Subgroups — #21419435

Order 7533.

460 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14578913	C₃	1	16	yes	maximal, complemented
2	#14578912	C₃	1	8	yes	maximal, complemented
3	#14578911	C₃	1	2	yes	maximal, complemented
4	C₃×#11719687	C₃	1	4	yes	maximal, complemented
5	3₋¹⁺²×C₃₁⋊C₃	C₃	1	8	yes	maximal, complemented
6	H(3,𝔽₃)×C₃₁⋊C₃	C₃	1	1	yes	maximal, characteristic
7	C₃₁×#737	C₃	1	1	yes	maximal, characteristic, complemented
8	C₃²×C₃₁⋊C₃	C₃²	1	4	yes	—
9	3₋¹⁺²×C₃₁	C₃²	1	8	yes	complemented
10	H(3,𝔽₃)×C₃₁	C₃²	1	1	yes	characteristic
11	C₃×C₉×C₃₁	C₃²	1	4	yes	complemented
12	C₃²⋊(C₃₁⋊C₃)	C₃²	1	8	yes	—
13	#11719689	C₃²	1	16	yes	complemented
14	C₃×C₃₁⋊C₉	C₃²	1	8	yes	complemented
15	#11719687	C₃²	1	8	yes	—
16	#11719687	C₃²	1	1	yes	characteristic
17	#11719687	C₃²	1	8	yes	complemented
18	#11719687	C₃²	1	16	yes	complemented
19	#11719686	C₃²	1	24	yes	complemented
20	C₉×C₃₁⋊C₃	C₃²	1	24	yes	complemented
21	C₃²×C₃₁	C₃³	1	4	yes	—
22	C₃×C₃₁⋊C₃	—	9	4	no	—
23	C₃×C₃₁⋊C₃	C₃³	1	8	yes	—
24	C₃×C₃₁⋊C₃	C₃³	1	1	yes	characteristic
25	C₉×C₃₁	C₃³	1	8	yes	—
26	C₉×C₃₁	C₃³	1	1	yes	characteristic
27	C₃₁⋊C₉	C₃³	1	16	yes	—
28	C₃₁⋊C₉	C₃³	1	2	yes	—
29	3₋¹⁺⁴	—	31	1	no	maximal, Sylow
30	C₃×C₃₁	—	3	4	no	—
31	C₃×C₃₁	C₃⁴	1	1	yes	characteristic
32	C₃₁⋊C₃	—	3	1	no	—
33	C₃₁⋊C₃	—	3	8	no	—
34	#737	—	31	16	no	—
35	#737	—	31	2	no	—
36	#737	—	31	8	no	—
37	#737	C₃₁⋊C₃	1	1	yes	characteristic, complemented
38	C₃×3₋¹⁺²	—	31	8	no	—
39	C₃×3₋¹⁺²	—	31	4	no	—
40	C₃×H(3,𝔽₃)	—	31	1	no	—
41	C₃₁	3₋¹⁺⁴	1	1	yes	characteristic, complemented, Sylow
42	C₃³	—	31	4	no	—
43	3₋¹⁺²	—	31	16	no	—
44	3₋¹⁺²	—	31	24	no	—
45	3₋¹⁺²	—	31	8	no	—
46	3₋¹⁺²	—	31	1	no	—
47	3₋¹⁺²	—	31	16	no	—
48	3₋¹⁺²	—	31	8	no	—
49	3₋¹⁺²	C₃×C₃₁⋊C₃	1	8	yes	complemented
50	H(3,𝔽₃)	—	31	8	no	—
51	H(3,𝔽₃)	C₃×C₃₁⋊C₃	1	1	yes	characteristic
52	C₃×C₉	—	31	8	no	—
53	C₃×C₉	—	31	24	no	—
54	C₃×C₉	C₃×C₃₁⋊C₃	1	4	yes	complemented
55	C₃²	—	279	4	no	—
56	C₃²	—	31	8	no	—
57	C₃²	—	31	1	no	—
58	C₃²	C₃²×C₃₁⋊C₃	1	4	yes	—
59	C₉	—	31	16	no	—
60	C₉	—	31	2	no	—
61	C₉	C₃²×C₃₁⋊C₃	1	8	yes	—
62	C₉	C₃²×C₃₁⋊C₃	1	1	yes	characteristic
63	C₃	—	3	4	no	—
64	C₃	—	93	1	no	—
65	C₃	—	93	8	no	—
66	C₃	C₃³×C₃₁⋊C₃	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₃	C₃³×C₃₁⋊C₃
commutator, socle	C₃×C₃₁	C₃⁴
Fitting	C₃₁×#737	C₃
hypercenter	#737	C₃₁⋊C₃
hypocenter	C₃₁	3₋¹⁺⁴
LCS	G ▷ C₃×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₃ ◁ #737

Complements

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

N	H ≅ G/N	type
1. #14578913	C₃	semidirect
2. #14578912	C₃	semidirect
3. #14578911	C₃	semidirect
4. C₃×#11719687	C₃	semidirect
5. 3₋¹⁺²×C₃₁⋊C₃	C₃	semidirect
7. C₃₁×#737	C₃	semidirect
9. 3₋¹⁺²×C₃₁	C₃²	semidirect
11. C₃×C₉×C₃₁	C₃²	semidirect
13. #11719689	C₃²	semidirect
14. C₃×C₃₁⋊C₉	C₃²	semidirect
17. #11719687	C₃²	semidirect
18. #11719687	C₃²	semidirect
19. #11719686	C₃²	semidirect
20. C₉×C₃₁⋊C₃	C₃²	semidirect
37. #737	C₃₁⋊C₃	semidirect
41. C₃₁	3₋¹⁺⁴	semidirect
49. 3₋¹⁺²	C₃×C₃₁⋊C₃	semidirect
54. C₃×C₉	C₃×C₃₁⋊C₃	semidirect