Subgroups — #21419419

Order 7533.

126 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14578912	C₃	1	3	yes	maximal, complemented
2	#14578902	C₃	1	2	yes	maximal, complemented
3	#14578900	C₃	1	1	yes	maximal, characteristic, complemented
4	#14578897	C₃	1	3	yes	maximal, complemented
5	#14578894	C₃	1	3	yes	maximal, complemented
6	C₃×3₋¹⁺²×C₃₁	C₃	1	1	yes	maximal, characteristic, complemented
7	C₃³×C₃₁	C₃²	1	1	yes	characteristic
8	3₋¹⁺²×C₃₁	—	3	3	no	—
9	C₃×C₉×C₃₁	C₃²	1	1	yes	characteristic
10	C₃×C₉×C₃₁	C₃²	1	1	yes	characteristic
11	C₃×C₉×C₃₁	C₃²	1	1	yes	characteristic
12	C₃²⋊(C₃₁⋊C₃)	C₃²	1	3	yes	—
13	#11719689	C₃²	1	6	yes	—
14	#11719687	—	3	3	no	—
15	#11719686	—	3	3	no	—
16	C₉×C₃₁⋊C₃	—	3	3	no	—
17	C₃²×C₃₁	—	3	1	no	—
18	C₃²×C₃₁	—	9	1	no	—
19	C₃²×C₃₁	C₃³	1	1	yes	characteristic
20	C₃×C₃₁⋊C₃	—	3	3	no	—
21	C₉×C₃₁	—	3	1	no	—
22	C₉×C₃₁	—	3	1	no	—
23	C₉×C₃₁	H(3,𝔽₃)	1	3	yes	—
24	C₃₁⋊C₉	—	3	6	no	—
25	#6913	—	31	1	no	maximal, Sylow
26	C₃×C₃₁	—	3	1	no	—
27	C₃×C₃₁	—	9	1	no	—
28	C₃×C₃₁	C₃×H(3,𝔽₃)	1	1	yes	characteristic
29	C₃₁⋊C₃	—	9	3	no	—
30	#737	—	31	3	no	—
31	C₃×3₋¹⁺²	C₃₁⋊C₃	1	1	yes	characteristic, complemented
32	#733	—	31	2	no	—
33	#732	—	31	1	no	—
34	#731	—	31	3	no	—
35	AΣL(1,𝔽₂₇)	—	31	3	no	—
36	C₃₁	#6913	1	1	yes	characteristic, complemented, Sylow
37	C₃³	C₃×C₃₁⋊C₃	1	1	yes	characteristic
38	3₋¹⁺²	—	3	3	no	—
39	3₋¹⁺²	—	31	6	no	—
40	3₋¹⁺²	—	93	3	no	—
41	3₋¹⁺²	—	93	3	no	—
42	H(3,𝔽₃)	—	31	3	no	—
43	C₃×C₉	—	93	3	no	—
44	C₃×C₉	C₃×C₃₁⋊C₃	1	1	yes	characteristic
45	C₃×C₉	C₃×C₃₁⋊C₃	1	1	yes	characteristic
46	C₃×C₉	C₃×C₃₁⋊C₃	1	1	yes	characteristic
47	C₃²	—	3	1	no	—
48	C₃²	—	9	1	no	—
49	C₃²	—	93	3	no	—
50	C₃²	C₃²×C₃₁⋊C₃	1	1	yes	characteristic
51	C₉	—	3	1	no	—
52	C₉	—	3	1	no	—
53	C₉	—	93	6	no	—
54	C₉	C₃²⋊(C₃₁⋊C₃)	1	3	yes	—
55	C₃	—	3	1	no	—
56	C₃	—	9	1	no	—
57	C₃	—	279	3	no	—
58	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	#6913
LCS	G ▷ C₃²×C₃₁ ▷ C₃×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₃ ◁ C₃×C₉ ◁ C₃×3₋¹⁺²

Complements

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

N	H ≅ G/N	type
1. #14578912	C₃	semidirect
2. #14578902	C₃	semidirect
3. #14578900	C₃	semidirect
4. #14578897	C₃	semidirect
5. #14578894	C₃	semidirect
6. C₃×3₋¹⁺²×C₃₁	C₃	semidirect
31. C₃×3₋¹⁺²	C₃₁⋊C₃	semidirect
36. C₃₁	#6913	semidirect