Subgroups — C₂×AGL(1,𝔽₇)×C₁₂₇⋊C₃ #44544799

Order 32004.

120 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×ΓL(1,𝔽₈)×C₁₂₇⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
2	AGL(1,𝔽₇)×C₁₂₇⋊C₃	C₂	1	2	yes	maximal, direct product factor
3	C₂×D₇×C₁₂₇⋊C₃	C₃	1	1	yes	maximal, characteristic, complemented
4	C₂×#19128629	C₃	1	1	yes	maximal, characteristic, complemented
5	C₂×#19128628	C₃	1	1	yes	maximal, characteristic, complemented
6	C₂×AGL(1,𝔽₇)×C₁₂₇	C₃	1	1	yes	maximal, characteristic, complemented
7	ΓL(1,𝔽₈)×C₁₂₇⋊C₃	C₂²	1	1	yes	characteristic, complemented, Hall
8	#19128629	C₂×C₃	1	2	yes	complemented
9	#19128628	C₂×C₃	1	2	yes	complemented
10	AGL(1,𝔽₇)×C₁₂₇	C₂×C₃	1	2	yes	complemented
11	D₇×C₁₂₇⋊C₃	C₂×C₃	1	2	yes	complemented
12	C₂×#14581887	C₂×C₃	1	1	yes	characteristic, complemented
13	C₂×#14581888	C₂×C₃	1	1	yes	characteristic, complemented
14	C₂×ΓL(1,𝔽₈)×C₁₂₇	C₂×C₃	1	1	yes	characteristic, complemented
15	C₂×C₇×C₁₂₇⋊C₃	C₂×C₃	1	1	yes	characteristic, complemented
16	C₂²×C₃×C₁₂₇⋊C₃	—	7	1	no	maximal, Hall
17	C₂×D₇×C₁₂₇	C₃²	1	1	yes	characteristic, complemented, Hall
18	#14581888	C₂²×C₃	1	1	yes	characteristic, complemented
19	#14581887	C₂²×C₃	1	1	yes	characteristic, complemented
20	C₇×C₁₂₇⋊C₃	C₂²×C₃	1	1	yes	characteristic, complemented
21	ΓL(1,𝔽₈)×C₁₂₇	C₂²×C₃	1	1	yes	characteristic, complemented
22	C₂×C₃×C₁₂₇⋊C₃	—	7	2	no	—
23	C₂×C₃×C₁₂₇⋊C₃	—	7	1	no	—
24	C₂×C₇×C₁₂₇	C₂×C₃²	1	1	yes	characteristic, complemented
25	D₇×C₁₂₇	C₂×C₃²	1	2	yes	complemented
26	C₂²×C₃×C₁₂₇	—	7	1	no	—
27	C₂²×C₁₂₇⋊C₃	—	7	1	no	—
28	C₂²×C₁₂₇⋊C₃	—	7	1	no	—
29	C₂²×C₁₂₇⋊C₃	—	7	1	no	—
30	C₃×C₁₂₇⋊C₃	—	7	1	no	Hall
31	C₇×C₁₂₇	C₂²×C₃²	1	1	yes	characteristic, complemented, Hall
32	C₂×C₃×C₁₂₇	—	7	2	no	—
33	C₂×C₃×C₁₂₇	—	7	1	no	—
34	C₂×C₁₂₇⋊C₃	—	7	2	no	—
35	C₂×C₁₂₇⋊C₃	—	7	2	no	—
36	C₂×C₁₂₇⋊C₃	—	7	1	no	—
37	C₂×C₁₂₇⋊C₃	—	7	1	no	—
38	C₂×C₁₂₇⋊C₃	—	7	2	no	—
