Subgroups — C₂×C₄⋊(C₅⋊(C₅⋊D₅)) #14522921

Order 2000.

1728 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅⋊(C₅⋊D₅)	C₂	1	2	yes	maximal, complemented
2	C₂×C₄×C₅³	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄⋊(C₅⋊(C₅⋊D₅))	C₂	1	4	yes	maximal, direct product factor
4	C₂²×C₅³	C₂²	1	1	yes	characteristic
5	C₂×C₅⋊(C₅⋊D₅)	—	2	4	no	—
6	C₂×C₅⋊(C₅⋊D₅)	C₂²	1	4	yes	complemented
7	C₄×C₅³	C₂²	1	2	yes	complemented
8	C₂×C₄⋊(C₅⋊D₅)	—	5	31	no	maximal
9	C₂×C₅³	D₄	1	2	yes	complemented
10	C₂×C₅³	C₂³	1	1	yes	characteristic
11	C₅⋊(C₅⋊D₅)	—	2	4	no	—
12	C₂²×C₅⋊D₅	—	5	62	no	—
13	C₂×C₄×C₅²	D₅	1	31	yes	complemented
14	C₄⋊(C₅⋊D₅)	—	5	124	no	—
15	C₅³	C₂×D₄	1	1	yes	characteristic, complemented, Sylow
16	C₂²×C₅²	C₂×D₅	1	31	yes	—
17	C₂×C₅⋊D₅	—	5	124	no	—
18	C₂×C₅⋊D₅	—	10	124	no	—
19	C₄×C₅²	C₂×D₅	1	62	yes	complemented
20	C₂×D₂₀	—	25	31	no	—
21	C₂×C₅²	D₂₀	1	62	yes	complemented
22	C₂×C₅²	C₂²×D₅	1	31	yes	—
23	C₅⋊D₅	—	10	124	no	—
24	C₂²×D₅	—	25	62	no	—
25	C₂×C₄×C₅	C₅⋊D₅	1	31	yes	complemented
26	D₂₀	—	25	124	no	—
27	C₅²	C₂×D₂₀	1	31	yes	complemented
28	C₂²×C₅	C₂×C₅⋊D₅	1	31	yes	—
29	C₂×D₅	—	25	124	no	—
30	C₂×D₅	—	50	124	no	—
31	C₄×C₅	C₂×C₅⋊D₅	1	62	yes	complemented
32	C₂×D₄	—	125	1	no	Sylow
33	C₂×C₅	C₄⋊(C₅⋊D₅)	1	62	yes	complemented
34	C₂×C₅	C₂²×C₅⋊D₅	1	31	yes	—
35	D₅	—	50	124	no	—
36	C₂³	—	125	2	no	—
37	D₄	—	125	4	no	—
38	C₂×C₄	C₅⋊(C₅⋊D₅)	1	1	yes	characteristic, complemented
39	C₅	C₂×C₄⋊(C₅⋊D₅)	1	31	yes	complemented
40	C₂²	—	125	4	no	—
41	C₂²	—	250	4	no	—
42	C₂²	C₂×C₅⋊(C₅⋊D₅)	1	1	yes	characteristic
43	C₄	C₂×C₅⋊(C₅⋊D₅)	1	2	yes	complemented
44	C₂	—	250	4	no	—
45	C₂	C₄⋊(C₅⋊(C₅⋊D₅))	1	2	yes	direct product factor
46	C₂	C₂²×C₅⋊(C₅⋊D₅)	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₅⋊(C₅⋊D₅)
commutator, cosocle	C₂×C₅³	C₂³
Frattini	C₂	C₂²×C₅⋊(C₅⋊D₅)
Fitting	C₂×C₄×C₅³	C₂
socle	C₂²×C₅³	C₂²
hypercenter	C₂×C₄	C₅⋊(C₅⋊D₅)
hypocenter	C₅³	C₂×D₄
LCS	G ▷ C₂×C₅³ ▷ C₅³
UCS	C₁ ◁ C₂² ◁ C₂×C₄

Complements

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

N	H ≅ G/N	type
1. C₂²×C₅⋊(C₅⋊D₅)	C₂	semidirect
2. C₂×C₄×C₅³	C₂	semidirect
3. C₄⋊(C₅⋊(C₅⋊D₅))	C₂	direct (N × H)
6. C₂×C₅⋊(C₅⋊D₅)	C₂²	semidirect
7. C₄×C₅³	C₂²	semidirect
9. C₂×C₅³	D₄	semidirect
13. C₂×C₄×C₅²	D₅	semidirect
15. C₅³	C₂×D₄	semidirect
19. C₄×C₅²	C₂×D₅	semidirect
21. C₂×C₅²	D₂₀	semidirect
25. C₂×C₄×C₅	C₅⋊D₅	semidirect
27. C₅²	C₂×D₂₀	semidirect
31. C₄×C₅	C₂×C₅⋊D₅	semidirect
33. C₂×C₅	C₄⋊(C₅⋊D₅)	semidirect
38. C₂×C₄	C₅⋊(C₅⋊D₅)	semidirect
39. C₅	C₂×C₄⋊(C₅⋊D₅)	semidirect
43. C₄	C₂×C₅⋊(C₅⋊D₅)	semidirect
45. C₂	C₄⋊(C₅⋊(C₅⋊D₅))	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×C₅⋊(C₅⋊D₅)	2
C₂×C₄×C₅³	2
C₄⋊(C₅⋊(C₅⋊D₅))	2
C₂×C₄⋊(C₅⋊D₅)	5

Subgroups — C2×C4⋊(C5⋊(C5⋊D5)) #14522921

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄⋊(C₅⋊(C₅⋊D₅)) #14522921