Subgroups — C₂×C₃×C₄×D₅ #6795

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	1	yes	maximal, characteristic
2	C₂×C₃×C₄×C₅	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×Dic₅	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×C₄×D₅	C₂	1	4	yes	maximal, direct product factor
5	C₂×C₄×D₅	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₂²×C₃×C₅	C₂²	1	1	yes	characteristic
7	C₂×C₃×D₅	C₄	1	4	yes	direct product factor
8	C₂×C₃×D₅	C₂²	1	2	yes	—
9	C₃×C₄×C₅	C₂²	1	2	yes	complemented
10	C₃×Dic₅	C₂²	1	2	yes	complemented
11	C₂²×C₃×C₄	—	5	1	no	maximal, Hall
12	C₂²×D₅	C₂×C₃	1	1	yes	characteristic
13	C₂×C₄×C₅	C₂×C₃	1	1	yes	characteristic, complemented
14	C₂×Dic₅	C₂×C₃	1	1	yes	characteristic, complemented
15	C₄×D₅	C₂×C₃	1	4	yes	direct product factor
16	C₂×C₃×C₅	C₂×C₄	1	2	yes	complemented
17	C₂×C₃×C₅	C₂³	1	1	yes	characteristic
18	C₃×D₅	C₂×C₄	1	4	yes	direct product factor
19	C₂³×C₃	—	5	1	no	—
20	C₂×C₃×C₄	—	5	1	no	—
21	C₂×C₃×C₄	—	5	4	no	—
22	C₂×C₃×C₄	D₅	1	1	yes	characteristic, direct product factor
23	C₂²×C₅	C₂²×C₃	1	1	yes	characteristic
24	C₂×D₅	C₃×C₄	1	4	yes	direct product factor
25	C₂×D₅	C₂²×C₃	1	2	yes	—
26	C₄×C₅	C₂²×C₃	1	2	yes	complemented
27	Dic₅	C₂²×C₃	1	2	yes	complemented
28	C₂²×C₄	—	5	1	no	Sylow
29	C₃×C₅	C₂²×C₄	1	1	yes	characteristic, complemented, Hall
30	C₂²×C₃	—	5	4	no	—
31	C₂²×C₃	—	5	2	no	—
32	C₂²×C₃	C₂×D₅	1	1	yes	characteristic
33	C₃×C₄	—	5	2	no	—
34	C₃×C₄	C₂×D₅	1	2	yes	direct product factor
35	C₂×C₅	C₂×C₃×C₄	1	2	yes	complemented
36	C₂×C₅	C₂³×C₃	1	1	yes	characteristic
37	D₅	C₂×C₃×C₄	1	4	yes	direct product factor
38	C₂³	—	5	1	no	—
39	C₂×C₄	—	5	1	no	—
40	C₂×C₄	—	5	4	no	—
41	C₂×C₄	C₃×D₅	1	1	yes	characteristic, direct product factor
42	C₂×C₃	—	5	4	no	—
43	C₂×C₃	C₄×D₅	1	2	yes	direct product factor
44	C₂×C₃	C₂²×D₅	1	1	yes	characteristic
45	C₅	C₂²×C₃×C₄	1	1	yes	characteristic, complemented, Sylow
46	C₂²	—	5	4	no	—
47	C₂²	—	5	2	no	—
48	C₂²	C₂×C₃×D₅	1	1	yes	characteristic
49	C₄	—	5	2	no	—
50	C₄	C₂×C₃×D₅	1	2	yes	direct product factor
51	C₃	C₂×C₄×D₅	1	1	yes	characteristic, direct product factor, Sylow
52	C₂	—	5	4	no	—
53	C₂	C₃×C₄×D₅	1	2	yes	direct product factor
54	C₂	C₂²×C₃×D₅	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂×C₃×C₄	D₅
commutator, hypocenter	C₅	C₂²×C₃×C₄
Frattini	C₂	C₂²×C₃×D₅
Fitting	C₂×C₃×C₄×C₅	C₂
socle	C₂²×C₃×C₅	C₂²
prime radical	C₂²×C₃×D₅	C₂
cosocle	C₂×C₅	C₂³×C₃
LCS	G ▷ C₅
UCS	C₁ ◁ C₂×C₃×C₄

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×C₃×Dic₅	C₂	semidirect
4. C₃×C₄×D₅	C₂	direct (N × H)
5. C₂×C₄×D₅	C₃	direct (N × H)
7. C₂×C₃×D₅	C₄	direct (N × H)
9. C₃×C₄×C₅	C₂²	semidirect
10. C₃×Dic₅	C₂²	semidirect
13. C₂×C₄×C₅	C₂×C₃	semidirect
14. C₂×Dic₅	C₂×C₃	semidirect
15. C₄×D₅	C₂×C₃	direct (N × H)
16. C₂×C₃×C₅	C₂×C₄	semidirect
18. C₃×D₅	C₂×C₄	direct (N × H)
22. C₂×C₃×C₄	D₅	direct (N × H)
24. C₂×D₅	C₃×C₄	direct (N × H)
26. C₄×C₅	C₂²×C₃	semidirect
27. Dic₅	C₂²×C₃	semidirect
29. C₃×C₅	C₂²×C₄	semidirect
34. C₃×C₄	C₂×D₅	direct (N × H)
35. C₂×C₅	C₂×C₃×C₄	semidirect
37. D₅	C₂×C₃×C₄	direct (N × H)
41. C₂×C₄	C₃×D₅	direct (N × H)
43. C₂×C₃	C₄×D₅	direct (N × H)
45. C₅	C₂²×C₃×C₄	semidirect
50. C₄	C₂×C₃×D₅	direct (N × H)
51. C₃	C₂×C₄×D₅	direct (N × H)
53. C₂	C₃×C₄×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
C₂×C₃×C₄×C₅	2
C₂×C₃×Dic₅	2
C₃×C₄×D₅	2
C₂×C₄×D₅	3
C₂²×C₃×C₄	5

Subgroups — C2×C3×C4×D5 #6795

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×C₄×D₅ #6795