Subgroups — #46798446

Order 33777.

94 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₃²⋊(C₁₃₉⋊C₃)	C₃	1	1	yes	maximal, characteristic, complemented
2	C₃²⋊(C₁₃₉⋊C₉)	C₃	1	2	yes	maximal, complemented
3	C₃×H(3,𝔽₃)×C₁₃₉	C₃	1	1	yes	maximal, characteristic, complemented
4	C₃²×C₁₃₉⋊C₃	—	3	1	no	—
5	C₃²⋊(C₁₃₉⋊C₃)	—	3	3	no	—
6	C₃×C₁₃₉⋊C₉	—	3	2	no	—
7	C₃³×C₁₃₉	—	3	1	no	—
8	C₃³×C₁₃₉	C₃²	1	1	yes	characteristic
9	H(3,𝔽₃)×C₁₃₉	—	3	3	no	—
10	C₃²×C₁₃₉	—	3	1	no	—
11	C₃²×C₁₃₉	—	3	3	no	—
12	C₃²×C₁₃₉	—	3	1	no	—
13	C₃²×C₁₃₉	—	3	2	no	—
14	C₃²×C₁₃₉	—	9	1	no	—
15	C₃²×C₁₃₉	—	9	2	no	—
16	C₃²×C₁₃₉	H(3,𝔽₃)	1	1	yes	characteristic
17	C₃×C₁₃₉⋊C₃	—	3	3	no	—
18	C₃×C₁₃₉⋊C₃	—	9	1	no	—
19	C₃×C₁₃₉⋊C₃	—	9	2	no	—
20	C₁₃₉⋊C₉	—	9	2	no	—
21	C₃×C₁₃₉	—	9	1	no	—
22	C₃×C₁₃₉	—	9	3	no	—
23	C₃×C₁₃₉	AΣL(1,𝔽₂₇)	1	1	yes	characteristic
24	C₃×C₁₃₉	AΣL(1,𝔽₂₇)	1	1	yes	characteristic
25	C₃×C₁₃₉	#732	1	2	yes	—
26	C₁₃₉⋊C₃	—	9	3	no	—
27	#6856	—	139	1	no	maximal, Sylow
28	C₁₃₉	#6856	1	1	yes	characteristic, complemented, Sylow
29	C₃×H(3,𝔽₃)	—	139	1	no	—
30	C₃×H(3,𝔽₃)	C₁₃₉⋊C₃	1	1	yes	characteristic, complemented
31	C₃²⋊C₉	—	139	2	no	—
32	C₃³	—	3	1	no	—
33	C₃³	—	417	1	no	—
34	C₃³	C₃×C₁₃₉⋊C₃	1	1	yes	characteristic
35	H(3,𝔽₃)	—	3	3	no	—
36	H(3,𝔽₃)	—	417	3	no	—
37	C₃×C₉	—	417	2	no	—
38	C₃²	—	3	1	no	—
39	C₃²	—	3	1	no	—
40	C₃²	—	3	3	no	—
41	C₃²	—	3	2	no	—
42	C₃²	—	9	1	no	—
43	C₃²	—	9	2	no	—
44	C₃²	—	417	3	no	—
45	C₃²	—	1251	1	no	—
46	C₃²	—	1251	2	no	—
47	C₃²	C₃²⋊(C₁₃₉⋊C₃)	1	1	yes	characteristic
48	C₉	—	1251	2	no	—
49	C₃	—	9	1	no	—
50	C₃	—	9	3	no	—
51	C₃	—	1251	3	no	—
52	C₃	#26014610	1	1	yes	characteristic
53	C₃	#26014611	1	1	yes	characteristic
54	C₃	#26014616	1	2	yes	—

	subgroup	G/H
center	C₃²	C₃²⋊(C₁₃₉⋊C₃)
commutator, cosocle	C₃³×C₁₃₉	C₃²
Frattini	C₃³	C₃×C₁₃₉⋊C₃
Fitting	C₃×H(3,𝔽₃)×C₁₃₉	C₃
socle	C₃²×C₁₃₉	H(3,𝔽₃)
hypercenter	C₃×H(3,𝔽₃)	C₁₃₉⋊C₃
hypocenter	C₁₃₉	#6856
LCS	G ▷ C₃³×C₁₃₉ ▷ C₃²×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. C₃×C₃²⋊(C₁₃₉⋊C₃)	C₃	semidirect
2. C₃²⋊(C₁₃₉⋊C₉)	C₃	semidirect
3. C₃×H(3,𝔽₃)×C₁₃₉	C₃	semidirect
28. C₁₃₉	#6856	semidirect
30. 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×C₃²⋊(C₁₃₉⋊C₃)	3
C₃²⋊(C₁₃₉⋊C₉)	3
C₃×H(3,𝔽₃)×C₁₃₉	3
#6856	139