Subgroups — C₂×C₄×C₅×D₃ #6805

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	1	yes	maximal, characteristic
2	C₂×C₃×C₄×C₅	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅×Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄×C₅×D₃	C₂	1	4	yes	maximal, direct product factor
5	C₂²×C₄×C₅	—	3	1	no	maximal, Hall
6	C₂²×C₃×C₅	C₂²	1	1	yes	characteristic
7	C₂×C₅×D₃	C₄	1	4	yes	direct product factor
8	C₂×C₅×D₃	C₂²	1	2	yes	—
9	C₃×C₄×C₅	C₂²	1	2	yes	complemented
10	C₅×Dic₃	C₂²	1	2	yes	complemented
11	C₂×C₄×D₃	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
12	C₂³×C₅	—	3	1	no	—
13	C₂×C₄×C₅	—	3	1	no	—
14	C₂×C₄×C₅	—	3	4	no	—
15	C₂×C₄×C₅	D₃	1	1	yes	characteristic, direct product factor
16	C₂×C₃×C₅	C₂×C₄	1	2	yes	complemented
17	C₂×C₃×C₅	C₂³	1	1	yes	characteristic
18	C₅×D₃	C₂×C₄	1	4	yes	direct product factor
19	C₂²×D₃	C₂×C₅	1	1	yes	characteristic
20	C₂×C₃×C₄	C₂×C₅	1	1	yes	characteristic, complemented
21	C₂×Dic₃	C₂×C₅	1	1	yes	characteristic, complemented
22	C₄×D₃	C₂×C₅	1	4	yes	direct product factor
23	C₂²×C₅	—	3	4	no	—
24	C₂²×C₅	—	3	2	no	—
25	C₂²×C₅	C₂×D₃	1	1	yes	characteristic
26	C₄×C₅	—	3	2	no	—
27	C₄×C₅	C₂×D₃	1	2	yes	direct product factor
28	C₂²×C₄	—	3	1	no	Sylow
29	C₃×C₅	C₂²×C₄	1	1	yes	characteristic, complemented, Hall
30	C₂²×C₃	C₂²×C₅	1	1	yes	characteristic
31	C₂×D₃	C₄×C₅	1	4	yes	direct product factor
32	C₂×D₃	C₂²×C₅	1	2	yes	—
33	C₃×C₄	C₂²×C₅	1	2	yes	complemented
34	Dic₃	C₂²×C₅	1	2	yes	complemented
35	C₂×C₅	—	3	4	no	—
36	C₂×C₅	C₄×D₃	1	2	yes	direct product factor
37	C₂×C₅	C₂²×D₃	1	1	yes	characteristic
38	C₂³	—	3	1	no	—
39	C₂×C₄	—	3	1	no	—
40	C₂×C₄	—	3	4	no	—
41	C₂×C₄	C₅×D₃	1	1	yes	characteristic, direct product factor
42	C₂×C₃	C₂×C₄×C₅	1	2	yes	complemented
43	C₂×C₃	C₂³×C₅	1	1	yes	characteristic
44	D₃	C₂×C₄×C₅	1	4	yes	direct product factor
45	C₅	C₂×C₄×D₃	1	1	yes	characteristic, direct product factor, Sylow
46	C₂²	—	3	4	no	—
47	C₂²	—	3	2	no	—
48	C₂²	C₂×C₅×D₃	1	1	yes	characteristic
49	C₄	—	3	2	no	—
50	C₄	C₂×C₅×D₃	1	2	yes	direct product factor
51	C₃	C₂²×C₄×C₅	1	1	yes	characteristic, complemented, Sylow
52	C₂	—	3	4	no	—
53	C₂	C₄×C₅×D₃	1	2	yes	direct product factor
54	C₂	C₂²×C₅×D₃	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂×C₄×C₅	D₃
commutator, hypocenter	C₃	C₂²×C₄×C₅
Frattini	C₂	C₂²×C₅×D₃
Fitting	C₂×C₃×C₄×C₅	C₂
socle	C₂²×C₃×C₅	C₂²
prime radical	C₂²×C₅×D₃	C₂
cosocle	C₂×C₃	C₂³×C₅
LCS	G ▷ C₃
UCS	C₁ ◁ C₂×C₄×C₅

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×C₄×C₅	C₂	semidirect
3. C₂×C₅×Dic₃	C₂	semidirect
4. C₄×C₅×D₃	C₂	direct (N × H)
7. C₂×C₅×D₃	C₄	direct (N × H)
9. C₃×C₄×C₅	C₂²	semidirect
10. C₅×Dic₃	C₂²	semidirect
11. C₂×C₄×D₃	C₅	direct (N × H)
15. C₂×C₄×C₅	D₃	direct (N × H)
16. C₂×C₃×C₅	C₂×C₄	semidirect
18. C₅×D₃	C₂×C₄	direct (N × H)
20. C₂×C₃×C₄	C₂×C₅	semidirect
21. C₂×Dic₃	C₂×C₅	semidirect
22. C₄×D₃	C₂×C₅	direct (N × H)
27. C₄×C₅	C₂×D₃	direct (N × H)
29. C₃×C₅	C₂²×C₄	semidirect
31. C₂×D₃	C₄×C₅	direct (N × H)
33. C₃×C₄	C₂²×C₅	semidirect
34. Dic₃	C₂²×C₅	semidirect
36. C₂×C₅	C₄×D₃	direct (N × H)
41. C₂×C₄	C₅×D₃	direct (N × H)
42. C₂×C₃	C₂×C₄×C₅	semidirect
44. D₃	C₂×C₄×C₅	direct (N × H)
45. C₅	C₂×C₄×D₃	direct (N × H)
50. C₄	C₂×C₅×D₃	direct (N × H)
51. C₃	C₂²×C₄×C₅	semidirect
53. C₂	C₄×C₅×D₃	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×C₅×D₃	2
C₂×C₃×C₄×C₅	2
C₂×C₅×Dic₃	2
C₄×C₅×D₃	2
C₂²×C₄×C₅	3
C₂×C₄×D₃	5

Subgroups — C2×C4×C5×D3 #6805

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×C₅×D₃ #6805