Subgroups — C₂×C₃×C₂²⋊D₅ #6803

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×C₃×D₅	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₂²⋊D₅	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₃×Dic₅	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₂²⋊D₅	C₃	1	1	yes	maximal, characteristic, direct product factor, 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×C₃×D₅	—	2	2	no	—
10	C₂×C₃×D₅	C₂²	1	2	yes	complemented
11	C₃×Dic₅	C₂²	1	2	yes	complemented
12	C₂×C₃×D₄	—	5	1	no	maximal, Hall
13	C₂³×C₅	C₂×C₃	1	1	yes	characteristic, complemented
14	C₂²×D₅	C₂×C₃	1	1	yes	characteristic, complemented
15	C₂²⋊D₅	C₂×C₃	1	4	yes	direct product factor
16	C₂×Dic₅	C₂×C₃	1	1	yes	characteristic, complemented
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	C₃×D₅	—	2	2	no	—
21	C₂³×C₃	—	5	1	no	—
22	C₂³×C₃	D₅	1	1	yes	characteristic, complemented
23	C₃×D₄	—	5	4	no	—
24	C₂×C₃×C₄	—	5	1	no	—
25	C₂²×C₅	—	2	2	no	—
26	C₂²×C₅	C₂²×C₃	1	1	yes	characteristic
27	C₂²×C₅	C₂²×C₃	1	2	yes	complemented
28	C₂×D₅	—	2	2	no	—
29	C₂×D₅	C₂²×C₃	1	2	yes	complemented
30	Dic₅	C₂²×C₃	1	2	yes	complemented
31	C₂×D₄	—	5	1	no	Sylow
32	C₃×C₅	C₂×D₄	1	1	yes	characteristic, complemented, Hall
33	C₂²×C₃	—	2	2	no	—
34	C₂²×C₃	—	5	2	no	—
35	C₂²×C₃	—	10	2	no	—
36	C₂²×C₃	C₂×D₅	1	1	yes	characteristic
37	C₂²×C₃	C₂×D₅	1	2	yes	complemented
38	C₃×C₄	—	5	2	no	—
39	C₂×C₅	—	2	2	no	—
40	C₂×C₅	C₃×D₄	1	2	yes	complemented
41	C₂×C₅	C₂³×C₃	1	1	yes	characteristic
42	D₅	—	2	2	no	—
43	C₂³	—	5	1	no	—
44	C₂³	C₃×D₅	1	1	yes	characteristic, complemented
45	D₄	—	5	4	no	—
46	C₂×C₄	—	5	1	no	—
47	C₂×C₃	—	2	2	no	—
48	C₂×C₃	—	10	2	no	—
49	C₂×C₃	C₂²⋊D₅	1	2	yes	direct product factor
50	C₂×C₃	C₂²×D₅	1	1	yes	characteristic
51	C₅	C₂×C₃×D₄	1	1	yes	characteristic, complemented, Sylow
52	C₂²	—	2	2	no	—
53	C₂²	—	5	2	no	—
54	C₂²	—	10	2	no	—
55	C₂²	C₂×C₃×D₅	1	1	yes	characteristic
56	C₂²	C₂×C₃×D₅	1	2	yes	complemented
57	C₄	—	5	2	no	—
58	C₃	C₂×C₂²⋊D₅	1	1	yes	characteristic, direct product factor, Sylow
59	C₂	—	2	2	no	—
60	C₂	—	10	2	no	—
61	C₂	C₃×C₂²⋊D₅	1	2	yes	direct product factor
62	C₂	C₂²×C₃×D₅	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. C₂³×C₃×C₅	C₂	semidirect
2. C₂²×C₃×D₅	C₂	semidirect
3. C₃×C₂²⋊D₅	C₂	direct (N × H)
4. C₂×C₃×Dic₅	C₂	semidirect
5. C₂×C₂²⋊D₅	C₃	direct (N × H)
8. C₂²×C₃×C₅	C₂²	semidirect
10. C₂×C₃×D₅	C₂²	semidirect
11. C₃×Dic₅	C₂²	semidirect
13. C₂³×C₅	C₂×C₃	semidirect
14. C₂²×D₅	C₂×C₃	semidirect
15. C₂²⋊D₅	C₂×C₃	direct (N × H)
16. C₂×Dic₅	C₂×C₃	semidirect
18. C₂×C₃×C₅	D₄	semidirect
22. C₂³×C₃	D₅	semidirect
27. C₂²×C₅	C₂²×C₃	semidirect
29. C₂×D₅	C₂²×C₃	semidirect
30. Dic₅	C₂²×C₃	semidirect
32. C₃×C₅	C₂×D₄	semidirect
37. C₂²×C₃	C₂×D₅	semidirect
40. C₂×C₅	C₃×D₄	semidirect
44. C₂³	C₃×D₅	semidirect
49. C₂×C₃	C₂²⋊D₅	direct (N × H)
51. C₅	C₂×C₃×D₄	semidirect
56. C₂²	C₂×C₃×D₅	semidirect
58. C₃	C₂×C₂²⋊D₅	direct (N × H)
61. C₂	C₃×C₂²⋊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
C₂³×C₃×C₅	2
C₂²×C₃×D₅	2
C₃×C₂²⋊D₅	2
C₂×C₃×Dic₅	2
C₂×C₂²⋊D₅	3
C₂×C₃×D₄	5

Subgroups — C2×C3×C22⋊D5 #6803

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×C₂²⋊D₅ #6803