NEW YORK — American Electric Power, one of the country’s largest utilities, gave $1 million last ­November to the Founding Fund, a new tax-exempt group that intends to raise most of its money from corporations and to push for limited government.

The giant insurer Aetna ­directed more than $3 million last year to the American ­Action Network, a Republican-leaning nonprofit organization that has spent millions of dollars attacking lawmakers who voted for President Obama’s health care bill, even as Aetna’s president publicly voiced support for the legislation.

Other major corporations — including Prudential Financial, Dow Chemical, and the drugmaker ­Merck — have poured millions of dollars more into the US Chamber of Commerce, a tax-exempt trade organization that has pledged to spend at least $50 million on political advertising this election cycle.

Two years after the Supreme Court’s Citizens United decision opened the door for corporate spending on elections, relatively little money has flowed from company treasuries into super PACs, which can accept unlimited contributions but must disclose their donors.

Instead, there is growing ­evidence that large corporations are trying to influence campaigns by donat­ing money to tax-exempt organizations that can spend millions of dollars without being subject to the disclosure requirements that apply to candidates, parties, and PACs.

The growing role of issue groups has prompted a rash of complaints and lawsuits.

The secrecy shrouding these groups makes it impossible to make a full accounting of corporate influ­ence on the electoral process.

But glimpses of their donors emerged in a New York Times review of corporate governance reports, tax returns filed by nonprofit organizations, and regulatory filings by insurers and labor unions.

The review found that corporate donations, many of them previously unreported, went to groups, large and small, dedicated to shaping public policy on both state and national levels. From a re­district­ing fight in Minnesota to the sprawling battleground of the 2012 presidential and congressional elections, corporations are opening their wallets and altering the political world.

Some of the biggest recipients of corporate money are ­organized under Section 501(c)(4) of the tax code, the federal designation for social welfare groups dedicated to advanc­ing broad community interests. Because they are not technically political organizations, they do not have to register with or disclose their donors to the Federal Election Commission, potentially shielding corporate contributors from shareholders or others unhappy with their political positions.

‘‘Companies want to be able to quietly push for their political agenda without being held accountable for it by their customers,’’ said Melanie Sloan, ­executive director of Citizens for Responsibility and Ethics in Washington, which has filed complaints against several ­issue groups. ‘‘I think the 501(c)(4)s are likely to outweigh super PAC spending, because so many donors want to remain anonymous.’’

Because social welfare groups are prohibited from devot­ing themselves primarily to political activity, many spend the bulk of their money on issue advertisements that purport to be educational in nature, rather than political.

In May, for example, Crossroads Grassroots Policy Strategies, a group cofounded by the Republican strategist Karl Rove, began a $25 million adver­tising campaign, carefully shaped with focus groups of undecided voters, that attacks Obama for increasing the federal deficit and urges him to cut spending.

The Internal Revenue Service has no clear test for determining what constitutes excessive political activity by a social welfare group. Tax-exempt groups are permitted to begin raising and spending money even before the IRS formally recognizes them. Two years ­after helping Republicans win control of the House with millions of dollars in issue advertising, the Crossroads GPS application for tax-exempt status is still pending.

During the 2010 midterm elections, tax-exempt groups outspent super PACs by a 3-to-2 margin, accord­ing to a recent study by the Center for Responsive Politics and the Center for Public Integrity, with most of that devoted to attacking Democrats or defending Republicans. Such groups have accounted for two-thirds of the political advertising bought by the biggest outside spenders so far in the 2012 election cycle, according to Kantar Media’s Campaign Media Analysis Group, with close to $100 million in issue ads.

The growing role of issue groups has prompted a rash of complaints and lawsuits from watchdog organ­izations accusing groups like the American Action Network, Crossroads, and the pro-Obama ­Priorities USA of operating as sham charities whose primary purpose is not the promotion of social welfare, but winning elections. ­Efforts in Congress to force more disclosure for politically active nonprofit organizations have been repeatedly stymied by Republicans, who have ­described the push as an ­assault on free speech.

Labor unions, which are among the beneficiaries of Citizens United, have also donated millions of dollars to national super PACs and state-level nonprofit groups involved in partisan battles over government spending, collective bargaining, and health care.

Donations from corporations and unions alike must be disclosed if they go to expressly political groups like super PACs.

Among the largest beneficiaries of corporate donations in recent years have been trade organizations like the US Chamber of Commerce, which largely backs Republican candidates. As a nonprofit ‘‘business league’’ under the tax code, the chamber does not have to disclose its supporters, who helped finance its $33 million in political ads in the 2010 midterm elections.