Subgroups — C₂×C₅×#67 #6813

Order 240.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

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

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

Complements

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

N	H ≅ G/N	type
1. C₂³×C₃×C₅	C₂	semidirect
2. C₂²×C₅×D₃	C₂	semidirect
3. C₅×#67	C₂	direct (N × H)
4. C₂×C₅×Dic₃	C₂	semidirect
8. C₂²×C₃×C₅	C₂²	semidirect
10. C₂×C₅×D₃	C₂²	semidirect
11. C₅×Dic₃	C₂²	semidirect
12. C₂×#67	C₅	direct (N × H)
14. C₂³×C₅	D₃	semidirect
18. C₂×C₃×C₅	D₄	semidirect
21. C₂³×C₃	C₂×C₅	semidirect
22. C₂²×D₃	C₂×C₅	semidirect
23. #67	C₂×C₅	direct (N × H)
24. C₂×Dic₃	C₂×C₅	semidirect
29. C₂²×C₅	C₂×D₃	semidirect
32. C₃×C₅	C₂×D₄	semidirect
35. C₂²×C₃	C₂²×C₅	semidirect
37. C₂×D₃	C₂²×C₅	semidirect
38. Dic₃	C₂²×C₅	semidirect
41. C₂×C₅	#67	direct (N × H)
44. C₂³	C₅×D₃	semidirect
48. C₂×C₃	C₅×D₄	semidirect
51. C₅	C₂×#67	direct (N × H)
56. C₂²	C₂×C₅×D₃	semidirect
58. C₃	C₂×C₅×D₄	semidirect
61. C₂	C₅×#67	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₃×C₅	2
C₂²×C₅×D₃	2
C₅×#67	2
C₂×C₅×Dic₃	2
C₂×C₅×D₄	3
C₂×#67	5

Subgroups — C2×C5×#67 #6813

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#67 #6813