Uncle Sam and a US soldier lift up the ?Polygamy House? and place a keg of ?Congress Powder? underneath. John Taylor and various women and children look out of the house in consternation. A group of males in robes look on; some are shaking their fists and one holds a book labeled, ?Protest by the 12 Apostles.? Behind the apostles a very large Brigham Young looks away. In the background the Salt Lake Tabernacle is visible and there is an obelisk with the words, ?In memory of Brigham Young? and ?Founder of the Mormon Church.? The accompanying text appears under the heading, ?The Situation in Utah,? on page 419. 
Oh, and one other thing: all the Mormon males have horns. ?Situation? is one of the few nineteenth-century images that show otherwise human Mormons with horns. There aren?t pointed ears or hooves or visible goat legs or even goatees: it?s just humans who happen to have very prominent horns. (Of course, any realism in the image is undercut by the presence of Uncle Sam, the three-fold difference in height between some of the figures, and context of a political cartoon in a satirical magazine.)
And now to my difficulty: I am finding it very difficult to match faces to names. Uncle Sam, John Taylor (in the house), and Brigham Young (to the right of the house), are clear to me. Nobody else is.
I assume the group near the book-bearer is supposed to represent the Twelve Apostles. In the group there are seven clearly-visible faces, two partially-visible but indistinct faces, and two sets of horns presumably attached to completely obscured faces, for a total of eleven. In 1879 February there were twelve living apostles, including John Taylor.  That the number of horn racks matches the number of apostles encourages me to think that the figures are supposed to be apostles. The detail below assigns a number to each face.
I think 4 is Orson Pratt; 5 could be Charles C Rich or George Q Cannon. The no-face horns are just to the right of the 3 and directly above the 8.
I count fourteen women in the image. I have no idea whether they are supposed to represent particular people or are just generic polygamist wives/mothers.
My first guess is that they are supposed to be John Taylor?s wives, though the number is wrong. Below I have collected putative images of Taylor’s wives, though I do not vouch for them: I googled the names and selected plausible images without any verification, regardless of the woman’s age in 1879 (or whether it was, you know, actually her). 
So? gentle reader: who are the people in the political cartoon? Anyone who renders assistance will be awarded 1,000 internets.
 Volume 3, number 131, pages 424-425.
 ?Behold there is weeping and wailing and gnashing of teeth, Rachael—no it ain?t either, it?s Ann Eliza—weepeth for her children and refuseth to be comforted. She weepeth also, a little, for herself. You see under the present regime it is comparatively an easy matter to get a sixteenth or a twentieth interest in a man; but, if this new heresy be adopted, or rather if Ann Eliza be compelled to return to the old orthodox principle that every woman is entitled to have a man to herself, the matter will not be so easy. It is all very well to say every dog should have his day, but, when there ain?t enough days to go around amongst them, what are you going to do? It is all very well for the law to say that every woman must have a man to herself, but, when there ain?t men enough to go around, what are you going to do? Why debar us from the privilege of having a half interest in one of the ?noblest works of God,? when we can?t get a whole one? Why deny us the privilege of having a fourth interest in a man when we can?t get a half interest? Thus sayeth Ann Eliza. [¶] The answer to Ann Eliza?s queries is brief but not particularly witty. It is because that mankind has been laboring for a number of years to raise itself above the level of the brute creation; laboring to cultivate and refine human nature up to a standard somewhat higher than that of the kine. [¶] Ann Eliza might aptly reply to this that mankind had not been singularly successful in its efforts, and there would be a good deal more of truth than poetry in her reply. But nevertheless it would be a somewhat lame argument to say that because the result of a number of centuries of labor had not been so satisfactory as it might have been the effort should not be given up. [¶] On the whole the situation in Utah is not at all reassuring for the much married men or the partly married women, and, unless Apostolic gold saves the structure the polygamous house is likely to be blown higher than a kite—an altitude which, kites being somewhat uncertain, we cannot be more definite about.?
 In the present-day Church there are usually (but not always) fifteen Apostles: twelve in the Quorum of the Twelve and three in the First Presidency. In 1877 there were two non-Apostle counselors in the First Presidency and one apostle not in the Quorum of the Twelve (Daniel H Wells), so when Brigham Young died in 1877 Aug, there were only thirteen living apostles. Orson Hyde died in 1878 and was not replaced until 1879 April and the First Presidency was not re-organized until 1880. The apostles in 1879 Feb were John Taylor, Wilford Woodruff, Orson Pratt, Charles C Rich, Lorenzo Snow, Erastus Snow, Franklin D Richards, George Q Cannon, Joseph F Smith, Brigham Young Jr, Albert Carrington, and Daniel H Wells.
 John Taylor?s wives with links to the images I used: Leonora Cannon (1796-1868; img); Elizabeth Kaighin (1811-1895; img); Jane Ballantyne (1813-1901; img, img); Mary Ann Oakley (1826-1911; img); Sophia Whitaker (1825-1887; img); Harriet Whitaker (1825-1882; img); Margaret Young (1837-1919; img); Josephine Elizabeth Roueche (1860-1943; img).