"He cleaned the dishes so that he had clean plates."
"so that" seems to function as a conjunction, because "that he had clean plates" is not an independent clause; "he had clean plates" is. Is this true?
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He cleaned the dishes [so that he had clean plates].
Although "so + that" may appear to be a single grammatical unit, i.e. a conjunction, it is best analysed as two distinct elements: the preposition "so" + the subordinator "that" introducing the subordinate clause "that he had clean plates" functioning as complement of "so".
The whole expression "so that he had clean plates" is thus a preposition phrase functioning as an adjunct of purpose (or possibly result).