Subgroups — D₈×AGL(1,𝔽₅) #65730

Order 320.

134 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₄×AGL(1,𝔽₅)	C₂	1	2	yes	maximal, complemented
2	D₅×D₈	C₂	1	1	yes	maximal, characteristic
3	#4007	C₂	1	2	yes	maximal, complemented
4	#3994	C₂	1	1	yes	maximal, characteristic, complemented
5	C₈×AGL(1,𝔽₅)	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×AGL(1,𝔽₅)	—	2	2	no	—
7	D₄×D₅	C₂²	1	2	yes	—
8	C₂²⋊AGL(1,𝔽₅)	—	2	2	no	—
9	C₄⋊AGL(1,𝔽₅)	C₂²	1	2	yes	—
10	C₄×AGL(1,𝔽₅)	C₂²	1	1	yes	characteristic
11	D₅⋊C₈	C₂²	1	1	yes	characteristic
12	C₅×D₈	C₄	1	1	yes	characteristic, complemented
13	#686	C₄	1	2	yes	complemented
14	D₄₀	C₄	1	1	yes	characteristic, complemented
15	C₈×D₅	C₂²	1	1	yes	characteristic
16	C₄×D₈	—	5	1	no	maximal, Sylow
17	C₂²×D₅	—	2	2	no	—
18	C₂×AGL(1,𝔽₅)	—	2	2	no	—
19	C₂×AGL(1,𝔽₅)	—	4	4	no	—
20	C₂×AGL(1,𝔽₅)	D₄	1	1	yes	characteristic
21	C₂×AGL(1,𝔽₅)	D₄	1	1	yes	characteristic
22	C₅×D₄	C₂×C₄	1	2	yes	complemented
23	C₂²⋊D₅	—	2	2	no	—
24	D₂₀	C₂×C₄	1	2	yes	complemented
25	C₄×D₅	C₂³	1	1	yes	characteristic
26	#170	—	2	1	no	—
27	C₅×C₈	C₂×C₄	1	1	yes	characteristic, complemented
28	#168	C₂×C₄	1	1	yes	characteristic, complemented
29	C₂×D₈	—	5	1	no	—
30	C₄×D₄	—	5	2	no	—
31	#107	—	5	1	no	—
32	#102	—	5	2	no	—
33	C₄×C₈	—	5	1	no	—
34	C₂²×C₅	—	2	2	no	—
35	C₂×D₅	—	2	2	no	—
36	C₂×D₅	—	4	2	no	—
37	C₂×D₅	—	4	2	no	—
38	C₂×D₅	C₂×D₄	1	1	yes	characteristic
39	AGL(1,𝔽₅)	—	2	1	no	—
40	AGL(1,𝔽₅)	—	4	2	no	—
41	AGL(1,𝔽₅)	D₈	1	2	yes	direct product factor
42	C₄×C₅	C₂²×C₄	1	1	yes	characteristic
43	Dic₅	#41	1	1	yes	characteristic
44	C₂×D₄	—	5	2	no	—
45	C₂²×C₄	—	10	2	no	—
46	D₈	—	5	2	no	—
47	D₈	—	5	1	no	—
48	D₈	AGL(1,𝔽₅)	1	1	yes	characteristic, direct product factor
49	C₂×C₈	—	5	1	no	—
50	C₂×C₈	—	5	1	no	—
51	C₄⋊C₄	—	5	2	no	—
52	C₂²⋊C₄	—	10	2	no	—
53	C₄²	—	5	1	no	—
54	C₂×C₅	—	4	2	no	—
55	C₂×C₅	C₄×D₄	1	1	yes	characteristic
56	D₅	—	4	2	no	—
57	D₅	C₂×D₈	1	1	yes	characteristic
58	D₅	#135	1	1	yes	characteristic
59	C₂³	—	10	2	no	—
60	D₄	—	5	2	no	—
61	D₄	—	10	2	no	—
62	D₄	C₂×AGL(1,𝔽₅)	1	2	yes	complemented
63	C₂×C₄	—	5	1	no	—
64	C₂×C₄	—	5	1	no	—
65	C₂×C₄	—	5	1	no	—
66	C₂×C₄	—	10	2	no	—
67	C₂×C₄	—	20	4	no	—
68	C₈	—	5	1	no	—
69	C₈	—	10	1	no	—
70	C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
71	C₅	C₄×D₈	1	1	yes	characteristic, complemented, Sylow
72	C₂²	—	2	2	no	—
73	C₂²	—	5	1	no	—
74	C₂²	—	10	2	no	—
75	C₂²	—	20	2	no	—
76	C₂²	—	20	2	no	—
77	C₄	—	5	2	no	—
78	C₄	—	5	1	no	—
79	C₄	—	10	1	no	—
80	C₄	—	20	2	no	—
81	C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
82	C₂	—	4	2	no	—
83	C₂	—	5	1	no	—
84	C₂	—	5	1	no	—
85	C₂	—	20	2	no	—
86	C₂	D₄×AGL(1,𝔽₅)	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₄×AGL(1,𝔽₅)
commutator	C₄×C₅	C₂²×C₄
Frattini	C₄	C₂²×AGL(1,𝔽₅)
Fitting	C₅×D₈	C₄
socle	C₂×C₅	C₄×D₄
prime radical	D₅×D₈	C₂
cosocle	C₄×D₅	C₂³
hypercenter	D₈	AGL(1,𝔽₅)
hypocenter	C₅	C₄×D₈
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂ ◁ C₄ ◁ D₈

Complements

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

N	H ≅ G/N	type
1. D₄×AGL(1,𝔽₅)	C₂	semidirect
3. #4007	C₂	semidirect
4. #3994	C₂	semidirect
5. C₈×AGL(1,𝔽₅)	C₂	semidirect
12. C₅×D₈	C₄	semidirect
13. #686	C₄	semidirect
14. D₄₀	C₄	semidirect
22. C₅×D₄	C₂×C₄	semidirect
24. D₂₀	C₂×C₄	semidirect
27. C₅×C₈	C₂×C₄	semidirect
28. #168	C₂×C₄	semidirect
41. AGL(1,𝔽₅)	D₈	direct (N × H)
48. D₈	AGL(1,𝔽₅)	direct (N × H)
62. D₄	C₂×AGL(1,𝔽₅)	semidirect
70. C₈	C₂×AGL(1,𝔽₅)	semidirect
71. C₅	C₄×D₈	semidirect

Maximal subgroups

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

subgroup	index
D₄×AGL(1,𝔽₅)	2
D₅×D₈	2
#4007	2
#3994	2
C₈×AGL(1,𝔽₅)	2
C₄×D₈	5

Subgroups — D8×AGL(1,𝔽5) #65730

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₈×AGL(1,𝔽₅) #65730