Subgroups — C₂×C₄×AGL(1,𝔽₅) #4128

Order 160.

108 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
2	C₂×C₄×D₅	C₂	1	1	yes	maximal, characteristic
3	C₄×AGL(1,𝔽₅)	C₂	1	4	yes	maximal, direct product factor
4	C₂²×D₅	C₂²	1	1	yes	characteristic
5	C₂×AGL(1,𝔽₅)	C₄	1	8	yes	direct product factor
6	C₂×AGL(1,𝔽₅)	C₂²	1	4	yes	—
7	C₂×C₄×C₅	C₄	1	1	yes	characteristic, complemented
8	C₂×Dic₅	C₄	1	1	yes	characteristic, complemented
9	C₄×D₅	C₄	1	2	yes	complemented
10	C₄×D₅	C₂²	1	2	yes	—
11	C₂×C₄²	—	5	1	no	maximal, Sylow
12	C₂²×C₅	C₂×C₄	1	1	yes	characteristic
13	C₂×D₅	C₂×C₄	1	1	yes	characteristic
14	C₂×D₅	C₂×C₄	1	4	yes	—
15	C₂×D₅	C₂³	1	1	yes	characteristic
16	AGL(1,𝔽₅)	C₂×C₄	1	8	yes	direct product factor
17	C₄×C₅	C₂×C₄	1	2	yes	complemented
18	Dic₅	C₂×C₄	1	2	yes	complemented
19	C₂²×C₄	—	5	2	no	—
20	C₂²×C₄	—	5	1	no	—
21	C₄²	—	5	4	no	—
22	C₂×C₅	C₄²	1	2	yes	complemented
23	C₂×C₅	C₂²×C₄	1	1	yes	characteristic
24	D₅	C₄²	1	2	yes	complemented
25	D₅	C₂²×C₄	1	2	yes	—
26	C₂³	—	5	1	no	—
27	C₂×C₄	—	5	4	no	—
28	C₂×C₄	—	5	8	no	—
29	C₂×C₄	—	5	1	no	—
30	C₂×C₄	—	5	2	no	—
31	C₂×C₄	—	5	2	no	—
32	C₂×C₄	AGL(1,𝔽₅)	1	1	yes	characteristic, direct product factor
33	C₅	C₂×C₄²	1	1	yes	characteristic, complemented, Sylow
34	C₂²	—	5	4	no	—
35	C₂²	—	5	1	no	—
36	C₂²	—	5	1	no	—
37	C₂²	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
38	C₄	—	5	8	no	—
39	C₄	—	5	2	no	—
40	C₄	C₂×AGL(1,𝔽₅)	1	2	yes	direct product factor
41	C₂	—	5	2	no	—
42	C₂	—	5	2	no	—
43	C₂	C₄×AGL(1,𝔽₅)	1	2	yes	direct product factor
44	C₂	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂×C₄	AGL(1,𝔽₅)
commutator, hypocenter	C₅	C₂×C₄²
Frattini	C₂	C₂²×AGL(1,𝔽₅)
Fitting	C₂×C₄×C₅	C₄
socle	C₂²×C₅	C₂×C₄
prime radical	C₂²×D₅	C₂²
cosocle	C₂×D₅	C₂³
LCS	G ▷ C₅
UCS	C₁ ◁ C₂×C₄

Complements

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

N	H ≅ G/N	type
3. C₄×AGL(1,𝔽₅)	C₂	direct (N × H)
5. C₂×AGL(1,𝔽₅)	C₄	direct (N × H)
7. C₂×C₄×C₅	C₄	semidirect
8. C₂×Dic₅	C₄	semidirect
9. C₄×D₅	C₄	semidirect
16. AGL(1,𝔽₅)	C₂×C₄	direct (N × H)
17. C₄×C₅	C₂×C₄	semidirect
18. Dic₅	C₂×C₄	semidirect
22. C₂×C₅	C₄²	semidirect
24. D₅	C₄²	semidirect
32. C₂×C₄	AGL(1,𝔽₅)	direct (N × H)
33. C₅	C₂×C₄²	semidirect
40. C₄	C₂×AGL(1,𝔽₅)	direct (N × H)
43. C₂	C₄×AGL(1,𝔽₅)	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,𝔽₅)	2
C₂×C₄×D₅	2
C₄×AGL(1,𝔽₅)	2
C₂×C₄²	5

Subgroups — C2×C4×AGL(1,𝔽5) #4128

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×AGL(1,𝔽₅) #4128