Subgroups — #46798641

Order 33777.

148 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#16836730	C₃	1	2	yes	maximal, complemented
2	C₃×C₃²⋊(C₁₃₉⋊C₃)	C₃	1	1	yes	maximal, characteristic, complemented
3	#26014610	C₃	1	9	yes	maximal, complemented
4	C₃×H(3,𝔽₃)×C₁₃₉	C₃	1	1	yes	maximal, characteristic, complemented
5	C₃²×C₁₃₉⋊C₃	C₃²	1	3	yes	complemented
6	#16836730	—	3	2	no	—
7	#16836730	—	3	2	no	—
8	#16836730	C₃²	1	6	yes	complemented
9	C₃²⋊(C₁₃₉⋊C₃)	—	3	2	no	—
10	C₃²⋊(C₁₃₉⋊C₃)	—	3	1	no	—
11	C₃×C₁₃₉⋊C₉	—	3	2	no	—
12	C₃³×C₁₃₉	—	3	1	no	—
13	C₃³×C₁₃₉	C₃²	1	1	yes	characteristic
14	H(3,𝔽₃)×C₁₃₉	—	3	2	no	—
15	H(3,𝔽₃)×C₁₃₉	C₃²	1	3	yes	—
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	C₃×C₁₃₉⋊C₃	—	3	3	no	—
23	C₃×C₁₃₉⋊C₃	—	9	3	no	—
24	C₁₃₉⋊C₉	—	3	6	no	—
25	C₃×C₁₃₉	—	3	1	no	—
26	C₃×C₁₃₉	—	3	1	no	—
27	C₃×C₁₃₉	—	3	2	no	—
28	C₃×C₁₃₉	—	9	3	no	—
29	C₃×C₁₃₉	C₃×H(3,𝔽₃)	1	1	yes	characteristic
30	C₁₃₉⋊C₃	—	9	3	no	—
31	#6909	—	139	1	no	maximal, Sylow
32	C₁₃₉	#6909	1	1	yes	characteristic, complemented, Sylow
33	C₃×3₋¹⁺²	—	139	2	no	—
34	C₃×H(3,𝔽₃)	—	139	1	no	—
35	C₃×H(3,𝔽₃)	C₁₃₉⋊C₃	1	1	yes	characteristic, complemented
36	AΣL(1,𝔽₂₇)	—	139	9	no	—
37	C₃³	—	3	1	no	—
38	C₃³	—	139	3	no	—
39	C₃³	C₃×C₁₃₉⋊C₃	1	1	yes	characteristic
40	3₋¹⁺²	—	139	6	no	—
41	3₋¹⁺²	—	417	2	no	—
42	3₋¹⁺²	—	417	2	no	—
43	H(3,𝔽₃)	—	3	2	no	—
44	H(3,𝔽₃)	—	417	2	no	—
45	H(3,𝔽₃)	—	417	1	no	—
46	H(3,𝔽₃)	C₃×C₁₃₉⋊C₃	1	3	yes	—
47	C₃×C₉	—	417	2	no	—
48	C₃²	—	3	3	no	—
49	C₃²	—	9	1	no	—
50	C₃²	—	9	3	no	—
51	C₃²	—	417	3	no	—
52	C₃²	—	1251	3	no	—
53	C₃²	C₃²⋊(C₁₃₉⋊C₃)	1	2	yes	—
54	C₃²	C₃²⋊(C₁₃₉⋊C₃)	1	1	yes	characteristic
55	C₃²	C₃²×C₁₃₉⋊C₃	1	1	yes	characteristic
56	C₉	—	417	6	no	—
57	C₃	—	3	1	no	—
58	C₃	—	3	1	no	—
59	C₃	—	3	2	no	—
60	C₃	—	9	3	no	—
61	C₃	—	1251	3	no	—
62	C₃	C₃×C₃²⋊(C₁₃₉⋊C₃)	1	1	yes	characteristic

	subgroup	G/H
center	C₃	C₃×C₃²⋊(C₁₃₉⋊C₃)
commutator, cosocle	C₃²×C₁₃₉	C₃³
Frattini	C₃²	C₃²×C₁₃₉⋊C₃
Fitting	C₃×H(3,𝔽₃)×C₁₃₉	C₃
socle	C₃×C₁₃₉	C₃×H(3,𝔽₃)
hypercenter	C₃×H(3,𝔽₃)	C₁₃₉⋊C₃
hypocenter	C₁₃₉	#6909
LCS	G ▷ C₃²×C₁₃₉ ▷ C₃×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₃ ◁ C₃³ ◁ 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. C₃×C₃²⋊(C₁₃₉⋊C₃)	C₃	semidirect
3. #26014610	C₃	semidirect
4. C₃×H(3,𝔽₃)×C₁₃₉	C₃	semidirect
5. C₃²×C₁₃₉⋊C₃	C₃²	semidirect
8. #16836730	C₃²	semidirect
32. C₁₃₉	#6909	semidirect
35. C₃×H(3,𝔽₃)	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×#16836730	3
C₃×C₃²⋊(C₁₃₉⋊C₃)	3
#26014610	3
C₃×H(3,𝔽₃)×C₁₃₉	3
#6909	139