“They’re going to defend this vigorously, and obviously all the facts have not come out,” Barry W. Agulnick, Mr. Shen’s lawyer, said after the arraignment.

The charge, theft from programs receiving federal funds, is punishable by up to 10 years in prison and a fine, said Robert Nardoza, a spokesman for the United States attorney for the Eastern District of New York. “As alleged in the complaint, this amounts to one of the largest lunch money thefts in history,” Mr. Nardoza said.

Red Apple Child Development Center runs six preschools in Queens, Brooklyn, Manhattan and Staten Island. Between 2002 and 2010, the preschools received $13.5 million from the Agriculture Department through a program intended to help institutions provide healthy meals and snacks to children and adults.

Audits by the New York State Health Department and the federal Agriculture Department since 2005 revealed a pattern of false submissions to the Agriculture Department, including lying about how many children were getting the meals, the complaint charges. For example, the complaint says, in January 2009, Ms. Fan submitted records claiming that 188 children had consumed 4,700 meals and snacks, when other records showed that 116 children ate 2,900 meals and snacks.