Conn. Gen. Stat. § 9-253
When a major or minor party is entitled to nominate two or more candidates for a particular office, the order of the names of its candidates for such office appearing on the voting machine ballot label shall be determined by the registrars of voters by lot in a ceremony which shall be open to the public, except as hereinafter provided. When such a candidate is nominated for the same office by more than one party, his name shall appear on each appropriate row on the voting machine ballot label in the same column in which it appears under the foregoing provision in either (1) the party row of the party with which he is enrolled or (2) the first party row on which his name is to appear if such candidate is an unaffiliated elector. The registrars of voters shall provide at least five days' public notice for each ceremony held under this section. The ballot order of nominating petition candidates for multiple-opening offices shall be as prescribed in section 9-453r.
Conn. Gen. Stat. § 9-253
(1955, S. 730d; 1957, P.A. 221, S. 1; P.A. 84-319, S. 36, 49; P.A. 87-197, S. 1, 3.)