Subgroups — C₂×C₄×C₅×D₅ #87888

Order 400.

124 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅×D₅	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₅²	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅×Dic₅	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄×C₅×D₅	C₂	1	4	yes	maximal, direct product factor
5	C₂²×C₅²	C₂²	1	1	yes	characteristic
6	C₂×C₅×D₅	C₄	1	4	yes	direct product factor
7	C₂×C₅×D₅	C₂²	1	2	yes	—
8	C₄×C₅²	C₂²	1	2	yes	complemented
9	C₅×Dic₅	C₂²	1	2	yes	complemented
10	C₂²×C₄×C₅	—	5	1	no	maximal
11	C₂×C₄×D₅	C₅	1	1	yes	maximal, characteristic, direct product factor
12	C₂×C₅²	C₂×C₄	1	2	yes	complemented
13	C₂×C₅²	C₂³	1	1	yes	characteristic
14	C₅×D₅	C₂×C₄	1	4	yes	direct product factor
15	C₂³×C₅	—	5	1	no	—
16	C₂²×D₅	C₂×C₅	1	1	yes	characteristic
17	C₂×C₄×C₅	—	2	2	no	—
18	C₂×C₄×C₅	—	5	1	no	—
19	C₂×C₄×C₅	—	5	4	no	—
20	C₂×C₄×C₅	D₅	1	1	yes	characteristic, direct product factor
21	C₂×C₄×C₅	C₂×C₅	1	1	yes	characteristic, complemented
22	C₂×Dic₅	C₂×C₅	1	1	yes	characteristic, complemented
23	C₄×D₅	C₂×C₅	1	4	yes	direct product factor
24	C₅²	C₂²×C₄	1	1	yes	characteristic, complemented, Sylow
25	C₂²×C₅	—	2	2	no	—
26	C₂²×C₅	—	5	4	no	—
27	C₂²×C₅	—	5	2	no	—
28	C₂²×C₅	C₂×D₅	1	1	yes	characteristic
29	C₂²×C₅	C₂²×C₅	1	1	yes	characteristic
30	C₂×D₅	C₄×C₅	1	4	yes	direct product factor
31	C₂×D₅	C₂²×C₅	1	2	yes	—
32	C₄×C₅	—	2	4	no	—
33	C₄×C₅	—	5	2	no	—
34	C₄×C₅	C₂×D₅	1	2	yes	direct product factor
35	C₄×C₅	C₂²×C₅	1	2	yes	complemented
36	Dic₅	C₂²×C₅	1	2	yes	complemented
37	C₂²×C₄	—	5	1	no	Sylow
38	C₂×C₅	—	2	4	no	—
39	C₂×C₅	—	2	2	no	—
40	C₂×C₅	—	5	4	no	—
41	C₂×C₅	C₄×D₅	1	2	yes	direct product factor
42	C₂×C₅	C₂×C₄×C₅	1	2	yes	complemented
43	C₂×C₅	C₂²×D₅	1	1	yes	characteristic
44	C₂×C₅	C₂³×C₅	1	1	yes	characteristic
45	D₅	C₂×C₄×C₅	1	4	yes	direct product factor
46	C₂³	—	5	1	no	—
47	C₂×C₄	—	5	1	no	—
48	C₂×C₄	—	5	4	no	—
49	C₂×C₄	C₅×D₅	1	1	yes	characteristic, direct product factor
50	C₅	—	2	2	no	—
51	C₅	C₂×C₄×D₅	1	1	yes	characteristic, direct product factor
52	C₅	C₂²×C₄×C₅	1	1	yes	characteristic, complemented
53	C₂²	—	5	4	no	—
54	C₂²	—	5	2	no	—
55	C₂²	C₂×C₅×D₅	1	1	yes	characteristic
56	C₄	—	5	2	no	—
57	C₄	C₂×C₅×D₅	1	2	yes	direct product factor
58	C₂	—	5	4	no	—
59	C₂	C₄×C₅×D₅	1	2	yes	direct product factor
60	C₂	C₂²×C₅×D₅	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂×C₄×C₅	D₅
commutator, hypocenter	C₅	C₂²×C₄×C₅
Frattini	C₂	C₂²×C₅×D₅
Fitting	C₂×C₄×C₅²	C₂
socle	C₂²×C₅²	C₂²
prime radical	C₂²×C₅×D₅	C₂
cosocle	C₂×C₅	C₂³×C₅
LCS	G ▷ C₅
UCS	C₁ ◁ C₂×C₄×C₅

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×C₅²	C₂	semidirect
3. C₂×C₅×Dic₅	C₂	semidirect
4. C₄×C₅×D₅	C₂	direct (N × H)
6. C₂×C₅×D₅	C₄	direct (N × H)
8. C₄×C₅²	C₂²	semidirect
9. C₅×Dic₅	C₂²	semidirect
11. C₂×C₄×D₅	C₅	direct (N × H)
12. C₂×C₅²	C₂×C₄	semidirect
14. C₅×D₅	C₂×C₄	direct (N × H)
20. C₂×C₄×C₅	D₅	direct (N × H)
21. C₂×C₄×C₅	C₂×C₅	semidirect
22. C₂×Dic₅	C₂×C₅	semidirect
23. C₄×D₅	C₂×C₅	direct (N × H)
24. C₅²	C₂²×C₄	semidirect
30. C₂×D₅	C₄×C₅	direct (N × H)
34. C₄×C₅	C₂×D₅	direct (N × H)
35. C₄×C₅	C₂²×C₅	semidirect
36. Dic₅	C₂²×C₅	semidirect
41. C₂×C₅	C₄×D₅	direct (N × H)
42. C₂×C₅	C₂×C₄×C₅	semidirect
45. D₅	C₂×C₄×C₅	direct (N × H)
49. C₂×C₄	C₅×D₅	direct (N × H)
51. C₅	C₂×C₄×D₅	direct (N × H)
52. C₅	C₂²×C₄×C₅	semidirect
57. C₄	C₂×C₅×D₅	direct (N × H)
59. C₂	C₄×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×D₅	2
C₂×C₄×C₅²	2
C₂×C₅×Dic₅	2
C₄×C₅×D₅	2
C₂²×C₄×C₅	5
C₂×C₄×D₅	5

Subgroups — C2×C4×C5×D5 #87888

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×C₅×D₅ #87888