39	C₂×C₁₂₇⋊C₃	AGL(1,𝔽₇)	1	1	yes	characteristic, direct product factor
40	C₂²×C₁₂₇	—	7	1	no	Hall
41	C₃×C₁₂₇	—	7	1	no	—
42	C₁₂₇⋊C₃	—	7	1	no	—
43	C₁₂₇⋊C₃	—	7	1	no	—
44	C₁₂₇⋊C₃	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic, direct product factor
45	C₂×C₁₂₇	—	7	2	no	—
46	C₂×C₁₂₇	C₃×AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
47	C₂×C₃×AGL(1,𝔽₇)	—	127	1	no	maximal, Hall
48	C₁₂₇	C₂×C₃×AGL(1,𝔽₇)	1	1	yes	characteristic, complemented, Sylow
49	C₂×C₃×ΓL(1,𝔽₈)	—	127	1	no	—
50	C₃×AGL(1,𝔽₇)	—	127	2	no	—
51	C₂×C₃×D₇	—	127	1	no	—
52	C₂×AGL(1,𝔽₇)	—	127	1	no	—
53	C₂×AGL(1,𝔽₇)	—	127	1	no	—
54	C₂×AGL(1,𝔽₇)	C₁₂₇⋊C₃	1	1	yes	characteristic, direct product factor
55	C₃×ΓL(1,𝔽₈)	—	127	1	no	Hall
56	C₂×C₃×C₇	—	127	1	no	—
57	C₃×D₇	—	127	2	no	—
58	C₂×ΓL(1,𝔽₈)	—	127	1	no	—
59	C₂×ΓL(1,𝔽₈)	—	127	1	no	—
60	C₂×ΓL(1,𝔽₈)	C₂×C₁₂₇⋊C₃	1	1	yes	characteristic, complemented
61	AGL(1,𝔽₇)	—	127	2	no	—
62	AGL(1,𝔽₇)	—	127	2	no	—
63	AGL(1,𝔽₇)	C₂×C₁₂₇⋊C₃	1	2	yes	direct product factor
64	C₂²×C₃²	—	889	1	no	Hall
65	C₂×D₇	C₃×C₁₂₇⋊C₃	1	1	yes	characteristic, complemented, Hall
66	C₃×C₇	—	127	1	no	—
67	ΓL(1,𝔽₈)	—	127	1	no	—
68	ΓL(1,𝔽₈)	—	127	1	no	—
69	ΓL(1,𝔽₈)	C₂²×C₁₂₇⋊C₃	1	1	yes	characteristic, complemented
70	C₂×C₃²	—	889	2	no	—
71	C₂×C₃²	—	889	1	no	—
72	C₂×C₇	C₂×C₃×C₁₂₇⋊C₃	1	1	yes	characteristic, complemented
73	D₇	C₂×C₃×C₁₂₇⋊C₃	1	2	yes	complemented
74	C₂²×C₃	—	7	1	no	—
75	C₂²×C₃	—	889	1	no	—
76	C₂²×C₃	—	889	1	no	—
77	C₂²×C₃	—	889	1	no	—
78	C₃²	—	889	1	no	Sylow
79	C₇	C₂²×C₃×C₁₂₇⋊C₃	1	1	yes	characteristic, complemented, Sylow
80	C₂×C₃	—	7	2	no	—
81	C₂×C₃	—	7	1	no	—
82	C₂×C₃	—	889	2	no	—
83	C₂×C₃	—	889	2	no	—
84	C₂×C₃	—	889	2	no	—
85	C₂×C₃	—	889	1	no	—
86	C₂×C₃	—	889	1	no	—
87	C₂×C₃	—	127	1	no	—
88	C₂²	—	7	1	no	Sylow
89	C₃	—	7	1	no	—
90	C₃	—	889	1	no	—
91	C₃	—	889	1	no	—
92	C₃	—	127	1	no	—
93	C₂	—	7	2	no	—
94	C₂	AGL(1,𝔽₇)×C₁₂₇⋊C₃	1	1	yes	characteristic, direct product factor

