Subgroups — C₂×C₂²⋊AGL(1,𝔽₅) #4137

Order 160.

132 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×D₅	C₂	1	1	yes	maximal, characteristic
2	C₂²×AGL(1,𝔽₅)	C₂	1	2	yes	maximal, complemented
3	C₂²⋊AGL(1,𝔽₅)	C₂	1	4	yes	maximal, direct product factor
4	C₂³×C₅	C₄	1	1	yes	characteristic, complemented
5	C₂²×D₅	—	2	4	no	—
6	C₂²×D₅	C₄	1	2	yes	complemented
7	C₂²×D₅	C₄	1	1	yes	characteristic, complemented
8	C₂²×D₅	C₂²	1	2	yes	—
9	C₂²×D₅	C₂²	1	1	yes	characteristic
10	C₂×AGL(1,𝔽₅)	—	2	4	no	—
11	C₂×AGL(1,𝔽₅)	C₂²	1	4	yes	complemented
12	C₂×C₂²⋊C₄	—	5	1	no	maximal, Sylow
13	C₂²×C₅	—	2	2	no	—
14	C₂²×C₅	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₅	C₂×C₄	1	2	yes	complemented
16	C₂×D₅	—	2	2	no	—
17	C₂×D₅	—	2	4	no	—
18	C₂×D₅	—	2	4	no	—
19	C₂×D₅	C₂×C₄	1	1	yes	characteristic
20	C₂×D₅	C₂×C₄	1	2	yes	complemented
21	C₂×D₅	D₄	1	4	yes	—
22	C₂×D₅	C₂³	1	1	yes	characteristic
23	AGL(1,𝔽₅)	—	2	4	no	—
24	C₂⁴	—	5	1	no	—
25	C₂²×C₄	—	5	2	no	—
26	C₂²⋊C₄	—	5	4	no	—
27	C₂×C₅	—	2	2	no	—
28	C₂×C₅	C₂²⋊C₄	1	2	yes	complemented
29	C₂×C₅	C₂²×C₄	1	1	yes	characteristic
30	D₅	—	2	2	no	—
31	D₅	C₂²⋊C₄	1	2	yes	complemented
32	D₅	C₂×D₄	1	2	yes	—
33	C₂³	—	5	1	no	—
34	C₂³	—	5	1	no	—
35	C₂³	—	5	2	no	—
36	C₂³	—	5	2	no	—
37	C₂³	—	10	4	no	—
38	C₂³	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
39	C₂×C₄	—	5	4	no	—
40	C₂×C₄	—	10	4	no	—
41	C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic, complemented, Sylow
42	C₂²	—	2	2	no	—
43	C₂²	—	5	4	no	—
44	C₂²	—	5	1	no	—
45	C₂²	—	5	1	no	—
46	C₂²	—	5	2	no	—
47	C₂²	—	10	4	no	—
48	C₂²	—	10	4	no	—
49	C₂²	—	10	2	no	—
50	C₂²	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
51	C₂²	C₂×AGL(1,𝔽₅)	1	2	yes	complemented
52	C₄	—	10	4	no	—
53	C₂	—	2	2	no	—
54	C₂	—	5	2	no	—
55	C₂	—	5	2	no	—
56	C₂	—	10	2	no	—
57	C₂	C₂²⋊AGL(1,𝔽₅)	1	2	yes	direct product factor
58	C₂	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
2. C₂²×AGL(1,𝔽₅)	C₂	semidirect
3. C₂²⋊AGL(1,𝔽₅)	C₂	direct (N × H)
4. C₂³×C₅	C₄	semidirect
6. C₂²×D₅	C₄	semidirect
7. C₂²×D₅	C₄	semidirect
11. C₂×AGL(1,𝔽₅)	C₂²	semidirect
15. C₂²×C₅	C₂×C₄	semidirect
20. C₂×D₅	C₂×C₄	semidirect
28. C₂×C₅	C₂²⋊C₄	semidirect
31. D₅	C₂²⋊C₄	semidirect
38. C₂³	AGL(1,𝔽₅)	semidirect
41. C₅	C₂×C₂²⋊C₄	semidirect
51. C₂²	C₂×AGL(1,𝔽₅)	semidirect
57. C₂	C₂²⋊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
C₂³×D₅	2
C₂²×AGL(1,𝔽₅)	2
C₂²⋊AGL(1,𝔽₅)	2
C₂×C₂²⋊C₄	5

Subgroups — C2×C22⋊AGL(1,𝔽5) #4137

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂²⋊AGL(1,𝔽₅) #4137