Subgroups — D₄×AGL(1,𝔽₈) #90601

Order 448.

127 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×AGL(1,𝔽₈)	C₂	1	2	yes	maximal, complemented
2	C₄×AGL(1,𝔽₈)	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×AGL(1,𝔽₈)	—	2	2	no	—
4	C₂×AGL(1,𝔽₈)	C₂²	1	1	yes	characteristic
5	C₂³×D₄	C₇	1	1	yes	maximal, characteristic, complemented, Sylow
6	AGL(1,𝔽₈)	D₄	1	1	yes	characteristic, direct product factor
7	C₇×D₄	—	8	1	no	maximal
8	C₂⁵	C₂×C₇	1	2	yes	complemented
9	C₂²×D₄	—	7	1	no	—
10	C₂²×D₄	—	7	1	no	—
11	C₂²×D₄	—	7	2	no	—
12	C₂³×C₄	C₂×C₇	1	1	yes	characteristic, complemented
13	C₂²×C₇	—	8	2	no	—
14	C₄×C₇	—	8	1	no	—
15	C₂⁴	—	2	2	no	—
16	C₂⁴	—	7	2	no	—
17	C₂⁴	—	7	2	no	—
18	C₂⁴	—	14	2	no	—
19	C₂⁴	C₂²×C₇	1	1	yes	characteristic
20	C₂×D₄	—	7	6	no	—
21	C₂×D₄	—	7	1	no	—
22	C₂×D₄	—	7	3	no	—
23	C₂×D₄	—	7	6	no	—
24	C₂²×C₄	—	7	1	no	—
25	C₂²×C₄	—	7	1	no	—
26	C₂×C₇	—	8	1	no	—
27	C₂×C₇	—	16	2	no	—
28	C₂³	—	7	1	no	—
29	C₂³	—	7	1	no	—
30	C₂³	—	7	6	no	—
31	C₂³	—	7	2	no	—
32	C₂³	—	14	2	no	—
33	C₂³	—	14	2	no	—
34	C₂³	—	14	6	no	—
35	C₂³	—	14	6	no	—
36	C₂³	C₇×D₄	1	1	yes	characteristic, complemented
37	D₄	—	7	6	no	—
38	D₄	—	7	2	no	—
39	D₄	—	7	1	no	—
40	D₄	AGL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
41	C₂×C₄	—	7	1	no	—
42	C₂×C₄	—	7	3	no	—
43	C₇	—	8	1	no	Sylow
44	C₂²	—	7	1	no	—
45	C₂²	—	7	3	no	—
46	C₂²	—	7	1	no	—
47	C₂²	—	7	2	no	—
48	C₂²	—	14	2	no	—
49	C₂²	—	14	2	no	—
50	C₂²	—	14	6	no	—
51	C₂²	—	14	6	no	—
52	C₂²	C₂×AGL(1,𝔽₈)	1	2	yes	complemented
53	C₄	—	7	1	no	—
54	C₄	C₂×AGL(1,𝔽₈)	1	1	yes	characteristic, complemented
55	C₂	—	2	2	no	—
56	C₂	—	7	1	no	—
57	C₂	—	7	1	no	—
58	C₂	—	14	2	no	—
59	C₂	C₂²×AGL(1,𝔽₈)	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂²×AGL(1,𝔽₈)
commutator, socle, cosocle	C₂⁴	C₂²×C₇
Fitting	C₂³×D₄	C₇
hypercenter	D₄	AGL(1,𝔽₈)
hypocenter	C₂³	C₇×D₄
LCS	G ▷ C₂⁴ ▷ C₂³
UCS	C₁ ◁ C₂ ◁ D₄

Complements

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

N	H ≅ G/N	type
1. C₂²×AGL(1,𝔽₈)	C₂	semidirect
2. C₄×AGL(1,𝔽₈)	C₂	semidirect
5. C₂³×D₄	C₇	semidirect
6. AGL(1,𝔽₈)	D₄	direct (N × H)
8. C₂⁵	C₂×C₇	semidirect
12. C₂³×C₄	C₂×C₇	semidirect
36. C₂³	C₇×D₄	semidirect
40. D₄	AGL(1,𝔽₈)	direct (N × H)
52. C₂²	C₂×AGL(1,𝔽₈)	semidirect
54. C₄	C₂×AGL(1,𝔽₈)	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,𝔽₈)	2
C₄×AGL(1,𝔽₈)	2
C₂³×D₄	7
C₇×D₄	8

Subgroups — D4×AGL(1,𝔽8) #90601

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₄×AGL(1,𝔽₈) #90601