Subgroups — C₂²×C₄×AGL(1,𝔽₅) #66252

Order 320.

498 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×AGL(1,𝔽₅)	C₂	1	2	yes	maximal
2	C₂²×C₄×D₅	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×AGL(1,𝔽₅)	C₂	1	12	yes	maximal, direct product factor
4	C₂³×D₅	C₂²	1	1	yes	characteristic
5	C₂²×AGL(1,𝔽₅)	C₄	1	16	yes	direct product factor
6	C₂²×AGL(1,𝔽₅)	C₂²	1	12	yes	—
7	C₂²×C₄×C₅	C₄	1	1	yes	characteristic, complemented
8	C₂²×Dic₅	C₄	1	1	yes	characteristic, complemented
9	C₂×C₄×D₅	C₄	1	6	yes	complemented
10	C₂×C₄×D₅	C₂²	1	6	yes	—
11	C₄×AGL(1,𝔽₅)	C₂²	1	16	yes	direct product factor
12	C₂²×C₄²	—	5	1	no	maximal, Sylow
13	C₂³×C₅	C₂×C₄	1	1	yes	characteristic
14	C₂²×D₅	C₂×C₄	1	3	yes	—
15	C₂²×D₅	C₂×C₄	1	8	yes	—
16	C₂²×D₅	C₂³	1	3	yes	—
17	C₂×AGL(1,𝔽₅)	C₂×C₄	1	48	yes	direct product factor
18	C₂×AGL(1,𝔽₅)	C₂³	1	8	yes	—
19	C₂×C₄×C₅	C₂×C₄	1	6	yes	complemented
20	C₂×Dic₅	C₂×C₄	1	6	yes	complemented
21	C₄×D₅	C₂×C₄	1	12	yes	complemented
22	C₄×D₅	C₂³	1	4	yes	—
23	C₂³×C₄	—	5	2	no	—
24	C₂³×C₄	—	5	1	no	—
25	C₂×C₄²	—	5	12	no	—
26	C₂²×C₅	C₄²	1	4	yes	complemented
27	C₂²×C₅	C₂²×C₄	1	3	yes	—
28	C₂×D₅	C₄²	1	12	yes	complemented
29	C₂×D₅	C₂²×C₄	1	3	yes	—
30	C₂×D₅	C₂²×C₄	1	12	yes	—
31	C₂×D₅	C₂⁴	1	1	yes	characteristic
32	AGL(1,𝔽₅)	C₂²×C₄	1	16	yes	direct product factor
33	C₄×C₅	C₂²×C₄	1	4	yes	complemented
34	Dic₅	C₂²×C₄	1	4	yes	complemented
35	C₂⁴	—	5	1	no	—
36	C₂²×C₄	—	5	12	no	—
37	C₂²×C₄	—	5	16	no	—
38	C₂²×C₄	—	5	1	no	—
39	C₂²×C₄	—	5	6	no	—
40	C₂²×C₄	—	5	6	no	—
41	C₂²×C₄	AGL(1,𝔽₅)	1	1	yes	characteristic, direct product factor
42	C₄²	—	5	16	no	—
43	C₂×C₅	C₂×C₄²	1	6	yes	complemented
44	C₂×C₅	C₂³×C₄	1	1	yes	characteristic
45	D₅	C₂×C₄²	1	6	yes	complemented
46	D₅	C₂³×C₄	1	2	yes	—
47	C₂³	—	5	8	no	—
48	C₂³	—	5	3	no	—
49	C₂³	—	5	3	no	—
50	C₂³	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
51	C₂×C₄	—	5	8	no	—
52	C₂×C₄	—	5	48	no	—
53	C₂×C₄	—	5	6	no	—
54	C₂×C₄	—	5	4	no	—
55	C₂×C₄	—	5	12	no	—
56	C₂×C₄	C₂×AGL(1,𝔽₅)	1	6	yes	direct product factor
57	C₅	C₂²×C₄²	1	1	yes	characteristic, complemented, Sylow
58	C₂²	—	5	12	no	—
59	C₂²	—	5	12	no	—
60	C₂²	—	5	1	no	—
61	C₂²	—	5	3	no	—
62	C₂²	C₄×AGL(1,𝔽₅)	1	4	yes	direct product factor
63	C₂²	C₂²×AGL(1,𝔽₅)	1	3	yes	—
64	C₄	—	5	16	no	—
65	C₄	—	5	4	no	—
66	C₄	C₂²×AGL(1,𝔽₅)	1	4	yes	direct product factor
67	C₂	—	5	2	no	—
68	C₂	—	5	6	no	—
69	C₂	C₂×C₄×AGL(1,𝔽₅)	1	6	yes	direct product factor
70	C₂	C₂³×AGL(1,𝔽₅)	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂²×C₄	AGL(1,𝔽₅)
commutator, hypocenter	C₅	C₂²×C₄²
Frattini	C₂	C₂³×AGL(1,𝔽₅)
Fitting	C₂²×C₄×C₅	C₄
socle	C₂³×C₅	C₂×C₄
prime radical	C₂³×D₅	C₂²
cosocle	C₂×D₅	C₂⁴
LCS	G ▷ C₅
UCS	C₁ ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
3. C₂×C₄×AGL(1,𝔽₅)	C₂	direct (N × H)
5. C₂²×AGL(1,𝔽₅)	C₄	direct (N × H)
7. C₂²×C₄×C₅	C₄	semidirect
8. C₂²×Dic₅	C₄	semidirect
9. C₂×C₄×D₅	C₄	semidirect
11. C₄×AGL(1,𝔽₅)	C₂²	direct (N × H)
17. C₂×AGL(1,𝔽₅)	C₂×C₄	direct (N × H)
19. C₂×C₄×C₅	C₂×C₄	semidirect
20. C₂×Dic₅	C₂×C₄	semidirect
21. C₄×D₅	C₂×C₄	semidirect
26. C₂²×C₅	C₄²	semidirect
28. C₂×D₅	C₄²	semidirect
32. AGL(1,𝔽₅)	C₂²×C₄	direct (N × H)
33. C₄×C₅	C₂²×C₄	semidirect
34. Dic₅	C₂²×C₄	semidirect
41. C₂²×C₄	AGL(1,𝔽₅)	direct (N × H)
43. C₂×C₅	C₂×C₄²	semidirect
45. D₅	C₂×C₄²	semidirect
56. C₂×C₄	C₂×AGL(1,𝔽₅)	direct (N × H)
57. C₅	C₂²×C₄²	semidirect
62. C₂²	C₄×AGL(1,𝔽₅)	direct (N × H)
66. C₄	C₂²×AGL(1,𝔽₅)	direct (N × H)
69. C₂	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
C₂³×AGL(1,𝔽₅)	2
C₂²×C₄×D₅	2
C₂×C₄×AGL(1,𝔽₅)	2
C₂²×C₄²	5

Subgroups — C22×C4×AGL(1,𝔽5) #66252

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₄×AGL(1,𝔽₅) #66252