Subgroups — AΣL(1,𝔽₂₇)×C₁₂₇⋊C₃ #44525089

Order 30861.

252 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃³×C₁₂₇⋊C₃	C₃	1	1	yes	maximal, characteristic, complemented
2	3₋¹⁺²×C₁₂₇⋊C₃	C₃	1	2	yes	maximal, complemented
3	H(3,𝔽₃)×C₁₂₇⋊C₃	C₃	1	1	yes	maximal, characteristic, complemented
4	#23779843	C₃	1	4	yes	maximal, complemented
5	#23779842	C₃	1	2	yes	maximal, complemented
6	#23779841	C₃	1	2	yes	maximal, complemented
7	AΣL(1,𝔽₂₇)×C₁₂₇	C₃	1	1	yes	maximal, characteristic, complemented
8	C₃²×C₁₂₇⋊C₃	—	3	2	no	—
9	C₃²×C₁₂₇⋊C₃	—	3	1	no	—
10	C₃²×C₁₂₇⋊C₃	—	3	9	no	—
11	C₃²×C₁₂₇⋊C₃	—	3	1	no	—
12	C₃²×C₁₂₇⋊C₃	C₃²	1	1	yes	characteristic
13	C₃²×C₁₂₇⋊C₃	C₃²	1	2	yes	complemented
14	#16812184	C₃²	1	4	yes	complemented
15	#16812183	—	3	2	no	—
16	#16812182	—	3	2	no	—
17	C₃²⋊(C₁₂₇⋊C₃)	—	3	2	no	—
18	C₃²⋊(C₁₂₇⋊C₃)	C₃²	1	2	yes	complemented
19	C₉×C₁₂₇⋊C₃	—	3	2	no	—
20	C₃³×C₁₂₇	C₃²	1	1	yes	characteristic, complemented
21	3₋¹⁺²×C₁₂₇	C₃²	1	2	yes	complemented
22	H(3,𝔽₃)×C₁₂₇	C₃²	1	1	yes	characteristic, complemented
23	C₃²×C₁₂₇	—	3	3	no	—
24	C₃²×C₁₂₇	—	3	1	no	—
25	C₃²×C₁₂₇	—	3	1	no	—
26	C₃²×C₁₂₇	C₃³	1	1	yes	characteristic
27	C₃×C₁₂₇⋊C₃	—	3	18	no	—
28	C₃×C₁₂₇⋊C₃	—	3	2	no	—
29	C₃×C₁₂₇⋊C₃	—	3	9	no	—
30	C₃×C₁₂₇⋊C₃	—	3	3	no	—
31	C₃×C₁₂₇⋊C₃	—	3	6	no	—
32	C₃×C₁₂₇⋊C₃	—	3	2	no	—
33	C₃×C₁₂₇⋊C₃	—	9	3	no	—
34	C₃×C₁₂₇⋊C₃	H(3,𝔽₃)	1	2	yes	—
35	C₃×C₁₂₇⋊C₃	H(3,𝔽₃)	1	1	yes	characteristic
36	C₉×C₁₂₇	—	3	2	no	—
37	#11765917	—	3	4	no	—
38	C₃×C₁₂₇	—	3	3	no	—
39	C₃×C₁₂₇	—	3	1	no	—
40	C₃×C₁₂₇	—	9	1	no	—
41	C₃×C₁₂₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
42	C₁₂₇⋊C₃	—	3	6	no	—
43	C₁₂₇⋊C₃	—	3	2	no	—
44	C₁₂₇⋊C₃	—	9	2	no	—
45	C₁₂₇⋊C₃	AΣL(1,𝔽₂₇)	1	3	yes	direct product factor
46	C₃×AΣL(1,𝔽₂₇)	—	127	1	no	maximal, Sylow
47	C₁₂₇	C₃×AΣL(1,𝔽₂₇)	1	1	yes	characteristic, complemented, Sylow
48	C₃⁴	—	127	1	no	—
49	C₃×3₋¹⁺²	—	127	2	no	—
50	C₃×H(3,𝔽₃)	—	127	1	no	—
51	AΣL(1,𝔽₂₇)	—	127	2	no	—
52	AΣL(1,𝔽₂₇)	—	127	4	no	—
53	AΣL(1,𝔽₂₇)	—	127	2	no	—
54	AΣL(1,𝔽₂₇)	C₁₂₇⋊C₃	1	1	yes	characteristic, direct product factor
55	C₃³	—	381	1	no	—
56	C₃³	—	381	9	no	—
57	C₃³	—	381	2	no	—
58	C₃³	—	381	1	no	—
59	C₃³	—	127	2	no	—
60	C₃³	—	127	1	no	—
61	C₃³	C₃×C₁₂₇⋊C₃	1	1	yes	characteristic, complemented
62	3₋¹⁺²	—	381	2	no	—
63	3₋¹⁺²	—	381	2	no	—
64	3₋¹⁺²	—	127	4	no	—
65	3₋¹⁺²	C₃×C₁₂₇⋊C₃	1	2	yes	complemented
66	H(3,𝔽₃)	—	381	2	no	—
67	H(3,𝔽₃)	—	127	2	no	—
68	H(3,𝔽₃)	C₃×C₁₂₇⋊C₃	1	1	yes	characteristic, complemented
69	C₃×C₉	—	381	2	no	—
70	C₃²	—	3	3	no	—
71	C₃²	—	3	1	no	—
72	C₃²	—	3	1	no	—
73	C₃²	—	1143	3	no	—
74	C₃²	—	381	6	no	—
75	C₃²	—	381	18	no	—
76	C₃²	—	381	3	no	—
77	C₃²	—	381	9	no	—
78	C₃²	—	381	2	no	—
79	C₃²	—	381	2	no	—
80	C₃²	—	127	2	no	—
81	C₃²	—	127	1	no	—
82	C₃²	C₃²×C₁₂₇⋊C₃	1	1	yes	characteristic
83	C₉	—	3	2	no	—
84	C₉	—	381	4	no	—
85	C₃	—	3	1	no	—
86	C₃	—	3	3	no	—
87	C₃	—	9	1	no	—
88	C₃	—	1143	2	no	—
89	C₃	—	381	2	no	—
90	C₃	—	381	6	no	—
91	C₃	—	127	3	no	—
92	C₃	H(3,𝔽₃)×C₁₂₇⋊C₃	1	1	yes	characteristic

