Subgroups — C₃×#26014610 #46798620

Order 33777.

252 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃³×C₁₃₉⋊C₃	C₃	1	1	yes	maximal, characteristic, complemented
2	C₃×#16836730	C₃	1	2	yes	maximal, complemented
3	#26014610	C₃	1	9	yes	maximal, direct product factor
4	C₃×H(3,𝔽₃)×C₁₃₉	C₃	1	1	yes	maximal, characteristic, complemented
5	C₃²×C₁₃₉⋊C₃	—	3	2	no	—
6	C₃²×C₁₃₉⋊C₃	—	3	1	no	—
7	C₃²×C₁₃₉⋊C₃	—	3	9	no	—
8	C₃²×C₁₃₉⋊C₃	C₃²	1	3	yes	complemented
9	#16836730	—	3	4	no	—
10	#16836730	C₃²	1	6	yes	complemented
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	complemented
16	C₃²×C₁₃₉	—	3	3	no	—
17	C₃²×C₁₃₉	—	3	3	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	18	no	—
23	C₃×C₁₃₉⋊C₃	—	3	9	no	—
24	C₃×C₁₃₉⋊C₃	—	3	3	no	—
25	C₃×C₁₃₉⋊C₃	—	3	9	no	—
26	C₁₃₉⋊C₉	—	3	6	no	—
27	C₃×C₁₃₉	—	3	2	no	—
28	C₃×C₁₃₉	—	3	1	no	—
29	C₃×C₁₃₉	—	9	3	no	—
30	C₃×C₁₃₉	AΣL(1,𝔽₂₇)	1	3	yes	complemented
31	C₃×C₁₃₉	C₃×H(3,𝔽₃)	1	1	yes	characteristic
32	C₁₃₉⋊C₃	—	3	9	no	—
33	C₃×AΣL(1,𝔽₂₇)	—	139	1	no	maximal, Sylow
34	C₁₃₉	C₃×AΣL(1,𝔽₂₇)	1	1	yes	characteristic, complemented, Sylow
35	C₃⁴	—	139	1	no	—
36	C₃×3₋¹⁺²	—	139	2	no	—
37	C₃×H(3,𝔽₃)	C₁₃₉⋊C₃	1	1	yes	characteristic, complemented
38	AΣL(1,𝔽₂₇)	—	139	9	no	—
39	C₃³	—	3	1	no	—
40	C₃³	—	139	3	no	—
41	C₃³	—	417	9	no	—
42	C₃³	—	417	2	no	—
43	C₃³	—	417	1	no	—
44	C₃³	C₃×C₁₃₉⋊C₃	1	1	yes	characteristic
45	3₋¹⁺²	—	139	6	no	—
46	3₋¹⁺²	—	417	4	no	—
47	H(3,𝔽₃)	—	3	2	no	—
48	H(3,𝔽₃)	C₃×C₁₃₉⋊C₃	1	3	yes	complemented
49	C₃×C₉	—	417	2	no	—
50	C₃²	—	3	3	no	—
51	C₃²	—	3	3	no	—
52	C₃²	—	9	3	no	—
53	C₃²	—	417	9	no	—
54	C₃²	—	417	18	no	—
55	C₃²	—	417	9	no	—
56	C₃²	—	417	3	no	—
57	C₃²	C₃²⋊(C₁₃₉⋊C₃)	1	2	yes	—
58	C₃²	C₃²⋊(C₁₃₉⋊C₃)	1	1	yes	characteristic
59	C₃²	C₃²×C₁₃₉⋊C₃	1	1	yes	characteristic
60	C₉	—	417	6	no	—
61	C₃	—	3	2	no	—
62	C₃	—	3	1	no	—
63	C₃	—	9	3	no	—
64	C₃	—	417	9	no	—
65	C₃	#26014610	1	3	yes	direct product factor
66	C₃	C₃×C₃²⋊(C₁₃₉⋊C₃)	1	1	yes	characteristic

	subgroup	G/H
center	C₃²	C₃²⋊(C₁₃₉⋊C₃)
commutator, cosocle	C₃²×C₁₃₉	C₃³
Frattini	C₃²	C₃²×C₁₃₉⋊C₃
Fitting	C₃×H(3,𝔽₃)×C₁₃₉	C₃
socle	C₃²×C₁₃₉	H(3,𝔽₃)
hypercenter	C₃×H(3,𝔽₃)	C₁₃₉⋊C₃
hypocenter	C₁₃₉	C₃×AΣL(1,𝔽₂₇)
LCS	G ▷ C₃²×C₁₃₉ ▷ C₃×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₃² ◁ C₃³ ◁ C₃×H(3,𝔽₃)

Complements

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

N	H ≅ G/N	type
1. C₃³×C₁₃₉⋊C₃	C₃	semidirect
2. C₃×#16836730	C₃	semidirect
3. #26014610	C₃	direct (N × H)
4. C₃×H(3,𝔽₃)×C₁₃₉	C₃	semidirect
8. C₃²×C₁₃₉⋊C₃	C₃²	semidirect
10. #16836730	C₃²	semidirect
15. H(3,𝔽₃)×C₁₃₉	C₃²	semidirect
30. C₃×C₁₃₉	AΣL(1,𝔽₂₇)	semidirect
34. C₁₃₉	C₃×AΣL(1,𝔽₂₇)	semidirect
37. C₃×H(3,𝔽₃)	C₁₃₉⋊C₃	semidirect
48. H(3,𝔽₃)	C₃×C₁₃₉⋊C₃	semidirect
65. C₃	#26014610	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃³×C₁₃₉⋊C₃	3
C₃×#16836730	3
#26014610	3
C₃×H(3,𝔽₃)×C₁₃₉	3
C₃×AΣL(1,𝔽₂₇)	139

Subgroups — C3×#26014610 #46798620

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#26014610 #46798620