Subgroups — #41008631

Order 26973.

154 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22582717	C₃	1	3	yes	maximal, complemented
2	#22582716	C₃	1	3	yes	maximal, complemented
3	#22582715	C₃	1	2	yes	maximal, complemented
4	(C₃₇⋊C₃)⋊C₈₁	C₃	1	3	yes	maximal, complemented
5	C₃×#15688534	C₃	1	1	yes	maximal, characteristic, complemented
6	#22582614	C₃	1	1	yes	maximal, characteristic
7	#15688534	C₃²	1	1	yes	characteristic
8	#15688534	C₃²	1	1	yes	characteristic
9	#15688534	C₃²	1	1	yes	characteristic
10	#15688534	C₃²	1	6	yes	—
11	#15688531	C₉	1	1	yes	characteristic
12	#15688511	C₉	1	1	yes	characteristic
13	#15688510	C₉	1	3	yes	—
14	#15688509	C₉	1	3	yes	—
15	C₃×#11758601	C₃²	1	1	yes	characteristic
16	C₂₇×C₃₇⋊C₃	C₃²	1	3	yes	—
17	C₃₇×C₂₇⋊C₃	C₉	1	1	yes	characteristic
18	C₃×C₉×C₃₇	C₃×C₉	1	1	yes	characteristic
19	C₃²⋊(C₃₇⋊C₃)	C₂₇	1	1	yes	characteristic
20	#11758608	C₂₇	1	2	yes	—
21	#11758606	C₂₇	1	3	yes	—
22	#11758605	C₂₇	1	3	yes	—
23	C₉×C₃₇⋊C₃	C₃×C₉	1	3	yes	—
24	C₂₇×C₃₇	C₃×C₉	1	3	yes	—
25	#11758601	C₃×C₉	1	3	yes	—
26	#11758601	C₃×C₉	1	1	yes	characteristic
27	#11758601	C₃×C₉	1	1	yes	characteristic
28	#11758601	C₃³	1	1	yes	characteristic
29	#10624833	—	37	1	no	maximal, Sylow
30	C₃²×C₃₇	C₃×C₂₇	1	1	yes	characteristic
31	C₃×C₃₇⋊C₃	C₃×C₂₇	1	3	yes	—
32	C₉×C₃₇	C₃×C₂₇	1	1	yes	characteristic
33	C₉×C₃₇	C₃×C₂₇	1	1	yes	characteristic
34	C₉×C₃₇	C₃²×C₉	1	1	yes	characteristic
35	#66524	C₃×C₂₇	1	6	yes	—
36	#6903	—	37	1	no	—
37	C₈₁⋊C₃	—	37	2	no	—
38	C₈₁⋊C₃	—	37	3	no	—
39	C₈₁⋊C₃	—	37	3	no	—
40	C₃×C₈₁	—	37	1	no	—
41	C₃×C₈₁	—	37	3	no	—
42	C₃×C₃₇	—	3	1	no	—
43	C₃×C₃₇	C₃²×C₂₇	1	1	yes	characteristic
44	C₃₇⋊C₃	—	3	3	no	—
45	#737	—	37	1	no	—
46	C₂₇⋊C₃	—	37	1	no	—
47	C₂₇⋊C₃	—	37	3	no	—
48	C₂₇⋊C₃	—	37	3	no	—
49	C₂₇⋊C₃	#66526	1	1	yes	characteristic
50	C₃×C₂₇	—	37	1	no	—
51	C₃×C₂₇	—	37	3	no	—
52	C₈₁	—	37	1	no	—
53	C₈₁	—	37	1	no	—
54	C₈₁	—	37	1	no	—
55	C₈₁	—	37	6	no	—
56	C₃₇	#10624833	1	1	yes	characteristic, complemented, Sylow
57	3₋¹⁺²	—	37	2	no	—
58	3₋¹⁺²	—	37	3	no	—
59	3₋¹⁺²	—	37	3	no	—
60	H(3,𝔽₃)	—	37	1	no	—
61	C₃×C₉	—	37	3	no	—
62	C₃×C₉	C₃×#66526	1	1	yes	characteristic
63	C₂₇	—	37	3	no	—
64	C₂₇	—	37	1	no	—
65	C₂₇	—	37	1	no	—
66	C₂₇	—	37	1	no	—
67	C₂₇	C₃×#66526	1	3	yes	—
68	C₃²	—	37	3	no	—
69	C₃²	(C₃₇⋊C₃)⋊C₂₇	1	1	yes	characteristic
70	C₉	—	37	6	no	—
71	C₉	(C₃₇⋊C₃)⋊C₂₇	1	1	yes	characteristic
72	C₉	(C₃₇⋊C₃)⋊C₂₇	1	1	yes	characteristic
73	C₉	C₃²×#66526	1	1	yes	characteristic
74	C₃	—	3	1	no	—
75	C₃	—	111	3	no	—
76	C₃	C₃×(C₃₇⋊C₃)⋊C₂₇	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₉	C₃²×#66526
commutator, socle	C₃×C₃₇	C₃²×C₂₇
Fitting	C₃₇×C₂₇⋊C₃	C₉
prime radical	C₃²⋊(C₃₇⋊C₃)	C₂₇
cosocle	#11758601	C₃³
hypercenter	C₂₇⋊C₃	#66526
hypocenter	C₃₇	#10624833
LCS	G ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₉ ◁ C₂₇⋊C₃

Complements

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

N	H ≅ G/N	type
1. #22582717	C₃	semidirect
2. #22582716	C₃	semidirect
3. #22582715	C₃	semidirect
4. (C₃₇⋊C₃)⋊C₈₁	C₃	semidirect
5. C₃×#15688534	C₃	semidirect
56. C₃₇	#10624833	semidirect

Maximal subgroups

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

subgroup	index
#22582717	3
#22582716	3
#22582715	3
(C₃₇⋊C₃)⋊C₈₁	3
C₃×#15688534	3
#22582614	3
#10624833	37