Subgroups — #23748563

Order 9477.

978 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#15691698	C₃	1	1	yes	maximal, characteristic, complemented
2	C₁₃⋊H(5,𝔽₃)	C₃	1	1	yes	maximal, characteristic
3	C₃×#11760452	C₃	1	1	yes	maximal, characteristic, complemented
4	#15691662	C₃	1	24	yes	maximal, complemented
5	C₃×#11760439	C₃	1	12	yes	maximal, complemented
6	C₁₃×3₋¹⁺⁴	C₃	1	1	yes	maximal, characteristic, complemented
7	#11760452	—	3	9	no	—
8	#11760452	—	3	9	no	—
9	C₃×#66851	C₃²	1	12	yes	complemented
10	C₃×#66851	C₃²	1	8	yes	complemented
11	C₃×C₃²⋊(C₁₃⋊C₃)	C₃²	1	12	yes	—
12	H(3,𝔽₃)×C₁₃⋊C₃	—	3	9	no	—
13	C₃²×C₁₃⋊C₉	C₃²	1	4	yes	complemented
14	#11760439	C₃²	1	9	yes	complemented
15	#11760439	C₃²	1	72	yes	complemented
16	C₁₃×#737	—	3	9	no	—
17	C₃×C₁₃×3₋¹⁺²	C₃²	1	12	yes	complemented
18	C₃×C₁₃×H(3,𝔽₃)	C₃²	1	1	yes	characteristic
19	#10624931	—	13	1	no	maximal, Sylow
20	C₃³×C₁₃	C₃³	1	4	yes	—
21	C₃²×C₁₃⋊C₃	—	3	12	no	—
22	C₁₃×3₋¹⁺²	—	3	24	no	—
23	C₁₃×3₋¹⁺²	C₃³	1	9	yes	—
24	C₁₃×H(3,𝔽₃)	H(3,𝔽₃)	1	9	yes	—
25	C₃×C₉×C₁₃	—	3	12	no	—
26	C₃²⋊(C₁₃⋊C₃)	—	3	24	no	—
27	C₃²⋊(C₁₃⋊C₃)	C₃³	1	9	yes	—
28	#66851	—	3	24	no	—
29	#66851	—	3	24	no	—
30	#66851	C₃³	1	9	yes	—
31	C₃×C₁₃⋊C₉	—	3	12	no	—
32	C₃×C₁₃⋊C₉	—	3	12	no	—
33	C₃×C₁₃⋊C₉	C₃³	1	1	yes	characteristic
34	C₃×C₁₃⋊C₉	C₃³	1	8	yes	—
35	3₋¹⁺⁴	—	13	1	no	—
36	3₋¹⁺⁴	C₁₃⋊C₃	1	1	yes	characteristic, complemented
37	H(5,𝔽₃)	—	13	1	no	—
38	C₃×#737	—	13	1	no	—
39	#6910	—	13	24	no	—
40	C₃×#731	—	13	12	no	—
41	C₃²×C₁₃	—	9	4	no	—
42	C₃²×C₁₃	C₃×H(3,𝔽₃)	1	12	yes	—
43	C₃²×C₁₃	C₃⁴	1	1	yes	characteristic
44	C₃×C₁₃⋊C₃	—	3	9	no	—
45	C₃×C₁₃⋊C₃	—	27	12	no	—
46	C₉×C₁₃	—	3	9	no	—
47	C₁₃⋊C₉	—	3	1	no	—
48	C₁₃⋊C₉	—	3	9	no	—
49	C₁₃⋊C₉	—	3	8	no	—
50	#737	—	3	9	no	—
51	#737	—	39	9	no	—
52	#737	—	39	9	no	—
53	C₃×3₋¹⁺²	—	13	8	no	—
54	C₃×3₋¹⁺²	—	13	12	no	—
55	C₃×3₋¹⁺²	C₃×C₁₃⋊C₃	1	12	yes	complemented
56	C₃×H(3,𝔽₃)	—	13	12	no	—
57	C₃×H(3,𝔽₃)	—	39	9	no	—
58	C₃×H(3,𝔽₃)	C₃×C₁₃⋊C₃	1	1	yes	characteristic
59	C₃²×C₉	—	13	4	no	—
60	#731	—	13	9	no	—
61	#731	—	13	72	no	—
62	C₃×C₁₃	—	3	12	no	—
63	C₃×C₁₃	—	3	1	no	—
64	C₃×C₁₃	C₃²×H(3,𝔽₃)	1	1	yes	characteristic
65	C₁₃⋊C₃	—	9	9	no	—
66	C₃³	—	39	12	no	—
67	C₃³	C₃²×C₁₃⋊C₃	1	4	yes	—
68	3₋¹⁺²	—	3	24	no	—
69	3₋¹⁺²	—	13	9	no	—
70	3₋¹⁺²	—	39	24	no	—
71	3₋¹⁺²	—	39	24	no	—
72	3₋¹⁺²	C₃²×C₁₃⋊C₃	1	9	yes	—
73	H(3,𝔽₃)	—	13	9	no	—
74	H(3,𝔽₃)	—	39	24	no	—
75	H(3,𝔽₃)	C₃²⋊(C₁₃⋊C₃)	1	9	yes	—
76	C₃×C₉	—	3	12	no	—
77	C₃×C₉	—	13	1	no	—
78	C₃×C₉	—	13	8	no	—
79	C₃×C₉	—	39	12	no	—
80	C₃×C₉	—	39	12	no	—
81	C₁₃	#10624931	1	1	yes	characteristic, complemented, Sylow
82	C₃²	—	9	4	no	—
83	C₃²	—	39	9	no	—
84	C₃²	—	351	12	no	—
85	C₃²	C₃×C₃²⋊(C₁₃⋊C₃)	1	12	yes	—
86	C₃²	C₃³×C₁₃⋊C₃	1	1	yes	characteristic
87	C₉	—	3	9	no	—
88	C₉	—	39	1	no	—
89	C₉	—	39	8	no	—
90	C₉	—	39	9	no	—
91	C₃	—	3	1	no	—
92	C₃	—	3	12	no	—
93	C₃	—	117	9	no	—
94	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	C₁₃×3₋¹⁺⁴	C₃
socle	C₃×C₁₃	C₃²×H(3,𝔽₃)
prime radical	C₁₃⋊H(5,𝔽₃)	C₃
hypercenter	3₋¹⁺⁴	C₁₃⋊C₃
hypocenter	C₁₃	#10624931
LCS	G ▷ C₃²×C₁₃ ▷ C₃×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₃ ◁ C₃×H(3,𝔽₃) ◁ 3₋¹⁺⁴

Complements

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

N	H ≅ G/N	type
1. #15691698	C₃	semidirect
3. C₃×#11760452	C₃	semidirect
4. #15691662	C₃	semidirect
5. C₃×#11760439	C₃	semidirect
6. C₁₃×3₋¹⁺⁴	C₃	semidirect
9. C₃×#66851	C₃²	semidirect
10. C₃×#66851	C₃²	semidirect
13. C₃²×C₁₃⋊C₉	C₃²	semidirect
14. #11760439	C₃²	semidirect
15. #11760439	C₃²	semidirect
17. C₃×C₁₃×3₋¹⁺²	C₃²	semidirect
36. 3₋¹⁺⁴	C₁₃⋊C₃	semidirect
55. C₃×3₋¹⁺²	C₃×C₁₃⋊C₃	semidirect
81. C₁₃	#10624931	semidirect