Subgroups — D₄×AGL(1,𝔽₅) #4132

Order 160.

94 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×AGL(1,𝔽₅)	C₂	1	2	yes	maximal, complemented
2	D₄×D₅	C₂	1	1	yes	maximal, characteristic
3	C₂²⋊AGL(1,𝔽₅)	C₂	1	2	yes	maximal, complemented
4	C₄⋊AGL(1,𝔽₅)	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄×AGL(1,𝔽₅)	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₅	C₂²	1	2	yes	—
7	C₂×AGL(1,𝔽₅)	—	2	4	no	—
8	C₂×AGL(1,𝔽₅)	C₂²	1	1	yes	characteristic
9	C₂×AGL(1,𝔽₅)	C₂²	1	2	yes	—
10	C₂×AGL(1,𝔽₅)	C₂²	1	1	yes	characteristic
11	C₅×D₄	C₄	1	1	yes	characteristic, complemented
12	C₂²⋊D₅	C₄	1	2	yes	complemented
13	D₂₀	C₄	1	1	yes	characteristic, complemented
14	C₄×D₅	C₂²	1	1	yes	characteristic
15	C₄×D₄	—	5	1	no	maximal, Sylow
16	C₂²×C₅	C₂×C₄	1	2	yes	complemented
17	C₂×D₅	—	2	2	no	—
18	C₂×D₅	—	2	2	no	—
19	C₂×D₅	C₂×C₄	1	2	yes	complemented
20	C₂×D₅	C₂³	1	1	yes	characteristic
21	AGL(1,𝔽₅)	—	2	2	no	—
22	AGL(1,𝔽₅)	—	2	1	no	—
23	AGL(1,𝔽₅)	D₄	1	2	yes	direct product factor
24	C₄×C₅	C₂×C₄	1	1	yes	characteristic, complemented
25	Dic₅	C₂×C₄	1	1	yes	characteristic, complemented
26	C₂×D₄	—	5	1	no	—
27	C₂²×C₄	—	5	2	no	—
28	C₄⋊C₄	—	5	1	no	—
29	C₂²⋊C₄	—	5	2	no	—
30	C₄²	—	5	1	no	—
31	C₂×C₅	—	2	2	no	—
32	C₂×C₅	C₂²×C₄	1	1	yes	characteristic
33	D₅	—	2	2	no	—
34	D₅	C₂×D₄	1	1	yes	characteristic
35	D₅	#41	1	1	yes	characteristic
36	C₂³	—	5	2	no	—
37	D₄	—	5	2	no	—
38	D₄	—	5	1	no	—
39	D₄	AGL(1,𝔽₅)	1	1	yes	characteristic, direct product factor
40	C₂×C₄	—	5	1	no	—
41	C₂×C₄	—	5	1	no	—
42	C₂×C₄	—	5	1	no	—
43	C₂×C₄	—	5	2	no	—
44	C₂×C₄	—	10	4	no	—
45	C₅	C₄×D₄	1	1	yes	characteristic, complemented, Sylow
46	C₂²	—	5	1	no	—
47	C₂²	—	5	2	no	—
48	C₂²	—	10	2	no	—
49	C₂²	—	10	2	no	—
50	C₂²	C₂×AGL(1,𝔽₅)	1	2	yes	complemented
51	C₄	—	5	2	no	—
52	C₄	—	5	1	no	—
53	C₄	—	10	2	no	—
54	C₄	—	10	1	no	—
55	C₄	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
56	C₂	—	2	2	no	—
57	C₂	—	5	1	no	—
58	C₂	—	5	1	no	—
59	C₂	—	10	2	no	—
60	C₂	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. C₂²×AGL(1,𝔽₅)	C₂	semidirect
3. C₂²⋊AGL(1,𝔽₅)	C₂	semidirect
4. C₄⋊AGL(1,𝔽₅)	C₂	semidirect
5. C₄×AGL(1,𝔽₅)	C₂	semidirect
11. C₅×D₄	C₄	semidirect
12. C₂²⋊D₅	C₄	semidirect
13. D₂₀	C₄	semidirect
16. C₂²×C₅	C₂×C₄	semidirect
19. C₂×D₅	C₂×C₄	semidirect
23. AGL(1,𝔽₅)	D₄	direct (N × H)
24. C₄×C₅	C₂×C₄	semidirect
25. Dic₅	C₂×C₄	semidirect
39. D₄	AGL(1,𝔽₅)	direct (N × H)
45. C₅	C₄×D₄	semidirect
50. C₂²	C₂×AGL(1,𝔽₅)	semidirect
55. C₄	C₂×AGL(1,𝔽₅)	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,𝔽₅)	2
D₄×D₅	2
C₂²⋊AGL(1,𝔽₅)	2
C₄⋊AGL(1,𝔽₅)	2
C₄×AGL(1,𝔽₅)	2
C₄×D₄	5

Subgroups — D4×AGL(1,𝔽5) #4132

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₄×AGL(1,𝔽₅) #4132