Subgroups — #22582641

Order 8991.

136 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#11758605	C₃	1	1	yes	maximal, characteristic, complemented
2	3₋¹⁺²×C₃₇⋊C₃	C₃	1	2	yes	maximal, complemented
3	H(3,𝔽₃)×C₃₇⋊C₃	C₃	1	1	yes	maximal, characteristic
4	#15688517	C₃	1	4	yes	maximal, complemented
5	#15688516	C₃	1	2	yes	maximal, complemented
6	#15688515	C₃	1	2	yes	maximal, complemented
7	C₃₇×#731	C₃	1	1	yes	maximal, characteristic, complemented
8	C₃²×C₃₇⋊C₃	—	3	1	no	—
9	C₃²×C₃₇⋊C₃	C₃²	1	1	yes	characteristic
10	3₋¹⁺²×C₃₇	C₃²	1	2	yes	complemented
11	H(3,𝔽₃)×C₃₇	C₃²	1	1	yes	characteristic
12	C₃×C₉×C₃₇	C₃²	1	1	yes	characteristic, complemented
13	C₃²⋊(C₃₇⋊C₃)	—	3	2	no	—
14	C₃²⋊(C₃₇⋊C₃)	C₃²	1	2	yes	—
15	#11758608	C₃²	1	4	yes	complemented
16	C₃×#66524	C₃²	1	2	yes	complemented
17	#11758606	—	3	2	no	—
18	#11758605	—	3	2	no	—
19	#11758605	—	3	2	no	—
20	#11758605	—	3	1	no	—
21	C₉×C₃₇⋊C₃	—	3	2	no	—
22	C₃²×C₃₇	—	3	1	no	—
23	C₃²×C₃₇	C₃³	1	1	yes	characteristic
24	C₃×C₃₇⋊C₃	—	3	2	no	—
25	C₃×C₃₇⋊C₃	—	9	1	no	—
26	C₃×C₃₇⋊C₃	—	9	3	no	—
27	C₃×C₃₇⋊C₃	H(3,𝔽₃)	1	2	yes	—
28	C₃×C₃₇⋊C₃	H(3,𝔽₃)	1	1	yes	characteristic
29	C₉×C₃₇	—	3	2	no	—
30	C₉×C₃₇	—	3	1	no	—
31	#66524	—	3	4	no	—
32	#66524	—	3	2	no	—
33	#6910	—	37	1	no	maximal, Sylow
34	C₃×C₃₇	—	3	1	no	—
35	C₃×C₃₇	—	9	1	no	—
36	C₃×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
37	C₃₇⋊C₃	—	3	1	no	—
38	C₃₇⋊C₃	—	3	2	no	—
39	C₃₇⋊C₃	—	9	2	no	—
40	C₃×3₋¹⁺²	—	37	1	no	—
41	C₃×3₋¹⁺²	—	37	2	no	—
42	C₃×H(3,𝔽₃)	—	37	1	no	—
43	#731	—	37	4	no	—
44	#731	—	37	2	no	—
45	#731	—	37	2	no	—
46	#731	C₃₇⋊C₃	1	1	yes	characteristic, complemented
47	C₃₇	#6910	1	1	yes	characteristic, complemented, Sylow
48	C₃³	—	37	1	no	—
49	C₃³	—	111	1	no	—
50	3₋¹⁺²	—	37	4	no	—
51	3₋¹⁺²	—	111	2	no	—
52	3₋¹⁺²	—	111	2	no	—
53	3₋¹⁺²	—	111	1	no	—
54	3₋¹⁺²	—	111	2	no	—
55	3₋¹⁺²	C₃×C₃₇⋊C₃	1	2	yes	complemented
56	H(3,𝔽₃)	—	37	2	no	—
57	H(3,𝔽₃)	—	111	2	no	—
58	H(3,𝔽₃)	C₃×C₃₇⋊C₃	1	1	yes	characteristic
59	C₃×C₉	—	37	2	no	—
60	C₃×C₉	—	111	2	no	—
61	C₃×C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic, complemented
62	C₃²	—	3	1	no	—
63	C₃²	—	37	2	no	—
64	C₃²	—	37	1	no	—
65	C₃²	—	333	1	no	—
66	C₃²	—	333	3	no	—
67	C₃²	—	111	2	no	—
68	C₃²	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
69	C₉	—	3	2	no	—
70	C₉	—	3	1	no	—
71	C₉	—	111	2	no	—
72	C₉	—	111	4	no	—
73	C₃	—	3	1	no	—
74	C₃	—	9	1	no	—
75	C₃	—	333	2	no	—
76	C₃	—	111	1	no	—
77	C₃	—	111	2	no	—
78	C₃	H(3,𝔽₃)×C₃₇⋊C₃	1	1	yes	characteristic

	subgroup	G/H
center	C₃	H(3,𝔽₃)×C₃₇⋊C₃
commutator	C₃²×C₃₇	C₃³
Frattini	C₃²	C₃²×C₃₇⋊C₃
Fitting	C₃₇×#731	C₃
socle	C₃×C₃₇	C₃×H(3,𝔽₃)
hypercenter	#731	C₃₇⋊C₃
hypocenter	C₃₇	#6910
LCS	G ▷ C₃²×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃ ◁ C₃² ◁ #731

Complements

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

N	H ≅ G/N	type
1. C₃×#11758605	C₃	semidirect
2. 3₋¹⁺²×C₃₇⋊C₃	C₃	semidirect
4. #15688517	C₃	semidirect
5. #15688516	C₃	semidirect
6. #15688515	C₃	semidirect
7. C₃₇×#731	C₃	semidirect
10. 3₋¹⁺²×C₃₇	C₃²	semidirect
12. C₃×C₉×C₃₇	C₃²	semidirect
15. #11758608	C₃²	semidirect
16. C₃×#66524	C₃²	semidirect
46. #731	C₃₇⋊C₃	semidirect
47. C₃₇	#6910	semidirect
55. 3₋¹⁺²	C₃×C₃₇⋊C₃	semidirect
61. C₃×C₉	C₃×C₃₇⋊C₃	semidirect