Subgroups — C₈×AGL(1,𝔽₇) #66538

Order 336.

44 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×AGL(1,𝔽₇)	C₂	1	1	yes	maximal, characteristic
2	C₈×ΓL(1,𝔽₈)	C₂	1	1	yes	maximal, characteristic, complemented
3	ΓL(1,𝔽₈)⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₈×D₇	C₃	1	1	yes	maximal, characteristic, complemented, Hall
5	C₂×AGL(1,𝔽₇)	C₄	1	1	yes	characteristic
6	C₄×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
7	ΓL(1,𝔽₈)⋊C₄	C₄	1	1	yes	characteristic
8	C₄×D₇	C₂×C₃	1	1	yes	characteristic
9	C₇×C₈	C₂×C₃	1	1	yes	characteristic, complemented
10	C₇⋊C₈	C₂×C₃	1	1	yes	characteristic, complemented
11	C₂×C₃×C₈	—	7	1	no	maximal, Hall
12	C₂×ΓL(1,𝔽₈)	C₂×C₄	1	1	yes	characteristic
13	AGL(1,𝔽₇)	C₈	1	2	yes	direct product factor
14	C₂×D₇	C₃×C₄	1	1	yes	characteristic
15	C₄×C₇	C₂²×C₃	1	1	yes	characteristic
16	Dic₇	C₃×C₄	1	1	yes	characteristic
17	C₂×C₃×C₄	—	7	1	no	—
18	C₃×C₈	—	7	1	no	—
19	C₃×C₈	—	7	1	no	—
20	ΓL(1,𝔽₈)	C₂×C₈	1	1	yes	characteristic, complemented, Hall
21	C₂×C₈	—	7	1	no	Sylow
22	C₂×C₇	C₂×C₃×C₄	1	1	yes	characteristic
23	D₇	C₃×C₈	1	2	yes	complemented
24	C₂²×C₃	—	7	1	no	—
25	C₃×C₄	—	7	1	no	—
26	C₃×C₄	—	7	1	no	—
27	C₂×C₄	—	7	1	no	—
28	C₈	—	7	1	no	—
29	C₈	AGL(1,𝔽₇)	1	1	yes	characteristic, direct product factor
30	C₇	C₂×C₃×C₈	1	1	yes	characteristic, complemented, Sylow
31	C₂×C₃	—	7	2	no	—
32	C₂×C₃	—	7	1	no	—
33	C₂²	—	7	1	no	—
34	C₄	—	7	1	no	—
35	C₄	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic
36	C₃	—	7	1	no	Sylow
37	C₂	—	7	2	no	—
38	C₂	C₄×AGL(1,𝔽₇)	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₈	AGL(1,𝔽₇)
commutator, hypocenter	C₇	C₂×C₃×C₈
Frattini	C₄	C₂×AGL(1,𝔽₇)
Fitting	C₇×C₈	C₂×C₃
socle	C₂×C₇	C₂×C₃×C₄
prime radical	C₂×AGL(1,𝔽₇)	C₄
cosocle	C₄×C₇	C₂²×C₃
LCS	G ▷ C₇
UCS	C₁ ◁ C₈

Complements

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

N	H ≅ G/N	type
2. C₈×ΓL(1,𝔽₈)	C₂	semidirect
3. ΓL(1,𝔽₈)⋊C₈	C₂	semidirect
4. C₈×D₇	C₃	semidirect
9. C₇×C₈	C₂×C₃	semidirect
10. C₇⋊C₈	C₂×C₃	semidirect
13. AGL(1,𝔽₇)	C₈	direct (N × H)
20. ΓL(1,𝔽₈)	C₂×C₈	semidirect
23. D₇	C₃×C₈	semidirect
29. C₈	AGL(1,𝔽₇)	direct (N × H)
30. C₇	C₂×C₃×C₈	semidirect

Maximal subgroups

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

subgroup	index
C₄×AGL(1,𝔽₇)	2
C₈×ΓL(1,𝔽₈)	2
ΓL(1,𝔽₈)⋊C₈	2
C₈×D₇	3
C₂×C₃×C₈	7

Subgroups — C8×AGL(1,𝔽7) #66538

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈×AGL(1,𝔽₇) #66538