Subgroups — C₅²×C₂²⋊AGL(1,𝔽₅) #14522787

Order 2000.

332 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅²×D₅	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅²×AGL(1,𝔽₅)	C₂	1	2	yes	maximal, complemented
3	C₂²×C₅³	C₄	1	1	yes	characteristic, complemented
4	C₂×C₅²×D₅	—	2	2	no	—
5	C₂×C₅²×D₅	C₄	1	1	yes	characteristic, complemented
6	C₂×C₅²×D₅	C₂²	1	1	yes	characteristic
7	C₅²×AGL(1,𝔽₅)	—	2	2	no	—
8	C₅×C₂²⋊AGL(1,𝔽₅)	C₅	1	6	yes	maximal, direct product factor
9	C₅²×C₂²⋊C₄	—	5	1	no	maximal
10	C₂×C₅³	—	2	1	no	—
11	C₂×C₅³	C₂×C₄	1	1	yes	characteristic
12	C₅²×D₅	—	2	1	no	—
13	C₅²×D₅	D₄	1	2	yes	—
14	C₂³×C₅²	—	5	1	no	—
15	C₂²×C₅×D₅	C₂×C₅	1	6	yes	—
16	C₂×C₅×AGL(1,𝔽₅)	C₂×C₅	1	12	yes	complemented
17	C₂×C₄×C₅²	—	5	2	no	—
18	C₅³	C₂²⋊C₄	1	1	yes	characteristic, complemented, Sylow
19	C₂²×C₅²	—	4	6	no	—
20	C₂²×C₅²	—	5	1	no	—
21	C₂²×C₅²	—	5	1	no	—
22	C₂²×C₅²	—	10	2	no	—
23	C₂²×C₅²	C₄×C₅	1	6	yes	complemented
24	C₂²×C₅²	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
25	C₂×C₅×D₅	—	2	12	no	—
26	C₂×C₅×D₅	C₄×C₅	1	6	yes	complemented
27	C₂×C₅×D₅	C₂²×C₅	1	6	yes	—
28	C₅×AGL(1,𝔽₅)	—	2	12	no	—
29	C₄×C₅²	—	10	2	no	—
30	C₂²⋊AGL(1,𝔽₅)	C₅²	1	1	yes	characteristic, direct product factor
31	C₅×C₂²⋊C₄	—	5	6	no	—
32	C₂×C₅²	—	2	1	no	—
33	C₂×C₅²	—	2	6	no	—
34	C₂×C₅²	—	4	12	no	—
35	C₂×C₅²	—	4	6	no	—
36	C₂×C₅²	—	5	2	no	—
37	C₂×C₅²	—	10	1	no	—
38	C₂×C₅²	C₂×C₄×C₅	1	6	yes	—
39	C₂×C₅²	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
40	C₅×D₅	—	2	6	no	—
41	C₅×D₅	C₅×D₄	1	12	yes	—
42	C₂³×C₅	—	5	6	no	—
43	C₂²×D₅	C₂×C₅²	1	1	yes	characteristic
44	C₂×AGL(1,𝔽₅)	C₂×C₅²	1	2	yes	complemented
45	C₂×C₄×C₅	—	5	12	no	—
46	C₅²	—	4	6	no	—
47	C₅²	C₅×C₂²⋊C₄	1	6	yes	complemented
48	C₅²	C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic, direct product factor
49	C₂²×C₅	—	4	6	no	—
50	C₂²×C₅	—	5	6	no	—
51	C₂²×C₅	—	5	6	no	—
52	C₂²×C₅	—	10	12	no	—
53	C₂²×C₅	C₄×C₅²	1	1	yes	characteristic, complemented
54	C₂²×C₅	C₅×AGL(1,𝔽₅)	1	6	yes	complemented
55	C₂×D₅	—	2	2	no	—
56	C₂×D₅	C₄×C₅²	1	1	yes	characteristic, complemented
57	C₂×D₅	C₂²×C₅²	1	1	yes	characteristic
58	AGL(1,𝔽₅)	—	2	2	no	—
59	C₄×C₅	—	10	12	no	—
60	C₂²⋊C₄	—	5	1	no	Sylow
61	C₂×C₅	—	2	1	no	—
62	C₂×C₅	—	2	6	no	—
63	C₂×C₅	—	4	12	no	—
64	C₂×C₅	—	4	6	no	—
65	C₂×C₅	—	5	12	no	—
66	C₂×C₅	—	10	6	no	—
67	C₂×C₅	C₂×C₄×C₅²	1	1	yes	characteristic
68	C₂×C₅	C₂×C₅×AGL(1,𝔽₅)	1	6	yes	—
69	D₅	—	2	1	no	—
70	D₅	C₅²×D₄	1	2	yes	—
71	C₂³	—	5	1	no	—
72	C₂×C₄	—	5	2	no	—
73	C₅	—	4	6	no	—
74	C₅	C₅²×C₂²⋊C₄	1	1	yes	characteristic, complemented
75	C₅	C₅×C₂²⋊AGL(1,𝔽₅)	1	6	yes	direct product factor
76	C₂²	—	5	1	no	—
77	C₂²	—	5	1	no	—
78	C₂²	—	10	2	no	—
79	C₂²	C₅²×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
80	C₄	—	10	2	no	—
81	C₂	—	2	1	no	—
82	C₂	—	5	2	no	—
83	C₂	—	10	1	no	—
84	C₂	C₂×C₅²×AGL(1,𝔽₅)	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₅²	C₂×AGL(1,𝔽₅)
commutator	C₂×C₅	C₂×C₄×C₅²
Frattini	C₂	C₂×C₅²×AGL(1,𝔽₅)
Fitting	C₂²×C₅³	C₄
socle	C₂×C₅³	C₂×C₄
prime radical	C₂²×C₅²×D₅	C₂
cosocle	C₂×D₅	C₂²×C₅²
hypercenter	C₂²×C₅²	AGL(1,𝔽₅)
hypocenter	C₅	C₅²×C₂²⋊C₄
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₅² ◁ C₂²×C₅²

