Republicans broadly share a belief in smaller government, but they are sharply divided over issues including perceptions of Wall Street, the power of big corporations, the value of immigration and free trade, according to a new study of the contours of the American electorate.

The findings provide timely insight into the political battle that unfolded this week in Mississippi, where Republican Sen. Thad Cochran narrowly defeated conservative state Sen. Chris McDaniel in a bitterly fought runoff that exposed the broader tensions within the GOP coalition.

That election pitted the business and establishment wing of the Republican Party against the populist-conservative wing. As it has through a series of elections this spring, the establishment wing prevailed, but the new study from the Pew Research Center suggests that the GOP faces continued instability because of profoundly different views on some issues held by those who identify with the party.

The study breaks down the population into eight groups, seven of them engaged in politics at least to a degree and the other mostly on the sidelines. Three are highly ideological and politically engaged — two that lean to the Republicans, one to the Democrats. Four other groups are “less partisan and less predictable” in their political views, what the study calls a “fragmented center” that poses challenges for both major parties.

The most loyal followers of the Republican Party account for about one-fifth of the total population, more than a quarter of registered voters and more than one-third of politically engaged Americans, according to Pew.

The Pew study labels these two Republican groups as “business conservatives” and “steadfast conservatives.” Almost nine in 10 people in each group are white, and about six in 10 in each group are men. Two-thirds of steadfast conservatives are 50 or older, compared with 53 percent of business conservatives. Only very small percentages of each say they disagree with the tea party movement.

The two GOP groups find common ground in their overwhelming disapproval of President Obama’s job performance and the Affordable Care Act. More than nine in 10 disapprove of both. Nine in 10 also say government is almost always wasteful and inefficient.

The GOP core groups are also united in their opposition to the Common Core State Standards, their support for a strong military, their belief that there is no solid evidence of global warming, and their perception that efforts to protect the environment have gone too far and that environmental laws have cost the economy too many jobs.

But among these conservative Republicans, the divisions are widespread.

On business issues, an overwhelming majority of steadfast conservatives say too much power is concentrated in the hands of a few large companies and are split evenly on whether the economic system unfairly favors the powerful.

But only about one-third of business conservatives say big corporations have too much power and by 2-to-1 say the economic system is fair to most people rather than tilted in favor of the powerful. Business conservatives are significantly more likely to agree with the statement that Wall Street helps the economy more than it hurts.

Immigration provides another fault line, according to the study. A big majority of steadfast conservatives say immigrants put a burden on the country and take away jobs, housing and health care from others. Eight in 10 of these conservatives say that growing numbers of newcomers from other countries threaten traditional values.

In contrast, business conservatives say newcomers strengthen American society and see immigrants as bolstering the country through hard work. They also strongly support a path to citizenship for illegal immigrants who meet certain eligibility requirements. Steadfast conservatives are evenly split on this most controversial aspect of the debate over immigration reform legislation.

The two core Republican groups also part company in their assessment of America’s role in the world. The business wing strongly agrees that it is best for the country to be active in the world. They also say problems in the world would be worse without U.S. participation.

By even greater numbers, steadfast conservatives say it’s time for the country to focus on problems at home rather than elsewhere in the world. A majority agrees with the proposition that U.S. involvement makes world problems worse. These conservatives are the only group in the study to say that free-trade agreements are bad for the country.

Both, however, say that military strength is the path to ensuring peace, rather than relying primarily on diplomacy, and that overwhelming force is the best way to defeat terrorism. This puts them at odds with the most loyal Democrats, who say too much force creates hatred and therefore more terrorism.

Democrats also are divided, although in practice the divisions have produced little of the intraparty warfare experienced among Republicans in recent years.

Pew calls the most ideological of the Democratic groups “solid liberals.” Other Democratic­leaning groups are called “faith-and-family left,” the “next-generation left” and “hard-pressed skeptics.”

All four groups backed Obama over Mitt Romney in 2012 by margins ranging from 40 to 88 points, according to the study.

The biggest differences come over some moral questions and social issues, including same-sex marriage. As their label implies, those in the “faith-and-family left” have strong religious convictions while a plurality of solid liberals have no religious affiliation.

The other Republican group is labeled “young outsiders,” and as the name suggests, they are generally younger, more independent and more likely to express negative views of both parties.

The study is the second report based on a survey of more than 10,000 adults conducted from January to March. Pew began its political typology studies 27 years ago in an effort to provide a more textured look at the politics of the population than is provided by “red-blue” analyses of the country.