Subgroups — #46798654

Order 33777.

124 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#16836728	C₃	1	1	yes	maximal, characteristic
2	3₋¹⁺²×C₁₃₉⋊C₃	C₃	1	3	yes	maximal
3	#26014619	C₃	1	2	yes	maximal, complemented
4	#26014618	C₃	1	6	yes	maximal, complemented
5	#733×C₁₃₉	C₃	1	1	yes	maximal, characteristic, complemented
6	C₃²×C₁₃₉⋊C₃	C₃²	1	1	yes	characteristic
7	#16836730	C₃²	1	6	yes	—
8	#16836729	—	3	3	no	—
9	#16836728	—	3	3	no	—
10	#16836728	—	3	3	no	—
11	C₃×C₁₃₉⋊C₉	C₃²	1	2	yes	—
12	C₉×C₁₃₉⋊C₃	—	3	3	no	—
13	3₋¹⁺²×C₁₃₉	C₃²	1	3	yes	—
14	C₃×C₉×C₁₃₉	C₃²	1	1	yes	characteristic
15	C₃²×C₁₃₉	C₃³	1	1	yes	characteristic
16	C₃×C₁₃₉⋊C₃	—	9	1	no	—
17	C₃×C₁₃₉⋊C₃	H(3,𝔽₃)	1	3	yes	—
18	C₉×C₁₃₉	—	3	1	no	—
19	C₉×C₁₃₉	—	3	3	no	—
20	C₁₃₉⋊C₉	—	3	6	no	—
21	C₁₃₉⋊C₉	—	3	2	no	—
22	C₃×C₁₃₉	—	3	1	no	—
23	C₃×C₁₃₉	C₃×H(3,𝔽₃)	1	1	yes	characteristic
24	C₁₃₉⋊C₃	—	3	3	no	—
25	#6912	—	139	1	no	maximal, Sylow
26	C₁₃₉	#6912	1	1	yes	characteristic, complemented, Sylow
27	C₃×3₋¹⁺²	—	139	1	no	—
28	C₃×3₋¹⁺²	—	139	3	no	—
29	#733	—	139	6	no	—
30	#733	—	139	2	no	—
31	#733	C₁₃₉⋊C₃	1	1	yes	characteristic, complemented
32	C₃³	—	139	1	no	—
33	3₋¹⁺²	—	139	6	no	—
34	3₋¹⁺²	—	417	3	no	—
35	3₋¹⁺²	—	417	3	no	—
36	3₋¹⁺²	—	417	3	no	—
37	3₋¹⁺²	C₃×C₁₃₉⋊C₃	1	3	yes	—
38	C₃×C₉	—	139	2	no	—
39	C₃×C₉	—	417	3	no	—
40	C₃×C₉	C₃×C₁₃₉⋊C₃	1	1	yes	characteristic
41	C₃²	—	139	3	no	—
42	C₃²	—	1251	1	no	—
43	C₃²	C₃²×C₁₃₉⋊C₃	1	1	yes	characteristic
44	C₉	—	3	3	no	—
45	C₉	—	3	1	no	—
46	C₉	—	417	2	no	—
47	C₉	—	417	6	no	—
48	C₃	—	3	1	no	—
49	C₃	—	417	3	no	—
50	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)
prime radical	C₃²×C₁₃₉⋊C₃	C₃²
hypercenter	#733	C₁₃₉⋊C₃
hypocenter	C₁₃₉	#6912
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
3. #26014619	C₃	semidirect
4. #26014618	C₃	semidirect
5. #733×C₁₃₉	C₃	semidirect
26. C₁₃₉	#6912	semidirect
31. #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
C₃×#16836728	3
3₋¹⁺²×C₁₃₉⋊C₃	3
#26014619	3
#26014618	3
#733×C₁₃₉	3
#6912	139