Subgroups — D₅×S₄ #6833

Order 240.

66 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₅×A₄	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅⋊S₄	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×S₄	C₂	1	1	yes	maximal, characteristic, complemented
4	D₄×D₅	—	3	1	no	maximal, Hall
5	C₅×A₄	C₂²	1	1	yes	characteristic, complemented
6	D₃×D₅	—	4	1	no	maximal
7	C₂×S₄	—	5	1	no	maximal, Hall
8	C₂²×D₅	—	3	1	no	—
9	C₂²×D₅	D₃	1	1	yes	characteristic, complemented
10	C₅×D₄	—	3	1	no	—
11	C₂²⋊D₅	—	3	1	no	—
12	C₂²⋊D₅	—	3	1	no	—
13	D₂₀	—	3	1	no	—
14	C₄×D₅	—	3	1	no	—
15	D₁₅	—	4	1	no	—
16	C₃×D₅	—	4	1	no	—
17	C₅×D₃	—	4	1	no	—
18	C₂×A₄	—	5	1	no	—
19	S₄	—	5	1	no	—
20	S₄	D₅	1	1	yes	characteristic, direct product factor
21	C₂²×C₅	—	3	1	no	—
22	C₂²×C₅	C₂×D₃	1	1	yes	characteristic, complemented
23	C₂×D₅	—	3	1	no	—
24	C₂×D₅	—	3	1	no	—
25	C₂×D₅	—	3	1	no	—
26	C₂×D₅	—	6	1	no	—
27	C₂×D₅	—	6	1	no	—
28	C₄×C₅	—	3	1	no	—
29	Dic₅	—	3	1	no	—
30	C₂×D₄	—	15	1	no	Sylow
31	C₃×C₅	—	4	1	no	Hall
32	C₂×D₃	—	20	1	no	—
33	A₄	C₂×D₅	1	1	yes	characteristic, complemented
34	C₂×C₅	—	3	1	no	—
35	C₂×C₅	—	6	1	no	—
36	D₅	—	3	1	no	—
37	D₅	—	6	1	no	—
38	D₅	S₄	1	1	yes	characteristic, direct product factor
39	C₂³	—	5	1	no	—
40	C₂³	—	15	1	no	—
41	D₄	—	3	1	no	—
42	D₄	—	15	1	no	—
43	D₄	—	15	1	no	—
44	D₄	—	15	1	no	—
45	C₂×C₄	—	15	1	no	—
46	C₂×C₃	—	20	1	no	—
47	D₃	—	4	1	no	—
48	D₃	—	20	1	no	—
49	C₅	C₂×S₄	1	1	yes	characteristic, complemented, Sylow
50	C₂²	—	3	1	no	—
51	C₂²	—	15	1	no	—
52	C₂²	—	15	1	no	—
53	C₂²	—	15	1	no	—
54	C₂²	—	30	1	no	—
55	C₂²	—	30	1	no	—
56	C₂²	D₃×D₅	1	1	yes	characteristic, complemented
57	C₄	—	3	1	no	—
58	C₄	—	15	1	no	—
59	C₃	—	4	1	no	Sylow
60	C₂	—	3	1	no	—
61	C₂	—	5	1	no	—
62	C₂	—	6	1	no	—
63	C₂	—	15	1	no	—
64	C₂	—	30	1	no	—

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, cosocle, hypocenter	C₅×A₄	C₂²
Fitting, socle	C₂²×C₅	C₂×D₃
LCS	G ▷ C₅×A₄
UCS	C₁

Complements

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

N	H ≅ G/N	type
1. D₅×A₄	C₂	semidirect
2. C₅⋊S₄	C₂	semidirect
3. C₅×S₄	C₂	semidirect
5. C₅×A₄	C₂²	semidirect
9. C₂²×D₅	D₃	semidirect
20. S₄	D₅	direct (N × H)
22. C₂²×C₅	C₂×D₃	semidirect
33. A₄	C₂×D₅	semidirect
38. D₅	S₄	direct (N × H)
49. C₅	C₂×S₄	semidirect
56. C₂²	D₃×D₅	semidirect

Maximal subgroups

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

subgroup	index
D₅×A₄	2
C₅⋊S₄	2
C₅×S₄	2
D₄×D₅	3
D₃×D₅	4
C₂×S₄	5

Subgroups — D5×S4 #6833

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₅×S₄ #6833