	subgroup	G/H
center	C₃	H(3,𝔽₃)×C₁₂₇⋊C₃
commutator, cosocle	C₃²×C₁₂₇	C₃³
Frattini	C₃²	C₃²×C₁₂₇⋊C₃
Fitting	AΣL(1,𝔽₂₇)×C₁₂₇	C₃
socle	C₃×C₁₂₇	C₃×H(3,𝔽₃)
hypercenter	AΣL(1,𝔽₂₇)	C₁₂₇⋊C₃
hypocenter	C₁₂₇	C₃×AΣL(1,𝔽₂₇)
LCS	G ▷ C₃²×C₁₂₇ ▷ C₃×C₁₂₇ ▷ C₁₂₇
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×C₁₂₇⋊C₃	C₃	semidirect
2. 3₋¹⁺²×C₁₂₇⋊C₃	C₃	semidirect
3. H(3,𝔽₃)×C₁₂₇⋊C₃	C₃	semidirect
4. #23779843	C₃	semidirect
5. #23779842	C₃	semidirect
6. #23779841	C₃	semidirect
7. AΣL(1,𝔽₂₇)×C₁₂₇	C₃	semidirect
13. C₃²×C₁₂₇⋊C₃	C₃²	semidirect
14. #16812184	C₃²	semidirect
18. C₃²⋊(C₁₂₇⋊C₃)	C₃²	semidirect
20. C₃³×C₁₂₇	C₃²	semidirect
21. 3₋¹⁺²×C₁₂₇	C₃²	semidirect
22. H(3,𝔽₃)×C₁₂₇	C₃²	semidirect
45. C₁₂₇⋊C₃	AΣL(1,𝔽₂₇)	direct (N × H)
47. C₁₂₇	C₃×AΣL(1,𝔽₂₇)	semidirect
54. AΣL(1,𝔽₂₇)	C₁₂₇⋊C₃	direct (N × H)
61. C₃³	C₃×C₁₂₇⋊C₃	semidirect
65. 3₋¹⁺²	C₃×C₁₂₇⋊C₃	semidirect
68. H(3,𝔽₃)	C₃×C₁₂₇⋊C₃	semidirect

Maximal subgroups

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

subgroup	index
C₃³×C₁₂₇⋊C₃	3
3₋¹⁺²×C₁₂₇⋊C₃	3
H(3,𝔽₃)×C₁₂₇⋊C₃	3
#23779843	3
#23779842	3
#23779841	3
AΣL(1,𝔽₂₇)×C₁₂₇	3
C₃×AΣL(1,𝔽₂₇)	127

Subgroups — AΣL(1,𝔽27)×C127⋊C3 #44525089

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — AΣL(1,𝔽₂₇)×C₁₂₇⋊C₃ #44525089