Subgroups — #21419404

Order 7533.

160 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	H(3,𝔽₃)×C₃₁⋊C₃	C₃	1	3	yes	maximal, complemented
3	#14578900	C₃	1	2	yes	maximal, complemented
4	#14578899	C₃	1	6	yes	maximal, complemented
5	C₃₁×#732	C₃	1	1	yes	maximal, characteristic, complemented
6	C₃²×C₃₁⋊C₃	—	3	3	no	—
7	C₃²×C₃₁⋊C₃	C₃²	1	1	yes	characteristic
8	H(3,𝔽₃)×C₃₁	C₃²	1	3	yes	complemented
9	C₃×C₉×C₃₁	C₃²	1	1	yes	characteristic, complemented
10	C₃²⋊(C₃₁⋊C₃)	—	3	6	no	—
11	C₃²⋊(C₃₁⋊C₃)	C₃²	1	6	yes	complemented
12	C₃×C₃₁⋊C₉	C₃²	1	2	yes	complemented
13	#11719687	—	3	3	no	—
14	C₃²×C₃₁	—	3	3	no	—
15	C₃²×C₃₁	C₃³	1	1	yes	characteristic
16	C₃×C₃₁⋊C₃	—	3	6	no	—
17	C₃×C₃₁⋊C₃	—	9	1	no	—
18	C₃×C₃₁⋊C₃	—	9	9	no	—
19	C₃×C₃₁⋊C₃	H(3,𝔽₃)	1	3	yes	—
20	C₉×C₃₁	—	3	1	no	—
21	C₃₁⋊C₉	—	3	2	no	—
22	#6911	—	31	1	no	maximal, Sylow
23	C₃×C₃₁	—	3	1	no	—
24	C₃×C₃₁	—	9	3	no	—
25	C₃×C₃₁	C₃×H(3,𝔽₃)	1	1	yes	characteristic
26	C₃₁⋊C₃	—	3	3	no	—
27	C₃₁⋊C₃	—	9	6	no	—
28	C₃×3₋¹⁺²	—	31	1	no	—
29	C₃×H(3,𝔽₃)	—	31	3	no	—
30	#732	—	31	6	no	—
31	#732	—	31	2	no	—
32	#732	C₃₁⋊C₃	1	1	yes	characteristic, complemented
33	C₃₁	#6911	1	1	yes	characteristic, complemented, Sylow
34	C₃³	—	31	1	no	—
35	C₃³	—	93	3	no	—
36	3₋¹⁺²	—	93	3	no	—
37	H(3,𝔽₃)	—	31	6	no	—
38	H(3,𝔽₃)	—	93	6	no	—
39	H(3,𝔽₃)	C₃×C₃₁⋊C₃	1	3	yes	complemented
40	C₃×C₉	—	31	2	no	—
41	C₃×C₉	C₃×C₃₁⋊C₃	1	1	yes	characteristic, complemented
42	C₃²	—	3	3	no	—
43	C₃²	—	279	1	no	—
44	C₃²	—	279	9	no	—
45	C₃²	—	31	3	no	—
46	C₃²	—	93	6	no	—
47	C₃²	C₃²×C₃₁⋊C₃	1	1	yes	characteristic
48	C₉	—	3	1	no	—
49	C₉	—	93	2	no	—
50	C₃	—	3	1	no	—
51	C₃	—	9	3	no	—
52	C₃	—	279	6	no	—
53	C₃	—	93	3	no	—
54	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×#732	C₃
socle	C₃×C₃₁	C₃×H(3,𝔽₃)
hypercenter	#732	C₃₁⋊C₃
hypocenter	C₃₁	#6911
LCS	G ▷ C₃²×C₃₁ ▷ C₃×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₃ ◁ C₃² ◁ #732

Complements

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

N	H ≅ G/N	type
1. C₃×#11719687	C₃	semidirect
2. H(3,𝔽₃)×C₃₁⋊C₃	C₃	semidirect
3. #14578900	C₃	semidirect
4. #14578899	C₃	semidirect
5. C₃₁×#732	C₃	semidirect
8. H(3,𝔽₃)×C₃₁	C₃²	semidirect
9. C₃×C₉×C₃₁	C₃²	semidirect
11. C₃²⋊(C₃₁⋊C₃)	C₃²	semidirect
12. C₃×C₃₁⋊C₉	C₃²	semidirect
32. #732	C₃₁⋊C₃	semidirect
33. C₃₁	#6911	semidirect
39. H(3,𝔽₃)	C₃×C₃₁⋊C₃	semidirect
41. C₃×C₉	C₃×C₃₁⋊C₃	semidirect