Vulgar: the basic meaning is low class, not refined.
Profane: the basic meaning is insulting to God or religion, to desecrate
Obscene: the basic meaning is dirty, offensive to modesty, or decency.
Using that word would certainly be vulgar, it is not the sort of word you would use when you visited the queen. It would most likely be obscene in many settings, since decency and modesty require abstaining from referring to sex, especially in a harsh way. Of course in some settings this might be considered complete appropriate, in which case it would not be obscene.
The word is not profane intrinsically since it is not religious in nature (such as "God damn", or "Jesus H. Christ" might be.) However, in a church setting when appropriate modesty and decency might be expected, the occupants of the church might find the language insulting to the sanctified nature of the place, and consequently profane.
FWIW, profanity and swearing in general usually falls into one of three categories in English: sexual references, references to bodily functions, and blasphemous language. Beside this there is a lot of swearing that is basically a minced form of this, such as corblimey, gosh darnit, or my favorite new one: "shut the front door."