Subgroups — D₄×AGL(1,𝔽₇) #66656

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	D₄×ΓL(1,𝔽₈)	C₂	1	1	yes	maximal, characteristic, complemented
3	(C₂²⋊D₇)⋊C₃	C₂	1	2	yes	maximal, complemented
4	C₄⋊AGL(1,𝔽₇)	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄×AGL(1,𝔽₇)	C₂	1	1	yes	maximal, characteristic, complemented
6	D₄×D₇	C₃	1	1	yes	maximal, characteristic, complemented, Hall
7	C₂²×ΓL(1,𝔽₈)	C₂²	1	2	yes	complemented
8	C₂×AGL(1,𝔽₇)	—	2	4	no	—
9	C₂×AGL(1,𝔽₇)	C₂²	1	1	yes	characteristic
10	C₂×AGL(1,𝔽₇)	C₂²	1	2	yes	complemented
11	C₄×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic, complemented
12	ΓL(1,𝔽₈)⋊C₄	C₂²	1	1	yes	characteristic, complemented
13	C₂²×D₇	C₂×C₃	1	2	yes	complemented
14	C₇×D₄	C₂×C₃	1	1	yes	characteristic, complemented
15	C₂²⋊D₇	C₂×C₃	1	2	yes	complemented
16	D₂₈	C₂×C₃	1	1	yes	characteristic, complemented
17	C₄×D₇	C₂×C₃	1	1	yes	characteristic, complemented
18	C₂×C₃×D₄	—	7	1	no	maximal, Hall
19	C₂×ΓL(1,𝔽₈)	—	2	2	no	—
20	C₂×ΓL(1,𝔽₈)	C₂³	1	1	yes	characteristic
21	AGL(1,𝔽₇)	—	2	2	no	—
22	AGL(1,𝔽₇)	D₄	1	2	yes	direct product factor
23	C₂²×C₇	C₂²×C₃	1	2	yes	complemented
24	C₂×D₇	—	2	4	no	—
25	C₂×D₇	C₂²×C₃	1	1	yes	characteristic
26	C₂×D₇	C₂²×C₃	1	2	yes	complemented
27	C₄×C₇	C₂²×C₃	1	1	yes	characteristic, complemented
28	Dic₇	C₂²×C₃	1	1	yes	characteristic, complemented
29	C₂³×C₃	—	7	2	no	—
30	C₃×D₄	—	7	1	no	—
31	C₃×D₄	—	7	2	no	—
32	C₃×D₄	—	7	1	no	—
33	C₂×C₃×C₄	—	7	1	no	—
34	ΓL(1,𝔽₈)	C₂×D₄	1	1	yes	characteristic, complemented, Hall
35	C₂×D₄	—	7	1	no	Sylow
36	C₂×C₇	—	2	2	no	—
37	C₂×C₇	C₂³×C₃	1	1	yes	characteristic
38	D₇	—	2	2	no	—
39	D₇	C₃×D₄	1	2	yes	complemented
40	C₂²×C₃	—	7	1	no	—
41	C₂²×C₃	—	7	2	no	—
42	C₂²×C₃	—	7	2	no	—
43	C₂²×C₃	—	14	4	no	—
44	C₃×C₄	—	7	1	no	—
45	C₃×C₄	—	7	1	no	—
46	C₂³	—	7	2	no	—
47	D₄	—	7	2	no	—
48	D₄	—	7	1	no	—
49	D₄	AGL(1,𝔽₇)	1	1	yes	characteristic, direct product factor
50	C₂×C₄	—	7	1	no	—
51	C₇	C₂×C₃×D₄	1	1	yes	characteristic, complemented, Sylow
52	C₂×C₃	—	7	2	no	—
53	C₂×C₃	—	7	1	no	—
54	C₂×C₃	—	14	2	no	—
55	C₂×C₃	—	14	2	no	—
56	C₂²	—	7	1	no	—
57	C₂²	—	7	2	no	—
58	C₂²	—	14	4	no	—
59	C₂²	C₂×AGL(1,𝔽₇)	1	2	yes	complemented
60	C₄	—	7	1	no	—
61	C₄	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
62	C₃	—	7	1	no	Sylow
63	C₂	—	2	2	no	—
64	C₂	—	7	2	no	—
65	C₂	—	14	2	no	—
66	C₂	C₂²×AGL(1,𝔽₇)	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. C₂²×AGL(1,𝔽₇)	C₂	semidirect
2. D₄×ΓL(1,𝔽₈)	C₂	semidirect
3. (C₂²⋊D₇)⋊C₃	C₂	semidirect
4. C₄⋊AGL(1,𝔽₇)	C₂	semidirect
5. C₄×AGL(1,𝔽₇)	C₂	semidirect
6. D₄×D₇	C₃	semidirect
7. C₂²×ΓL(1,𝔽₈)	C₂²	semidirect
10. C₂×AGL(1,𝔽₇)	C₂²	semidirect
11. C₄×ΓL(1,𝔽₈)	C₂²	semidirect
12. ΓL(1,𝔽₈)⋊C₄	C₂²	semidirect
13. C₂²×D₇	C₂×C₃	semidirect
14. C₇×D₄	C₂×C₃	semidirect
15. C₂²⋊D₇	C₂×C₃	semidirect
16. D₂₈	C₂×C₃	semidirect
17. C₄×D₇	C₂×C₃	semidirect
22. AGL(1,𝔽₇)	D₄	direct (N × H)
23. C₂²×C₇	C₂²×C₃	semidirect
26. C₂×D₇	C₂²×C₃	semidirect
27. C₄×C₇	C₂²×C₃	semidirect
28. Dic₇	C₂²×C₃	semidirect
34. ΓL(1,𝔽₈)	C₂×D₄	semidirect
39. D₇	C₃×D₄	semidirect
49. D₄	AGL(1,𝔽₇)	direct (N × H)
51. C₇	C₂×C₃×D₄	semidirect
59. C₂²	C₂×AGL(1,𝔽₇)	semidirect
61. C₄	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
C₂²×AGL(1,𝔽₇)	2
D₄×ΓL(1,𝔽₈)	2
(C₂²⋊D₇)⋊C₃	2
C₄⋊AGL(1,𝔽₇)	2
C₄×AGL(1,𝔽₇)	2
D₄×D₇	3
C₂×C₃×D₄	7

Subgroups — D4×AGL(1,𝔽7) #66656

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₄×AGL(1,𝔽₇) #66656