Subgroups — C₃×D₄×D₅ #6798

Order 240.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

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

	subgroup	G/H
center	C₂×C₃	C₂²×D₅
commutator, cosocle	C₂×C₅	C₂³×C₃
Frattini	C₂	C₂²×C₃×D₅
Fitting	C₃×C₅×D₄	C₂
socle	C₂×C₃×C₅	C₂³
hypercenter	C₃×D₄	D₅
hypocenter	C₅	C₂×C₃×D₄
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₃ ◁ C₃×D₄

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃×D₅	C₂	semidirect
2. C₃×C₅×D₄	C₂	semidirect
3. C₃×C₂²⋊D₅	C₂	semidirect
4. C₃×D₂₀	C₂	semidirect
5. C₃×C₄×D₅	C₂	semidirect
6. D₄×D₅	C₃	direct (N × H)
7. C₂²×C₃×C₅	C₂²	semidirect
10. C₂×C₃×D₅	C₂²	semidirect
11. C₃×C₄×C₅	C₂²	semidirect
12. C₃×Dic₅	C₂²	semidirect
14. C₂²×D₅	C₂×C₃	semidirect
15. C₅×D₄	C₂×C₃	semidirect
16. C₂²⋊D₅	C₂×C₃	semidirect
17. D₂₀	C₂×C₃	semidirect
18. C₄×D₅	C₂×C₃	semidirect
22. C₃×D₅	D₄	direct (N × H)
26. C₃×D₄	D₅	direct (N × H)
28. C₂²×C₅	C₂²×C₃	semidirect
31. C₂×D₅	C₂²×C₃	semidirect
32. C₄×C₅	C₂²×C₃	semidirect
33. Dic₅	C₂²×C₃	semidirect
35. C₃×C₅	C₂×D₄	semidirect
39. C₂²×C₃	C₂×D₅	semidirect
41. C₃×C₄	C₂×D₅	semidirect
45. D₅	C₃×D₄	direct (N × H)
49. D₄	C₃×D₅	direct (N × H)
55. C₅	C₂×C₃×D₄	semidirect
59. C₂²	C₂×C₃×D₅	semidirect
61. C₄	C₂×C₃×D₅	semidirect
62. C₃	D₄×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×C₃×D₅	2
C₃×C₅×D₄	2
C₃×C₂²⋊D₅	2
C₃×D₂₀	2
C₃×C₄×D₅	2
D₄×D₅	3
C₂×C₃×D₄	5

Subgroups — C3×D4×D5 #6798

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×D₄×D₅ #6798