Subway does not have a ticker symbol because it is a privately-held company, according to the company website. As Investopedia explains, companies that are privately held and thus not traded on the market do not have stock ticker symbols.
According to Bloomberg Businessweek, Subway is owned by Doctor's Associates, Inc., which is also a private company. Although privately-held companies may issue shares, bonds and stocks, these shares do not trade on the open market, according to Investopedia.
Investopedia describes a ticker symbol as the shorthand used to identify a company's stock for trade on the market. Since privately-held companies do not trade on the market, they have no need for ticker symbols. Ticker symbols are also known as stock symbols.