Subgroups — C₂×C₄⋊AGL(1,𝔽₇) #66654

Order 336.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×AGL(1,𝔽₇)	C₂	1	2	yes	maximal, complemented
2	C₂×C₄×ΓL(1,𝔽₈)	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄⋊AGL(1,𝔽₇)	C₂	1	4	yes	maximal, direct product factor
4	C₂×D₂₈	C₃	1	1	yes	maximal, characteristic, complemented, Hall
5	C₂²×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
6	C₂×AGL(1,𝔽₇)	—	2	4	no	—
7	C₂×AGL(1,𝔽₇)	C₂²	1	4	yes	complemented
8	C₄×ΓL(1,𝔽₈)	C₂²	1	2	yes	complemented
9	C₂²×D₇	C₂×C₃	1	2	yes	complemented
10	C₂×C₄×C₇	C₂×C₃	1	1	yes	characteristic, complemented
11	D₂₈	C₂×C₃	1	4	yes	complemented
12	C₂×C₃×D₄	—	7	1	no	maximal, Hall
13	C₂×ΓL(1,𝔽₈)	D₄	1	2	yes	complemented
14	C₂×ΓL(1,𝔽₈)	C₂³	1	1	yes	characteristic
15	AGL(1,𝔽₇)	—	2	4	no	—
16	C₂²×C₇	C₂²×C₃	1	1	yes	characteristic
17	C₂×D₇	—	2	4	no	—
18	C₂×D₇	C₂²×C₃	1	4	yes	complemented
19	C₄×C₇	C₂²×C₃	1	2	yes	complemented
20	C₂³×C₃	—	7	2	no	—
21	C₃×D₄	—	7	4	no	—
22	C₂×C₃×C₄	—	7	1	no	—
23	ΓL(1,𝔽₈)	C₂×D₄	1	1	yes	characteristic, complemented, Hall
24	C₂×D₄	—	7	1	no	Sylow
25	C₂×C₇	C₃×D₄	1	2	yes	complemented
26	C₂×C₇	C₂³×C₃	1	1	yes	characteristic
27	D₇	—	2	4	no	—
28	C₂²×C₃	—	7	4	no	—
29	C₂²×C₃	—	7	1	no	—
30	C₂²×C₃	—	14	4	no	—
31	C₃×C₄	—	7	2	no	—
32	C₂³	—	7	2	no	—
33	D₄	—	7	4	no	—
34	C₂×C₄	AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
35	C₇	C₂×C₃×D₄	1	1	yes	characteristic, complemented, Sylow
36	C₂×C₃	—	7	2	no	—
37	C₂×C₃	—	7	1	no	—
38	C₂×C₃	—	14	4	no	—
39	C₂²	—	7	4	no	—
40	C₂²	—	14	4	no	—
41	C₂²	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic
42	C₄	C₂×AGL(1,𝔽₇)	1	2	yes	complemented
43	C₃	—	7	1	no	Sylow
44	C₂	—	14	4	no	—
45	C₂	C₄⋊AGL(1,𝔽₇)	1	2	yes	direct product factor
46	C₂	C₂²×AGL(1,𝔽₇)	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. C₂²×AGL(1,𝔽₇)	C₂	semidirect
2. C₂×C₄×ΓL(1,𝔽₈)	C₂	semidirect
3. C₄⋊AGL(1,𝔽₇)	C₂	direct (N × H)
4. C₂×D₂₈	C₃	semidirect
7. C₂×AGL(1,𝔽₇)	C₂²	semidirect
8. C₄×ΓL(1,𝔽₈)	C₂²	semidirect
9. C₂²×D₇	C₂×C₃	semidirect
10. C₂×C₄×C₇	C₂×C₃	semidirect
11. D₂₈	C₂×C₃	semidirect
13. C₂×ΓL(1,𝔽₈)	D₄	semidirect
18. C₂×D₇	C₂²×C₃	semidirect
19. C₄×C₇	C₂²×C₃	semidirect
23. ΓL(1,𝔽₈)	C₂×D₄	semidirect
25. C₂×C₇	C₃×D₄	semidirect
34. C₂×C₄	AGL(1,𝔽₇)	semidirect
35. C₇	C₂×C₃×D₄	semidirect
42. C₄	C₂×AGL(1,𝔽₇)	semidirect
45. C₂	C₄⋊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×AGL(1,𝔽₇)	2
C₂×C₄×ΓL(1,𝔽₈)	2
C₄⋊AGL(1,𝔽₇)	2
C₂×D₂₈	3
C₂×C₃×D₄	7

Subgroups — C2×C4⋊AGL(1,𝔽7) #66654

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄⋊AGL(1,𝔽₇) #66654