Subgroups — C₅²×C₄⋊Dic₅ #14522615

Order 2000.

400 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₅³	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅²×Dic₅	C₂	1	2	yes	maximal
3	C₂²×C₅³	C₂²	1	1	yes	characteristic
4	C₄×C₅³	C₄	1	2	yes	complemented
5	C₅²×Dic₅	—	2	2	no	—
6	C₅²×C₄⋊C₄	—	5	1	no	maximal
7	C₅×C₄⋊Dic₅	C₅	1	6	yes	maximal, direct product factor
8	C₂×C₅³	C₂×C₄	1	1	yes	characteristic
9	C₂×C₅³	D₄	1	1	yes	characteristic
10	C₂×C₅³	Q₈	1	1	yes	characteristic
11	C₂×C₄×C₅²	—	2	12	no	—
12	C₂×C₄×C₅²	—	5	2	no	—
13	C₂×C₄×C₅²	D₅	1	1	yes	characteristic
14	C₂×C₄×C₅²	C₂×C₅	1	6	yes	—
15	C₂×C₅×Dic₅	C₂×C₅	1	12	yes	—
16	C₅³	C₄⋊C₄	1	1	yes	characteristic, complemented, Sylow
17	C₂²×C₅²	—	2	12	no	—
18	C₂²×C₅²	C₂×D₅	1	1	yes	characteristic
19	C₂²×C₅²	C₂²×C₅	1	6	yes	—
20	C₄×C₅²	—	2	24	no	—
21	C₄×C₅²	—	10	2	no	—
22	C₄×C₅²	Dic₅	1	2	yes	complemented
23	C₄×C₅²	C₄×C₅	1	12	yes	complemented
24	C₅×Dic₅	—	2	12	no	—
25	C₅×C₄⋊C₄	—	5	6	no	—
26	C₄⋊Dic₅	C₅²	1	1	yes	characteristic, direct product factor
27	C₂×C₅²	—	2	12	no	—
28	C₂×C₅²	—	2	12	no	—
29	C₂×C₅²	—	2	12	no	—
30	C₂×C₅²	Dic₁₀	1	1	yes	characteristic
31	C₂×C₅²	D₂₀	1	1	yes	characteristic
32	C₂×C₅²	C₂×Dic₅	1	1	yes	characteristic
33	C₂×C₅²	C₂×C₄×C₅	1	6	yes	—
34	C₂×C₅²	C₅×D₄	1	6	yes	—
35	C₂×C₅²	C₅×Q₈	1	6	yes	—
36	C₂×C₄×C₅	—	2	12	no	—
37	C₂×C₄×C₅	—	5	12	no	—
38	C₂×C₄×C₅	C₅×D₅	1	6	yes	—
39	C₂×C₄×C₅	C₂×C₅²	1	1	yes	characteristic
40	C₂×Dic₅	C₂×C₅²	1	2	yes	—
41	C₅²	—	2	12	no	—
42	C₅²	C₄⋊Dic₅	1	1	yes	characteristic, direct product factor
43	C₅²	C₅×C₄⋊C₄	1	6	yes	complemented
44	C₂²×C₅	—	2	12	no	—
45	C₂²×C₅	C₂×C₅×D₅	1	6	yes	—
46	C₂²×C₅	C₂²×C₅²	1	1	yes	characteristic
47	C₄×C₅	—	2	24	no	—
48	C₄×C₅	—	10	12	no	—
49	C₄×C₅	C₅×Dic₅	1	12	yes	complemented
50	C₄×C₅	C₄×C₅²	1	2	yes	complemented
51	Dic₅	—	2	2	no	—
52	C₄⋊C₄	—	5	1	no	Sylow
53	C₂×C₅	—	2	12	no	—
54	C₂×C₅	—	2	12	no	—
55	C₂×C₅	—	2	12	no	—
56	C₂×C₅	C₅×Dic₁₀	1	6	yes	—
57	C₂×C₅	C₅×D₂₀	1	6	yes	—
58	C₂×C₅	C₂×C₅×Dic₅	1	6	yes	—
59	C₂×C₅	C₂×C₄×C₅²	1	1	yes	characteristic
60	C₂×C₅	C₅²×D₄	1	1	yes	characteristic
61	C₂×C₅	C₅²×Q₈	1	1	yes	characteristic
62	C₂×C₄	—	5	2	no	—
63	C₂×C₄	C₅²×D₅	1	1	yes	characteristic
64	C₅	—	2	12	no	—
65	C₅	C₅×C₄⋊Dic₅	1	6	yes	direct product factor
66	C₅	C₅²×C₄⋊C₄	1	1	yes	characteristic, complemented
67	C₂²	C₂×C₅²×D₅	1	1	yes	characteristic
68	C₄	—	10	2	no	—
69	C₄	C₅²×Dic₅	1	2	yes	complemented
70	C₂	C₅²×Dic₁₀	1	1	yes	characteristic
71	C₂	C₅²×D₂₀	1	1	yes	characteristic
72	C₂	C₂×C₅²×Dic₅	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
4. C₄×C₅³	C₄	semidirect
7. C₅×C₄⋊Dic₅	C₅	direct (N × H)
16. C₅³	C₄⋊C₄	semidirect
22. C₄×C₅²	Dic₅	semidirect
23. C₄×C₅²	C₄×C₅	semidirect
26. C₄⋊Dic₅	C₅²	direct (N × H)
42. C₅²	C₄⋊Dic₅	direct (N × H)
43. C₅²	C₅×C₄⋊C₄	semidirect
49. C₄×C₅	C₅×Dic₅	semidirect
50. C₄×C₅	C₄×C₅²	semidirect
65. C₅	C₅×C₄⋊Dic₅	direct (N × H)
66. C₅	C₅²×C₄⋊C₄	semidirect
69. C₄	C₅²×Dic₅	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×C₅³	2
C₂×C₅²×Dic₅	2
C₅²×C₄⋊C₄	5
C₅×C₄⋊Dic₅	5

Subgroups — C52×C4⋊Dic5 #14522615

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅²×C₄⋊Dic₅ #14522615