	subgroup	G/H
center, hypercenter	C₂	AGL(1,𝔽₇)×C₁₂₇⋊C₃
commutator, cosocle, hypocenter	C₇×C₁₂₇	C₂²×C₃²
Frattini	C₁	G
Fitting, socle	C₂×C₇×C₁₂₇	C₂×C₃²
LCS	G ▷ C₇×C₁₂₇
UCS	C₁ ◁ C₂

Complements

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

N	H ≅ G/N	type
1. C₂×ΓL(1,𝔽₈)×C₁₂₇⋊C₃	C₂	semidirect
2. AGL(1,𝔽₇)×C₁₂₇⋊C₃	C₂	direct (N × H)
3. C₂×D₇×C₁₂₇⋊C₃	C₃	semidirect
4. C₂×#19128629	C₃	semidirect
5. C₂×#19128628	C₃	semidirect
6. C₂×AGL(1,𝔽₇)×C₁₂₇	C₃	semidirect
7. ΓL(1,𝔽₈)×C₁₂₇⋊C₃	C₂²	semidirect
8. #19128629	C₂×C₃	semidirect
9. #19128628	C₂×C₃	semidirect
10. AGL(1,𝔽₇)×C₁₂₇	C₂×C₃	semidirect
11. D₇×C₁₂₇⋊C₃	C₂×C₃	semidirect
12. C₂×#14581887	C₂×C₃	semidirect
13. C₂×#14581888	C₂×C₃	semidirect
14. C₂×ΓL(1,𝔽₈)×C₁₂₇	C₂×C₃	semidirect
15. C₂×C₇×C₁₂₇⋊C₃	C₂×C₃	semidirect
17. C₂×D₇×C₁₂₇	C₃²	semidirect
18. #14581888	C₂²×C₃	semidirect
19. #14581887	C₂²×C₃	semidirect
20. C₇×C₁₂₇⋊C₃	C₂²×C₃	semidirect
21. ΓL(1,𝔽₈)×C₁₂₇	C₂²×C₃	semidirect
24. C₂×C₇×C₁₂₇	C₂×C₃²	semidirect
25. D₇×C₁₂₇	C₂×C₃²	semidirect
31. C₇×C₁₂₇	C₂²×C₃²	semidirect
39. C₂×C₁₂₇⋊C₃	AGL(1,𝔽₇)	direct (N × H)
44. C₁₂₇⋊C₃	C₂×AGL(1,𝔽₇)	direct (N × H)
46. C₂×C₁₂₇	C₃×AGL(1,𝔽₇)	semidirect
48. C₁₂₇	C₂×C₃×AGL(1,𝔽₇)	semidirect
54. C₂×AGL(1,𝔽₇)	C₁₂₇⋊C₃	direct (N × H)
60. C₂×ΓL(1,𝔽₈)	C₂×C₁₂₇⋊C₃	semidirect
63. AGL(1,𝔽₇)	C₂×C₁₂₇⋊C₃	direct (N × H)
65. C₂×D₇	C₃×C₁₂₇⋊C₃	semidirect
69. ΓL(1,𝔽₈)	C₂²×C₁₂₇⋊C₃	semidirect
72. C₂×C₇	C₂×C₃×C₁₂₇⋊C₃	semidirect
73. D₇	C₂×C₃×C₁₂₇⋊C₃	semidirect
79. C₇	C₂²×C₃×C₁₂₇⋊C₃	semidirect
94. C₂	AGL(1,𝔽₇)×C₁₂₇⋊C₃	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×ΓL(1,𝔽₈)×C₁₂₇⋊C₃	2
AGL(1,𝔽₇)×C₁₂₇⋊C₃	2
C₂×D₇×C₁₂₇⋊C₃	3
C₂×#19128629	3
C₂×#19128628	3
C₂×AGL(1,𝔽₇)×C₁₂₇	3
C₂²×C₃×C₁₂₇⋊C₃	7
C₂×C₃×AGL(1,𝔽₇)	127

Subgroups — C2×AGL(1,𝔽7)×C127⋊C3 #44544799

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×AGL(1,𝔽₇)×C₁₂₇⋊C₃ #44544799