Subgroups — C₅×D₃×D₄ #6808

Order 240.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅×D₃	C₂	1	2	yes	maximal, complemented
2	C₃×C₅×D₄	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#67	C₂	1	2	yes	maximal, complemented
4	C₅×D₁₂	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄×C₅×D₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₅×D₄	—	3	1	no	maximal, Hall
7	C₂²×C₃×C₅	C₂²	1	2	yes	complemented
8	C₂×C₅×D₃	—	2	4	no	—
9	C₂×C₅×D₃	C₂²	1	1	yes	characteristic
10	C₂×C₅×D₃	C₂²	1	2	yes	complemented
11	C₃×C₄×C₅	C₂²	1	1	yes	characteristic, complemented
12	C₅×Dic₃	C₂²	1	1	yes	characteristic, complemented
13	D₃×D₄	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
14	C₂³×C₅	—	3	2	no	—
15	C₅×D₄	—	3	2	no	—
16	C₅×D₄	—	3	1	no	—
17	C₅×D₄	D₃	1	1	yes	characteristic, direct product factor
18	C₂×C₄×C₅	—	3	1	no	—
19	C₂×C₃×C₅	—	2	2	no	—
20	C₂×C₃×C₅	C₂³	1	1	yes	characteristic
21	C₅×D₃	—	2	2	no	—
22	C₅×D₃	D₄	1	2	yes	direct product factor
23	C₂²×D₃	C₂×C₅	1	2	yes	complemented
24	C₃×D₄	C₂×C₅	1	1	yes	characteristic, complemented
25	#67	C₂×C₅	1	2	yes	complemented
26	D₁₂	C₂×C₅	1	1	yes	characteristic, complemented
27	C₄×D₃	C₂×C₅	1	1	yes	characteristic, complemented
28	C₂²×C₅	—	3	1	no	—
29	C₂²×C₅	—	3	2	no	—
30	C₂²×C₅	—	6	4	no	—
31	C₂²×C₅	C₂×D₃	1	2	yes	complemented
32	C₄×C₅	—	3	1	no	—
33	C₄×C₅	C₂×D₃	1	1	yes	characteristic, complemented
34	C₂×D₄	—	3	1	no	Sylow
35	C₃×C₅	C₂×D₄	1	1	yes	characteristic, complemented, Hall
36	C₂²×C₃	C₂²×C₅	1	2	yes	complemented
37	C₂×D₃	—	2	4	no	—
38	C₂×D₃	C₂²×C₅	1	1	yes	characteristic
39	C₂×D₃	C₂²×C₅	1	2	yes	complemented
40	C₃×C₄	C₂²×C₅	1	1	yes	characteristic, complemented
41	Dic₃	C₂²×C₅	1	1	yes	characteristic, complemented
42	C₂×C₅	—	2	2	no	—
43	C₂×C₅	—	3	2	no	—
44	C₂×C₅	—	6	2	no	—
45	C₂×C₅	C₂²×D₃	1	1	yes	characteristic
46	C₂³	—	3	2	no	—
47	D₄	—	3	2	no	—
48	D₄	—	3	1	no	—
49	D₄	C₅×D₃	1	1	yes	characteristic, direct product factor
50	C₂×C₄	—	3	1	no	—
51	C₂×C₃	—	2	2	no	—
52	C₂×C₃	C₂³×C₅	1	1	yes	characteristic
53	D₃	—	2	2	no	—
54	D₃	C₅×D₄	1	2	yes	direct product factor
55	C₅	D₃×D₄	1	1	yes	characteristic, direct product factor, Sylow
56	C₂²	—	3	1	no	—
57	C₂²	—	3	2	no	—
58	C₂²	—	6	4	no	—
59	C₂²	C₂×C₅×D₃	1	2	yes	complemented
60	C₄	—	3	1	no	—
61	C₄	C₂×C₅×D₃	1	1	yes	characteristic, complemented
62	C₃	C₂×C₅×D₄	1	1	yes	characteristic, complemented, Sylow
63	C₂	—	2	2	no	—
64	C₂	—	3	2	no	—
65	C₂	—	6	2	no	—
66	C₂	C₂²×C₅×D₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₅	C₂²×D₃
commutator, cosocle	C₂×C₃	C₂³×C₅
Frattini	C₂	C₂²×C₅×D₃
Fitting	C₃×C₅×D₄	C₂
socle	C₂×C₃×C₅	C₂³
hypercenter	C₅×D₄	D₃
hypocenter	C₃	C₂×C₅×D₄
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₅ ◁ C₅×D₄

Complements

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

N	H ≅ G/N	type
1. C₂²×C₅×D₃	C₂	semidirect
2. C₃×C₅×D₄	C₂	semidirect
3. C₅×#67	C₂	semidirect
4. C₅×D₁₂	C₂	semidirect
5. C₄×C₅×D₃	C₂	semidirect
7. C₂²×C₃×C₅	C₂²	semidirect
10. C₂×C₅×D₃	C₂²	semidirect
11. C₃×C₄×C₅	C₂²	semidirect
12. C₅×Dic₃	C₂²	semidirect
13. D₃×D₄	C₅	direct (N × H)
17. C₅×D₄	D₃	direct (N × H)
22. C₅×D₃	D₄	direct (N × H)
23. C₂²×D₃	C₂×C₅	semidirect
24. C₃×D₄	C₂×C₅	semidirect
25. #67	C₂×C₅	semidirect
26. D₁₂	C₂×C₅	semidirect
27. C₄×D₃	C₂×C₅	semidirect
31. C₂²×C₅	C₂×D₃	semidirect
33. C₄×C₅	C₂×D₃	semidirect
35. C₃×C₅	C₂×D₄	semidirect
36. C₂²×C₃	C₂²×C₅	semidirect
39. C₂×D₃	C₂²×C₅	semidirect
40. C₃×C₄	C₂²×C₅	semidirect
41. Dic₃	C₂²×C₅	semidirect
49. D₄	C₅×D₃	direct (N × H)
54. D₃	C₅×D₄	direct (N × H)
55. C₅	D₃×D₄	direct (N × H)
59. C₂²	C₂×C₅×D₃	semidirect
61. C₄	C₂×C₅×D₃	semidirect
62. C₃	C₂×C₅×D₄	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₅×D₃	2
C₃×C₅×D₄	2
C₅×#67	2
C₅×D₁₂	2
C₄×C₅×D₃	2
C₂×C₅×D₄	3
D₃×D₄	5

Subgroups — C5×D3×D4 #6808

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×D₃×D₄ #6808