Subgroups — C₂×C₄×C₅⋊(C₅⋊D₅) #14522920

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	1	yes	maximal, characteristic
2	C₂×C₄×C₅³	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#92553	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄×C₅⋊(C₅⋊D₅)	C₂	1	4	yes	maximal, direct product factor
5	C₂²×C₅³	C₂²	1	1	yes	characteristic
6	C₂×C₅⋊(C₅⋊D₅)	C₄	1	4	yes	direct product factor
7	C₂×C₅⋊(C₅⋊D₅)	C₂²	1	2	yes	—
8	C₄×C₅³	C₂²	1	2	yes	complemented
9	#92553	C₂²	1	2	yes	complemented
10	C₂×C₄×C₅⋊D₅	—	5	31	no	maximal
11	C₂×C₅³	C₂×C₄	1	2	yes	complemented
12	C₂×C₅³	C₂³	1	1	yes	characteristic
13	C₅⋊(C₅⋊D₅)	C₂×C₄	1	4	yes	direct product factor
14	C₂²×C₅⋊D₅	—	5	31	no	—
15	C₂×C₄×C₅²	D₅	1	31	yes	complemented
16	C₂×#1039	—	5	31	no	—
17	C₄×C₅⋊D₅	—	5	124	no	—
18	C₅³	C₂²×C₄	1	1	yes	characteristic, complemented, Sylow
19	C₂²×C₅²	C₂×D₅	1	31	yes	—
20	C₂×C₅⋊D₅	—	5	124	no	—
21	C₂×C₅⋊D₅	—	5	62	no	—
22	C₄×C₅²	C₂×D₅	1	62	yes	complemented
23	#1039	—	5	62	no	—
24	C₂×C₄×D₅	—	25	31	no	—
25	C₂×C₅²	C₄×D₅	1	62	yes	complemented
26	C₂×C₅²	C₂²×D₅	1	31	yes	—
27	C₅⋊D₅	—	5	124	no	—
28	C₂²×D₅	—	25	31	no	—
29	C₂×C₄×C₅	C₅⋊D₅	1	31	yes	complemented
30	C₂×Dic₅	—	25	31	no	—
31	C₄×D₅	—	25	124	no	—
32	C₅²	C₂×C₄×D₅	1	31	yes	complemented
33	C₂²×C₅	C₂×C₅⋊D₅	1	31	yes	—
34	C₂×D₅	—	25	124	no	—
35	C₂×D₅	—	25	62	no	—
36	C₄×C₅	C₂×C₅⋊D₅	1	62	yes	complemented
37	Dic₅	—	25	62	no	—
38	C₂²×C₄	—	125	1	no	Sylow
39	C₂×C₅	C₄×C₅⋊D₅	1	62	yes	complemented
40	C₂×C₅	C₂²×C₅⋊D₅	1	31	yes	—
41	D₅	—	25	124	no	—
42	C₂³	—	125	1	no	—
43	C₂×C₄	—	125	1	no	—
44	C₂×C₄	—	125	4	no	—
45	C₂×C₄	C₅⋊(C₅⋊D₅)	1	1	yes	characteristic, direct product factor
46	C₅	C₂×C₄×C₅⋊D₅	1	31	yes	complemented
47	C₂²	—	125	4	no	—
48	C₂²	—	125	2	no	—
49	C₂²	C₂×C₅⋊(C₅⋊D₅)	1	1	yes	characteristic
50	C₄	—	125	2	no	—
51	C₄	C₂×C₅⋊(C₅⋊D₅)	1	2	yes	direct product factor
52	C₂	—	125	4	no	—
53	C₂	C₄×C₅⋊(C₅⋊D₅)	1	2	yes	direct product factor
54	C₂	C₂²×C₅⋊(C₅⋊D₅)	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×C₅³	C₂	semidirect
3. C₂×#92553	C₂	semidirect
4. C₄×C₅⋊(C₅⋊D₅)	C₂	direct (N × H)
6. C₂×C₅⋊(C₅⋊D₅)	C₄	direct (N × H)
8. C₄×C₅³	C₂²	semidirect
9. #92553	C₂²	semidirect
11. C₂×C₅³	C₂×C₄	semidirect
13. C₅⋊(C₅⋊D₅)	C₂×C₄	direct (N × H)
15. C₂×C₄×C₅²	D₅	semidirect
18. C₅³	C₂²×C₄	semidirect
22. C₄×C₅²	C₂×D₅	semidirect
25. C₂×C₅²	C₄×D₅	semidirect
29. C₂×C₄×C₅	C₅⋊D₅	semidirect
32. C₅²	C₂×C₄×D₅	semidirect
36. C₄×C₅	C₂×C₅⋊D₅	semidirect
39. C₂×C₅	C₄×C₅⋊D₅	semidirect
45. C₂×C₄	C₅⋊(C₅⋊D₅)	direct (N × H)
46. C₅	C₂×C₄×C₅⋊D₅	semidirect
51. C₄	C₂×C₅⋊(C₅⋊D₅)	direct (N × H)
53. 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
C₂×#92553	2
C₄×C₅⋊(C₅⋊D₅)	2
C₂×C₄×C₅⋊D₅	5

Subgroups — C2×C4×C5⋊(C5⋊D5) #14522920

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×C₅⋊(C₅⋊D₅) #14522920