Subgroups — #23748560

Order 9477.

978 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#15691698	C₃	1	2	yes	maximal, complemented
2	C₃×#11760452	C₃	1	1	yes	maximal, characteristic, complemented
3	#15691660	C₃	1	24	yes	maximal, complemented
4	C₃×#11760437	C₃	1	12	yes	maximal, complemented
5	C₁₃×H(5,𝔽₃)	C₃	1	1	yes	maximal, characteristic
6	#11760452	—	3	9	no	—
7	#11760452	—	3	18	no	—
8	C₃×#66851	C₃²	1	24	yes	complemented
9	C₃×#66851	C₃²	1	8	yes	complemented
10	C₃²×C₁₃⋊C₉	C₃²	1	4	yes	complemented
11	#11760437	C₃²	1	9	yes	complemented
12	#11760437	C₃²	1	72	yes	complemented
13	C₃×C₁₃×H(3,𝔽₃)	—	3	9	no	—
14	C₃×C₁₃×H(3,𝔽₃)	C₃²	1	12	yes	—
15	C₃×C₁₃×H(3,𝔽₃)	C₃²	1	1	yes	characteristic
16	#10624931	—	13	1	no	maximal, Sylow
17	C₃³×C₁₃	—	3	12	no	—
18	C₃³×C₁₃	C₃³	1	4	yes	—
19	C₁₃×H(3,𝔽₃)	—	3	24	no	—
20	C₁₃×H(3,𝔽₃)	H(3,𝔽₃)	1	9	yes	—
21	C₁₃×H(3,𝔽₃)	C₃³	1	9	yes	—
22	#66851	—	3	48	no	—
23	#66851	—	3	24	no	—
24	#66851	C₃³	1	18	yes	—
25	C₃×C₁₃⋊C₉	—	3	12	no	—
26	C₃×C₁₃⋊C₉	—	3	24	no	—
27	C₃×C₁₃⋊C₉	C₃³	1	1	yes	characteristic
28	C₃×C₁₃⋊C₉	C₃³	1	8	yes	—
29	3₋¹⁺⁴	—	13	2	no	—
30	H(5,𝔽₃)	C₁₃⋊C₃	1	1	yes	characteristic
31	C₃×#737	—	13	1	no	—
32	#6910	—	13	24	no	—
33	C₃×#731	—	13	12	no	—
34	C₃²×C₁₃	—	3	9	no	—
35	C₃²×C₁₃	—	9	4	no	—
36	C₃²×C₁₃	—	27	12	no	—
37	C₃²×C₁₃	C₃×H(3,𝔽₃)	1	12	yes	—
38	C₃²×C₁₃	C₃⁴	1	1	yes	characteristic
39	C₁₃⋊C₉	—	3	1	no	—
40	C₁₃⋊C₉	—	3	18	no	—
41	C₁₃⋊C₉	—	3	8	no	—
42	#737	—	39	9	no	—
43	#737	—	39	18	no	—
44	C₃×3₋¹⁺²	—	13	8	no	—
45	C₃×3₋¹⁺²	—	13	24	no	—
46	C₃×H(3,𝔽₃)	—	3	9	no	—
47	C₃×H(3,𝔽₃)	C₃×C₁₃⋊C₃	1	12	yes	—
48	C₃×H(3,𝔽₃)	C₃×C₁₃⋊C₃	1	1	yes	characteristic
49	C₃²×C₉	—	13	4	no	—
50	#731	—	13	9	no	—
51	#731	—	13	72	no	—
52	C₃×C₁₃	—	3	1	no	—
53	C₃×C₁₃	—	3	12	no	—
54	C₃×C₁₃	—	9	9	no	—
55	C₃×C₁₃	C₃²×H(3,𝔽₃)	1	1	yes	characteristic
56	C₃³	—	3	12	no	—
57	C₃³	C₃²×C₁₃⋊C₃	1	4	yes	—
58	3₋¹⁺²	—	13	18	no	—
59	3₋¹⁺²	—	39	24	no	—
60	3₋¹⁺²	—	39	48	no	—
61	H(3,𝔽₃)	—	3	24	no	—
62	H(3,𝔽₃)	C₃²⋊(C₁₃⋊C₃)	1	9	yes	—
63	H(3,𝔽₃)	C₃²×C₁₃⋊C₃	1	9	yes	—
64	C₃×C₉	—	13	1	no	—
65	C₃×C₉	—	13	8	no	—
66	C₃×C₉	—	39	12	no	—
67	C₃×C₉	—	39	24	no	—
68	C₁₃	#10624931	1	1	yes	characteristic, complemented, Sylow
69	C₃²	—	3	9	no	—
70	C₃²	—	9	4	no	—
71	C₃²	—	27	12	no	—
72	C₃²	C₃×C₃²⋊(C₁₃⋊C₃)	1	12	yes	—
73	C₃²	C₃³×C₁₃⋊C₃	1	1	yes	characteristic
74	C₉	—	39	1	no	—
75	C₉	—	39	8	no	—
76	C₉	—	39	18	no	—
77	C₃	—	3	1	no	—
78	C₃	—	3	12	no	—
79	C₃	—	9	9	no	—
80	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, prime radical	C₁₃×H(5,𝔽₃)	C₃
socle	C₃×C₁₃	C₃²×H(3,𝔽₃)
hypercenter	H(5,𝔽₃)	C₁₃⋊C₃
hypocenter	C₁₃	#10624931
LCS	G ▷ C₃²×C₁₃ ▷ C₃×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₃ ◁ C₃×H(3,𝔽₃) ◁ H(5,𝔽₃)

Complements

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

N	H ≅ G/N	type
1. #15691698	C₃	semidirect
2. C₃×#11760452	C₃	semidirect
3. #15691660	C₃	semidirect
4. C₃×#11760437	C₃	semidirect
8. C₃×#66851	C₃²	semidirect
9. C₃×#66851	C₃²	semidirect
10. C₃²×C₁₃⋊C₉	C₃²	semidirect
11. #11760437	C₃²	semidirect
12. #11760437	C₃²	semidirect
68. C₁₃	#10624931	semidirect