Subgroups — C₂²×D₃×D₅ #6841

Order 240.

268 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₁₅	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₅×D₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₃×D₅	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₃×D₅	C₂	1	12	yes	maximal, direct product factor
5	C₂³×D₅	—	3	1	no	maximal, Hall
6	C₂²×C₃×C₅	C₂²	1	1	yes	characteristic, complemented
7	C₂×D₁₅	C₂²	1	6	yes	complemented
8	C₂×C₅×D₃	C₂²	1	6	yes	complemented
9	C₂×C₃×D₅	C₂²	1	6	yes	complemented
10	D₃×D₅	C₂²	1	16	yes	direct product factor
11	C₂³×D₃	—	5	1	no	maximal, Hall
12	C₂³×C₅	—	3	1	no	—
13	C₂²×D₅	—	3	1	no	—
14	C₂²×D₅	—	3	12	no	—
15	C₂²×D₅	D₃	1	1	yes	characteristic, direct product factor
16	C₂×C₃×C₅	C₂³	1	3	yes	complemented
17	D₁₅	C₂³	1	4	yes	complemented
18	C₃×D₅	C₂³	1	4	yes	complemented
19	C₅×D₃	C₂³	1	4	yes	complemented
20	C₂³×C₃	—	5	1	no	—
21	C₂²×D₃	—	5	1	no	—
22	C₂²×D₃	—	5	12	no	—
23	C₂²×D₃	D₅	1	1	yes	characteristic, direct product factor
24	C₂²×C₅	—	3	6	no	—
25	C₂²×C₅	C₂×D₃	1	1	yes	characteristic, complemented
26	C₂×D₅	—	3	6	no	—
27	C₂×D₅	—	3	16	no	—
28	C₂×D₅	C₂×D₃	1	6	yes	direct product factor
29	C₂⁴	—	15	1	no	Sylow
30	C₃×C₅	C₂⁴	1	1	yes	characteristic, complemented, Hall
31	C₂²×C₃	—	5	6	no	—
32	C₂²×C₃	C₂×D₅	1	1	yes	characteristic, complemented
33	C₂×D₃	—	5	6	no	—
34	C₂×D₃	—	5	16	no	—
35	C₂×D₃	C₂×D₅	1	6	yes	direct product factor
36	C₂×C₅	—	3	4	no	—
37	C₂×C₅	C₂²×D₃	1	3	yes	complemented
38	D₅	—	3	4	no	—
39	D₅	C₂²×D₃	1	4	yes	direct product factor
40	C₂³	—	3	1	no	—
41	C₂³	—	5	1	no	—
42	C₂³	—	15	12	no	—
43	C₂³	—	15	1	no	—
44	C₂×C₃	—	5	4	no	—
45	C₂×C₃	C₂²×D₅	1	3	yes	complemented
46	D₃	—	5	4	no	—
47	D₃	C₂²×D₅	1	4	yes	direct product factor
48	C₅	C₂³×D₃	1	1	yes	characteristic, complemented, Sylow
49	C₂²	—	3	6	no	—
50	C₂²	—	5	6	no	—
51	C₂²	—	15	16	no	—
52	C₂²	—	15	6	no	—
53	C₂²	D₃×D₅	1	1	yes	characteristic, direct product factor
54	C₃	C₂³×D₅	1	1	yes	characteristic, complemented, Sylow
55	C₂	—	3	4	no	—
56	C₂	—	5	4	no	—
57	C₂	—	15	4	no	—
58	C₂	C₂×D₃×D₅	1	3	yes	direct product factor

	subgroup	G/H
center, hypercenter	C₂²	D₃×D₅
commutator, cosocle, hypocenter	C₃×C₅	C₂⁴
Frattini	C₁	G
Fitting, socle	C₂²×C₃×C₅	C₂²
LCS	G ▷ C₃×C₅
UCS	C₁ ◁ C₂²

Complements

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

N	H ≅ G/N	type
1. C₂²×D₁₅	C₂	semidirect
2. C₂²×C₅×D₃	C₂	semidirect
3. C₂²×C₃×D₅	C₂	semidirect
4. C₂×D₃×D₅	C₂	direct (N × H)
6. C₂²×C₃×C₅	C₂²	semidirect
7. C₂×D₁₅	C₂²	semidirect
8. C₂×C₅×D₃	C₂²	semidirect
9. C₂×C₃×D₅	C₂²	semidirect
10. D₃×D₅	C₂²	direct (N × H)
15. C₂²×D₅	D₃	direct (N × H)
16. C₂×C₃×C₅	C₂³	semidirect
17. D₁₅	C₂³	semidirect
18. C₃×D₅	C₂³	semidirect
19. C₅×D₃	C₂³	semidirect
23. C₂²×D₃	D₅	direct (N × H)
25. C₂²×C₅	C₂×D₃	semidirect
28. C₂×D₅	C₂×D₃	direct (N × H)
30. C₃×C₅	C₂⁴	semidirect
32. C₂²×C₃	C₂×D₅	semidirect
35. C₂×D₃	C₂×D₅	direct (N × H)
37. C₂×C₅	C₂²×D₃	semidirect
39. D₅	C₂²×D₃	direct (N × H)
45. C₂×C₃	C₂²×D₅	semidirect
47. D₃	C₂²×D₅	direct (N × H)
48. C₅	C₂³×D₃	semidirect
53. C₂²	D₃×D₅	direct (N × H)
54. C₃	C₂³×D₅	semidirect
58. C₂	C₂×D₃×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×D₁₅	2
C₂²×C₅×D₃	2
C₂²×C₃×D₅	2
C₂×D₃×D₅	2
C₂³×D₅	3
C₂³×D₃	5

Subgroups — C22×D3×D5 #6841

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×D₃×D₅ #6841