Subgroups — #46798560

Order 33777.

166 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#16836730	C₃	1	3	yes	maximal, complemented
2	C₃²×C₁₃₉⋊C₉	C₃	1	1	yes	maximal, characteristic, complemented
3	C₃²⋊(C₁₃₉⋊C₉)	C₃	1	6	yes	maximal, complemented
4	C₃²⋊(C₁₃₉⋊C₉)	C₃	1	2	yes	maximal, complemented
5	C₃×H(3,𝔽₃)×C₁₃₉	C₃	1	1	yes	maximal, characteristic
6	#16836730	—	3	9	no	—
7	C₃×C₁₃₉⋊C₉	—	3	3	no	—
8	C₃×C₁₃₉⋊C₉	C₃²	1	2	yes	—
9	C₃×C₁₃₉⋊C₉	C₃²	1	6	yes	—
10	C₃×C₁₃₉⋊C₉	C₃²	1	1	yes	characteristic
11	C₃³×C₁₃₉	C₃²	1	3	yes	—
12	C₃³×C₁₃₉	C₃²	1	1	yes	characteristic
13	H(3,𝔽₃)×C₁₃₉	—	3	3	no	—
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	C₁₃₉⋊C₉	—	3	2	no	—
22	C₁₃₉⋊C₉	—	3	6	no	—
23	C₁₃₉⋊C₉	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)	—	139	1	no	maximal, Sylow
30	C₁₃₉	C₉⋊H(3,𝔽₃)	1	1	yes	characteristic, complemented, Sylow
31	C₃×3₋¹⁺²	—	139	3	no	—
32	C₃×H(3,𝔽₃)	C₁₃₉⋊C₃	1	1	yes	characteristic
33	C₃²×C₉	—	139	1	no	—
34	C₃²⋊C₉	—	139	6	no	—
35	C₃²⋊C₉	—	139	2	no	—
36	C₃³	C₃×C₁₃₉⋊C₃	1	3	yes	—
37	C₃³	C₃×C₁₃₉⋊C₃	1	1	yes	characteristic
38	3₋¹⁺²	—	417	9	no	—
39	H(3,𝔽₃)	—	3	3	no	—
40	C₃×C₉	—	139	1	no	—
41	C₃×C₉	—	139	2	no	—
42	C₃×C₉	—	139	6	no	—
43	C₃×C₉	—	417	3	no	—
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	#16836730	1	3	yes	complemented
50	C₃²	C₃²×C₁₃₉⋊C₃	1	1	yes	characteristic
51	C₉	—	139	3	no	—
52	C₉	—	417	2	no	—
53	C₉	—	417	6	no	—
54	C₃	—	3	3	no	—
55	C₃	—	9	3	no	—
56	C₃	H(3,𝔽₃)×C₁₃₉⋊C₃	1	1	yes	characteristic
57	C₃	C₃×#16836730	1	1	yes	characteristic
58	C₃	#26014628	1	2	yes	—

	subgroup	G/H
center, Frattini	C₃²	C₃²×C₁₃₉⋊C₃
commutator, socle, cosocle	C₃²×C₁₃₉	C₃³
Fitting, prime radical	C₃×H(3,𝔽₃)×C₁₃₉	C₃
hypercenter	C₃×H(3,𝔽₃)	C₁₃₉⋊C₃
hypocenter	C₁₃₉	C₉⋊H(3,𝔽₃)
LCS	G ▷ C₃²×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₃² ◁ C₃×H(3,𝔽₃)

Complements

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

N	H ≅ G/N	type
1. C₃×#16836730	C₃	semidirect
2. C₃²×C₁₃₉⋊C₉	C₃	semidirect
3. C₃²⋊(C₁₃₉⋊C₉)	C₃	semidirect
4. C₃²⋊(C₁₃₉⋊C₉)	C₃	semidirect
19. C₃²×C₁₃₉	3₋¹⁺²	semidirect
23. C₁₃₉⋊C₉	H(3,𝔽₃)	semidirect
30. C₁₃₉	C₉⋊H(3,𝔽₃)	semidirect
49. C₃²	#16836730	semidirect

Maximal subgroups

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

subgroup	index
C₃×#16836730	3
C₃²×C₁₃₉⋊C₉	3
C₃²⋊(C₁₃₉⋊C₉)	3
C₃×H(3,𝔽₃)×C₁₃₉	3
C₉⋊H(3,𝔽₃)	139