Subgroups — #21419402

Order 7533.

136 conjugacy classes of subgroups.

Conjugacy classes of subgroups

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

	subgroup	G/H
center	C₃	C₃×C₃²⋊(C₃₁⋊C₃)
commutator	C₃²×C₃₁	C₃³
Frattini	C₃²	C₃²×C₃₁⋊C₃
Fitting	C₃×3₋¹⁺²×C₃₁	C₃
socle	C₃×C₃₁	C₃×H(3,𝔽₃)
hypercenter	C₃×3₋¹⁺²	C₃₁⋊C₃
hypocenter	C₃₁	#6910
LCS	G ▷ C₃²×C₃₁ ▷ C₃×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₃ ◁ C₃³ ◁ C₃×3₋¹⁺²

Complements

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

N	H ≅ G/N	type
1. C₃×#11719689	C₃	semidirect
2. C₃×#11719689	C₃	semidirect
4. #14578898	C₃	semidirect
5. C₃×3₋¹⁺²×C₃₁	C₃	semidirect
10. 3₋¹⁺²×C₃₁	C₃²	semidirect
20. #11719689	C₃²	semidirect
22. C₃×C₃₁⋊C₉	C₃²	semidirect
42. C₃×3₋¹⁺²	C₃₁⋊C₃	semidirect
45. C₃₁	#6910	semidirect
56. 3₋¹⁺²	C₃×C₃₁⋊C₃	semidirect