Subgroups — #21419309

Order 7533.

252 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃³×C₃₁⋊C₃	C₃	1	1	yes	maximal, characteristic
2	C₃×#11719689	C₃	1	2	yes	maximal, complemented
3	(C₃×C₃₁⋊C₃)⋊C₉	C₃	1	9	yes	maximal, complemented
4	C₃×3₋¹⁺²×C₃₁	C₃	1	1	yes	maximal, characteristic, complemented
5	C₃³×C₃₁	C₃²	1	1	yes	characteristic
6	C₃²×C₃₁⋊C₃	—	3	3	no	—
7	C₃²×C₃₁⋊C₃	—	3	9	no	—
8	C₃²×C₃₁⋊C₃	C₃²	1	3	yes	—
9	3₋¹⁺²×C₃₁	—	3	3	no	—
10	C₃×C₉×C₃₁	C₃²	1	3	yes	complemented
11	#11719689	—	3	6	no	—
12	C₃×C₃₁⋊C₉	C₃²	1	6	yes	complemented
13	C₃²×C₃₁	—	3	3	no	—
14	C₃²×C₃₁	H(3,𝔽₃)	1	3	yes	—
15	C₃²×C₃₁	C₃³	1	1	yes	characteristic
16	C₃×C₃₁⋊C₃	—	3	27	no	—
17	C₃×C₃₁⋊C₃	—	3	3	no	—
18	C₃×C₃₁⋊C₃	—	3	6	no	—
19	C₃×C₃₁⋊C₃	3₋¹⁺²	1	9	yes	complemented
20	C₉×C₃₁	—	3	3	no	—
21	C₃₁⋊C₉	—	3	6	no	—
22	#6891	—	31	1	no	maximal, Sylow
23	C₃×C₃₁	—	3	3	no	—
24	C₃×C₃₁	C₃×H(3,𝔽₃)	1	1	yes	characteristic
25	C₃×C₃₁	C₃×3₋¹⁺²	1	3	yes	—
26	C₃₁⋊C₃	—	3	9	no	—
27	C₃⁴	—	31	1	no	—
28	C₃×3₋¹⁺²	—	31	2	no	—
29	C₃×3₋¹⁺²	C₃₁⋊C₃	1	1	yes	characteristic, complemented
30	C₃²⋊C₉	—	31	9	no	—
31	C₃₁	#6891	1	1	yes	characteristic, complemented, Sylow
32	C₃³	—	31	3	no	—
33	C₃³	—	93	9	no	—
34	C₃³	—	93	3	no	—
35	C₃³	C₃×C₃₁⋊C₃	1	1	yes	characteristic
36	3₋¹⁺²	—	3	3	no	—
37	3₋¹⁺²	—	93	6	no	—
38	C₃×C₉	—	31	6	no	—
39	C₃×C₉	C₃×C₃₁⋊C₃	1	3	yes	complemented
40	C₃²	—	3	3	no	—
41	C₃²	—	31	9	no	—
42	C₃²	—	93	27	no	—
43	C₃²	—	93	3	no	—
44	C₃²	—	93	6	no	—
45	C₃²	C₃²⋊(C₃₁⋊C₃)	1	3	yes	—
46	C₃²	C₃²×C₃₁⋊C₃	1	1	yes	characteristic
47	C₉	—	3	3	no	—
48	C₉	—	93	6	no	—
49	C₃	—	3	3	no	—
50	C₃	—	93	9	no	—
51	C₃	C₃×C₃²⋊(C₃₁⋊C₃)	1	1	yes	characteristic
52	C₃	3₋¹⁺²×C₃₁⋊C₃	1	3	yes	—

	subgroup	G/H
center, Frattini	C₃²	C₃²×C₃₁⋊C₃
commutator, socle	C₃²×C₃₁	C₃³
Fitting	C₃×3₋¹⁺²×C₃₁	C₃
hypercenter	C₃×3₋¹⁺²	C₃₁⋊C₃
hypocenter	C₃₁	#6891
LCS	G ▷ C₃²×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₃² ◁ C₃×3₋¹⁺²

Complements

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

N	H ≅ G/N	type
2. C₃×#11719689	C₃	semidirect
3. (C₃×C₃₁⋊C₃)⋊C₉	C₃	semidirect
4. C₃×3₋¹⁺²×C₃₁	C₃	semidirect
10. C₃×C₉×C₃₁	C₃²	semidirect
12. C₃×C₃₁⋊C₉	C₃²	semidirect
19. C₃×C₃₁⋊C₃	3₋¹⁺²	semidirect
29. C₃×3₋¹⁺²	C₃₁⋊C₃	semidirect
31. C₃₁	#6891	semidirect
39. C₃×C₉	C₃×C₃₁⋊C₃	semidirect