A point on grammar (because language constrains thought, it is useful to get this right!). The noun "infinity" is rarely used to refer to something for which it would make sense to ask if it has a "subset". "Infinity" is usually used in a context to refer to a particular sort of quantity -- e.g. the extended real number $+\infty$ of real analysis.
I don't know of any naming scheme that would use the word "infinity" to refer to a particular set -- and if it did, it would follow the usual rules of set theory: any proper subset of that set would be a different set.
What you want, I think, is the adjective "infinite" as applied to sets or to cardinal numbers. i.e. "Is a subset of an infinite set still infinite?" In this case, it depends in the particular subset -- an infinite set has many subsets, some of finite size, and some of infinite size.