atlantis jointly - retired from politics. It helped that Anna and Augustus actually got along really well and worked great together.

Anyways, as Augustus and Anna are the monarchs, they are de jure leaders of Atlantis, so they are the most powerful; Aurora, however, is de facto matriarch because she is the eldest living member of the family, so she has to get accorded some respect; Mara wields great power due to her elected position, but also because of the personal influence she wields in the family. Like in a lot of cases in real life, regardless, it is not so much a person or two that dominate a family, or a state for that matter, but a network of people with shared goals and interests. It would not be uncommon historically, for instance, for a monarch whose most powerful advisors or ministers were relatives from junior branches of the family who may or may not have been older than him. Still, the Lieu clan has been extremely successful in not only spreading their members all over the government, military, and society, but also remaining relatively united in their outlook and goals, which is why they are so powerful and effective. As a side note, inheritence in real life was also not always first born child (depending on culture and/or polity, you could have wildly different laws of succession, ranging from youngest child inheriting, splitting things evenly, family members electing the next leader, etc.). Even in places which emphasizes senior branches of the family, or firstborn children, how much an emphasis that is can vary.