# gps_subgroup_search downloaded from the LMFDB on 13 July 2026.
# Search link: https://www.lmfdb.org/Groups/Abstract/?search_type=Subgroups&ambient=80.31&characteristic=yes
# Query "{'characteristic': True, 'ambient': '80.31'}" returned 12 gps_subgroup_searchs, sorted by ambient order.

# Each entry in the following data list has the form:
#    [Label, Subgroup, Ambient, Quotient]
# For more details, see the definitions at the bottom of the file.



"80.31.1.a1.a1"	[["80.31", "C_{20}:C_4"], 80, 0, true, true, false, false, false, false, false, true, true, false, false, false, false, true, NULL]	[["80.31", "C_{20}:C_4"], 80]	[["1.1", "C_1"], 1, false, true, true, true, true, true, false, true, true]
"80.31.2.a1.a1"	[["40.5", "C_4\\times D_5"], 40, 0, true, true, true, false, false, false, false, true, true, false, false, true, false, true, NULL]	[["80.31", "C_{20}:C_4"], 80]	[["2.1", "C_2"], 2, false, true, true, true, true, true, true, true, true]
"80.31.4.a1.a1"	[["20.4", "D_{10}"], 20, 0, true, true, false, false, false, false, false, true, true, false, false, true, false, true, NULL]	[["80.31", "C_{20}:C_4"], 80]	[["4.2", "C_2^2"], 4, false, false, true, true, true, true, false, true, true]
"80.31.4.b1.a1"	[["20.2", "C_{20}"], 20, 0, true, true, false, false, true, true, true, true, true, false, false, true, true, true, true]	[["80.31", "C_{20}:C_4"], 80]	[["4.1", "C_4"], 4, false, true, true, true, true, true, false, true, true]
"80.31.4.c1.a1"	[["20.1", "C_5:C_4"], 20, 0, true, true, false, false, false, false, false, true, true, false, false, true, true, true, true]	[["80.31", "C_{20}:C_4"], 80]	[["4.1", "C_4"], 4, false, true, true, true, true, true, false, true, true]
"80.31.8.a1.a1"	[["10.2", "C_{10}"], 10, 0, true, true, false, false, true, true, true, true, true, false, false, true, true, true, true]	[["80.31", "C_{20}:C_4"], 80]	[["8.2", "C_2\\times C_4"], 8, false, false, true, true, true, true, false, true, true]
"80.31.8.b1.a1"	[["10.1", "D_5"], 10, 0, true, true, false, false, false, false, false, true, true, false, false, true, true, true, true]	[["80.31", "C_{20}:C_4"], 80]	[["8.4", "Q_8"], 8, false, false, false, true, true, true, false, false, true]
"80.31.8.c1.a1"	[["10.1", "D_5"], 10, 0, true, true, false, false, false, false, false, true, true, false, false, true, true, true, true]	[["80.31", "C_{20}:C_4"], 80]	[["8.3", "D_4"], 8, false, false, false, true, true, true, false, false, true]
"80.31.16.a1.a1"	[["5.1", "C_5"], 5, 5, true, true, false, false, true, true, true, true, true, false, true, true, true, true, true]	[["80.31", "C_{20}:C_4"], 80]	[["16.4", "C_4:C_4"], 16, true, false, false, true, true, true, false, false, true]
"80.31.20.a1.a1"	[["4.1", "C_4"], 4, 0, true, true, false, false, true, true, true, true, true, false, false, true, true, true, true]	[["80.31", "C_{20}:C_4"], 80]	[["20.3", "F_5"], 20, false, false, false, false, true, true, false, true, true]
"80.31.40.a1.a1"	[["2.1", "C_2"], 2, 0, true, true, false, true, true, true, true, true, true, false, true, true, true, true, true]	[["80.31", "C_{20}:C_4"], 80]	[["40.12", "C_2\\times F_5"], 40, true, false, false, false, true, true, false, true, true]
"80.31.80.a1.a1"	[["1.1", "C_1"], 1, 1, true, true, false, true, true, true, true, true, true, true, false, true, true, true, true]	[["80.31", "C_{20}:C_4"], 80]	[["80.31", "C_{20}:C_4"], 80, false, false, false, false, true, true, false, false, true]


# Label --
#    The full label of a subgroup is of the form $\mathtt{N.i.m.a}$ or $\mathtt{N.i.m.a.j}$.  In either case, $\mathtt{N.i}$ is the label of the ambient group, $\mathtt{m}$ is the index of the subgroup in the ambient group, $\mathtt{a}$ distinguishes subgroups up to automorphism and $\mathtt{j}$ distinguishes subgroups up to conjugacy.  Both $\mathtt{a}$ and $\mathtt{j}$ take the form of a sequence of letters followed by an ordinal. The letters are a base 26 (a=0, z=25) encoding of the ordering up to Gassmann-equivalence, and the integer following distinguishes groups in the same Gassmann class.  (For details on how we label Gassmann classes and order subgroups within a Gassmann class, see Computing labels of subgroups.) For some ambient groups, we only compute subgroups up to automorphism (in which case the first form is used, omitting the $\mathtt{j}$.

#    When the ambient group is clear from context, sometimes the short label $\mathtt{m.a}$ or $\mathtt{m.a.j}$ is used instead.