Complements

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

N	H ≅ G/N	type
2. C₂×C₅²×AGL(1,𝔽₅)	C₂	semidirect
3. C₂²×C₅³	C₄	semidirect
5. C₂×C₅²×D₅	C₄	semidirect
8. C₅×C₂²⋊AGL(1,𝔽₅)	C₅	direct (N × H)
16. C₂×C₅×AGL(1,𝔽₅)	C₂×C₅	semidirect
18. C₅³	C₂²⋊C₄	semidirect
23. C₂²×C₅²	C₄×C₅	semidirect
24. C₂²×C₅²	AGL(1,𝔽₅)	semidirect
26. C₂×C₅×D₅	C₄×C₅	semidirect
30. C₂²⋊AGL(1,𝔽₅)	C₅²	direct (N × H)
44. C₂×AGL(1,𝔽₅)	C₂×C₅²	semidirect
47. C₅²	C₅×C₂²⋊C₄	semidirect
48. C₅²	C₂²⋊AGL(1,𝔽₅)	direct (N × H)
53. C₂²×C₅	C₄×C₅²	semidirect
54. C₂²×C₅	C₅×AGL(1,𝔽₅)	semidirect
56. C₂×D₅	C₄×C₅²	semidirect
74. C₅	C₅²×C₂²⋊C₄	semidirect
75. C₅	C₅×C₂²⋊AGL(1,𝔽₅)	direct (N × H)
79. C₂²	C₅²×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×C₅²×D₅	2
C₂×C₅²×AGL(1,𝔽₅)	2
C₅×C₂²⋊AGL(1,𝔽₅)	5
C₅²×C₂²⋊C₄	5

Subgroups — C52×C22⋊AGL(1,𝔽5) #14522787

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅²×C₂²⋊AGL(1,𝔽₅) #14522787