I've been researching information about our predecessors from before the war and about the societies of the outside world. I think that the Brotherhood comes closest in organization to the Knights Templar of the so-called 'Dark Ages'.
I've come to the conclusion that the Brotherhood functions mostly like a pseudo-religious organization. We have the strong military hierarchy, combined with codes of honor passed down from our ancestors. Combine this with our technical skills and I think that there are two paths of development for the Brotherhood: a techno-religious dictatorship or a benevolent research society.
Ultimately we have a command hierarchy, with the Elders at the top. The most influential positions, then, are the Elders and the High Elder. A charismatic High Elder could direct the Brotherhood almost completely. We need to be careful to make sure that we have a leader who is both strong and visionary. Thankfully,
I think that General Maxson fits that description.
According to my research, the Knights Templar were a military brotherhood which focused upon the acquisition of wealth and knowledge. Although they had some religious overtones in their structure, their quest for knowledge ultimately made them a secular organization. I think that they would have been a more theocratic organization if they had a greater interest in wealth and power, since the church of that era was primarily involved in control of gold and land.