Subgroups — #46798645

Order 33777.

136 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#16836729	C₃	1	1	yes	maximal, characteristic, complemented
2	3₋¹⁺²×C₁₃₉⋊C₃	C₃	1	2	yes	maximal, complemented
3	H(3,𝔽₃)×C₁₃₉⋊C₃	C₃	1	1	yes	maximal, characteristic
4	#26014615	C₃	1	4	yes	maximal, complemented
5	#26014614	C₃	1	2	yes	maximal, complemented
6	#26014613	C₃	1	2	yes	maximal, complemented
7	#731×C₁₃₉	C₃	1	1	yes	maximal, characteristic, complemented
8	C₃²×C₁₃₉⋊C₃	—	3	1	no	—
9	C₃²×C₁₃₉⋊C₃	C₃²	1	1	yes	characteristic
10	#16836730	C₃²	1	4	yes	complemented
11	#16836729	—	3	2	no	—
12	#16836729	—	3	2	no	—
13	#16836729	—	3	1	no	—
14	#16836728	—	3	2	no	—
15	C₃²⋊(C₁₃₉⋊C₃)	—	3	2	no	—
16	C₃²⋊(C₁₃₉⋊C₃)	C₃²	1	2	yes	—
17	C₃×C₁₃₉⋊C₉	C₃²	1	2	yes	complemented
18	C₉×C₁₃₉⋊C₃	—	3	2	no	—
19	3₋¹⁺²×C₁₃₉	C₃²	1	2	yes	complemented
20	H(3,𝔽₃)×C₁₃₉	C₃²	1	1	yes	characteristic
21	C₃×C₉×C₁₃₉	C₃²	1	1	yes	characteristic, complemented
22	C₃²×C₁₃₉	—	3	1	no	—
23	C₃²×C₁₃₉	C₃³	1	1	yes	characteristic
24	C₃×C₁₃₉⋊C₃	—	3	2	no	—
25	C₃×C₁₃₉⋊C₃	—	9	1	no	—
26	C₃×C₁₃₉⋊C₃	—	9	3	no	—
27	C₃×C₁₃₉⋊C₃	H(3,𝔽₃)	1	2	yes	—
28	C₃×C₁₃₉⋊C₃	H(3,𝔽₃)	1	1	yes	characteristic
29	C₉×C₁₃₉	—	3	1	no	—
30	C₉×C₁₃₉	—	3	2	no	—
31	C₁₃₉⋊C₉	—	3	4	no	—
32	C₁₃₉⋊C₉	—	3	2	no	—
33	C₃×C₁₃₉	—	3	1	no	—
34	C₃×C₁₃₉	—	9	1	no	—
35	C₃×C₁₃₉	C₃×H(3,𝔽₃)	1	1	yes	characteristic
36	C₁₃₉⋊C₃	—	3	1	no	—
37	C₁₃₉⋊C₃	—	3	2	no	—
38	C₁₃₉⋊C₃	—	9	2	no	—
39	#6910	—	139	1	no	maximal, Sylow
40	C₁₃₉	#6910	1	1	yes	characteristic, complemented, Sylow
41	C₃×3₋¹⁺²	—	139	1	no	—
42	C₃×3₋¹⁺²	—	139	2	no	—
43	C₃×H(3,𝔽₃)	—	139	1	no	—
44	#731	—	139	4	no	—
45	#731	—	139	2	no	—
46	#731	—	139	2	no	—
47	#731	C₁₃₉⋊C₃	1	1	yes	characteristic, complemented
48	C₃³	—	139	1	no	—
49	C₃³	—	417	1	no	—
50	3₋¹⁺²	—	139	4	no	—
51	3₋¹⁺²	—	417	2	no	—
52	3₋¹⁺²	—	417	2	no	—
53	3₋¹⁺²	—	417	2	no	—
54	3₋¹⁺²	—	417	1	no	—
55	3₋¹⁺²	C₃×C₁₃₉⋊C₃	1	2	yes	complemented
56	H(3,𝔽₃)	—	139	2	no	—
57	H(3,𝔽₃)	—	417	2	no	—
58	H(3,𝔽₃)	C₃×C₁₃₉⋊C₃	1	1	yes	characteristic
59	C₃×C₉	—	139	2	no	—
60	C₃×C₉	—	417	2	no	—
61	C₃×C₉	C₃×C₁₃₉⋊C₃	1	1	yes	characteristic, complemented
62	C₃²	—	3	1	no	—
63	C₃²	—	139	2	no	—
64	C₃²	—	139	1	no	—
65	C₃²	—	417	2	no	—
66	C₃²	—	1251	1	no	—
67	C₃²	—	1251	3	no	—
68	C₃²	C₃²×C₁₃₉⋊C₃	1	1	yes	characteristic
69	C₉	—	3	2	no	—
70	C₉	—	3	1	no	—
71	C₉	—	417	2	no	—
72	C₉	—	417	4	no	—
73	C₃	—	3	1	no	—
74	C₃	—	9	1	no	—
75	C₃	—	417	1	no	—
76	C₃	—	417	2	no	—
77	C₃	—	1251	2	no	—
78	C₃	H(3,𝔽₃)×C₁₃₉⋊C₃	1	1	yes	characteristic

	subgroup	G/H
center	C₃	H(3,𝔽₃)×C₁₃₉⋊C₃
commutator, cosocle	C₃²×C₁₃₉	C₃³
Frattini	C₃²	C₃²×C₁₃₉⋊C₃
Fitting	#731×C₁₃₉	C₃
socle	C₃×C₁₃₉	C₃×H(3,𝔽₃)
prime radical	H(3,𝔽₃)×C₁₃₉⋊C₃	C₃
hypercenter	#731	C₁₃₉⋊C₃
hypocenter	C₁₃₉	#6910
LCS	G ▷ C₃²×C₁₃₉ ▷ C₃×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₃ ◁ C₃² ◁ #731

Complements

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

N	H ≅ G/N	type
1. C₃×#16836729	C₃	semidirect
2. 3₋¹⁺²×C₁₃₉⋊C₃	C₃	semidirect
4. #26014615	C₃	semidirect
5. #26014614	C₃	semidirect
6. #26014613	C₃	semidirect
7. #731×C₁₃₉	C₃	semidirect
10. #16836730	C₃²	semidirect
17. C₃×C₁₃₉⋊C₉	C₃²	semidirect
19. 3₋¹⁺²×C₁₃₉	C₃²	semidirect
21. C₃×C₉×C₁₃₉	C₃²	semidirect
40. C₁₃₉	#6910	semidirect
47. #731	C₁₃₉⋊C₃	semidirect
55. 3₋¹⁺²	C₃×C₁₃₉⋊C₃	semidirect
61. C₃×C₉	C₃×C₁₃₉⋊C₃	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₃×#16836729	3
3₋¹⁺²×C₁₃₉⋊C₃	3
H(3,𝔽₃)×C₁₃₉⋊C₃	3
#26014615	3
#26014614	3
#26014613	3
#731×C₁₃₉	3
#6910	139