Subgroups — C₂×C₅×S₄ #6835

Order 240.

66 conjugacy classes of subgroups.

Conjugacy classes of subgroups

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

	subgroup	G/H
center, hypercenter	C₂×C₅	S₄
commutator, cosocle, hypocenter	A₄	C₂²×C₅
Frattini	C₁	G
Fitting, socle	C₂³×C₅	D₃
LCS	G ▷ A₄
UCS	C₁ ◁ C₂×C₅

Complements

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

N	H ≅ G/N	type
1. C₂×C₅×A₄	C₂	semidirect
2. C₅×S₄	C₂	direct (N × H)
5. C₅×A₄	C₂²	semidirect
6. C₂×S₄	C₅	direct (N × H)
8. C₂³×C₅	D₃	semidirect
14. C₂×A₄	C₂×C₅	semidirect
15. S₄	C₂×C₅	direct (N × H)
21. C₂²×C₅	C₂×D₃	semidirect
26. A₄	C₂²×C₅	semidirect
30. C₂×C₅	S₄	direct (N × H)
32. C₂³	C₅×D₃	semidirect
38. C₅	C₂×S₄	direct (N × H)
44. C₂²	C₂×C₅×D₃	semidirect
50. C₂	C₅×S₄	direct (N × H)

Maximal subgroups

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

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

Subgroups — C2×C5×S4 #6835

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×S₄ #6835