Subgroups — #22582552

Order 8991.

252 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃³×C₃₇⋊C₃	C₃	1	1	yes	maximal, characteristic
2	C₃×#11758606	C₃	1	3	yes	maximal, complemented
3	C₃²⋊#66524	C₃	1	2	yes	maximal, complemented
4	#15688502	C₃	1	6	yes	maximal, complemented
5	C₃₇×C₃²⋊C₉	C₃	1	1	yes	maximal, characteristic, complemented
6	C₃³×C₃₇	C₃²	1	1	yes	characteristic
7	C₃²×C₃₇⋊C₃	—	3	3	no	—
8	C₃²×C₃₇⋊C₃	—	3	3	no	—
9	C₃²×C₃₇⋊C₃	—	3	6	no	—
10	C₃²×C₃₇⋊C₃	C₃²	1	1	yes	characteristic
11	C₃²×C₃₇⋊C₃	C₃²	1	2	yes	—
12	C₃×C₉×C₃₇	C₃²	1	3	yes	complemented
13	C₃×#66524	C₃²	1	6	yes	complemented
14	#11758606	—	3	9	no	—
15	C₃²×C₃₇	—	3	1	no	—
16	C₃²×C₃₇	—	3	2	no	—
17	C₃²×C₃₇	3₋¹⁺²	1	3	yes	complemented
18	C₃²×C₃₇	C₃³	1	1	yes	characteristic
19	C₃×C₃₇⋊C₃	—	3	27	no	—
20	C₃×C₃₇⋊C₃	—	3	2	no	—
21	C₃×C₃₇⋊C₃	—	3	2	no	—
22	C₃×C₃₇⋊C₃	—	3	2	no	—
23	C₃×C₃₇⋊C₃	—	3	1	no	—
24	C₃×C₃₇⋊C₃	—	3	2	no	—
25	C₃×C₃₇⋊C₃	H(3,𝔽₃)	1	3	yes	—
26	C₃×C₃₇⋊C₃	3₋¹⁺²	1	6	yes	complemented
27	C₉×C₃₇	—	3	3	no	—
28	#66524	—	3	6	no	—
29	#6891	—	37	1	no	maximal, Sylow
30	C₃×C₃₇	—	3	3	no	—
31	C₃×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
32	C₃×C₃₇	C₃×3₋¹⁺²	1	2	yes	—
33	C₃×C₃₇	C₃×3₋¹⁺²	1	1	yes	characteristic
34	C₃₇⋊C₃	—	3	3	no	—
35	C₃₇⋊C₃	—	3	6	no	—
36	C₃⁴	—	37	1	no	—
37	C₃×3₋¹⁺²	—	37	3	no	—
38	C₃²⋊C₉	—	37	2	no	—
39	C₃²⋊C₉	—	37	6	no	—
40	C₃²⋊C₉	C₃₇⋊C₃	1	1	yes	characteristic, complemented
41	C₃₇	#6891	1	1	yes	characteristic, complemented, Sylow
42	C₃³	—	37	2	no	—
43	C₃³	—	37	1	no	—
44	C₃³	—	111	3	no	—
45	C₃³	—	111	6	no	—
46	C₃³	—	111	3	no	—
47	C₃³	C₃×C₃₇⋊C₃	1	1	yes	characteristic
48	3₋¹⁺²	—	111	9	no	—
49	C₃×C₉	—	37	6	no	—
50	C₃×C₉	C₃×C₃₇⋊C₃	1	3	yes	complemented
51	C₃²	—	3	1	no	—
52	C₃²	—	3	2	no	—
53	C₃²	—	37	3	no	—
54	C₃²	—	37	6	no	—
55	C₃²	—	111	27	no	—
56	C₃²	—	111	2	no	—
57	C₃²	—	111	2	no	—
58	C₃²	—	111	2	no	—
59	C₃²	—	111	2	no	—
60	C₃²	—	111	1	no	—
61	C₃²	#11758606	1	3	yes	complemented
62	C₃²	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
63	C₉	—	3	3	no	—
64	C₉	—	111	6	no	—
65	C₃	—	3	3	no	—
66	C₃	—	111	3	no	—
67	C₃	—	111	6	no	—
68	C₃	H(3,𝔽₃)×C₃₇⋊C₃	1	1	yes	characteristic
69	C₃	3₋¹⁺²×C₃₇⋊C₃	1	2	yes	—
70	C₃	C₃×#11758606	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₃²	C₃²×C₃₇⋊C₃
commutator, socle	C₃²×C₃₇	C₃³
Fitting	C₃₇×C₃²⋊C₉	C₃
hypercenter	C₃²⋊C₉	C₃₇⋊C₃
hypocenter	C₃₇	#6891
LCS	G ▷ C₃²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃² ◁ C₃²⋊C₉

Complements

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

N	H ≅ G/N	type
2. C₃×#11758606	C₃	semidirect
3. C₃²⋊#66524	C₃	semidirect
4. #15688502	C₃	semidirect
5. C₃₇×C₃²⋊C₉	C₃	semidirect
12. C₃×C₉×C₃₇	C₃²	semidirect
13. C₃×#66524	C₃²	semidirect
17. C₃²×C₃₇	3₋¹⁺²	semidirect
26. C₃×C₃₇⋊C₃	3₋¹⁺²	semidirect
40. C₃²⋊C₉	C₃₇⋊C₃	semidirect
41. C₃₇	#6891	semidirect
50. C₃×C₉	C₃×C₃₇⋊C₃	semidirect
61. C₃²	#11758606	semidirect