To best address the "hierarchal, patriarchal" order in Athens, it is best to look at what these terms mean. "Hierarchal" means:
...a person having high position or considerable authority
"Patriarchal" is defined as:
This is seen in the beginning of Shakespeare's A Midsummer Night's Dream, when Egeus, Hermia's father, drags his daughter in front of Theseus, the Duke of Athens, invoking his right under the law to force his daughter Hermia to marry Egeus' choice, Demetrius, or be put to death.
Be it so she will not here before your Grace
Consent to marry with Demetrius,
I beg the ancient privilege of Athens:
As she is mine, I may dispose of her;
Which shall be either to this gentleman
Or to her death, according to our law (I.i.40-45)
"Hierarchal" is reflected in Theseus' position as the one who is not only familiar with Athenian law, but has the power to carry it out.
"Patriarchal" is reflected in society's expectation (as well as power granted by the law) that Hermia must follow the dictates of her father's will rather than her own—as Theseus explains.
Because the law is clear and Hermia has little choice but to comply with her father's demands if she is to live—or live in Athens—she and Lysander (the man she truly loves) plot to meet in the forest and flee Athens and the reach of not only the "patriarchal" influence of Egeus, but also the "hierarchal" laws of Athens, by living in another town where Athenian law cannot touch them.
In addressing the "hierarchal" and "patriarchal," Shakespeare introduces the initial conflict in his play. However, it is the lack of such control in the forest—a place of disorder—that creates the humor of the story. For in the forest, there is one important element that exists, which steps out of the realm of Athenian and parental control—that is the supernatural: the fairy realm.
The fairy realm is not without its own laws, which are hierarchal and patriarchal as well: Oberon is the king of the fairy world (hierarchal); the word of the male carries more weight (patriarchal), as it does in the human world. However, where magic and mischief are present— and on a plane of existence that has its own variety of rules—chaos quickly ensues.
Elizabethans believed that after dark, fairies ruled the forest. The pandemonium is a result of humans in the woods after the sun goes down: both with the lovers, and with Bottom and the players. The disorder is not the result of Athenian law, but is caused by the supernatural (things that take place beyond what is natural or normal). Disorder is brought about by the behavior of the members of the fairy world—by Oberon, his henchman Puck, and Titania, the queen of the fairies. The mayhem is the result first from the antics of Puck, having fun at the expense of humans—
Lord, what fools these mortals be! (III.ii.115)
Secondly, more trouble comes because of Oberon's desire to see Helena happily married to Demetrius. Puck changes Bottom's head into that of a donkey and puts "love juice" on Titania's eyes so she falls in love with Bottom. Oberon directs Puck to use the same potion on Demetrius so he falls in love with Helena, but Puck uses it on Lysander by mistake.
Finally, Oberon steps in. When the lovers "wake," all is as it should be, even to the satisfaction of Athenian law. Love triumphs, and order is restored.