Subgroups — AΣL(1,𝔽₂₇)×C₁₃₉⋊C₃ #46798619

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	3₋¹⁺²×C₁₃₉⋊C₃	C₃	1	2	yes	maximal, complemented
3	H(3,𝔽₃)×C₁₃₉⋊C₃	C₃	1	1	yes	maximal, characteristic, complemented
4	#26014612	C₃	1	4	yes	maximal, complemented
5	#26014611	C₃	1	2	yes	maximal, complemented
6	#26014610	C₃	1	2	yes	maximal, complemented
7	AΣL(1,𝔽₂₇)×C₁₃₉	C₃	1	1	yes	maximal, characteristic, complemented
8	C₃²×C₁₃₉⋊C₃	—	3	2	no	—
9	C₃²×C₁₃₉⋊C₃	—	3	1	no	—
10	C₃²×C₁₃₉⋊C₃	—	3	9	no	—
11	C₃²×C₁₃₉⋊C₃	—	3	1	no	—
12	C₃²×C₁₃₉⋊C₃	C₃²	1	1	yes	characteristic
13	C₃²×C₁₃₉⋊C₃	C₃²	1	2	yes	complemented
14	#16836730	C₃²	1	4	yes	complemented
15	#16836729	—	3	2	no	—
16	#16836728	—	3	2	no	—
17	C₃²⋊(C₁₃₉⋊C₃)	—	3	2	no	—
18	C₃²⋊(C₁₃₉⋊C₃)	C₃²	1	2	yes	complemented
19	C₉×C₁₃₉⋊C₃	—	3	2	no	—
20	C₃³×C₁₃₉	C₃²	1	1	yes	characteristic, complemented
21	3₋¹⁺²×C₁₃₉	C₃²	1	2	yes	complemented
22	H(3,𝔽₃)×C₁₃₉	C₃²	1	1	yes	characteristic, complemented
23	C₃²×C₁₃₉	—	3	3	no	—
24	C₃²×C₁₃₉	—	3	1	no	—
25	C₃²×C₁₃₉	—	3	1	no	—
26	C₃²×C₁₃₉	C₃³	1	1	yes	characteristic
27	C₃×C₁₃₉⋊C₃	—	3	18	no	—
28	C₃×C₁₃₉⋊C₃	—	3	2	no	—
29	C₃×C₁₃₉⋊C₃	—	3	9	no	—
30	C₃×C₁₃₉⋊C₃	—	3	3	no	—
31	C₃×C₁₃₉⋊C₃	—	3	6	no	—
32	C₃×C₁₃₉⋊C₃	—	3	2	no	—
33	C₃×C₁₃₉⋊C₃	—	9	3	no	—
34	C₃×C₁₃₉⋊C₃	H(3,𝔽₃)	1	2	yes	—
35	C₃×C₁₃₉⋊C₃	H(3,𝔽₃)	1	1	yes	characteristic
36	C₉×C₁₃₉	—	3	2	no	—
37	C₁₃₉⋊C₉	—	3	4	no	—
38	C₃×C₁₃₉	—	3	3	no	—
39	C₃×C₁₃₉	—	3	1	no	—
40	C₃×C₁₃₉	—	9	1	no	—
41	C₃×C₁₃₉	C₃×H(3,𝔽₃)	1	1	yes	characteristic
42	C₁₃₉⋊C₃	—	3	6	no	—
43	C₁₃₉⋊C₃	—	3	2	no	—
44	C₁₃₉⋊C₃	—	9	2	no	—
45	C₁₃₉⋊C₃	AΣL(1,𝔽₂₇)	1	3	yes	direct product factor
46	C₃×AΣL(1,𝔽₂₇)	—	139	1	no	maximal, Sylow
47	C₁₃₉	C₃×AΣL(1,𝔽₂₇)	1	1	yes	characteristic, complemented, Sylow
48	C₃⁴	—	139	1	no	—
49	C₃×3₋¹⁺²	—	139	2	no	—
50	C₃×H(3,𝔽₃)	—	139	1	no	—
51	AΣL(1,𝔽₂₇)	—	139	2	no	—
52	AΣL(1,𝔽₂₇)	—	139	4	no	—
53	AΣL(1,𝔽₂₇)	—	139	2	no	—
54	AΣL(1,𝔽₂₇)	C₁₃₉⋊C₃	1	1	yes	characteristic, direct product factor
55	C₃³	—	139	1	no	—
56	C₃³	—	139	2	no	—
57	C₃³	—	417	1	no	—
58	C₃³	—	417	2	no	—
59	C₃³	—	417	1	no	—
60	C₃³	—	417	9	no	—
61	C₃³	C₃×C₁₃₉⋊C₃	1	1	yes	characteristic, complemented
62	3₋¹⁺²	—	139	4	no	—
63	3₋¹⁺²	—	417	2	no	—
64	3₋¹⁺²	—	417	2	no	—
65	3₋¹⁺²	C₃×C₁₃₉⋊C₃	1	2	yes	complemented
66	H(3,𝔽₃)	—	139	2	no	—
67	H(3,𝔽₃)	—	417	2	no	—
68	H(3,𝔽₃)	C₃×C₁₃₉⋊C₃	1	1	yes	characteristic, complemented
69	C₃×C₉	—	417	2	no	—
70	C₃²	—	3	3	no	—
71	C₃²	—	3	1	no	—
72	C₃²	—	3	1	no	—
73	C₃²	—	139	1	no	—
74	C₃²	—	139	2	no	—
75	C₃²	—	417	3	no	—
76	C₃²	—	417	9	no	—
77	C₃²	—	417	6	no	—
78	C₃²	—	417	18	no	—
79	C₃²	—	417	2	no	—
80	C₃²	—	417	2	no	—
81	C₃²	—	1251	3	no	—
82	C₃²	C₃²×C₁₃₉⋊C₃	1	1	yes	characteristic
83	C₉	—	3	2	no	—
84	C₉	—	417	4	no	—
85	C₃	—	3	1	no	—
86	C₃	—	3	3	no	—
87	C₃	—	9	1	no	—
88	C₃	—	139	3	no	—
89	C₃	—	417	2	no	—
90	C₃	—	417	6	no	—
91	C₃	—	1251	2	no	—
92	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	AΣL(1,𝔽₂₇)×C₁₃₉	C₃
socle	C₃×C₁₃₉	C₃×H(3,𝔽₃)
hypercenter	AΣL(1,𝔽₂₇)	C₁₃₉⋊C₃
hypocenter	C₁₃₉	C₃×AΣL(1,𝔽₂₇)
LCS	G ▷ C₃²×C₁₃₉ ▷ C₃×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₃ ◁ C₃² ◁ AΣL(1,𝔽₂₇)

