Subgroups — C₄²×C₅×AGL(1,𝔽₅) #13104445

Order 1600.

531 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₅×AGL(1,𝔽₅)	C₂	1	6	yes	maximal
2	C₄²×C₅×D₅	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₅×AGL(1,𝔽₅)	C₂²	1	4	yes	—
4	C₂×C₄×C₅×D₅	C₂²	1	3	yes	—
5	C₄×C₅×AGL(1,𝔽₅)	C₄	1	24	yes	direct product factor
6	C₄²×C₅²	C₄	1	1	yes	characteristic, complemented
7	C₄×C₅×Dic₅	C₄	1	3	yes	complemented
8	C₄²×AGL(1,𝔽₅)	C₅	1	1	yes	maximal, characteristic, direct product factor
9	C₄³×C₅	—	5	1	no	maximal
10	C₂²×C₅×D₅	C₂³	1	1	yes	characteristic
11	C₂×C₅×AGL(1,𝔽₅)	C₂×C₄	1	24	yes	—
12	C₂×C₄×C₅²	C₂×C₄	1	3	yes	—
13	C₂×C₅×Dic₅	C₂×C₄	1	3	yes	—
14	C₄×C₅×D₅	C₂×C₄	1	12	yes	—
15	C₂×C₄×AGL(1,𝔽₅)	C₂×C₅	1	6	yes	—
16	C₂×C₄²×C₅	—	5	6	no	—
17	C₂×C₄²×C₅	—	5	1	no	—
18	C₄²×D₅	C₂×C₅	1	1	yes	characteristic
19	C₂²×C₅²	C₂²×C₄	1	1	yes	characteristic
20	C₂×C₅×D₅	C₂²×C₄	1	6	yes	—
21	C₅×AGL(1,𝔽₅)	C₄²	1	16	yes	direct product factor
22	C₄×C₅²	C₄²	1	6	yes	complemented
23	C₅×Dic₅	C₄²	1	6	yes	complemented
24	C₂²×AGL(1,𝔽₅)	C₂²×C₅	1	4	yes	—
25	C₂²×C₄×C₅	—	5	4	no	—
26	C₂²×C₄×C₅	—	5	3	no	—
27	C₂×C₄×D₅	C₂²×C₅	1	3	yes	—
28	C₄×AGL(1,𝔽₅)	C₄×C₅	1	24	yes	direct product factor
29	C₄²×C₅	—	4	1	no	—
30	C₄²×C₅	—	5	24	no	—
31	C₄²×C₅	—	5	3	no	—
32	C₄²×C₅	C₄×C₅	1	1	yes	characteristic, complemented
33	C₄²×C₅	AGL(1,𝔽₅)	1	1	yes	characteristic, direct product factor
34	C₄×Dic₅	C₄×C₅	1	3	yes	complemented
35	C₄³	—	5	1	no	Sylow
36	C₂×C₅²	C₂×C₄²	1	3	yes	—
37	C₅×D₅	C₂×C₄²	1	4	yes	—
38	C₂³×C₅	—	5	1	no	—
39	C₂²×D₅	C₂³×C₅	1	1	yes	characteristic
40	C₂×AGL(1,𝔽₅)	C₂×C₄×C₅	1	24	yes	—
41	C₂×C₄×C₅	—	4	3	no	—
42	C₂×C₄×C₅	—	5	24	no	—
43	C₂×C₄×C₅	—	5	3	no	—
44	C₂×C₄×C₅	—	5	12	no	—
45	C₂×C₄×C₅	C₂×C₄×C₅	1	3	yes	—
46	C₂×C₄×C₅	C₂×AGL(1,𝔽₅)	1	3	yes	—
47	C₂×Dic₅	C₂×C₄×C₅	1	3	yes	—
48	C₄×D₅	C₂×C₄×C₅	1	12	yes	—
49	C₂×C₄²	—	5	6	no	—
50	C₂×C₄²	—	5	1	no	—
51	C₅²	C₄³	1	1	yes	characteristic, complemented, Sylow
52	C₂²×C₅	—	4	1	no	—
53	C₂²×C₅	—	5	6	no	—
54	C₂²×C₅	C₂²×C₄×C₅	1	1	yes	characteristic
55	C₂²×C₅	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
56	C₂×D₅	C₂²×C₄×C₅	1	6	yes	—
57	AGL(1,𝔽₅)	C₄²×C₅	1	16	yes	direct product factor
58	C₄×C₅	—	4	6	no	—
59	C₄×C₅	—	5	16	no	—
60	C₄×C₅	—	5	6	no	—
61	C₄×C₅	C₄²×C₅	1	6	yes	complemented
62	C₄×C₅	C₄×AGL(1,𝔽₅)	1	6	yes	direct product factor
63	Dic₅	C₄²×C₅	1	6	yes	complemented
64	C₂²×C₄	—	5	4	no	—
65	C₂²×C₄	—	5	3	no	—
66	C₄²	—	5	24	no	—
67	C₄²	—	5	3	no	—
68	C₄²	C₅×AGL(1,𝔽₅)	1	1	yes	characteristic, direct product factor
69	C₂×C₅	—	4	3	no	—
70	C₂×C₅	—	5	4	no	—
71	C₂×C₅	C₂×C₄²×C₅	1	3	yes	—
72	C₂×C₅	C₂×C₄×AGL(1,𝔽₅)	1	3	yes	—
73	D₅	C₂×C₄²×C₅	1	4	yes	—
74	C₂³	—	5	1	no	—
75	C₂×C₄	—	5	24	no	—
76	C₂×C₄	—	5	3	no	—
77	C₂×C₄	—	5	12	no	—
78	C₂×C₄	C₂×C₅×AGL(1,𝔽₅)	1	3	yes	—
79	C₅	—	4	1	no	—
80	C₅	C₄³×C₅	1	1	yes	characteristic, complemented
81	C₅	C₄²×AGL(1,𝔽₅)	1	1	yes	characteristic, direct product factor
82	C₂²	—	5	6	no	—
83	C₂²	C₂²×C₅×AGL(1,𝔽₅)	1	1	yes	characteristic
84	C₄	—	5	16	no	—
85	C₄	—	5	6	no	—
86	C₄	C₄×C₅×AGL(1,𝔽₅)	1	6	yes	direct product factor
87	C₂	—	5	4	no	—
88	C₂	C₂×C₄×C₅×AGL(1,𝔽₅)	1	3	yes	—

