Subgroups — #23748555

Order 9477.

1118 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×H(3,𝔽₃)×C₁₃⋊C₃	C₃	1	1	yes	maximal, characteristic, complemented
3	#15691655	C₃	1	24	yes	maximal, complemented
4	C₃×#11760434	C₃	1	12	yes	maximal, complemented
5	C₁₃×H(5,𝔽₃)	C₃	1	1	yes	maximal, characteristic, complemented
6	C₃³×C₁₃⋊C₃	C₃²	1	4	yes	complemented
7	#11760452	—	3	18	no	—
8	C₃×#66851	C₃²	1	24	yes	complemented
9	C₃×C₃²⋊(C₁₃⋊C₃)	C₃²	1	8	yes	complemented
10	H(3,𝔽₃)×C₁₃⋊C₃	—	3	9	no	—
11	#11760434	C₃²	1	9	yes	complemented
12	#11760434	C₃²	1	72	yes	complemented
13	C₃×C₁₃×H(3,𝔽₃)	—	3	9	no	—
14	C₃×C₁₃×H(3,𝔽₃)	C₃²	1	1	yes	characteristic
15	C₃×C₁₃×H(3,𝔽₃)	C₃²	1	12	yes	complemented
16	#10624930	—	13	1	no	maximal, Sylow
17	C₃³×C₁₃	—	3	12	no	—
18	C₃³×C₁₃	C₃³	1	4	yes	—
19	C₃²×C₁₃⋊C₃	—	3	12	no	—
20	C₃²×C₁₃⋊C₃	—	9	12	no	—
21	C₃²×C₁₃⋊C₃	C₃³	1	1	yes	characteristic
22	C₃²×C₁₃⋊C₃	C₃³	1	8	yes	—
23	C₁₃×H(3,𝔽₃)	—	3	24	no	—
24	C₁₃×H(3,𝔽₃)	H(3,𝔽₃)	1	9	yes	—
25	C₁₃×H(3,𝔽₃)	C₃³	1	9	yes	—
26	C₃²⋊(C₁₃⋊C₃)	—	3	24	no	—
27	#66851	—	3	48	no	—
28	#66851	C₃³	1	18	yes	—
29	C₃×C₁₃⋊C₉	—	3	24	no	—
30	3₋¹⁺⁴	—	13	2	no	—
31	H(5,𝔽₃)	C₁₃⋊C₃	1	1	yes	characteristic, complemented
32	C₃²×H(3,𝔽₃)	—	13	1	no	—
33	#6909	—	13	24	no	—
34	C₃×AΣL(1,𝔽₂₇)	—	13	12	no	—
35	C₃²×C₁₃	—	3	9	no	—
36	C₃²×C₁₃	—	9	4	no	—
37	C₃²×C₁₃	—	27	12	no	—
38	C₃²×C₁₃	C₃×H(3,𝔽₃)	1	12	yes	—
39	C₃²×C₁₃	C₃⁴	1	1	yes	characteristic
40	C₃×C₁₃⋊C₃	—	3	1	no	—
41	C₃×C₁₃⋊C₃	—	3	3	no	—
42	C₃×C₁₃⋊C₃	—	3	8	no	—
43	C₃×C₁₃⋊C₃	—	9	36	no	—
44	C₃×C₁₃⋊C₃	—	9	8	no	—
45	C₁₃⋊C₉	—	3	18	no	—
46	C₃⁴	—	13	4	no	—
47	#737	—	39	18	no	—
48	C₃×3₋¹⁺²	—	13	24	no	—
49	C₃×H(3,𝔽₃)	—	3	9	no	—
50	C₃×H(3,𝔽₃)	—	13	8	no	—
51	C₃×H(3,𝔽₃)	—	39	9	no	—
52	C₃×H(3,𝔽₃)	C₃×C₁₃⋊C₃	1	1	yes	characteristic
53	C₃×H(3,𝔽₃)	C₃×C₁₃⋊C₃	1	12	yes	complemented
54	AΣL(1,𝔽₂₇)	—	13	9	no	—
55	AΣL(1,𝔽₂₇)	—	13	72	no	—
56	C₃×C₁₃	—	3	1	no	—
57	C₃×C₁₃	—	3	12	no	—
58	C₃×C₁₃	—	9	9	no	—
59	C₃×C₁₃	C₃²×H(3,𝔽₃)	1	1	yes	characteristic
60	C₁₃⋊C₃	—	3	3	no	—
61	C₁₃⋊C₃	—	9	8	no	—
62	C₃³	—	3	12	no	—
63	C₃³	—	13	1	no	—
64	C₃³	—	13	8	no	—
65	C₃³	—	39	12	no	—
66	C₃³	—	117	12	no	—
67	C₃³	C₃²×C₁₃⋊C₃	1	4	yes	—
68	3₋¹⁺²	—	13	18	no	—
69	3₋¹⁺²	—	39	48	no	—
70	H(3,𝔽₃)	—	3	24	no	—
71	H(3,𝔽₃)	—	39	24	no	—
72	H(3,𝔽₃)	C₃²⋊(C₁₃⋊C₃)	1	9	yes	—
73	H(3,𝔽₃)	C₃²×C₁₃⋊C₃	1	9	yes	—
74	C₃×C₉	—	39	24	no	—
75	C₁₃	#10624930	1	1	yes	characteristic, complemented, Sylow
76	C₃²	—	3	9	no	—
77	C₃²	—	9	4	no	—
78	C₃²	—	27	12	no	—
79	C₃²	—	39	3	no	—
80	C₃²	—	39	1	no	—
81	C₃²	—	39	8	no	—
82	C₃²	—	117	36	no	—
83	C₃²	—	117	8	no	—
84	C₃²	C₃×C₃²⋊(C₁₃⋊C₃)	1	12	yes	—
85	C₃²	C₃³×C₁₃⋊C₃	1	1	yes	characteristic
86	C₉	—	39	18	no	—
87	C₃	—	3	1	no	—
88	C₃	—	3	12	no	—
89	C₃	—	9	9	no	—
90	C₃	—	39	3	no	—
91	C₃	—	117	8	no	—
92	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×H(5,𝔽₃)	C₃
socle	C₃×C₁₃	C₃²×H(3,𝔽₃)
hypercenter	H(5,𝔽₃)	C₁₃⋊C₃
hypocenter	C₁₃	#10624930
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×H(3,𝔽₃)×C₁₃⋊C₃	C₃	semidirect
3. #15691655	C₃	semidirect
4. C₃×#11760434	C₃	semidirect
5. C₁₃×H(5,𝔽₃)	C₃	semidirect
6. C₃³×C₁₃⋊C₃	C₃²	semidirect
8. C₃×#66851	C₃²	semidirect
9. C₃×C₃²⋊(C₁₃⋊C₃)	C₃²	semidirect
11. #11760434	C₃²	semidirect
12. #11760434	C₃²	semidirect
15. C₃×C₁₃×H(3,𝔽₃)	C₃²	semidirect
31. H(5,𝔽₃)	C₁₃⋊C₃	semidirect
53. C₃×H(3,𝔽₃)	C₃×C₁₃⋊C₃	semidirect
75. C₁₃	#10624930	semidirect