。
She looked through the list to see if her name was on it.
Literal
She [topic-は] her own name [subject-が] exists [question-か] roster [object-を] looked-through.
An embedded yes-no question without ~かどうか — just a clause ending in か, used as the content of what she was checking. 目を通す is a fixed idiom literally 'to pass one's eyes through' = 'to skim, to look over (a document, list, etc.).' Note the choice of ある, not いる, for the existence of one's name on the list — names are inanimate items even when they belong to people, so the existential split between animate (いる) and inanimate (ある) places them on the inanimate side.