Subgroups — C₂×C₄²×D₅ #65805

Order 320.

498 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×D₅	C₂	1	3	yes	maximal
2	C₂×C₄²×C₅	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₄×Dic₅	C₂	1	3	yes	maximal, complemented
4	C₄²×D₅	C₂	1	8	yes	maximal, direct product factor
5	C₂³×D₅	C₂²	1	1	yes	characteristic
6	C₂²×C₄×C₅	C₂²	1	3	yes	—
7	C₂²×Dic₅	C₂²	1	3	yes	—
8	C₂×C₄×D₅	C₄	1	24	yes	direct product factor
9	C₂×C₄×D₅	C₂²	1	12	yes	—
10	C₄²×C₅	C₂²	1	4	yes	complemented
11	C₄×Dic₅	C₂²	1	12	yes	complemented
12	C₂²×C₄²	—	5	1	no	maximal, Sylow
13	C₂³×C₅	C₂³	1	1	yes	characteristic
14	C₂²×D₅	C₂×C₄	1	12	yes	—
15	C₂²×D₅	C₂³	1	2	yes	—
16	C₂×C₄×C₅	C₂×C₄	1	12	yes	complemented
17	C₂×C₄×C₅	C₂³	1	6	yes	—
18	C₂×Dic₅	C₂×C₄	1	12	yes	complemented
19	C₂×Dic₅	C₂³	1	6	yes	—
20	C₄×D₅	C₂×C₄	1	48	yes	direct product factor
21	C₂³×C₄	—	5	3	no	—
22	C₂×C₄²	—	5	3	no	—
23	C₂×C₄²	—	5	8	no	—
24	C₂×C₄²	D₅	1	1	yes	characteristic, direct product factor
25	C₂²×C₅	C₂²×C₄	1	6	yes	—
26	C₂²×C₅	C₂⁴	1	1	yes	characteristic
27	C₂×D₅	C₄²	1	16	yes	direct product factor
28	C₂×D₅	C₂²×C₄	1	12	yes	—
29	C₄×C₅	C₂²×C₄	1	12	yes	complemented
30	Dic₅	C₂²×C₄	1	12	yes	complemented
31	C₂⁴	—	5	1	no	—
32	C₂²×C₄	—	5	3	no	—
33	C₂²×C₄	—	5	24	no	—
34	C₂²×C₄	—	5	12	no	—
35	C₂²×C₄	C₂×D₅	1	3	yes	—
36	C₄²	—	5	12	no	—
37	C₄²	C₂×D₅	1	4	yes	direct product factor
38	C₂×C₅	C₂×C₄²	1	4	yes	complemented
39	C₂×C₅	C₂³×C₄	1	3	yes	—
40	D₅	C₂×C₄²	1	8	yes	direct product factor
41	C₂³	—	5	12	no	—
42	C₂³	—	5	2	no	—
43	C₂³	C₂²×D₅	1	1	yes	characteristic
44	C₂×C₄	—	5	12	no	—
45	C₂×C₄	—	5	6	no	—
46	C₂×C₄	—	5	48	no	—
47	C₂×C₄	C₄×D₅	1	12	yes	direct product factor
48	C₂×C₄	C₂²×D₅	1	6	yes	—
49	C₅	C₂²×C₄²	1	1	yes	characteristic, complemented, Sylow
50	C₂²	—	5	12	no	—
51	C₂²	—	5	16	no	—
52	C₂²	C₂×C₄×D₅	1	6	yes	—
53	C₂²	C₂³×D₅	1	1	yes	characteristic
54	C₄	—	5	12	no	—
55	C₄	C₂×C₄×D₅	1	12	yes	direct product factor
56	C₂	—	5	8	no	—
57	C₂	C₄²×D₅	1	4	yes	direct product factor
58	C₂	C₂²×C₄×D₅	1	3	yes	—

	subgroup	G/H
center, hypercenter	C₂×C₄²	D₅
commutator, hypocenter	C₅	C₂²×C₄²
Frattini	C₂²	C₂³×D₅
Fitting	C₂×C₄²×C₅	C₂
socle	C₂³×C₅	C₂³
prime radical	C₂³×D₅	C₂²
cosocle	C₂²×C₅	C₂⁴
LCS	G ▷ C₅
UCS	C₁ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
2. C₂×C₄²×C₅	C₂	semidirect
3. C₂×C₄×Dic₅	C₂	semidirect
4. C₄²×D₅	C₂	direct (N × H)
8. C₂×C₄×D₅	C₄	direct (N × H)
10. C₄²×C₅	C₂²	semidirect
11. C₄×Dic₅	C₂²	semidirect
16. C₂×C₄×C₅	C₂×C₄	semidirect
18. C₂×Dic₅	C₂×C₄	semidirect
20. C₄×D₅	C₂×C₄	direct (N × H)
24. C₂×C₄²	D₅	direct (N × H)
27. C₂×D₅	C₄²	direct (N × H)
29. C₄×C₅	C₂²×C₄	semidirect
30. Dic₅	C₂²×C₄	semidirect
37. C₄²	C₂×D₅	direct (N × H)
38. C₂×C₅	C₂×C₄²	semidirect
40. D₅	C₂×C₄²	direct (N × H)
47. C₂×C₄	C₄×D₅	direct (N × H)
49. C₅	C₂²×C₄²	semidirect
55. C₄	C₂×C₄×D₅	direct (N × H)
57. C₂	C₄²×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×C₄×D₅	2
C₂×C₄²×C₅	2
C₂×C₄×Dic₅	2
C₄²×D₅	2
C₂²×C₄²	5

Subgroups — C2×C42×D5 #65805

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄²×D₅ #65805