Subgroups — #22582679

Order 8991.

460 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#15688532	C₃	1	18	yes	maximal, complemented
2	#15688531	C₃	1	9	yes	maximal, complemented
3	C₃×#11758608	C₃	1	2	yes	maximal, complemented
4	3₋¹⁺²×C₃₇⋊C₃	C₃	1	9	yes	maximal, complemented
5	C₃×C₃²⋊(C₃₇⋊C₃)	C₃	1	1	yes	maximal, characteristic
6	C₃×3₋¹⁺²×C₃₇	C₃	1	1	yes	maximal, characteristic, complemented
7	C₃³×C₃₇	C₃²	1	1	yes	characteristic
8	C₃²×C₃₇⋊C₃	C₃²	1	3	yes	—
9	3₋¹⁺²×C₃₇	C₃²	1	3	yes	—
10	3₋¹⁺²×C₃₇	C₃²	1	6	yes	complemented
11	C₃×C₉×C₃₇	C₃²	1	3	yes	complemented
12	C₃²⋊(C₃₇⋊C₃)	C₃²	1	3	yes	—
13	C₃²⋊(C₃₇⋊C₃)	C₃²	1	6	yes	—
14	#11758608	C₃²	1	6	yes	—
15	#11758608	C₃²	1	6	yes	complemented
16	#11758608	C₃²	1	6	yes	complemented
17	C₃×#66524	C₃²	1	6	yes	complemented
18	#11758606	C₃²	1	27	yes	complemented
19	#11758605	C₃²	1	27	yes	complemented
20	C₉×C₃₇⋊C₃	C₃²	1	27	yes	complemented
21	C₃²×C₃₇	—	9	1	no	—
22	C₃²×C₃₇	C₃³	1	2	yes	—
23	C₃²×C₃₇	C₃³	1	1	yes	characteristic
24	C₃²×C₃₇	C₃³	1	1	yes	characteristic
25	C₃×C₃₇⋊C₃	—	9	3	no	—
26	C₃×C₃₇⋊C₃	C₃³	1	9	yes	—
27	C₉×C₃₇	C₃³	1	9	yes	—
28	#66524	C₃³	1	18	yes	—
29	3₋¹⁺⁴	—	37	1	no	maximal, Sylow
30	C₃×C₃₇	—	3	1	no	—
31	C₃×C₃₇	—	3	2	no	—
32	C₃×C₃₇	—	3	1	no	—
33	C₃×C₃₇	C₃⁴	1	1	yes	characteristic
34	C₃₇⋊C₃	—	3	9	no	—
35	#737	—	37	18	no	—
36	#737	—	37	9	no	—
37	C₃×3₋¹⁺²	—	37	2	no	—
38	C₃×3₋¹⁺²	—	37	9	no	—
39	C₃×3₋¹⁺²	C₃₇⋊C₃	1	1	yes	characteristic, complemented
40	C₃×H(3,𝔽₃)	—	37	1	no	—
41	C₃₇	3₋¹⁺⁴	1	1	yes	characteristic, complemented, Sylow
42	C₃³	—	37	3	no	—
43	C₃³	C₃×C₃₇⋊C₃	1	1	yes	characteristic
44	3₋¹⁺²	—	37	6	no	—
45	3₋¹⁺²	—	37	6	no	—
46	3₋¹⁺²	—	37	6	no	—
47	3₋¹⁺²	—	37	27	no	—
48	3₋¹⁺²	—	37	27	no	—
49	3₋¹⁺²	C₃×C₃₇⋊C₃	1	3	yes	—
50	3₋¹⁺²	C₃×C₃₇⋊C₃	1	6	yes	complemented
51	H(3,𝔽₃)	—	37	6	no	—
52	H(3,𝔽₃)	—	37	3	no	—
53	C₃×C₉	—	37	6	no	—
54	C₃×C₉	—	37	27	no	—
55	C₃×C₉	C₃×C₃₇⋊C₃	1	3	yes	complemented
56	C₃²	—	9	1	no	—
57	C₃²	—	37	9	no	—
58	C₃²	—	333	3	no	—
59	C₃²	C₃²×C₃₇⋊C₃	1	2	yes	—
60	C₃²	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
61	C₃²	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
62	C₉	—	37	18	no	—
63	C₉	C₃²×C₃₇⋊C₃	1	9	yes	—
64	C₃	—	3	1	no	—
65	C₃	—	3	2	no	—
66	C₃	—	3	1	no	—
67	C₃	—	111	9	no	—
68	C₃	C₃³×C₃₇⋊C₃	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₃	C₃³×C₃₇⋊C₃
commutator, socle	C₃×C₃₇	C₃⁴
Fitting	C₃×3₋¹⁺²×C₃₇	C₃
hypercenter	C₃×3₋¹⁺²	C₃₇⋊C₃
hypocenter	C₃₇	3₋¹⁺⁴
LCS	G ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃ ◁ C₃×3₋¹⁺²

Complements

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

N	H ≅ G/N	type
1. #15688532	C₃	semidirect
2. #15688531	C₃	semidirect
3. C₃×#11758608	C₃	semidirect
4. 3₋¹⁺²×C₃₇⋊C₃	C₃	semidirect
6. C₃×3₋¹⁺²×C₃₇	C₃	semidirect
10. 3₋¹⁺²×C₃₇	C₃²	semidirect
11. C₃×C₉×C₃₇	C₃²	semidirect
15. #11758608	C₃²	semidirect
16. #11758608	C₃²	semidirect
17. C₃×#66524	C₃²	semidirect
18. #11758606	C₃²	semidirect
19. #11758605	C₃²	semidirect
20. C₉×C₃₇⋊C₃	C₃²	semidirect
39. C₃×3₋¹⁺²	C₃₇⋊C₃	semidirect
41. C₃₇	3₋¹⁺⁴	semidirect
50. 3₋¹⁺²	C₃×C₃₇⋊C₃	semidirect
55. C₃×C₉	C₃×C₃₇⋊C₃	semidirect