Subgroups — #46798667

Order 33777.

126 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26014630	C₃	1	3	yes	maximal, complemented
2	C₃×#16836728	C₃	1	1	yes	maximal, characteristic, complemented
3	#26014619	C₃	1	1	yes	maximal, characteristic, complemented
4	#26014617	C₃	1	1	yes	maximal, characteristic, complemented
5	#26014615	C₃	1	3	yes	maximal, complemented
6	#26014612	C₃	1	3	yes	maximal, complemented
7	#733×C₁₃₉	C₃	1	1	yes	maximal, characteristic, complemented
8	C₃²×C₁₃₉⋊C₃	C₃²	1	1	yes	characteristic
9	#16836730	C₃²	1	3	yes	—
10	#16836729	—	3	3	no	—
11	#16836728	—	3	3	no	—
12	#16836728	—	3	3	no	—
13	C₃²⋊(C₁₃₉⋊C₃)	C₃²	1	3	yes	—
14	C₃×C₁₃₉⋊C₉	C₃²	1	1	yes	characteristic
15	C₃×C₁₃₉⋊C₉	C₃²	1	1	yes	characteristic
16	C₉×C₁₃₉⋊C₃	—	3	3	no	—
17	3₋¹⁺²×C₁₃₉	C₃²	1	3	yes	—
18	C₃×C₉×C₁₃₉	C₃²	1	1	yes	characteristic
19	C₃²×C₁₃₉	C₃³	1	1	yes	characteristic
20	C₃×C₁₃₉⋊C₃	—	3	3	no	—
21	C₃×C₁₃₉⋊C₃	—	3	1	no	—
22	C₃×C₁₃₉⋊C₃	—	9	1	no	—
23	C₉×C₁₃₉	—	3	1	no	—
24	C₉×C₁₃₉	—	3	3	no	—
25	C₁₃₉⋊C₉	—	3	3	no	—
26	C₁₃₉⋊C₉	—	3	1	no	—
27	C₁₃₉⋊C₉	H(3,𝔽₃)	1	3	yes	—
28	C₃×C₁₃₉	—	3	1	no	—
29	C₃×C₁₃₉	C₃×H(3,𝔽₃)	1	1	yes	characteristic
30	C₁₃₉⋊C₃	—	9	3	no	—
31	C₁₃₉⋊C₃	—	9	1	no	—
32	#6913	—	139	1	no	maximal, Sylow
33	C₁₃₉	#6913	1	1	yes	characteristic, complemented, Sylow
34	#737	—	139	3	no	—
35	C₃×3₋¹⁺²	—	139	1	no	—
36	#733	—	139	1	no	—
37	#733	C₁₃₉⋊C₃	1	1	yes	characteristic, complemented
38	#732	—	139	1	no	—
39	#731	—	139	3	no	—
40	AΣL(1,𝔽₂₇)	—	139	3	no	—
41	C₃³	—	139	1	no	—
42	3₋¹⁺²	—	139	3	no	—
43	3₋¹⁺²	—	417	3	no	—
44	3₋¹⁺²	—	417	3	no	—
45	3₋¹⁺²	—	417	3	no	—
46	3₋¹⁺²	C₃×C₁₃₉⋊C₃	1	3	yes	—
47	H(3,𝔽₃)	—	139	3	no	—
48	C₃×C₉	—	139	1	no	—
49	C₃×C₉	—	139	1	no	—
50	C₃×C₉	—	417	3	no	—
51	C₃×C₉	C₃×C₁₃₉⋊C₃	1	1	yes	characteristic
52	C₃²	—	417	3	no	—
53	C₃²	—	417	1	no	—
54	C₃²	—	1251	1	no	—
55	C₃²	C₃²×C₁₃₉⋊C₃	1	1	yes	characteristic
56	C₉	—	3	3	no	—
57	C₉	—	3	1	no	—
58	C₉	—	139	3	no	—
59	C₉	—	417	1	no	—
60	C₉	—	417	3	no	—
61	C₃	—	3	1	no	—
62	C₃	—	1251	1	no	—
63	C₃	—	1251	3	no	—
64	C₃	H(3,𝔽₃)×C₁₃₉⋊C₃	1	1	yes	characteristic

	subgroup	G/H
center	C₃	H(3,𝔽₃)×C₁₃₉⋊C₃
commutator, cosocle	C₃²×C₁₃₉	C₃³
Frattini	C₃²	C₃²×C₁₃₉⋊C₃
Fitting	#733×C₁₃₉	C₃
socle	C₃×C₁₃₉	C₃×H(3,𝔽₃)
hypercenter	#733	C₁₃₉⋊C₃
hypocenter	C₁₃₉	#6913
LCS	G ▷ C₃²×C₁₃₉ ▷ C₃×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₃ ◁ C₃² ◁ #733

Complements

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

N	H ≅ G/N	type
1. #26014630	C₃	semidirect
2. C₃×#16836728	C₃	semidirect
3. #26014619	C₃	semidirect
4. #26014617	C₃	semidirect
5. #26014615	C₃	semidirect
6. #26014612	C₃	semidirect
7. #733×C₁₃₉	C₃	semidirect
33. C₁₃₉	#6913	semidirect
37. #733	C₁₃₉⋊C₃	semidirect

Maximal subgroups

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

subgroup	index
#26014630	3
C₃×#16836728	3
#26014619	3
#26014617	3
#26014615	3
#26014612	3
#733×C₁₃₉	3
#6913	139