I would like to be able to draw the lattices of subgroups of certain groups. I thought it would be easy when I already know the structure, but I've never done it before, and it turns out it's harder than it seems. Are there any good ways of making the picture as readable as possible?
Let's say I want to draw the lattice of subgroups of $G=\Bbb Z/1176\ \Bbb Z.$ I have $1176=2^3\cdot 3\cdot 7^2$ so there are $(3+1)\cdot(1+1)\cdot(2+1)=24$ subgroups of $G$, each generated by one of the $24$ divisors of $1176.$ The largest is $\langle 1\rangle$, and immediately below it are $\langle 2\rangle$, $\langle 3\rangle$ and $\langle 7\rangle.$ This is easy to draw. But now it seems that I can't avoid a self-intersection of the graph below that. The next row should consist of $\langle 2^2\rangle,$ $\langle2\cdot3\rangle,$ $\langle3\cdot7\rangle,\ \langle2\cdot7\rangle$ and $\langle7^2\rangle.$ What is the best way to draw it? By best I mean most readable and tidy-looking.
Additionally, is there some kind of free software that would find the most readable way of drawing the lattice?
There are several ways how to define "nice" in this context. For example, you can want your diagram to
The problem is that the conditions contradict each other. Even as simple poset as $2^{\{1,2,3\}}$ has three nice diagrams.
Usually, I order the elements by their (lower) rank; then I try to minimize the number of intersections; then I try to preserve as much symmetry as possible. If the number of elements is not too big, there is usually a reasonably nice picture.
Some software links: