Subgroups — C₂×C₄⋊(C₅⋊AGL(1,𝔽₇)) #13132577

Order 1680.

216 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅⋊AGL(1,𝔽₇)	C₂	1	2	yes	maximal, complemented
2	C₂×C₄×C₅×ΓL(1,𝔽₈)	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄⋊(C₅⋊AGL(1,𝔽₇))	C₂	1	4	yes	maximal, direct product factor
4	C₂×D₁₄₀	C₃	1	1	yes	maximal, characteristic, complemented, Hall
5	C₂²×C₅×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
6	C₂×C₅⋊AGL(1,𝔽₇)	—	2	4	no	—
7	C₂×C₅⋊AGL(1,𝔽₇)	C₂²	1	4	yes	complemented
8	C₄×C₅×ΓL(1,𝔽₈)	C₂²	1	2	yes	complemented
9	C₂×C₄⋊AGL(1,𝔽₇)	—	5	1	no	maximal, Hall
10	C₂²×D₃₅	C₂×C₃	1	2	yes	complemented
11	C₂×C₄×C₅×C₇	C₂×C₃	1	1	yes	characteristic, complemented
12	D₁₄₀	C₂×C₃	1	4	yes	complemented
13	C₂×C₃×D₂₀	—	7	1	no	maximal, Hall
14	C₂×C₅×ΓL(1,𝔽₈)	D₄	1	2	yes	complemented
15	C₂×C₅×ΓL(1,𝔽₈)	C₂³	1	1	yes	characteristic
16	C₅⋊AGL(1,𝔽₇)	—	2	4	no	—
17	C₂²×AGL(1,𝔽₇)	—	5	2	no	—
18	C₂×C₄×ΓL(1,𝔽₈)	D₅	1	1	yes	characteristic, complemented
19	C₄⋊AGL(1,𝔽₇)	—	5	4	no	—
20	C₂²×C₅×C₇	C₂²×C₃	1	1	yes	characteristic
21	C₂×D₃₅	—	2	4	no	—
22	C₂×D₃₅	C₂²×C₃	1	4	yes	complemented
23	C₄×C₅×C₇	C₂²×C₃	1	2	yes	complemented
24	C₂²×C₃×D₅	—	7	2	no	—
25	C₂×C₃×C₄×C₅	—	7	1	no	—
26	C₃×D₂₀	—	7	4	no	—
27	C₂×D₂₈	—	5	1	no	Hall
28	C₅×ΓL(1,𝔽₈)	C₂×D₄	1	1	yes	characteristic, complemented, Hall
29	C₂²×ΓL(1,𝔽₈)	C₂×D₅	1	1	yes	characteristic
30	C₂×AGL(1,𝔽₇)	—	5	4	no	—
31	C₂×AGL(1,𝔽₇)	—	10	4	no	—
32	C₄×ΓL(1,𝔽₈)	C₂×D₅	1	2	yes	complemented
33	C₂×D₂₀	—	7	1	no	Hall
34	C₂×C₅×C₇	C₃×D₄	1	2	yes	complemented
35	C₂×C₅×C₇	C₂³×C₃	1	1	yes	characteristic
36	D₃₅	—	2	4	no	—
37	C₂²×C₃×C₅	—	7	1	no	—
38	C₂×C₃×D₅	—	7	4	no	—
39	C₂×C₃×D₅	—	14	4	no	—
40	C₃×C₄×C₅	—	7	2	no	—
41	C₂²×D₇	—	5	2	no	—
42	C₂×C₄×C₇	C₃×D₅	1	1	yes	characteristic, complemented
43	D₂₈	—	5	4	no	—
44	C₂×C₃×D₄	—	35	1	no	Hall
45	C₂×ΓL(1,𝔽₈)	D₂₀	1	2	yes	complemented
46	C₂×ΓL(1,𝔽₈)	C₂²×D₅	1	1	yes	characteristic
47	AGL(1,𝔽₇)	—	10	4	no	—
48	C₂²×D₅	—	7	2	no	—
49	C₂×C₄×C₅	AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
50	D₂₀	—	7	4	no	—
51	C₅×C₇	C₂×C₃×D₄	1	1	yes	characteristic, complemented, Hall
52	C₂×C₃×C₅	—	7	2	no	—
53	C₂×C₃×C₅	—	7	1	no	—
54	C₃×D₅	—	14	4	no	—
55	C₂²×C₇	C₂×C₃×D₅	1	1	yes	characteristic
56	C₂×D₇	—	5	4	no	—
57	C₂×D₇	—	10	4	no	—
58	C₄×C₇	C₂×C₃×D₅	1	2	yes	complemented
59	C₂³×C₃	—	35	2	no	—
60	C₃×D₄	—	35	4	no	—
61	C₂×C₃×C₄	—	7	1	no	—
62	ΓL(1,𝔽₈)	C₂×D₂₀	1	1	yes	characteristic, complemented, Hall
63	C₂²×C₅	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic
64	C₂×D₅	—	7	4	no	—
65	C₂×D₅	—	14	4	no	—
66	C₄×C₅	C₂×AGL(1,𝔽₇)	1	2	yes	complemented
67	C₂×D₄	—	35	1	no	Sylow
68	C₃×C₅	—	7	1	no	Hall
69	C₂×C₇	C₃×D₂₀	1	2	yes	complemented
70	C₂×C₇	C₂²×C₃×D₅	1	1	yes	characteristic
71	D₇	—	10	4	no	—
72	C₂²×C₃	—	7	1	no	—
73	C₂²×C₃	—	35	4	no	—
74	C₂²×C₃	—	70	4	no	—
75	C₃×C₄	—	7	2	no	—
76	C₂×C₅	C₄⋊AGL(1,𝔽₇)	1	2	yes	complemented
77	C₂×C₅	C₂²×AGL(1,𝔽₇)	1	1	yes	characteristic
78	D₅	—	14	4	no	—
79	C₂³	—	35	2	no	—
80	D₄	—	35	4	no	—
81	C₂×C₄	C₅⋊AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
82	C₇	C₂×C₃×D₂₀	1	1	yes	characteristic, complemented, Sylow
83	C₂×C₃	—	7	2	no	—
84	C₂×C₃	—	7	1	no	—
85	C₂×C₃	—	70	4	no	—
86	C₅	C₂×C₄⋊AGL(1,𝔽₇)	1	1	yes	characteristic, complemented, Sylow
87	C₂²	—	35	4	no	—
88	C₂²	—	70	4	no	—
89	C₂²	C₂×C₅⋊AGL(1,𝔽₇)	1	1	yes	characteristic
90	C₄	C₂×C₅⋊AGL(1,𝔽₇)	1	2	yes	complemented
91	C₃	—	7	1	no	Sylow
92	C₂	—	70	4	no	—
93	C₂	C₄⋊(C₅⋊AGL(1,𝔽₇))	1	2	yes	direct product factor
94	C₂	C₂²×C₅⋊AGL(1,𝔽₇)	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₅⋊AGL(1,𝔽₇)
commutator, cosocle	C₂×C₅×C₇	C₂³×C₃
Frattini	C₂	C₂²×C₅⋊AGL(1,𝔽₇)
Fitting	C₂×C₄×C₅×C₇	C₂×C₃
socle	C₂²×C₅×C₇	C₂²×C₃
hypercenter	C₂×C₄	C₅⋊AGL(1,𝔽₇)
hypocenter	C₅×C₇	C₂×C₃×D₄
LCS	G ▷ C₂×C₅×C₇ ▷ C₅×C₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄

