To a certain extent, Miss Prism and Dr. Chasuble are also satirical figures through whom Wilde attacks British institutions, namely education and the Church of England. Explain briefly what aspects of these institutions Wilde is satirizing.

Miss Prism and Dr. Chasuble are characters in Wilde's The Importance of Being Earnest. Miss Prism is Cecily's governess, while Dr. Chasuble is basically the parish reverend. Both, though, are associated with education. While they may seem less absurd than the main characters, Jack and Algernon, Prism and Chasuble also make some ridiculous statements that indicate they are also targets of Wilde's satire.
Chasuble is made fun of, even by Miss Prism, for his obedience to "the primitive church." Clearly, this early version of the church is no longer followed in the Victorian Era, but Chasuble uses it as an excuse to explain why he remains unmarried. Even when he finally decides to admit his love for Miss Prism and marry her, he claims that the primitive church must've been wrong. Chasuble looks ridiculous, and his beliefs seem backward and outdated, which could imply that the church is the same, in Wilde's estimation. Chasuble also seems pretty dense when it comes to the issue of christening Jack and Algernon (so they can change their names to Ernest); he doesn't catch why they are doing it and even suggests they join in at a time where he is christening babies. The church is thus painted as frivolous and lacking perception or intelligence while just going through the motions and rituals prescribed.
Education is also a target of satire throughout the play, such as when Lady Bracknell claims that education in England has no effect and that this is for the best, because the upper classes would be in danger if the lower classes became educated. Miss Prism, as a governess, is the character who is used to most strongly satirize education, though. Prism tries to get Cecily to study but is not effective at doing so; she is also depicted as a silly, vacuous, and careless person. She is the one, of course, who accidentally switched a baby and a manuscript and lost baby Jack/Ernest at the train station. Further attacks on education can be seen in the way Dr. Chasuble and Miss Prism talk to each other "metaphorically" to hide behind "education" in order to mask their feelings. Their metaphors are often nonsensical and pretentious. Also, Cecily says that Dr. Chasuble must be extremely intelligent and must know so much because of how few books he has published. This instance of irony may poke fun at the alignment between expertise and production in terms of published treatises.
All in all, the characters in Earnest are superficial and frivolous, even those associated with education and religion. Therefore, Wilde's play satirizes these institutions and their failures while also critiquing marriage customs.