Subgroups — C₃×#26014616 #46798628

Order 33777.

168 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₃²⋊(C₁₃₉⋊C₃)	C₃	1	2	yes	maximal, complemented
2	C₃²×C₁₃₉⋊C₉	C₃	1	1	yes	maximal, characteristic, complemented
3	#26014616	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	4	no	—
7	C₃²⋊(C₁₃₉⋊C₃)	C₃²	1	6	yes	complemented
8	C₃×C₁₃₉⋊C₉	—	3	2	no	—
9	C₃×C₁₃₉⋊C₉	—	3	1	no	—
10	C₃×C₁₃₉⋊C₉	C₃²	1	3	yes	complemented
11	C₃³×C₁₃₉	—	3	1	no	—
12	C₃³×C₁₃₉	C₃²	1	1	yes	characteristic
13	H(3,𝔽₃)×C₁₃₉	—	3	2	no	—
14	H(3,𝔽₃)×C₁₃₉	C₃²	1	3	yes	complemented
15	C₃²×C₁₃₉	—	3	3	no	—
16	C₃²×C₁₃₉	—	3	3	no	—
17	C₃²×C₁₃₉	—	9	3	no	—
18	C₃²×C₁₃₉	H(3,𝔽₃)	1	2	yes	—
19	C₃²×C₁₃₉	H(3,𝔽₃)	1	1	yes	characteristic
20	C₃²×C₁₃₉	C₃³	1	1	yes	characteristic
21	C₃×C₁₃₉⋊C₃	—	3	6	no	—
22	C₃×C₁₃₉⋊C₃	—	9	6	no	—
23	C₁₃₉⋊C₉	—	3	3	no	—
24	C₃×C₁₃₉	—	3	2	no	—
25	C₃×C₁₃₉	—	3	1	no	—
26	C₃×C₁₃₉	—	9	3	no	—
27	C₃×C₁₃₉	#732	1	3	yes	complemented
28	C₃×C₁₃₉	C₃×H(3,𝔽₃)	1	1	yes	characteristic
29	C₁₃₉⋊C₃	—	9	6	no	—
30	C₃×#732	—	139	1	no	maximal, Sylow
31	C₁₃₉	C₃×#732	1	1	yes	characteristic, complemented, Sylow
32	C₃×H(3,𝔽₃)	—	139	2	no	—
33	C₃×H(3,𝔽₃)	C₁₃₉⋊C₃	1	1	yes	characteristic, complemented
34	C₃²×C₉	—	139	1	no	—
35	#732	—	139	9	no	—
36	C₃³	—	3	1	no	—
37	C₃³	—	417	2	no	—
38	C₃³	C₃×C₁₃₉⋊C₃	1	1	yes	characteristic
39	H(3,𝔽₃)	—	3	2	no	—
40	H(3,𝔽₃)	—	139	6	no	—
41	H(3,𝔽₃)	—	417	4	no	—
42	H(3,𝔽₃)	C₃×C₁₃₉⋊C₃	1	3	yes	complemented
43	C₃×C₉	—	139	3	no	—
44	C₃×C₉	—	417	2	no	—
45	C₃×C₉	—	417	1	no	—
46	C₃²	—	3	3	no	—
47	C₃²	—	3	3	no	—
48	C₃²	—	9	3	no	—
49	C₃²	—	417	6	no	—
50	C₃²	—	1251	6	no	—
51	C₃²	C₃²⋊(C₁₃₉⋊C₃)	1	2	yes	—
52	C₃²	C₃²⋊(C₁₃₉⋊C₃)	1	1	yes	characteristic
53	C₃²	C₃²×C₁₃₉⋊C₃	1	1	yes	characteristic
54	C₉	—	417	3	no	—
55	C₃	—	3	2	no	—
56	C₃	—	3	1	no	—
57	C₃	—	9	3	no	—
58	C₃	—	1251	6	no	—
59	C₃	#26014616	1	3	yes	direct product factor
60	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×#732
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. C₃×C₃²⋊(C₁₃₉⋊C₃)	C₃	semidirect
2. C₃²×C₁₃₉⋊C₉	C₃	semidirect
3. #26014616	C₃	direct (N × H)
4. C₃×H(3,𝔽₃)×C₁₃₉	C₃	semidirect
7. C₃²⋊(C₁₃₉⋊C₃)	C₃²	semidirect
10. C₃×C₁₃₉⋊C₉	C₃²	semidirect
14. H(3,𝔽₃)×C₁₃₉	C₃²	semidirect
27. C₃×C₁₃₉	#732	semidirect
31. C₁₃₉	C₃×#732	semidirect
33. C₃×H(3,𝔽₃)	C₁₃₉⋊C₃	semidirect
42. H(3,𝔽₃)	C₃×C₁₃₉⋊C₃	semidirect
59. C₃	#26014616	direct (N × H)

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
#26014616	3
C₃×H(3,𝔽₃)×C₁₃₉	3
C₃×#732	139

Subgroups — C3×#26014616 #46798628

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#26014616 #46798628