Your equation is polynomial of degree 5 in the unknown R, with huge symbolic coefficients.
Such equations usually do not have explicit solutions (see Abel–Ruffini theorem theorem and Galois theory). 

The worksheet is corrupted, or contains some hidden fields.
If the .mw is exported as .mpl and then opened in a fresh session, it runs OK.

The documentation of SubgroupLattice says:

The algorithm used does not find any perfect subgroups of G, other than potentially G itself, its soluble residual, and the trivial group. If the soluble residual of G is of size at least 360, then it may contain other nontrivial perfect subgroups. These will not be found by the algorithm.

Maybe there are other omissions.
 

The denominator has many very small values ==> large roundoff errors.
Increasing Digits (e.g. Digits:=15) solves the problem.

Just replace in param1,...,param4:  alpha*gamma=0.004;   with   gamma=0.004/alpha;
(or use algsubs).

Advices:
- don't load packages which are not needed
- gamma is a Maple constant, don't use it as a variable
- linalg is deprecated; use LinearAlgebra instead.

 

convert(EllipticK(x), Int): 
eval(%,x=1); value(%);

          infinity

with(GroupTheory):
g:=DrawSubgroupLattice(SmallGroup(200, 31), labels = ids, output=graph):
GraphTheory:-DrawGraph(g, size=[1200,800]);

Execute

pp:=interface(prettyprint=1);

before Generators(...), then copy.
To restore the old settings:

interface(prettyprint=pp):

The simplest solution is to use

DrawSubgroupLattice(SymmetricGroup(4), labels=ids);

This way the labels will be `n/d` i.e. the full identifier of the (small) group, n being its order.
It is possible to manipulate the PLOT structure to eliminate `/d`, but I think `n/d` is better.
Edit
If you really want n only, use:

with(GroupTheory):
P:=DrawSubgroupLattice(SymmetricGroup(4), labels=ids):
subsindets(P, name,  proc(a) local t:=SearchText("/",a); `if`(t>0,substring(a,1..t-1),a) end proc); 

Why not use mtaylor?
Or, the next proc which is even faster:

T:=(x,y,x0,y0,N) -> eval(series(f(x0+t*x,y0+t*y),t,N),t=1);

with(GroupTheory):
G := SmallGroup(48, 8);
       G :=  < a permutation group on 48 letters with 5 generators > 

gen:=NonRedundantGenerators(G);
gen := [(1283)(4122218)(5112117)(614713)(9163120)(10153219)(2336

  4744)(24354843)(25344542)(26334641)(27402938)(28393037), (14721

  82265)(21114183171312)(923294531472725)(1024304632482826)(1533

  394319413735)(1634404420423836), (1910)(21516)(31920)(42324)(5

  2526)(62728)(72930)(83132)(113334)(123536)(133738)(143940)(1741

  42)(184344)(214546)(224748)]

lprint(gen);
[Perm([[1, 2, 8, 3], [4, 12, 22, 18], [5, 11, 21, 17], [6, 14, 7, 13], [9, 16,
31, 20], [10, 15, 32, 19], [23, 36, 47, 44], [24, 35, 48, 43], [25, 34, 45, 42]
, [26, 33, 46, 41], [27, 40, 29, 38], [28, 39, 30, 37]]), Perm([[1, 4, 7, 21, 8
, 22, 6, 5], [2, 11, 14, 18, 3, 17, 13, 12], [9, 23, 29, 45, 31, 47, 27, 25], [
10, 24, 30, 46, 32, 48, 28, 26], [15, 33, 39, 43, 19, 41, 37, 35], [16, 34, 40,
44, 20, 42, 38, 36]]), Perm([[1, 9, 10], [2, 15, 16], [3, 19, 20], [4, 23, 24],
[5, 25, 26], [6, 27, 28], [7, 29, 30], [8, 31, 32], [11, 33, 34], [12, 35, 36],
[13, 37, 38], [14, 39, 40], [17, 41, 42], [18, 43, 44], [21, 45, 46], [22, 47,
48]])]

nops(gen); # number of generators
        3
 

It is easy to find the soluble groups using the builtin database:

for n in [60, 120, 168, 180] do SearchSmallGroups( 'order' = n, soluble) od;

Finding a polynomial in Z[X]  for such groups is another problem; I don't think Maple or other CAS can do that efficiently.

C₆ is a NOT a subgroup of S₅. It is just isomorphic to a subgroup of S₅.