Subgroups — #21419405

Order 7533.

160 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×C₃²⋊(C₃₁⋊C₃)	C₃	1	2	yes	maximal, complemented
3	#14578899	C₃	1	9	yes	maximal, complemented
4	C₃×H(3,𝔽₃)×C₃₁	C₃	1	1	yes	maximal, characteristic, complemented
5	C₃³×C₃₁	—	3	1	no	—
6	C₃³×C₃₁	C₃²	1	1	yes	characteristic
7	C₃²×C₃₁⋊C₃	—	3	2	no	—
8	H(3,𝔽₃)×C₃₁	—	3	2	no	—
9	H(3,𝔽₃)×C₃₁	C₃²	1	3	yes	complemented
10	C₃²⋊(C₃₁⋊C₃)	—	3	2	no	—
11	C₃²⋊(C₃₁⋊C₃)	—	3	2	no	—
12	C₃²⋊(C₃₁⋊C₃)	C₃²	1	6	yes	complemented
13	#11719689	—	3	1	no	—
14	#11719689	—	3	2	no	—
15	C₃×C₃₁⋊C₉	C₃²	1	3	yes	complemented
16	C₃²×C₃₁	—	3	3	no	—
17	C₃²×C₃₁	—	9	1	no	—
18	C₃²×C₃₁	—	9	3	no	—
19	C₃²×C₃₁	H(3,𝔽₃)	1	1	yes	characteristic
20	C₃²×C₃₁	H(3,𝔽₃)	1	2	yes	—
21	C₃²×C₃₁	C₃³	1	1	yes	characteristic
22	C₃×C₃₁⋊C₃	—	3	6	no	—
23	C₃×C₃₁⋊C₃	—	9	6	no	—
24	C₃₁⋊C₉	—	3	3	no	—
25	#6911	—	31	1	no	maximal, Sylow
26	C₃×C₃₁	—	3	2	no	—
27	C₃×C₃₁	—	3	1	no	—
28	C₃×C₃₁	—	3	1	no	—
29	C₃×C₃₁	—	9	3	no	—
30	C₃×C₃₁	C₃×H(3,𝔽₃)	1	1	yes	characteristic
31	C₃₁⋊C₃	—	9	6	no	—
32	C₃×3₋¹⁺²	—	31	1	no	—
33	C₃×H(3,𝔽₃)	—	31	2	no	—
34	C₃×H(3,𝔽₃)	C₃₁⋊C₃	1	1	yes	characteristic, complemented
35	#732	—	31	9	no	—
36	C₃₁	#6911	1	1	yes	characteristic, complemented, Sylow
37	C₃³	—	3	1	no	—
38	C₃³	—	93	2	no	—
39	C₃³	C₃×C₃₁⋊C₃	1	1	yes	characteristic
40	3₋¹⁺²	—	93	2	no	—
41	3₋¹⁺²	—	93	1	no	—
42	H(3,𝔽₃)	—	3	2	no	—
43	H(3,𝔽₃)	—	31	6	no	—
44	H(3,𝔽₃)	—	93	2	no	—
45	H(3,𝔽₃)	—	93	2	no	—
46	H(3,𝔽₃)	C₃×C₃₁⋊C₃	1	3	yes	complemented
47	C₃×C₉	—	31	3	no	—
48	C₃²	—	3	3	no	—
49	C₃²	—	9	1	no	—
50	C₃²	—	9	3	no	—
51	C₃²	—	279	6	no	—
52	C₃²	—	93	6	no	—
53	C₃²	C₃²⋊(C₃₁⋊C₃)	1	2	yes	—
54	C₃²	C₃²⋊(C₃₁⋊C₃)	1	1	yes	characteristic
55	C₃²	C₃²×C₃₁⋊C₃	1	1	yes	characteristic
56	C₉	—	93	3	no	—
57	C₃	—	3	1	no	—
58	C₃	—	3	2	no	—
59	C₃	—	3	1	no	—
60	C₃	—	9	3	no	—
61	C₃	—	279	6	no	—
62	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×H(3,𝔽₃)×C₃₁	C₃
socle	C₃×C₃₁	C₃×H(3,𝔽₃)
hypercenter	C₃×H(3,𝔽₃)	C₃₁⋊C₃
hypocenter	C₃₁	#6911
LCS	G ▷ C₃²×C₃₁ ▷ C₃×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₃ ◁ C₃³ ◁ C₃×H(3,𝔽₃)

Complements

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

N	H ≅ G/N	type
1. C₃×#11719689	C₃	semidirect
2. C₃×C₃²⋊(C₃₁⋊C₃)	C₃	semidirect
3. #14578899	C₃	semidirect
4. C₃×H(3,𝔽₃)×C₃₁	C₃	semidirect
9. H(3,𝔽₃)×C₃₁	C₃²	semidirect
12. C₃²⋊(C₃₁⋊C₃)	C₃²	semidirect
15. C₃×C₃₁⋊C₉	C₃²	semidirect
34. C₃×H(3,𝔽₃)	C₃₁⋊C₃	semidirect
36. C₃₁	#6911	semidirect
46. H(3,𝔽₃)	C₃×C₃₁⋊C₃	semidirect