Subgroups — #36381207

Order 22599.

196 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₉×#11719689	C₃	1	1	yes	maximal, characteristic
2	(C₃₁⋊C₉)⋊C₂₇	C₃	1	6	yes	maximal, complemented
3	C₃²⋊(C₃₁⋊C₂₇)	C₃	1	2	yes	maximal
4	#21419221	C₃	1	3	yes	maximal, complemented
5	C₃²×C₂₇×C₃₁	C₃	1	1	yes	maximal, characteristic
6	C₃×#11719689	C₉	1	1	yes	characteristic
7	C₃×C₃₁⋊C₂₇	C₃²	1	6	yes	—
8	(C₃₁⋊C₉)⋊C₉	C₉	1	6	yes	—
9	C₃²⋊(C₃₁⋊C₉)	C₉	1	2	yes	—
10	C₉×C₃₁⋊C₉	C₃²	1	3	yes	—
11	C₃²×C₉×C₃₁	C₃²	1	1	yes	characteristic
12	C₃×C₂₇×C₃₁	—	3	3	no	—
13	C₃×C₂₇×C₃₁	C₃²	1	2	yes	—
14	C₃×C₂₇×C₃₁	C₃²	1	1	yes	characteristic
15	C₃³×C₃₁	C₃×C₉	1	1	yes	characteristic
16	C₃×C₉×C₃₁	—	3	1	no	—
17	C₃×C₉×C₃₁	—	3	2	no	—
18	C₃×C₉×C₃₁	C₃×C₉	1	2	yes	—
19	C₃×C₉×C₃₁	C₃³	1	1	yes	characteristic
20	#11719689	—	3	3	no	—
21	C₃×C₃₁⋊C₉	C₃×C₉	1	6	yes	—
22	C₃×C₃₁⋊C₉	C₃×C₉	1	3	yes	—
23	C₂₇×C₃₁	—	3	2	no	—
24	C₂₇×C₃₁	3₋¹⁺²	1	3	yes	complemented
25	C₃₁⋊C₂₇	—	3	6	no	—
26	#10624727	—	31	1	no	maximal, Sylow
27	C₃²×C₃₁	—	3	1	no	—
28	C₃²×C₃₁	—	3	2	no	—
29	C₃²×C₃₁	C₂₇⋊C₃	1	3	yes	—
30	C₃²×C₃₁	C₃²×C₉	1	1	yes	characteristic
31	C₉×C₃₁	—	3	2	no	—
32	C₉×C₃₁	C₃×3₋¹⁺²	1	1	yes	characteristic
33	C₉×C₃₁	#737	1	2	yes	—
34	C₃₁⋊C₉	—	3	3	no	—
35	C₃₁⋊C₉	—	3	6	no	—
36	C₃²×C₂₇	C₃₁⋊C₃	1	1	yes	characteristic
37	C₉×3₋¹⁺²	—	31	1	no	—
38	C₉⋊C₂₇	—	31	6	no	—
39	C₃²⋊C₂₇	—	31	2	no	—
40	#6864	—	31	3	no	—
41	C₃×C₃₁	—	3	3	no	—
42	C₃×C₃₁	C₉×3₋¹⁺²	1	1	yes	characteristic
43	C₃×C₃₁	C₃×C₂₇⋊C₃	1	1	yes	characteristic
44	C₃×C₃₁	#6903	1	2	yes	—
45	C₃×3₋¹⁺²	—	31	1	no	—
46	C₃²×C₉	C₃×C₃₁⋊C₃	1	1	yes	characteristic
47	C₃×C₂₇	—	3	3	no	—
48	C₃×C₂₇	—	31	6	no	—
49	C₃×C₂₇	C₃×C₃₁⋊C₃	1	2	yes	—
50	C₃×C₂₇	C₃×C₃₁⋊C₃	1	1	yes	characteristic
51	C₉⋊C₉	—	31	6	no	—
52	C₃²⋊C₉	—	31	2	no	—
53	C₉²	—	31	3	no	—
54	C₃₁	#10624727	1	1	yes	characteristic, complemented, Sylow
55	C₃³	C₉×C₃₁⋊C₃	1	1	yes	characteristic
56	3₋¹⁺²	—	93	3	no	—
57	C₃×C₉	—	3	1	no	—
58	C₃×C₉	—	3	2	no	—
59	C₃×C₉	—	31	6	no	—
60	C₃×C₉	—	31	3	no	—
61	C₃×C₉	C₉×C₃₁⋊C₃	1	2	yes	—
62	C₃×C₉	C₃²×C₃₁⋊C₃	1	1	yes	characteristic
63	C₂₇	—	3	2	no	—
64	C₂₇	—	93	6	no	—
65	C₂₇	#11719689	1	3	yes	complemented
66	C₃²	—	3	1	no	—
67	C₃²	—	3	2	no	—
68	C₃²	#14578891	1	3	yes	—
69	C₃²	C₃×C₉×C₃₁⋊C₃	1	1	yes	characteristic
70	C₉	—	3	2	no	—
71	C₉	—	93	3	no	—
72	C₉	—	93	6	no	—
73	C₉	C₃×#11719689	1	1	yes	characteristic
74	C₉	#14578912	1	2	yes	—
75	C₃	—	3	3	no	—
76	C₃	C₉×#11719689	1	1	yes	characteristic
77	C₃	C₃×#14578891	1	1	yes	characteristic
78	C₃	#21419370	1	2	yes	—

	subgroup	G/H
center, Frattini	C₃×C₉	C₃²×C₃₁⋊C₃
commutator, socle	C₃²×C₃₁	C₃²×C₉
Fitting	C₃²×C₂₇×C₃₁	C₃
prime radical	C₃³×C₃₁	C₃×C₉
cosocle	C₃×C₉×C₃₁	C₃³
hypercenter	C₃²×C₂₇	C₃₁⋊C₃
hypocenter	C₃₁	#10624727
LCS	G ▷ C₃²×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₃×C₉ ◁ C₃²×C₂₇

Complements

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

N	H ≅ G/N	type
2. (C₃₁⋊C₉)⋊C₂₇	C₃	semidirect
4. #21419221	C₃	semidirect
24. C₂₇×C₃₁	3₋¹⁺²	semidirect
54. C₃₁	#10624727	semidirect
65. C₂₇	#11719689	semidirect

Maximal subgroups

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

subgroup	index
C₉×#11719689	3
(C₃₁⋊C₉)⋊C₂₇	3
C₃²⋊(C₃₁⋊C₂₇)	3
#21419221	3
C₃²×C₂₇×C₃₁	3
#10624727	31