Subgroups — #21419314

Order 7533.

166 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#11719689	C₃	1	3	yes	maximal, complemented
2	C₃²×C₃₁⋊C₉	C₃	1	1	yes	maximal, characteristic, complemented
3	C₃²⋊(C₃₁⋊C₉)	C₃	1	6	yes	maximal, complemented
4	C₃²⋊(C₃₁⋊C₉)	C₃	1	2	yes	maximal, complemented
5	C₃×H(3,𝔽₃)×C₃₁	C₃	1	1	yes	maximal, characteristic
6	C₃³×C₃₁	C₃²	1	3	yes	—
7	C₃³×C₃₁	C₃²	1	1	yes	characteristic
8	H(3,𝔽₃)×C₃₁	—	3	3	no	—
9	#11719689	—	3	9	no	—
10	C₃×C₃₁⋊C₉	—	3	3	no	—
11	C₃×C₃₁⋊C₉	C₃²	1	2	yes	—
12	C₃×C₃₁⋊C₉	C₃²	1	6	yes	—
13	C₃×C₃₁⋊C₉	C₃²	1	1	yes	characteristic
14	C₃²×C₃₁	—	3	1	no	—
15	C₃²×C₃₁	—	3	2	no	—
16	C₃²×C₃₁	—	3	3	no	—
17	C₃²×C₃₁	—	3	3	no	—
18	C₃²×C₃₁	—	3	6	no	—
19	C₃²×C₃₁	3₋¹⁺²	1	3	yes	complemented
20	C₃²×C₃₁	C₃³	1	1	yes	characteristic
21	C₃₁⋊C₉	—	3	2	no	—
22	C₃₁⋊C₉	—	3	6	no	—
23	C₃₁⋊C₉	H(3,𝔽₃)	1	3	yes	complemented
24	C₉⋊H(3,𝔽₃)	—	31	1	no	maximal, Sylow
25	C₃×C₃₁	—	3	3	no	—
26	C₃×C₃₁	—	9	3	no	—
27	C₃×C₃₁	C₃×H(3,𝔽₃)	1	1	yes	characteristic
28	C₃×C₃₁	C₃×3₋¹⁺²	1	1	yes	characteristic
29	C₃×C₃₁	#737	1	2	yes	—
30	C₃×3₋¹⁺²	—	31	3	no	—
31	C₃×H(3,𝔽₃)	C₃₁⋊C₃	1	1	yes	characteristic
32	C₃²×C₉	—	31	1	no	—
33	C₃²⋊C₉	—	31	6	no	—
34	C₃²⋊C₉	—	31	2	no	—
35	C₃₁	C₉⋊H(3,𝔽₃)	1	1	yes	characteristic, complemented, Sylow
36	C₃³	C₃×C₃₁⋊C₃	1	3	yes	—
37	C₃³	C₃×C₃₁⋊C₃	1	1	yes	characteristic
38	3₋¹⁺²	—	93	9	no	—
39	H(3,𝔽₃)	—	3	3	no	—
40	C₃×C₉	—	31	1	no	—
41	C₃×C₉	—	31	2	no	—
42	C₃×C₉	—	31	6	no	—
43	C₃×C₉	—	93	3	no	—
44	C₃²	—	3	1	no	—
45	C₃²	—	3	2	no	—
46	C₃²	—	3	3	no	—
47	C₃²	—	3	3	no	—
48	C₃²	—	3	6	no	—
49	C₃²	#11719689	1	3	yes	complemented
50	C₃²	C₃²×C₃₁⋊C₃	1	1	yes	characteristic
51	C₉	—	31	3	no	—
52	C₉	—	93	2	no	—
53	C₉	—	93	6	no	—
54	C₃	—	3	3	no	—
55	C₃	—	9	3	no	—
56	C₃	H(3,𝔽₃)×C₃₁⋊C₃	1	1	yes	characteristic
57	C₃	C₃×#11719689	1	1	yes	characteristic
58	C₃	#14578911	1	2	yes	—

	subgroup	G/H
center, Frattini	C₃²	C₃²×C₃₁⋊C₃
commutator, socle	C₃²×C₃₁	C₃³
Fitting	C₃×H(3,𝔽₃)×C₃₁	C₃
hypercenter	C₃×H(3,𝔽₃)	C₃₁⋊C₃
hypocenter	C₃₁	C₉⋊H(3,𝔽₃)
LCS	G ▷ C₃²×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₃² ◁ 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. C₃²×C₃₁⋊C₉	C₃	semidirect
3. C₃²⋊(C₃₁⋊C₉)	C₃	semidirect
4. C₃²⋊(C₃₁⋊C₉)	C₃	semidirect
19. C₃²×C₃₁	3₋¹⁺²	semidirect
23. C₃₁⋊C₉	H(3,𝔽₃)	semidirect
35. C₃₁	C₉⋊H(3,𝔽₃)	semidirect
49. C₃²	#11719689	semidirect