Lady Bracknell: Pardon me, you are not engaged to any one. When you do become engaged to some one, I, or your father, should his health permit him, will inform you of the fact. An engagement should come on a young girl as a surprise, pleasant or unpleasant, as the case may be. It is hardly a matter that she could be allowed to arrange for herself... And now I have a few questions to put to you, Mr. Worthing. (I.172)
Lady Bracknell’s comments suggest that girls are not capable or experienced enough to prudently choose husbands.
Gwendolen: What wonderfully blue eyes you have, Ernest! They are quite, quite, blue. I hope you will always look at me just like that, especially when there are other people present. (I.167)
That Gwendolen desires Ernest to "look at [her] just like that, especially when there are other people present" reveals her as a vain woman concerned about her appearance in the eyes of others. It is also telling that Gwendolen wants men to look at her in a desirous way, as if she specifically needs the male sex to validate her.
Jack: [In a very patronising manner] My dear fellow, the truth isn't quite the sort of thing one tells to a nice, sweet, refined girl. What extraordinary ideas you have about the way to behave to a woman! (I.236)
The implication here is that women are too pampered, idealistic, and fragile to have "the truth." This explains why Jack and Algernon don’t lose sleep over their lies to their beloveds. They truly believe they are protecting their women from a harsh society.
Algernon: Might I have a buttonhole first? I never have any appetite unless I have a buttonhole first.
Cecily: A Maréchal Niel? [Picks up scissors.]
Algernon: No, I'd sooner have a pink rose.
Cecily: Why? [Cuts a flower.]
Algernon: Because you are like a pink rose, Cousin Cecily.
Cecily: I don't think it can be right for you to talk to me like that. Miss Prism never says such things to me. (II.71-76)
Algernon’s line about Cecily’s being "like a pink rose" reveals that men flirt with women by praising their beauty. Although Cecily protests the propriety of Algernon’s comment, she secretly revels in it, as can be seen in a later scene, where she copies down all of Algernon’s compliments in her diary.
Gwendolen: Outside the family circle, papa, I am glad to say, is entirely unknown. I think that is quite as it should be. The home seems to me to be the proper sphere for the man. And certainly once a man begins to neglect his domestic duties he becomes painfully effeminate, does he not? And I don't like that. It makes men so very attractive. (II.266)
Here, Gwendolen is shown reversing the traditional roles of men and women. Gwendolen challenges the conventional idea that women should be the ones at home cooking, cleaning, and raising children. This is one of the few places where Wilde overtly shows that woman can occupy positions of power and usurp the traditional gender roles.
Gwendolen: [After a pause] They don't seem to notice us at all. Couldn't you cough?
Cecily: But I haven't got a cough.
Gwendolen: They're looking at us. What effrontery!
Cecily: They're approaching. That's very forward of them. (III.3-6)
This sudden solidarity in the face of dishonest men shows how quickly women can change sides. When insulted by the men they love, they are quick to turn against them, even if it means siding with former enemies. In a society where women have virtually no power, commanding a man’s gaze gives women a sense of empowerment.
Lady Bracknell: Her [Gwendolen’s] unhappy father is, I am glad to say, under the impression that she is attending a more than usually lengthy lecture by the University Extension Scheme on the Influence of a permanent income on Thought. I do not propose to undeceive him. Indeed I have never undeceived him on any question. I would consider it wrong. (III.44)
Lady Bracknell takes on a powerful role here by deceiving Lord Bracknell. Her claim that his daughter "is attending a more than usually lengthy lecture" instead of rendezvousing with her lover is certainly a deception. Here, we see Lady Bracknell "protecting" Lord Bracknell from the truth.