Subgroups — C₁₃₉×#6910 #46798433

Order 33777.

136 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×3₋¹⁺²×C₁₃₉	C₃	1	2	yes	maximal, complemented
2	C₃×3₋¹⁺²×C₁₃₉	C₃	1	1	yes	maximal, characteristic, complemented
3	C₃×H(3,𝔽₃)×C₁₃₉	C₃	1	1	yes	maximal, characteristic
4	#731×C₁₃₉	C₃	1	9	yes	maximal, complemented
5	C₃³×C₁₃₉	—	3	1	no	—
6	C₃³×C₁₃₉	C₃²	1	1	yes	characteristic
7	3₋¹⁺²×C₁₃₉	—	3	2	no	—
8	3₋¹⁺²×C₁₃₉	—	3	2	no	—
9	3₋¹⁺²×C₁₃₉	—	3	2	no	—
10	3₋¹⁺²×C₁₃₉	—	3	1	no	—
11	3₋¹⁺²×C₁₃₉	C₃²	1	6	yes	complemented
12	H(3,𝔽₃)×C₁₃₉	—	3	2	no	—
13	H(3,𝔽₃)×C₁₃₉	C₃²	1	3	yes	—
14	C₃×C₉×C₁₃₉	—	3	2	no	—
15	C₃×C₉×C₁₃₉	C₃²	1	3	yes	complemented
16	C₃²×C₁₃₉	—	3	3	no	—
17	C₃²×C₁₃₉	—	9	1	no	—
18	C₃²×C₁₃₉	—	9	3	no	—
19	C₃²×C₁₃₉	H(3,𝔽₃)	1	2	yes	—
20	C₃²×C₁₃₉	H(3,𝔽₃)	1	1	yes	characteristic
21	C₃²×C₁₃₉	C₃³	1	1	yes	characteristic
22	C₉×C₁₃₉	—	3	6	no	—
23	C₉×C₁₃₉	—	3	3	no	—
24	C₃×C₁₃₉	—	3	1	no	—
25	C₃×C₁₃₉	—	3	2	no	—
26	C₃×C₁₃₉	—	3	1	no	—
27	C₃×C₁₃₉	—	9	3	no	—
28	C₃×C₁₃₉	C₃×H(3,𝔽₃)	1	1	yes	characteristic
29	#6910	C₁₃₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
30	C₁₃₉	#6910	1	1	yes	characteristic, direct product factor, Sylow
31	C₃×3₋¹⁺²	C₃×C₁₃₉	1	2	yes	complemented
32	C₃×3₋¹⁺²	C₃×C₁₃₉	1	1	yes	characteristic, complemented
33	C₃×H(3,𝔽₃)	C₃×C₁₃₉	1	1	yes	characteristic
34	#731	C₃×C₁₃₉	1	9	yes	complemented
35	C₃³	—	3	1	no	—
36	C₃³	C₃²×C₁₃₉	1	1	yes	characteristic
37	3₋¹⁺²	—	3	2	no	—
38	3₋¹⁺²	—	3	2	no	—
39	3₋¹⁺²	—	3	2	no	—
40	3₋¹⁺²	—	3	1	no	—
41	3₋¹⁺²	C₃²×C₁₃₉	1	6	yes	complemented
42	H(3,𝔽₃)	—	3	2	no	—
43	H(3,𝔽₃)	C₃²×C₁₃₉	1	3	yes	—
44	C₃×C₉	—	3	2	no	—
45	C₃×C₉	C₃²×C₁₃₉	1	3	yes	complemented
46	C₃²	—	3	3	no	—
47	C₃²	—	9	1	no	—
48	C₃²	—	9	3	no	—
49	C₃²	H(3,𝔽₃)×C₁₃₉	1	2	yes	—
50	C₃²	H(3,𝔽₃)×C₁₃₉	1	1	yes	characteristic
51	C₃²	C₃³×C₁₃₉	1	1	yes	characteristic
52	C₉	—	3	6	no	—
53	C₉	—	3	3	no	—
54	C₃	—	3	1	no	—
55	C₃	—	3	2	no	—
56	C₃	—	3	1	no	—
57	C₃	—	9	3	no	—
58	C₃	C₃×H(3,𝔽₃)×C₁₃₉	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₃×C₁₃₉	C₃×H(3,𝔽₃)
commutator, Frattini, cosocle	C₃²	C₃³×C₁₃₉
prime radical	C₃×H(3,𝔽₃)×C₁₃₉	C₃
LCS	G ▷ C₃² ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃×C₁₃₉ ◁ C₃³×C₁₃₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃×3₋¹⁺²×C₁₃₉	C₃	semidirect
2. C₃×3₋¹⁺²×C₁₃₉	C₃	semidirect
4. #731×C₁₃₉	C₃	semidirect
11. 3₋¹⁺²×C₁₃₉	C₃²	semidirect
15. C₃×C₉×C₁₃₉	C₃²	semidirect
29. #6910	C₁₃₉	direct (N × H)
30. C₁₃₉	#6910	direct (N × H)
31. C₃×3₋¹⁺²	C₃×C₁₃₉	semidirect
32. C₃×3₋¹⁺²	C₃×C₁₃₉	semidirect
34. #731	C₃×C₁₃₉	semidirect
41. 3₋¹⁺²	C₃²×C₁₃₉	semidirect
45. C₃×C₉	C₃²×C₁₃₉	semidirect

Maximal subgroups

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

subgroup	index
C₃×3₋¹⁺²×C₁₃₉	3
C₃×H(3,𝔽₃)×C₁₃₉	3
#731×C₁₃₉	3
#6910	139

Subgroups — C139×#6910 #46798433

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₉×#6910 #46798433