	subgroup	G/H
center, hypercenter	C₄²×C₅	AGL(1,𝔽₅)
commutator, hypocenter	C₅	C₄³×C₅
Frattini	C₂²	C₂²×C₅×AGL(1,𝔽₅)
Fitting	C₄²×C₅²	C₄
socle	C₂²×C₅²	C₂²×C₄
prime radical	C₂²×C₅×D₅	C₂³
cosocle	C₂²×D₅	C₂³×C₅
LCS	G ▷ C₅
UCS	C₁ ◁ C₄²×C₅

Complements

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

N	H ≅ G/N	type
5. C₄×C₅×AGL(1,𝔽₅)	C₄	direct (N × H)
6. C₄²×C₅²	C₄	semidirect
7. C₄×C₅×Dic₅	C₄	semidirect
8. C₄²×AGL(1,𝔽₅)	C₅	direct (N × H)
21. C₅×AGL(1,𝔽₅)	C₄²	direct (N × H)
22. C₄×C₅²	C₄²	semidirect
23. C₅×Dic₅	C₄²	semidirect
28. C₄×AGL(1,𝔽₅)	C₄×C₅	direct (N × H)
32. C₄²×C₅	C₄×C₅	semidirect
33. C₄²×C₅	AGL(1,𝔽₅)	direct (N × H)
34. C₄×Dic₅	C₄×C₅	semidirect
51. C₅²	C₄³	semidirect
57. AGL(1,𝔽₅)	C₄²×C₅	direct (N × H)
61. C₄×C₅	C₄²×C₅	semidirect
62. C₄×C₅	C₄×AGL(1,𝔽₅)	direct (N × H)
63. Dic₅	C₄²×C₅	semidirect
68. C₄²	C₅×AGL(1,𝔽₅)	direct (N × H)
80. C₅	C₄³×C₅	semidirect
81. C₅	C₄²×AGL(1,𝔽₅)	direct (N × H)
86. C₄	C₄×C₅×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
C₂×C₄×C₅×AGL(1,𝔽₅)	2
C₄²×C₅×D₅	2
C₄²×AGL(1,𝔽₅)	5
C₄³×C₅	5

Subgroups — C42×C5×AGL(1,𝔽5) #13104445

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄²×C₅×AGL(1,𝔽₅) #13104445