Most of the time, if you add a prefix to a Spanish verb you know the meaning of you can have a pretty good idea of what the new verb means. But that isn't the case with the sumir family of verbs — while sumir typically means "to sink," verbs such as presumir and asumir don't seem to be even remotely related in meaning.
Sumir is derived from a Latin verb, sumere, which meant "to take," and most of the verbs related to sumir have some vague connection with that meaning. It's possible that the current meaning of sumir came about through the shortening of an unrelated verb, sumergir, which also refers to submerging.
In any case, here are the most common verbs related to sumir with usual meanings and an example of usage in a sentence. As you can see, many of them have meanings held by their English cognates, but not necessarily the most common meaning of the English word. In a sense, then, many of the -sumir verbs can become false friends:
Asumir: to take on a responsibility, to take on a characteristic, to come to terms with. Los comités de obreros asumieron el poder sobre las industrias que tenían importancia para la guerra. The worker committees assumed power over the industries that had importance to the war. (In Latin America, the word also has the meaning of "presuppose," likely due to due the influence of English.)
Consumir: to consume, to waste, to ruin. Todo lo que consumimos está contaminado. Everything we consume is contaminated.
Presumir: to show off, to brag, to boast, to promote. Microsoft presume de la seguridad de su navegador actualizado. Microsoft is boasting about the safety of its updated browser. (The meaning of "to presuppose" is sometimes used but is less common than the others.)
Resumir: to summarize, to sum up. Vamos a resumir algunas de las más destacadas leyendas de la mitología griega. We are going to summarize some of the most outstanding legends of Greek mythology.
Subsumir: to include in a larger category. La rotura de un cristal para robar una caja fuerte se subsume en el delito de robo con escalamiento. The breaking of a pane of glass in order to rob a safe is included in the crime of aggravated robbery.
