WINCHESTER Each hath his place and function to attend. I am left out; for me nothing remains. But long I will not be Jack-out-of-office. The King from Eltham I intend to steal, And sit at chiefest stern of public weal. (1.1.176-180)
Whew—not all is well with the in crowd. Winchester's pretty annoyed that he didn't get any jobs for taking care of the young king. He's plotting to get more power, though. Fortunately he doesn't succeed in kidnapping the king, but you can see how much he wants to be one of the cool kids.
PUCELLE Dauphin, I am by birth a shepherd's daughter, My wit untrained in any kind of art. Heaven and Our Lady gracious hath it pleased To shine on my contemptible estate. (1.2.73-76)
Joan's explaining here why the clique should take her seriously. Yes, she says, she's not part of the aristocracy. But she says that God and the Virgin Mary ("Our Lady") have favored her. She may not be part of the club yet, but she claims to have supernatural power—which the French could use, especially since they've just lost a battle to the English in this scene.
SOMERSET Was not thy father Richard, Earl of Cambridge, For treason executed in our late king's days? And, by his treason, stand'st not thou attainted, Corrupted, and exempt from ancient gentry? His trespass yet lives guilty in thy blood, And till thou be restored thou art a yeoman. (2.4.91-96)
Like any clique, the aristocracy sometimes kicks people out, and when it does, it isn't pretty. Richard's father was executed for treason, so his title and property can't pass to Richard. This is reversible, but it takes an act of Parliament a bit later in the play for Richard to get the family title back, and in the meantime, he's the target of jokes from Somerset, who's well established in the aristocracy.
KING HENRY Rise, Richard, like a true Plantagenet, And rise created princely Duke of York. (3.1.181-182)
The clique can be nice if they want to. The King is especially nice, actually—he seems genuinely happy to welcome Richard back to the title.
KING HENRY Therefore stand up; and for these good deserts We here create you Earl of Shrewsbury; And in our coronation take your place. (3.4.25-27)
Military power and loyalty sometimes translates to a higher standing in society. Lord Talbot was already a member of the aristocracy, but he gets another title for his faithful fighting for the crown. It's sort of like going from Jedi Knight to member of the Jedi Council.
TALBOT: I vowed, base knight, when I did meet thee next, To tear the Garter from thy craven's leg, (tearing it off) Which I have done, because unworthily Thou wast installed in that high degree.— (4.1.14-17)
The knights of the Garter are a special kind of knight, and Talbot says they originally had to be part of the aristocracy, as well as courageous and virtuous (4.1.33-35). Talbot says Fastolfe is a coward or "craven" and doesn't deserve to wear the garter, and King Henry VI, usually a gentle type, sends Fastolfe packing after what Talbot says (4.1.46). Titles don't count for everything, even to the aristocracy. You need to have courage to back it up, at least some of the time.
EXETER But howsoe'er, no simple man that sees This jarring discord of nobility, This shouldering of each other in the court, This factious bandying of their favorites, But that it doth presage some ill event. (4.1.188-192)
It's hard to get respect from other people if your pals are fighting in public. Exeter is afraid that if the nobility can't unite, they won't be able to impress others, not even a "simple man." That's a problem, because if the English people and their French enemies don't see a united front from the aristocracy, things aren't going to go well.
WINCHESTER Now Winchester will not submit, I trow, Or be inferior to the proudest peer. Humphrey of Gloucester, thou shalt well perceive That neither in birth or for authority The Bishop will be overborne by thee. (5.1.56-60)
In the time period, the Catholic Church was a major political force, often more powerful than whole countries. It was also seen as having the supernatural power to win God's favor, so the combination was a strong one. That's why Winchester says being a cardinal will give him more status in society.
PUCELLE Decrepit miser, base ignoble wretch! I am descended of a gentler blood. Thou art no father nor no friend of mine. (5.4.7-9)
Joan is now claiming that she's not a shepherd's daughter, but is instead from a higher place in society. Why do you think she wants to be identified with noble birth? Is it spending time with the nobility and seeing how easily nobles gain respect in the society around her? What's at stake for her?
GLOUCESTER Her [Margaret's] father is no better than an earl, Although in glorious titles he excel. (5.5.37-38)
Titles do count for a lot to the nobles, but they're pretty sharp when it comes to seeing whether a title is hollow. They know who actually has power and wealth, and most of them don't want Henry to marry someone who doesn't.