Subgroups — C₂³×C₅⋊AGL(1,𝔽₇) #13132954

Order 1680.

536 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₅×ΓL(1,𝔽₈)	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₅⋊AGL(1,𝔽₇)	C₂	1	14	yes	maximal, direct product factor
3	C₂³×D₃₅	C₃	1	1	yes	maximal, characteristic, complemented, Hall
4	C₂²×C₅×ΓL(1,𝔽₈)	C₂²	1	7	yes	complemented
5	C₂×C₅⋊AGL(1,𝔽₇)	C₂²	1	28	yes	direct product factor
6	C₂³×AGL(1,𝔽₇)	—	5	1	no	maximal, Hall
7	C₂³×C₅×C₇	C₂×C₃	1	1	yes	characteristic, complemented
8	C₂²×D₃₅	C₂×C₃	1	14	yes	complemented
9	C₂³×C₃×D₅	—	7	1	no	maximal, Hall
10	C₂×C₅×ΓL(1,𝔽₈)	C₂³	1	7	yes	complemented
11	C₅⋊AGL(1,𝔽₇)	C₂³	1	8	yes	direct product factor
12	C₂³×ΓL(1,𝔽₈)	D₅	1	1	yes	characteristic, complemented
13	C₂²×AGL(1,𝔽₇)	—	5	14	no	—
14	C₂²×C₅×C₇	C₂²×C₃	1	7	yes	complemented
15	C₂×D₃₅	C₂²×C₃	1	28	yes	complemented
16	C₂³×C₃×C₅	—	7	1	no	—
17	C₂²×C₃×D₅	—	7	14	no	—
18	C₂³×D₇	—	5	1	no	Hall
19	C₅×ΓL(1,𝔽₈)	C₂⁴	1	1	yes	characteristic, complemented, Hall
20	C₂²×ΓL(1,𝔽₈)	C₂×D₅	1	7	yes	complemented
21	C₂×AGL(1,𝔽₇)	—	5	28	no	—
22	C₂³×D₅	—	7	1	no	Hall
23	C₂×C₅×C₇	C₂³×C₃	1	7	yes	complemented
24	D₃₅	C₂³×C₃	1	8	yes	complemented
25	C₂²×C₃×C₅	—	7	7	no	—
26	C₂×C₃×D₅	—	7	28	no	—
27	C₂³×C₇	C₃×D₅	1	1	yes	characteristic, complemented
28	C₂²×D₇	—	5	14	no	—
29	C₂⁴×C₃	—	35	1	no	Hall
30	C₂×ΓL(1,𝔽₈)	C₂²×D₅	1	7	yes	complemented
31	AGL(1,𝔽₇)	—	5	8	no	—
32	C₂³×C₅	AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
33	C₂²×D₅	—	7	14	no	—
34	C₅×C₇	C₂⁴×C₃	1	1	yes	characteristic, complemented, Hall
35	C₂×C₃×C₅	—	7	7	no	—
36	C₃×D₅	—	7	8	no	—
37	C₂²×C₇	C₂×C₃×D₅	1	7	yes	complemented
38	C₂×D₇	—	5	28	no	—
39	C₂³×C₃	—	7	1	no	—
40	C₂³×C₃	—	35	14	no	—
41	ΓL(1,𝔽₈)	C₂³×D₅	1	1	yes	characteristic, complemented, Hall
42	C₂²×C₅	C₂×AGL(1,𝔽₇)	1	7	yes	complemented
43	C₂×D₅	—	7	28	no	—
44	C₂⁴	—	35	1	no	Sylow
45	C₃×C₅	—	7	1	no	Hall
46	C₂×C₇	C₂²×C₃×D₅	1	7	yes	complemented
47	D₇	—	5	8	no	—
48	C₂²×C₃	—	7	7	no	—
49	C₂²×C₃	—	35	28	no	—
50	C₂×C₅	C₂²×AGL(1,𝔽₇)	1	7	yes	complemented
51	D₅	—	7	8	no	—
52	C₂³	—	35	14	no	—
53	C₂³	C₅⋊AGL(1,𝔽₇)	1	1	yes	characteristic, direct product factor
54	C₇	C₂³×C₃×D₅	1	1	yes	characteristic, complemented, Sylow
55	C₂×C₃	—	7	7	no	—
56	C₂×C₃	—	35	8	no	—
57	C₅	C₂³×AGL(1,𝔽₇)	1	1	yes	characteristic, complemented, Sylow
58	C₂²	—	35	28	no	—
59	C₂²	C₂×C₅⋊AGL(1,𝔽₇)	1	7	yes	direct product factor
60	C₃	—	7	1	no	Sylow
61	C₂	—	35	8	no	—
62	C₂	C₂²×C₅⋊AGL(1,𝔽₇)	1	7	yes	direct product factor

