I think the answer you are looking for is yes but it is not necessarily the case. "C7" refers to the glass part of the bulb. It means cone shaped" and 7/8ths of an inch in diameter. The size of the base on most C7 bulbs is "candelabra". E12 is the same as a candelabra size base. The names of the base sizes are Mogel,, Medium, Intermediate, Candelabra, Midget, and Miniature. (Most "standard" household bulbs have a Medium base. Most C7 bulbs have a candelabra base. Most C9 bulbs have an intermediate base.)
TED