Complements

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

N	H ≅ G/N	type
1. C₂²×C₅⋊AGL(1,𝔽₇)	C₂	semidirect
2. C₂×C₄×C₅×ΓL(1,𝔽₈)	C₂	semidirect
3. C₄⋊(C₅⋊AGL(1,𝔽₇))	C₂	direct (N × H)
4. C₂×D₁₄₀	C₃	semidirect
7. C₂×C₅⋊AGL(1,𝔽₇)	C₂²	semidirect
8. C₄×C₅×ΓL(1,𝔽₈)	C₂²	semidirect
10. C₂²×D₃₅	C₂×C₃	semidirect
11. C₂×C₄×C₅×C₇	C₂×C₃	semidirect
12. D₁₄₀	C₂×C₃	semidirect
14. C₂×C₅×ΓL(1,𝔽₈)	D₄	semidirect
18. C₂×C₄×ΓL(1,𝔽₈)	D₅	semidirect
22. C₂×D₃₅	C₂²×C₃	semidirect
23. C₄×C₅×C₇	C₂²×C₃	semidirect
28. C₅×ΓL(1,𝔽₈)	C₂×D₄	semidirect
32. C₄×ΓL(1,𝔽₈)	C₂×D₅	semidirect
34. C₂×C₅×C₇	C₃×D₄	semidirect
42. C₂×C₄×C₇	C₃×D₅	semidirect
45. C₂×ΓL(1,𝔽₈)	D₂₀	semidirect
49. C₂×C₄×C₅	AGL(1,𝔽₇)	semidirect
51. C₅×C₇	C₂×C₃×D₄	semidirect
58. C₄×C₇	C₂×C₃×D₅	semidirect
62. ΓL(1,𝔽₈)	C₂×D₂₀	semidirect
66. C₄×C₅	C₂×AGL(1,𝔽₇)	semidirect
69. C₂×C₇	C₃×D₂₀	semidirect
76. C₂×C₅	C₄⋊AGL(1,𝔽₇)	semidirect
81. C₂×C₄	C₅⋊AGL(1,𝔽₇)	semidirect
82. C₇	C₂×C₃×D₂₀	semidirect
86. C₅	C₂×C₄⋊AGL(1,𝔽₇)	semidirect
90. C₄	C₂×C₅⋊AGL(1,𝔽₇)	semidirect
93. C₂	C₄⋊(C₅⋊AGL(1,𝔽₇))	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₅⋊AGL(1,𝔽₇)	2
C₂×C₄×C₅×ΓL(1,𝔽₈)	2
C₄⋊(C₅⋊AGL(1,𝔽₇))	2
C₂×D₁₄₀	3
C₂×C₄⋊AGL(1,𝔽₇)	5
C₂×C₃×D₂₀	7

Subgroups — C2×C4⋊(C5⋊AGL(1,𝔽7)) #13132577

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄⋊(C₅⋊AGL(1,𝔽₇)) #13132577