Subgroups — D₅×AGL(1,𝔽₇) #88291

Order 420.

40 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅⋊AGL(1,𝔽₇)	C₂	1	1	yes	maximal, characteristic, complemented
2	D₅×ΓL(1,𝔽₈)	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×AGL(1,𝔽₇)	C₂	1	1	yes	maximal, characteristic, complemented
4	D₅×D₇	C₃	1	1	yes	maximal, characteristic, complemented, Hall
5	C₅×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic, complemented, Hall
6	C₂×AGL(1,𝔽₇)	—	5	1	no	maximal, Hall
7	D₃₅	C₂×C₃	1	1	yes	characteristic, complemented
8	C₅×D₇	C₂×C₃	1	1	yes	characteristic, complemented
9	C₇×D₅	C₂×C₃	1	1	yes	characteristic, complemented
10	C₂×C₃×D₅	—	7	1	no	maximal, Hall
11	C₂×ΓL(1,𝔽₈)	—	5	1	no	—
12	AGL(1,𝔽₇)	—	5	1	no	—
13	AGL(1,𝔽₇)	D₅	1	1	yes	characteristic, direct product factor
14	C₅×C₇	C₂²×C₃	1	1	yes	characteristic, complemented, Hall
15	C₂×C₃×C₅	—	7	1	no	—
16	C₃×D₅	—	7	1	no	—
17	C₃×D₅	—	7	1	no	—
18	C₂×D₇	—	5	1	no	Hall
19	ΓL(1,𝔽₈)	C₂×D₅	1	1	yes	characteristic, complemented, Hall
20	C₂×D₅	—	7	1	no	Hall
21	C₃×C₅	—	7	1	no	Hall
22	C₂×C₇	—	5	1	no	—
23	D₇	—	5	1	no	—
24	D₇	C₃×D₅	1	1	yes	characteristic, complemented
25	C₂²×C₃	—	35	1	no	Hall
26	C₂×C₅	—	7	1	no	—
27	D₅	—	7	1	no	—
28	D₅	AGL(1,𝔽₇)	1	1	yes	characteristic, direct product factor
29	C₇	C₂×C₃×D₅	1	1	yes	characteristic, complemented, Sylow
30	C₂×C₃	—	7	1	no	—
31	C₂×C₃	—	35	1	no	—
32	C₂×C₃	—	35	1	no	—
33	C₅	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic, complemented, Sylow
34	C₂²	—	35	1	no	Sylow
35	C₃	—	7	1	no	Sylow
36	C₂	—	5	1	no	—
37	C₂	—	7	1	no	—
38	C₂	—	35	1	no	—

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₅×C₇	C₂²×C₃
LCS	G ▷ C₅×C₇
UCS	C₁

Complements

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

N	H ≅ G/N	type
1. C₅⋊AGL(1,𝔽₇)	C₂	semidirect
2. D₅×ΓL(1,𝔽₈)	C₂	semidirect
3. C₅×AGL(1,𝔽₇)	C₂	semidirect
4. D₅×D₇	C₃	semidirect
5. C₅×ΓL(1,𝔽₈)	C₂²	semidirect
7. D₃₅	C₂×C₃	semidirect
8. C₅×D₇	C₂×C₃	semidirect
9. C₇×D₅	C₂×C₃	semidirect
13. AGL(1,𝔽₇)	D₅	direct (N × H)
14. C₅×C₇	C₂²×C₃	semidirect
19. ΓL(1,𝔽₈)	C₂×D₅	semidirect
24. D₇	C₃×D₅	semidirect
28. D₅	AGL(1,𝔽₇)	direct (N × H)
29. C₇	C₂×C₃×D₅	semidirect
33. C₅	C₂×AGL(1,𝔽₇)	semidirect

Maximal subgroups

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

subgroup	index
C₅⋊AGL(1,𝔽₇)	2
D₅×ΓL(1,𝔽₈)	2
C₅×AGL(1,𝔽₇)	2
D₅×D₇	3
C₂×AGL(1,𝔽₇)	5
C₂×C₃×D₅	7

Subgroups — D5×AGL(1,𝔽7) #88291

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₅×AGL(1,𝔽₇) #88291