Subgroups — C₅²×C₄⋊AGL(1,𝔽₅) #14522784

Order 2000.

244 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅²×AGL(1,𝔽₅)	C₂	1	2	yes	maximal
2	C₄×C₅²×D₅	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅²×D₅	C₂²	1	1	yes	characteristic
4	C₅²×AGL(1,𝔽₅)	—	2	2	no	—
5	C₄×C₅³	C₄	1	1	yes	characteristic, complemented
6	C₅²×Dic₅	C₄	1	1	yes	characteristic, complemented
7	C₅×C₄⋊AGL(1,𝔽₅)	C₅	1	6	yes	maximal, direct product factor
8	C₅²×C₄⋊C₄	—	5	1	no	maximal
9	C₂×C₅³	C₂×C₄	1	1	yes	characteristic
10	C₅²×D₅	D₄	1	1	yes	characteristic
11	C₅²×D₅	Q₈	1	1	yes	characteristic
12	C₂×C₅×AGL(1,𝔽₅)	C₂×C₅	1	12	yes	—
13	C₂×C₄×C₅²	—	5	1	no	—
14	C₂×C₄×C₅²	—	5	2	no	—
15	C₄×C₅×D₅	C₂×C₅	1	6	yes	—
16	C₅³	C₄⋊C₄	1	1	yes	characteristic, complemented, Sylow
17	C₂²×C₅²	—	5	1	no	—
18	C₂×C₅×D₅	C₂²×C₅	1	6	yes	—
19	C₅×AGL(1,𝔽₅)	—	2	12	no	—
20	C₄×C₅²	—	4	6	no	—
21	C₄×C₅²	—	5	1	no	—
22	C₄×C₅²	—	10	2	no	—
23	C₄×C₅²	C₄×C₅	1	6	yes	complemented
24	C₄×C₅²	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
25	C₅×Dic₅	C₄×C₅	1	6	yes	complemented
26	C₄⋊AGL(1,𝔽₅)	C₅²	1	1	yes	characteristic, direct product factor
27	C₅×C₄⋊C₄	—	5	6	no	—
28	C₂×C₅²	—	4	6	no	—
29	C₂×C₅²	—	5	1	no	—
30	C₂×C₅²	—	5	1	no	—
31	C₂×C₅²	C₂×C₄×C₅	1	6	yes	—
32	C₂×C₅²	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
33	C₅×D₅	C₅×D₄	1	6	yes	—
34	C₅×D₅	C₅×Q₈	1	6	yes	—
35	C₂×AGL(1,𝔽₅)	C₂×C₅²	1	2	yes	—
36	C₂×C₄×C₅	—	5	6	no	—
37	C₂×C₄×C₅	—	5	12	no	—
38	C₄×D₅	C₂×C₅²	1	1	yes	characteristic
39	C₅²	—	4	6	no	—
40	C₅²	C₅×C₄⋊C₄	1	6	yes	complemented
41	C₅²	C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic, direct product factor
42	C₂²×C₅	—	5	6	no	—
43	C₂×D₅	C₂²×C₅²	1	1	yes	characteristic
44	AGL(1,𝔽₅)	—	2	2	no	—
45	C₄×C₅	—	4	6	no	—
46	C₄×C₅	—	5	6	no	—
47	C₄×C₅	—	10	12	no	—
48	C₄×C₅	C₄×C₅²	1	1	yes	characteristic, complemented
49	C₄×C₅	C₅×AGL(1,𝔽₅)	1	6	yes	complemented
50	Dic₅	C₄×C₅²	1	1	yes	characteristic, complemented
51	C₄⋊C₄	—	5	1	no	Sylow
52	C₂×C₅	—	4	6	no	—
53	C₂×C₅	—	5	6	no	—
54	C₂×C₅	—	5	6	no	—
55	C₂×C₅	C₂×C₄×C₅²	1	1	yes	characteristic
56	C₂×C₅	C₂×C₅×AGL(1,𝔽₅)	1	6	yes	—
57	D₅	C₅²×D₄	1	1	yes	characteristic
58	D₅	C₅²×Q₈	1	1	yes	characteristic
59	C₂×C₄	—	5	1	no	—
60	C₂×C₄	—	5	2	no	—
61	C₅	—	4	6	no	—
62	C₅	C₅²×C₄⋊C₄	1	1	yes	characteristic, complemented
63	C₅	C₅×C₄⋊AGL(1,𝔽₅)	1	6	yes	direct product factor
64	C₂²	—	5	1	no	—
65	C₄	—	5	1	no	—
66	C₄	—	10	2	no	—
67	C₄	C₅²×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
68	C₂	—	5	1	no	—
69	C₂	—	5	1	no	—
70	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	C₄×C₅³	C₄
socle	C₂×C₅³	C₂×C₄
prime radical	C₂×C₅²×D₅	C₂²
cosocle	C₂×D₅	C₂²×C₅²
hypercenter	C₄×C₅²	AGL(1,𝔽₅)
hypocenter	C₅	C₅²×C₄⋊C₄
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₅² ◁ C₄×C₅²

Complements

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

N	H ≅ G/N	type
5. C₄×C₅³	C₄	semidirect
6. C₅²×Dic₅	C₄	semidirect
7. C₅×C₄⋊AGL(1,𝔽₅)	C₅	direct (N × H)
16. C₅³	C₄⋊C₄	semidirect
23. C₄×C₅²	C₄×C₅	semidirect
24. C₄×C₅²	AGL(1,𝔽₅)	semidirect
25. C₅×Dic₅	C₄×C₅	semidirect
26. C₄⋊AGL(1,𝔽₅)	C₅²	direct (N × H)
40. C₅²	C₅×C₄⋊C₄	semidirect
41. C₅²	C₄⋊AGL(1,𝔽₅)	direct (N × H)
48. C₄×C₅	C₄×C₅²	semidirect
49. C₄×C₅	C₅×AGL(1,𝔽₅)	semidirect
50. Dic₅	C₄×C₅²	semidirect
62. C₅	C₅²×C₄⋊C₄	semidirect
63. C₅	C₅×C₄⋊AGL(1,𝔽₅)	direct (N × H)
67. C₄	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
C₂×C₅²×AGL(1,𝔽₅)	2
C₄×C₅²×D₅	2
C₅×C₄⋊AGL(1,𝔽₅)	5
C₅²×C₄⋊C₄	5

Subgroups — C52×C4⋊AGL(1,𝔽5) #14522784

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅²×C₄⋊AGL(1,𝔽₅) #14522784