Subgroups — C₂×C₃×C₅×D₄ #6825

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	2	yes	maximal, complemented
2	C₃×C₅×D₄	C₂	1	4	yes	maximal, direct product factor
3	C₂×C₃×C₄×C₅	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₅×D₄	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
5	C₂²×C₃×C₅	—	2	4	no	—
6	C₂²×C₃×C₅	C₂²	1	1	yes	characteristic
7	C₂²×C₃×C₅	C₂²	1	4	yes	complemented
8	C₃×C₄×C₅	C₂²	1	2	yes	complemented
9	C₂×C₃×D₄	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
10	C₂³×C₅	C₂×C₃	1	2	yes	complemented
11	C₅×D₄	C₂×C₃	1	4	yes	direct product factor
12	C₂×C₄×C₅	C₂×C₃	1	1	yes	characteristic, complemented
13	C₂×C₃×C₅	—	2	4	no	—
14	C₂×C₃×C₅	D₄	1	2	yes	direct product factor
15	C₂×C₃×C₅	C₂³	1	1	yes	characteristic
16	C₂³×C₃	C₂×C₅	1	2	yes	complemented
17	C₃×D₄	C₂×C₅	1	4	yes	direct product factor
18	C₂×C₃×C₄	C₂×C₅	1	1	yes	characteristic, complemented
19	C₂²×C₅	—	2	4	no	—
20	C₂²×C₅	C₂²×C₃	1	1	yes	characteristic
21	C₂²×C₅	C₂²×C₃	1	4	yes	complemented
22	C₄×C₅	C₂²×C₃	1	2	yes	complemented
23	C₂×D₄	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
24	C₃×C₅	C₂×D₄	1	1	yes	characteristic, direct product factor, Hall
25	C₂²×C₃	—	2	4	no	—
26	C₂²×C₃	C₂²×C₅	1	1	yes	characteristic
27	C₂²×C₃	C₂²×C₅	1	4	yes	complemented
28	C₃×C₄	C₂²×C₅	1	2	yes	complemented
29	C₂×C₅	—	2	4	no	—
30	C₂×C₅	C₃×D₄	1	2	yes	direct product factor
31	C₂×C₅	C₂³×C₃	1	1	yes	characteristic
32	C₂³	C₂×C₃×C₅	1	2	yes	complemented
33	D₄	C₂×C₃×C₅	1	4	yes	direct product factor
34	C₂×C₄	C₂×C₃×C₅	1	1	yes	characteristic, complemented
35	C₂×C₃	—	2	4	no	—
36	C₂×C₃	C₅×D₄	1	2	yes	direct product factor
37	C₂×C₃	C₂³×C₅	1	1	yes	characteristic
38	C₅	C₂×C₃×D₄	1	1	yes	characteristic, direct product factor, Sylow
39	C₂²	—	2	4	no	—
40	C₂²	C₂²×C₃×C₅	1	1	yes	characteristic
41	C₂²	C₂²×C₃×C₅	1	4	yes	complemented
42	C₄	C₂²×C₃×C₅	1	2	yes	complemented
43	C₃	C₂×C₅×D₄	1	1	yes	characteristic, direct product factor, Sylow
44	C₂	—	2	4	no	—
45	C₂	C₃×C₅×D₄	1	2	yes	direct product factor
46	C₂	C₂³×C₃×C₅	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. C₂³×C₃×C₅	C₂	semidirect
2. C₃×C₅×D₄	C₂	direct (N × H)
3. C₂×C₃×C₄×C₅	C₂	semidirect
4. C₂×C₅×D₄	C₃	direct (N × H)
7. C₂²×C₃×C₅	C₂²	semidirect
8. C₃×C₄×C₅	C₂²	semidirect
9. C₂×C₃×D₄	C₅	direct (N × H)
10. C₂³×C₅	C₂×C₃	semidirect
11. C₅×D₄	C₂×C₃	direct (N × H)
12. C₂×C₄×C₅	C₂×C₃	semidirect
14. C₂×C₃×C₅	D₄	direct (N × H)
16. C₂³×C₃	C₂×C₅	semidirect
17. C₃×D₄	C₂×C₅	direct (N × H)
18. C₂×C₃×C₄	C₂×C₅	semidirect
21. C₂²×C₅	C₂²×C₃	semidirect
22. C₄×C₅	C₂²×C₃	semidirect
23. C₂×D₄	C₃×C₅	direct (N × H)
24. C₃×C₅	C₂×D₄	direct (N × H)
27. C₂²×C₃	C₂²×C₅	semidirect
28. C₃×C₄	C₂²×C₅	semidirect
30. C₂×C₅	C₃×D₄	direct (N × H)
32. C₂³	C₂×C₃×C₅	semidirect
33. D₄	C₂×C₃×C₅	direct (N × H)
34. C₂×C₄	C₂×C₃×C₅	semidirect
36. C₂×C₃	C₅×D₄	direct (N × H)
38. C₅	C₂×C₃×D₄	direct (N × H)
41. C₂²	C₂²×C₃×C₅	semidirect
42. C₄	C₂²×C₃×C₅	semidirect
43. C₃	C₂×C₅×D₄	direct (N × H)
45. C₂	C₃×C₅×D₄	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
C₃×C₅×D₄	2
C₂×C₃×C₄×C₅	2
C₂×C₅×D₄	3
C₂×C₃×D₄	5

Subgroups — C2×C3×C5×D4 #6825

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×C₅×D₄ #6825