Complements

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

N	H ≅ G/N	type
1. C₃³×C₁₃₉⋊C₃	C₃	semidirect
2. 3₋¹⁺²×C₁₃₉⋊C₃	C₃	semidirect
3. H(3,𝔽₃)×C₁₃₉⋊C₃	C₃	semidirect
4. #26014612	C₃	semidirect
5. #26014611	C₃	semidirect
6. #26014610	C₃	semidirect
7. AΣL(1,𝔽₂₇)×C₁₃₉	C₃	semidirect
13. C₃²×C₁₃₉⋊C₃	C₃²	semidirect
14. #16836730	C₃²	semidirect
18. C₃²⋊(C₁₃₉⋊C₃)	C₃²	semidirect
20. C₃³×C₁₃₉	C₃²	semidirect
21. 3₋¹⁺²×C₁₃₉	C₃²	semidirect
22. H(3,𝔽₃)×C₁₃₉	C₃²	semidirect
45. C₁₃₉⋊C₃	AΣL(1,𝔽₂₇)	direct (N × H)
47. C₁₃₉	C₃×AΣL(1,𝔽₂₇)	semidirect
54. AΣL(1,𝔽₂₇)	C₁₃₉⋊C₃	direct (N × H)
61. C₃³	C₃×C₁₃₉⋊C₃	semidirect
65. 3₋¹⁺²	C₃×C₁₃₉⋊C₃	semidirect
68. H(3,𝔽₃)	C₃×C₁₃₉⋊C₃	semidirect

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
3₋¹⁺²×C₁₃₉⋊C₃	3
H(3,𝔽₃)×C₁₃₉⋊C₃	3
#26014612	3
#26014611	3
#26014610	3
AΣL(1,𝔽₂₇)×C₁₃₉	3
C₃×AΣL(1,𝔽₂₇)	139

Subgroups — AΣL(1,𝔽27)×C139⋊C3 #46798619

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — AΣL(1,𝔽₂₇)×C₁₃₉⋊C₃ #46798619