Subgroups — #46798665

Order 33777.

126 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26014629	C₃	1	3	yes	maximal, complemented
2	#26014619	C₃	1	2	yes	maximal, complemented
3	#26014617	C₃	1	1	yes	maximal, characteristic, complemented
4	#26014614	C₃	1	3	yes	maximal, complemented
5	#26014611	C₃	1	3	yes	maximal, complemented
6	C₃×3₋¹⁺²×C₁₃₉	C₃	1	1	yes	maximal, characteristic, complemented
7	#16836730	C₃²	1	6	yes	—
8	#16836729	—	3	3	no	—
9	#16836728	—	3	3	no	—
10	C₃²⋊(C₁₃₉⋊C₃)	C₃²	1	3	yes	—
11	C₉×C₁₃₉⋊C₃	—	3	3	no	—
12	C₃³×C₁₃₉	C₃²	1	1	yes	characteristic
13	3₋¹⁺²×C₁₃₉	—	3	3	no	—
14	C₃×C₉×C₁₃₉	C₃²	1	1	yes	characteristic
15	C₃×C₉×C₁₃₉	C₃²	1	1	yes	characteristic
16	C₃×C₉×C₁₃₉	C₃²	1	1	yes	characteristic
17	C₃²×C₁₃₉	—	3	1	no	—
18	C₃²×C₁₃₉	—	9	1	no	—
19	C₃²×C₁₃₉	C₃³	1	1	yes	characteristic
20	C₃×C₁₃₉⋊C₃	—	3	3	no	—
21	C₉×C₁₃₉	—	3	1	no	—
22	C₉×C₁₃₉	—	3	1	no	—
23	C₉×C₁₃₉	H(3,𝔽₃)	1	3	yes	—
24	C₁₃₉⋊C₉	—	3	6	no	—
25	C₃×C₁₃₉	—	3	1	no	—
26	C₃×C₁₃₉	—	9	1	no	—
27	C₃×C₁₃₉	C₃×H(3,𝔽₃)	1	1	yes	characteristic
28	C₁₃₉⋊C₃	—	9	3	no	—
29	#6913	—	139	1	no	maximal, Sylow
30	C₁₃₉	#6913	1	1	yes	characteristic, complemented, Sylow
31	#737	—	139	3	no	—
32	C₃×3₋¹⁺²	C₁₃₉⋊C₃	1	1	yes	characteristic, complemented
33	#733	—	139	2	no	—
34	#732	—	139	1	no	—
35	#731	—	139	3	no	—
36	AΣL(1,𝔽₂₇)	—	139	3	no	—
37	C₃³	C₃×C₁₃₉⋊C₃	1	1	yes	characteristic
38	3₋¹⁺²	—	3	3	no	—
39	3₋¹⁺²	—	139	6	no	—
40	3₋¹⁺²	—	417	3	no	—
41	3₋¹⁺²	—	417	3	no	—
42	H(3,𝔽₃)	—	139	3	no	—
43	C₃×C₉	—	417	3	no	—
44	C₃×C₉	C₃×C₁₃₉⋊C₃	1	1	yes	characteristic
45	C₃×C₉	C₃×C₁₃₉⋊C₃	1	1	yes	characteristic
46	C₃×C₉	C₃×C₁₃₉⋊C₃	1	1	yes	characteristic
47	C₃²	—	3	1	no	—
48	C₃²	—	9	1	no	—
49	C₃²	—	417	3	no	—
50	C₃²	C₃²×C₁₃₉⋊C₃	1	1	yes	characteristic
51	C₉	—	3	1	no	—
52	C₉	—	3	1	no	—
53	C₉	—	417	6	no	—
54	C₉	C₃²⋊(C₁₃₉⋊C₃)	1	3	yes	—
55	C₃	—	3	1	no	—
56	C₃	—	9	1	no	—
57	C₃	—	1251	3	no	—
58	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	C₃×3₋¹⁺²×C₁₃₉	C₃
socle	C₃×C₁₃₉	C₃×H(3,𝔽₃)
hypercenter	C₃×3₋¹⁺²	C₁₃₉⋊C₃
hypocenter	C₁₃₉	#6913
LCS	G ▷ C₃²×C₁₃₉ ▷ C₃×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₃ ◁ C₃×C₉ ◁ C₃×3₋¹⁺²

Complements

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

N	H ≅ G/N	type
1. #26014629	C₃	semidirect
2. #26014619	C₃	semidirect
3. #26014617	C₃	semidirect
4. #26014614	C₃	semidirect
5. #26014611	C₃	semidirect
6. C₃×3₋¹⁺²×C₁₃₉	C₃	semidirect
30. C₁₃₉	#6913	semidirect
32. C₃×3₋¹⁺²	C₁₃₉⋊C₃	semidirect

Maximal subgroups

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

subgroup	index
#26014629	3
#26014619	3
#26014617	3
#26014614	3
#26014611	3
C₃×3₋¹⁺²×C₁₃₉	3
#6913	139