Subgroups — #23748567

Order 9477.

1062 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃⋊H(5,𝔽₃)	C₃	1	3	yes	maximal, complemented
2	#15691666	C₃	1	24	yes	maximal, complemented
3	C₃×#11760441	C₃	1	12	yes	maximal, complemented
4	C₃×C₁₃×#737	C₃	1	1	yes	maximal, characteristic, complemented
5	C₃×C₃²⋊(C₁₃⋊C₃)	C₃²	1	36	yes	complemented
6	H(3,𝔽₃)×C₁₃⋊C₃	—	3	27	no	—
7	#11760441	C₃²	1	9	yes	complemented
8	#11760441	C₃²	1	72	yes	complemented
9	C₁₃×#737	—	3	9	no	—
10	C₃×C₁₃×3₋¹⁺²	C₃²	1	8	yes	complemented
11	C₃×C₁₃×H(3,𝔽₃)	C₃²	1	1	yes	characteristic
12	C₃²×C₉×C₁₃	C₃²	1	4	yes	complemented
13	#10624932	—	13	1	no	maximal, Sylow
14	C₃³×C₁₃	C₃³	1	4	yes	—
15	C₃²×C₁₃⋊C₃	—	3	36	no	—
16	C₁₃×3₋¹⁺²	—	3	24	no	—
17	C₁₃×H(3,𝔽₃)	H(3,𝔽₃)	1	9	yes	—
18	C₃×C₉×C₁₃	—	3	12	no	—
19	C₃×C₉×C₁₃	C₃³	1	1	yes	characteristic
20	C₃×C₉×C₁₃	C₃³	1	8	yes	—
21	C₃²⋊(C₁₃⋊C₃)	—	3	72	no	—
22	C₃²⋊(C₁₃⋊C₃)	C₃³	1	27	yes	—
23	H(5,𝔽₃)	—	13	3	no	—
24	C₃×#737	C₁₃⋊C₃	1	1	yes	characteristic, complemented
25	#6911	—	13	24	no	—
26	C₃×#732	—	13	12	no	—
27	C₃²×C₁₃	—	9	4	no	—
28	C₃²×C₁₃	C₃×H(3,𝔽₃)	1	12	yes	—
29	C₃²×C₁₃	C₃⁴	1	1	yes	characteristic
30	C₃×C₁₃⋊C₃	—	3	27	no	—
31	C₃×C₁₃⋊C₃	—	27	36	no	—
32	C₉×C₁₃	—	3	1	no	—
33	C₉×C₁₃	—	3	8	no	—
34	#737	—	3	9	no	—
35	C₃×3₋¹⁺²	C₃×C₁₃⋊C₃	1	8	yes	complemented
36	C₃×H(3,𝔽₃)	—	13	36	no	—
37	C₃×H(3,𝔽₃)	—	39	27	no	—
38	C₃×H(3,𝔽₃)	C₃×C₁₃⋊C₃	1	1	yes	characteristic
39	C₃²×C₉	C₃×C₁₃⋊C₃	1	4	yes	complemented
40	#732	—	13	9	no	—
41	#732	—	13	72	no	—
42	C₃×C₁₃	—	3	12	no	—
43	C₃×C₁₃	—	3	1	no	—
44	C₃×C₁₃	C₃²×H(3,𝔽₃)	1	1	yes	characteristic
45	C₁₃⋊C₃	—	9	27	no	—
46	C₃³	—	39	36	no	—
47	C₃³	C₃²×C₁₃⋊C₃	1	4	yes	—
48	3₋¹⁺²	—	3	24	no	—
49	H(3,𝔽₃)	—	13	27	no	—
50	H(3,𝔽₃)	—	39	72	no	—
51	H(3,𝔽₃)	C₃²⋊(C₁₃⋊C₃)	1	9	yes	—
52	C₃×C₉	—	3	12	no	—
53	C₃×C₉	C₃²×C₁₃⋊C₃	1	1	yes	characteristic
54	C₃×C₉	C₃²×C₁₃⋊C₃	1	8	yes	—
55	C₁₃	#10624932	1	1	yes	characteristic, complemented, Sylow
56	C₃²	—	9	4	no	—
57	C₃²	—	39	27	no	—
58	C₃²	—	351	36	no	—
59	C₃²	C₃×C₃²⋊(C₁₃⋊C₃)	1	12	yes	—
60	C₃²	C₃³×C₁₃⋊C₃	1	1	yes	characteristic
61	C₉	—	3	1	no	—
62	C₉	—	3	8	no	—
63	C₃	—	3	1	no	—
64	C₃	—	3	12	no	—
65	C₃	—	117	27	no	—
66	C₃	C₃²×C₃²⋊(C₁₃⋊C₃)	1	1	yes	characteristic

	subgroup	G/H
center	C₃	C₃²×C₃²⋊(C₁₃⋊C₃)
commutator, cosocle	C₃²×C₁₃	C₃⁴
Frattini	C₃²	C₃³×C₁₃⋊C₃
Fitting	C₃×C₁₃×#737	C₃
socle	C₃×C₁₃	C₃²×H(3,𝔽₃)
hypercenter	C₃×#737	C₁₃⋊C₃
hypocenter	C₁₃	#10624932
LCS	G ▷ C₃²×C₁₃ ▷ C₃×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₃ ◁ C₃×H(3,𝔽₃) ◁ C₃×#737

Complements

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

N	H ≅ G/N	type
1. C₁₃⋊H(5,𝔽₃)	C₃	semidirect
2. #15691666	C₃	semidirect
3. C₃×#11760441	C₃	semidirect
4. C₃×C₁₃×#737	C₃	semidirect
5. C₃×C₃²⋊(C₁₃⋊C₃)	C₃²	semidirect
7. #11760441	C₃²	semidirect
8. #11760441	C₃²	semidirect
10. C₃×C₁₃×3₋¹⁺²	C₃²	semidirect
12. C₃²×C₉×C₁₃	C₃²	semidirect
24. C₃×#737	C₁₃⋊C₃	semidirect
35. C₃×3₋¹⁺²	C₃×C₁₃⋊C₃	semidirect
39. C₃²×C₉	C₃×C₁₃⋊C₃	semidirect
55. C₁₃	#10624932	semidirect