Subgroups — C₁₃₉×C₉⋊H(3,𝔽₃) #46798415

Order 33777.

166 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×3₋¹⁺²×C₁₃₉	C₃	1	3	yes	maximal, complemented
2	C₃×H(3,𝔽₃)×C₁₃₉	C₃	1	1	yes	maximal, characteristic
3	C₃²×C₉×C₁₃₉	C₃	1	1	yes	maximal, characteristic, complemented
4	C₃²⋊C₉×C₁₃₉	C₃	1	6	yes	maximal, complemented
5	C₃²⋊C₉×C₁₃₉	C₃	1	2	yes	maximal, complemented
6	C₃³×C₁₃₉	C₃²	1	3	yes	—
7	C₃³×C₁₃₉	C₃²	1	1	yes	characteristic
8	3₋¹⁺²×C₁₃₉	—	3	9	no	—
9	H(3,𝔽₃)×C₁₃₉	—	3	3	no	—
10	C₃×C₉×C₁₃₉	—	3	3	no	—
11	C₃×C₉×C₁₃₉	C₃²	1	2	yes	—
12	C₃×C₉×C₁₃₉	C₃²	1	6	yes	—
13	C₃×C₉×C₁₃₉	C₃²	1	1	yes	characteristic
14	C₃²×C₁₃₉	—	3	1	no	—
15	C₃²×C₁₃₉	—	3	2	no	—
16	C₃²×C₁₃₉	—	3	3	no	—
17	C₃²×C₁₃₉	—	3	3	no	—
18	C₃²×C₁₃₉	—	3	6	no	—
19	C₃²×C₁₃₉	3₋¹⁺²	1	3	yes	complemented
20	C₃²×C₁₃₉	C₃³	1	1	yes	characteristic
21	C₉×C₁₃₉	—	3	2	no	—
22	C₉×C₁₃₉	—	3	6	no	—
23	C₉×C₁₃₉	H(3,𝔽₃)	1	3	yes	complemented
24	C₃×C₁₃₉	—	3	3	no	—
25	C₃×C₁₃₉	—	9	3	no	—
26	C₃×C₁₃₉	C₃×H(3,𝔽₃)	1	1	yes	characteristic
27	C₃×C₁₃₉	C₃×3₋¹⁺²	1	1	yes	characteristic
28	C₃×C₁₃₉	#737	1	2	yes	—
29	C₉⋊H(3,𝔽₃)	C₁₃₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
30	C₁₃₉	C₉⋊H(3,𝔽₃)	1	1	yes	characteristic, direct product factor, Sylow
31	C₃×3₋¹⁺²	C₃×C₁₃₉	1	3	yes	complemented
32	C₃×H(3,𝔽₃)	C₃×C₁₃₉	1	1	yes	characteristic
33	C₃²×C₉	C₃×C₁₃₉	1	1	yes	characteristic, complemented
34	C₃²⋊C₉	C₃×C₁₃₉	1	6	yes	complemented
35	C₃²⋊C₉	C₃×C₁₃₉	1	2	yes	complemented
36	C₃³	C₃²×C₁₃₉	1	3	yes	—
37	C₃³	C₃²×C₁₃₉	1	1	yes	characteristic
38	3₋¹⁺²	—	3	9	no	—
39	H(3,𝔽₃)	—	3	3	no	—
40	C₃×C₉	—	3	3	no	—
41	C₃×C₉	C₃²×C₁₃₉	1	2	yes	—
42	C₃×C₉	C₃²×C₁₃₉	1	6	yes	—
43	C₃×C₉	C₃²×C₁₃₉	1	1	yes	characteristic
44	C₃²	—	3	1	no	—
45	C₃²	—	3	2	no	—
46	C₃²	—	3	3	no	—
47	C₃²	—	3	3	no	—
48	C₃²	—	3	6	no	—
49	C₃²	3₋¹⁺²×C₁₃₉	1	3	yes	complemented
50	C₃²	C₃³×C₁₃₉	1	1	yes	characteristic
51	C₉	—	3	2	no	—
52	C₉	—	3	6	no	—
53	C₉	H(3,𝔽₃)×C₁₃₉	1	3	yes	complemented
54	C₃	—	3	3	no	—
55	C₃	—	9	3	no	—
56	C₃	C₃×H(3,𝔽₃)×C₁₃₉	1	1	yes	characteristic
57	C₃	C₃×3₋¹⁺²×C₁₃₉	1	1	yes	characteristic
58	C₃	#737×C₁₃₉	1	2	yes	—

	subgroup	G/H
center, socle	C₃²×C₁₃₉	C₃³
commutator, Frattini, cosocle	C₃²	C₃³×C₁₃₉
prime radical	C₃×H(3,𝔽₃)×C₁₃₉	C₃
LCS	G ▷ C₃² ▷ C₁
UCS	C₁ ◁ C₃²×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
3. C₃²×C₉×C₁₃₉	C₃	semidirect
4. C₃²⋊C₉×C₁₃₉	C₃	semidirect
5. C₃²⋊C₉×C₁₃₉	C₃	semidirect
19. C₃²×C₁₃₉	3₋¹⁺²	semidirect
23. C₉×C₁₃₉	H(3,𝔽₃)	semidirect
29. C₉⋊H(3,𝔽₃)	C₁₃₉	direct (N × H)
30. C₁₃₉	C₉⋊H(3,𝔽₃)	direct (N × H)
31. C₃×3₋¹⁺²	C₃×C₁₃₉	semidirect
33. C₃²×C₉	C₃×C₁₃₉	semidirect
34. C₃²⋊C₉	C₃×C₁₃₉	semidirect
35. C₃²⋊C₉	C₃×C₁₃₉	semidirect
49. C₃²	3₋¹⁺²×C₁₃₉	semidirect
53. C₉	H(3,𝔽₃)×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
C₃²×C₉×C₁₃₉	3
C₃²⋊C₉×C₁₃₉	3
C₉⋊H(3,𝔽₃)	139

Subgroups — C139×C9⋊H(3,𝔽3) #46798415

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₉×C₉⋊H(3,𝔽₃) #46798415