	subgroup	G/H
center, hypercenter	C₂³	C₅⋊AGL(1,𝔽₇)
commutator, cosocle, hypocenter	C₅×C₇	C₂⁴×C₃
Frattini	C₁	G
Fitting, socle	C₂³×C₅×C₇	C₂×C₃
LCS	G ▷ C₅×C₇
UCS	C₁ ◁ C₂³

Complements

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

N	H ≅ G/N	type
1. C₂³×C₅×ΓL(1,𝔽₈)	C₂	semidirect
2. C₂²×C₅⋊AGL(1,𝔽₇)	C₂	direct (N × H)
3. C₂³×D₃₅	C₃	semidirect
4. C₂²×C₅×ΓL(1,𝔽₈)	C₂²	semidirect
5. C₂×C₅⋊AGL(1,𝔽₇)	C₂²	direct (N × H)
7. C₂³×C₅×C₇	C₂×C₃	semidirect
8. C₂²×D₃₅	C₂×C₃	semidirect
10. C₂×C₅×ΓL(1,𝔽₈)	C₂³	semidirect
11. C₅⋊AGL(1,𝔽₇)	C₂³	direct (N × H)
12. C₂³×ΓL(1,𝔽₈)	D₅	semidirect
14. C₂²×C₅×C₇	C₂²×C₃	semidirect
15. C₂×D₃₅	C₂²×C₃	semidirect
19. C₅×ΓL(1,𝔽₈)	C₂⁴	semidirect
20. C₂²×ΓL(1,𝔽₈)	C₂×D₅	semidirect
23. C₂×C₅×C₇	C₂³×C₃	semidirect
24. D₃₅	C₂³×C₃	semidirect
27. C₂³×C₇	C₃×D₅	semidirect
30. C₂×ΓL(1,𝔽₈)	C₂²×D₅	semidirect
32. C₂³×C₅	AGL(1,𝔽₇)	semidirect
34. C₅×C₇	C₂⁴×C₃	semidirect
37. C₂²×C₇	C₂×C₃×D₅	semidirect
41. ΓL(1,𝔽₈)	C₂³×D₅	semidirect
42. C₂²×C₅	C₂×AGL(1,𝔽₇)	semidirect
46. C₂×C₇	C₂²×C₃×D₅	semidirect
50. C₂×C₅	C₂²×AGL(1,𝔽₇)	semidirect
53. C₂³	C₅⋊AGL(1,𝔽₇)	direct (N × H)
54. C₇	C₂³×C₃×D₅	semidirect
57. C₅	C₂³×AGL(1,𝔽₇)	semidirect
59. C₂²	C₂×C₅⋊AGL(1,𝔽₇)	direct (N × H)
62. C₂	C₂²×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
C₂³×C₅×ΓL(1,𝔽₈)	2
C₂²×C₅⋊AGL(1,𝔽₇)	2
C₂³×D₃₅	3
C₂³×AGL(1,𝔽₇)	5
C₂³×C₃×D₅	7

Subgroups — C23×C5⋊AGL(1,𝔽7) #13132954

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×C₅⋊AGL(1,𝔽₇) #13132954