Subgroups — #36380954

Order 22599.

290 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#14578892	C₃	1	2	yes	maximal
2	C₃²×C₃₁⋊C₂₇	C₃	1	1	yes	maximal, characteristic
3	#21419222	C₃	1	6	yes	maximal, complemented
4	C₉×C₃₁⋊C₂₇	C₃	1	3	yes	maximal, complemented
5	C₃₁×#6887	C₃	1	1	yes	maximal, characteristic
6	#14578892	C₉	1	18	yes	complemented
7	C₃×C₃₁⋊C₂₇	C₉	1	9	yes	complemented
8	C₃×C₃₁⋊C₂₇	C₃²	1	1	yes	characteristic
9	C₃×C₃₁⋊C₂₇	C₃²	1	2	yes	—
10	C₃×C₃₁⋊C₂₇	C₃²	1	6	yes	—
11	C₃²×C₉×C₃₁	C₃²	1	1	yes	characteristic
12	C₃₁×C₉⋊C₉	C₉	1	6	yes	—
13	C₃₁×C₃²⋊C₉	C₉	1	3	yes	—
14	C₉²×C₃₁	C₃²	1	3	yes	—
15	C₃³×C₃₁	C₃×C₉	1	1	yes	characteristic
16	C₃×C₉×C₃₁	C₃×C₉	1	9	yes	—
17	C₃×C₉×C₃₁	C₃×C₉	1	2	yes	—
18	C₃×C₉×C₃₁	C₃×C₉	1	3	yes	—
19	C₃×C₉×C₃₁	C₃×C₉	1	6	yes	—
20	C₃×C₉×C₃₁	C₃³	1	1	yes	characteristic
21	C₃₁⋊C₂₇	C₃×C₉	1	3	yes	—
22	C₃₁⋊C₂₇	C₃×C₉	1	6	yes	—
23	C₃₁⋊C₂₇	C₃×C₉	1	18	yes	—
24	#10624681	—	31	1	no	maximal, Sylow
25	C₃²×C₃₁	—	3	3	no	—
26	C₃²×C₃₁	C₉²	1	3	yes	—
27	C₃²×C₃₁	C₃²×C₉	1	1	yes	characteristic
28	C₉×C₃₁	—	3	9	no	—
29	C₉×C₃₁	C₉²	1	6	yes	—
30	C₉×C₃₁	C₃²×C₉	1	1	yes	characteristic
31	C₉×C₃₁	C₃²×C₉	1	2	yes	—
32	C₃×C₂₇⋊C₃	—	31	2	no	—
33	C₃²×C₂₇	—	31	1	no	—
34	#6887	C₃₁⋊C₃	1	1	yes	characteristic
35	#6864	—	31	6	no	—
36	C₉×C₂₇	—	31	3	no	—
37	C₃×C₃₁	—	3	3	no	—
38	C₃×C₃₁	C₃×C₉²	1	1	yes	characteristic
39	C₃×C₃₁	#6903	1	3	yes	—
40	C₃²×C₉	C₃×C₃₁⋊C₃	1	1	yes	characteristic
41	C₂₇⋊C₃	—	31	18	no	—
42	C₃×C₂₇	—	31	9	no	—
43	C₃×C₂₇	—	31	1	no	—
44	C₃×C₂₇	—	31	6	no	—
45	C₃×C₂₇	—	31	2	no	—
46	C₉⋊C₉	C₃₁⋊C₉	1	6	yes	—
47	C₃²⋊C₉	C₃₁⋊C₉	1	3	yes	—
48	C₉²	C₃×C₃₁⋊C₃	1	3	yes	—
49	C₃₁	#10624681	1	1	yes	characteristic, complemented, Sylow
50	C₃³	C₃×C₃₁⋊C₉	1	1	yes	characteristic
51	C₃×C₉	C₉×C₃₁⋊C₃	1	3	yes	—
52	C₃×C₉	C₉×C₃₁⋊C₃	1	6	yes	—
53	C₃×C₉	C₃×C₃₁⋊C₉	1	9	yes	—
54	C₃×C₉	C₃×C₃₁⋊C₉	1	2	yes	—
55	C₃×C₉	C₃²×C₃₁⋊C₃	1	1	yes	characteristic
56	C₂₇	—	31	3	no	—
57	C₂₇	—	31	6	no	—
58	C₂₇	—	31	18	no	—
59	C₃²	—	3	3	no	—
60	C₃²	C₉×C₃₁⋊C₉	1	3	yes	—
61	C₃²	C₃²×C₃₁⋊C₉	1	1	yes	characteristic
62	C₉	—	3	9	no	—
63	C₉	C₉×C₃₁⋊C₉	1	6	yes	—
64	C₉	C₃×C₉×C₃₁⋊C₃	1	1	yes	characteristic
65	C₉	C₃×C₉×C₃₁⋊C₃	1	2	yes	—
66	C₃	—	3	3	no	—
67	C₃	C₃×C₉×C₃₁⋊C₉	1	1	yes	characteristic
68	C₃	#21419371	1	3	yes	—

	subgroup	G/H
center, Frattini	C₃×C₉	C₃²×C₃₁⋊C₃
commutator	C₃×C₃₁	C₃×C₉²
Fitting	C₃₁×#6887	C₃
socle	C₃²×C₃₁	C₃²×C₉
prime radical	C₃³×C₃₁	C₃×C₉
cosocle	C₃×C₉×C₃₁	C₃³
hypercenter	#6887	C₃₁⋊C₃
hypocenter	C₃₁	#10624681
LCS	G ▷ C₃×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₃×C₉ ◁ #6887

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
3. #21419222	C₃	semidirect
4. C₉×C₃₁⋊C₂₇	C₃	semidirect
6. #14578892	C₉	semidirect
7. C₃×C₃₁⋊C₂₇	C₉	semidirect
49. C₃₁	#10624681	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₃×#14578892	3
C₃²×C₃₁⋊C₂₇	3
#21419222	3
C₉×C₃₁⋊C₂₇	3
C₃₁×#6887	3
#10624681	31