Subgroups — #21419398

Order 7533.

136 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#11719687	C₃	1	1	yes	maximal, characteristic, complemented
2	3₋¹⁺²×C₃₁⋊C₃	C₃	1	2	yes	maximal, complemented
3	H(3,𝔽₃)×C₃₁⋊C₃	C₃	1	1	yes	maximal, characteristic
4	#14578898	C₃	1	4	yes	maximal, complemented
5	#14578897	C₃	1	2	yes	maximal, complemented
6	#14578896	C₃	1	2	yes	maximal, complemented
7	C₃₁×#731	C₃	1	1	yes	maximal, characteristic, complemented
8	C₃²×C₃₁⋊C₃	—	3	1	no	—
9	C₃²×C₃₁⋊C₃	C₃²	1	1	yes	characteristic
10	3₋¹⁺²×C₃₁	C₃²	1	2	yes	complemented
11	H(3,𝔽₃)×C₃₁	C₃²	1	1	yes	characteristic
12	C₃×C₉×C₃₁	C₃²	1	1	yes	characteristic, complemented
13	C₃²⋊(C₃₁⋊C₃)	—	3	2	no	—
14	C₃²⋊(C₃₁⋊C₃)	C₃²	1	2	yes	—
15	#11719689	C₃²	1	4	yes	complemented
16	C₃×C₃₁⋊C₉	C₃²	1	2	yes	complemented
17	#11719687	—	3	2	no	—
18	#11719687	—	3	2	no	—
19	#11719687	—	3	1	no	—
20	#11719686	—	3	2	no	—
21	C₉×C₃₁⋊C₃	—	3	2	no	—
22	C₃²×C₃₁	—	3	1	no	—
23	C₃²×C₃₁	C₃³	1	1	yes	characteristic
24	C₃×C₃₁⋊C₃	—	3	2	no	—
25	C₃×C₃₁⋊C₃	—	9	1	no	—
26	C₃×C₃₁⋊C₃	—	9	3	no	—
27	C₃×C₃₁⋊C₃	H(3,𝔽₃)	1	2	yes	—
28	C₃×C₃₁⋊C₃	H(3,𝔽₃)	1	1	yes	characteristic
29	C₉×C₃₁	—	3	2	no	—
30	C₉×C₃₁	—	3	1	no	—
31	C₃₁⋊C₉	—	3	4	no	—
32	C₃₁⋊C₉	—	3	2	no	—
33	#6910	—	31	1	no	maximal, Sylow
34	C₃×C₃₁	—	3	1	no	—
35	C₃×C₃₁	—	9	1	no	—
36	C₃×C₃₁	C₃×H(3,𝔽₃)	1	1	yes	characteristic
37	C₃₁⋊C₃	—	3	1	no	—
38	C₃₁⋊C₃	—	3	2	no	—
39	C₃₁⋊C₃	—	9	2	no	—
40	C₃×3₋¹⁺²	—	31	1	no	—
41	C₃×3₋¹⁺²	—	31	2	no	—
42	C₃×H(3,𝔽₃)	—	31	1	no	—
43	#731	—	31	4	no	—
44	#731	—	31	2	no	—
45	#731	—	31	2	no	—
46	#731	C₃₁⋊C₃	1	1	yes	characteristic, complemented
47	C₃₁	#6910	1	1	yes	characteristic, complemented, Sylow
48	C₃³	—	31	1	no	—
49	C₃³	—	93	1	no	—
50	3₋¹⁺²	—	31	4	no	—
51	3₋¹⁺²	—	93	2	no	—
52	3₋¹⁺²	—	93	2	no	—
53	3₋¹⁺²	—	93	2	no	—
54	3₋¹⁺²	—	93	1	no	—
55	3₋¹⁺²	C₃×C₃₁⋊C₃	1	2	yes	complemented
56	H(3,𝔽₃)	—	31	2	no	—
57	H(3,𝔽₃)	—	93	2	no	—
58	H(3,𝔽₃)	C₃×C₃₁⋊C₃	1	1	yes	characteristic
59	C₃×C₉	—	31	2	no	—
60	C₃×C₉	—	93	2	no	—
61	C₃×C₉	C₃×C₃₁⋊C₃	1	1	yes	characteristic, complemented
62	C₃²	—	3	1	no	—
63	C₃²	—	279	1	no	—
64	C₃²	—	279	3	no	—
65	C₃²	—	31	2	no	—
66	C₃²	—	31	1	no	—
67	C₃²	—	93	2	no	—
68	C₃²	C₃²×C₃₁⋊C₃	1	1	yes	characteristic
69	C₉	—	3	2	no	—
70	C₉	—	3	1	no	—
71	C₉	—	93	2	no	—
72	C₉	—	93	4	no	—
73	C₃	—	3	1	no	—
74	C₃	—	9	1	no	—
75	C₃	—	279	2	no	—
76	C₃	—	93	1	no	—
77	C₃	—	93	2	no	—
78	C₃	H(3,𝔽₃)×C₃₁⋊C₃	1	1	yes	characteristic

	subgroup	G/H
center	C₃	H(3,𝔽₃)×C₃₁⋊C₃
commutator	C₃²×C₃₁	C₃³
Frattini	C₃²	C₃²×C₃₁⋊C₃
Fitting	C₃₁×#731	C₃
socle	C₃×C₃₁	C₃×H(3,𝔽₃)
hypercenter	#731	C₃₁⋊C₃
hypocenter	C₃₁	#6910
LCS	G ▷ C₃²×C₃₁ ▷ C₃×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₃ ◁ C₃² ◁ #731

Complements

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

N	H ≅ G/N	type
1. C₃×#11719687	C₃	semidirect
2. 3₋¹⁺²×C₃₁⋊C₃	C₃	semidirect
4. #14578898	C₃	semidirect
5. #14578897	C₃	semidirect
6. #14578896	C₃	semidirect
7. C₃₁×#731	C₃	semidirect
10. 3₋¹⁺²×C₃₁	C₃²	semidirect
12. C₃×C₉×C₃₁	C₃²	semidirect
15. #11719689	C₃²	semidirect
16. C₃×C₃₁⋊C₉	C₃²	semidirect
46. #731	C₃₁⋊C₃	semidirect
47. C₃₁	#6910	semidirect
55. 3₋¹⁺²	C₃×C₃₁⋊C₃	semidirect
61. C₃×C₉	C₃×C₃₁⋊C₃	semidirect