With practicality her philosophy and patronage her whip, Mayor Jane Byrne has forced Chicago's Democatic machine to toe the line since she was inaugurated six months ago.

At the same time, she has helped raise the sales tax and transit fares, demonstrated how she can punish a recalcitrant legislator, and maintained her popularity with the electorate.

Her first six months were capped by the publicity surrounding her decision to endorse Sen. Edward M. Kennedy (D-Mass.) over President Carter. That came just two weeks after she and Carter had stood together as 11,000 persons gathered here to pay her homage, eat filet mignon and ante up $1 million to her political coffers.

It ws a big change from a year ago, when she was waging a seemingly hopeless and underfinanced campaign against the most powerful urban political organization in the nation.

But the machine broke down in the Great Snow of 1979 and Bryne rode an avalanche of voter resentment into the mayor's office, where, some thought, she would be chewed to bits by machine politicians who hungered for power during almost a quarter-century of rule by the late Richard J. Daley and his lackluster successor, Michael A. Bilandic.

Six months after her April inauguration, those politicians, far from chewing her up, are earing out of her hand.

Most of the city's 50 aldermen were at her banquet. Bryne had brought them under control by mid-may, when Alderman Roman C. Pucinski remarked, "The mayor has declawed and defanged a lot of those who earlier threatened strong action."

The city's Democratic delegation to the state legislature, many of whom were at the dinner, took a little longer to whip into shape. But in October, one member who dared to disobey the mayor compared himself to the people who drank the poison at Jonestown.

A lot of city employees came to the dinner after getting letters from their superiors suggesting they order one or more $100 tickets. "Your support is necessary to make this occasion a success," said one letter. Some Top busnessmen, still unsure of where they stand with the mayor, where there n force. So were labor leaders, who had threatened to boycott building tradesmen was "negotiable."

Of course, 3 million Chicagoans were not at the dinner, but most of them approve of Jane Byrne, according to an Oct. 11 poll by television station WBBM. Sixty percent of those polled gave her an excellent or good rating, a loss of only two points since a poll on May 14.

What makes this remarkable is that in the intervening five months Byrne helped raise the sales tax and transit fares and successfully opposed the elimination of the sales tax on food and drugs.

Byrne allied herself with Republican Gov. James R. Thompson on those issues, and her lobbying provides a vivid example of how she operates.

The sales tax increase, designed to raise money for mass transit, failed on the first go-round in the legislature, and Byrne blamed House Democratic leader Michael Madigan, an ambitious young politician from the city's southwest side.

She gave Madigan a public tongue-lashing, suggesting he was unfit for leadership. Then she fired the president of his ward's political oranization from his $43,000-a-year city job. Then she took $22,000 of city legal business away from Madigan and gave half to a more tractble legislator and half to the law partner of two other legislators.

Madigan got the message. He helped pass the tax increase the next chance he got. Later, on orders from Byrne, Madigan convinced almost all the machine Democrats to sustain Thompson's veto of the bill exemping food and drugs from the sales tax, a bill all the Democrats had supported only three months earlier.

Byrne praised Madigan's leadership and quietly rehired his political assocate. And nobody will be surprised if she throws some legal business his way.

Byrne opposed sales tax relief partly because of the potential loss of state aid to the cit and partly because she wanted to humiliate its leading proponent, state Sen. Richard M. Daley, a man who might parlay his revered name into a mayoral candidacy in 1983.

In her campaign, Byrne portrayed herself as a faithful follower of Daley's father, who gave Byrne her first public job, as the city's consumer sales commissioner.She has turned around and fired, at last count 24 patronage workers from young Daley's ward, many of whom were hired by his father.

Byrne has grabbed headlines by firing "loafing" city employees and by vowing to cut 1,000 jobs from the city payroll. At the same time, she got her 22-year-old daughter a $17,500-a-year job as a writer for the Chicago Transit Authority and put a college classmate of her daughter on her own staff at $18,000 a year.

"I don't feel the least bit of remorse or embarrassment about having a qualified member of my family work for the CTA," Byrne snapped. "So anybody trying to draw blood or irritate me will not be successful."

But her use of patronage has drawn remarkably little criticism from the city's weak, but usually vocal, independent movement. Said Charles Bernardini, former head of the Independent Votors of Illinois, "She used reformers and independents to get elected and we used her to cause disruption in the machine. We'll continue to use each other when it suits our purposes."

Byrne appealed to reformers by promising to conduct a nationwide search for top administrative talent. But aside from Donald H. Haider, a deputy U.S. Treasury secretary whom she named budget director, most of her talent if homegrown.

Some of Byrne's proposals seem to be based more on whim than on study and have been blurted out at impromptu news conferences when reporters corner her in hallways. She used that forum to propse a city-run gambling casino, a new subway on the southwest side and a downtown monorail. She long ago abandoned three big Daley-Bilandic capital projects: a new downtown subway, an expressway through the south and west sides of town and a domed stadium.

Right now, Byrne's immediate concern is balancing her 1980 budget, dur Nov. 14.

Though Chicago is in better financial health than many other big cities, Standard & Poor reduced the city's bond rating from Aa to A plus in September, largely because of questionable budgetary practices by Bryne's predecessors.

Her budget could suffer if, as many here expected, Carter retaliates for the Kennedy endorsement by reducing federal aid to Chicago. But Byrne says she doesn't think that will happen.

"I don't think he's a vindicitive person," she said.