Subgroups — C₂×C₃×C₅×D₅ #64501

Order 300.

48 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅²	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×C₅×D₅	C₂	1	2	yes	maximal, direct product factor
3	C₂×C₅×D₅	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
4	C₃×C₅²	C₂²	1	1	yes	characteristic, complemented, Hall
5	C₂²×C₃×C₅	—	5	1	no	maximal
6	C₂×C₃×D₅	C₅	1	1	yes	maximal, characteristic, direct product factor
7	C₂×C₅²	C₂×C₃	1	1	yes	characteristic, complemented
8	C₅×D₅	C₂×C₃	1	2	yes	direct product factor
9	C₂×C₃×C₅	—	2	2	no	—
10	C₂×C₃×C₅	—	5	2	no	—
11	C₂×C₃×C₅	D₅	1	1	yes	characteristic, direct product factor
12	C₂×C₃×C₅	C₂×C₅	1	1	yes	characteristic, complemented
13	C₃×D₅	C₂×C₅	1	2	yes	direct product factor
14	C₅²	C₂²×C₃	1	1	yes	characteristic, complemented, Sylow
15	C₂²×C₅	—	5	1	no	—
16	C₂×D₅	C₃×C₅	1	1	yes	characteristic, direct product factor
17	C₃×C₅	—	2	2	no	—
18	C₃×C₅	C₂×D₅	1	1	yes	characteristic, direct product factor
19	C₃×C₅	C₂²×C₅	1	1	yes	characteristic, complemented
20	C₂²×C₃	—	5	1	no	Hall
21	C₂×C₅	—	2	2	no	—
22	C₂×C₅	—	5	2	no	—
23	C₂×C₅	C₃×D₅	1	1	yes	characteristic, direct product factor
24	C₂×C₅	C₂×C₃×C₅	1	1	yes	characteristic, complemented
25	D₅	C₂×C₃×C₅	1	2	yes	direct product factor
26	C₂×C₃	—	5	2	no	—
27	C₂×C₃	C₅×D₅	1	1	yes	characteristic, direct product factor
28	C₅	—	2	2	no	—
29	C₅	C₂×C₃×D₅	1	1	yes	characteristic, direct product factor
30	C₅	C₂²×C₃×C₅	1	1	yes	characteristic, complemented
31	C₂²	—	5	1	no	Sylow
32	C₃	C₂×C₅×D₅	1	1	yes	characteristic, direct product factor, Sylow
33	C₂	—	5	2	no	—
34	C₂	C₃×C₅×D₅	1	1	yes	characteristic, direct product factor

	subgroup	G/H
center, hypercenter	C₂×C₃×C₅	D₅
commutator, cosocle, hypocenter	C₅	C₂²×C₃×C₅
Frattini	C₁	G
Fitting, socle	C₂×C₃×C₅²	C₂
LCS	G ▷ C₅
UCS	C₁ ◁ C₂×C₃×C₅

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×C₅²	C₂	semidirect
2. C₃×C₅×D₅	C₂	direct (N × H)
3. C₂×C₅×D₅	C₃	direct (N × H)
4. C₃×C₅²	C₂²	semidirect
6. C₂×C₃×D₅	C₅	direct (N × H)
7. C₂×C₅²	C₂×C₃	semidirect
8. C₅×D₅	C₂×C₃	direct (N × H)
11. C₂×C₃×C₅	D₅	direct (N × H)
12. C₂×C₃×C₅	C₂×C₅	semidirect
13. C₃×D₅	C₂×C₅	direct (N × H)
14. C₅²	C₂²×C₃	semidirect
16. C₂×D₅	C₃×C₅	direct (N × H)
18. C₃×C₅	C₂×D₅	direct (N × H)
19. C₃×C₅	C₂²×C₅	semidirect
23. C₂×C₅	C₃×D₅	direct (N × H)
24. C₂×C₅	C₂×C₃×C₅	semidirect
25. D₅	C₂×C₃×C₅	direct (N × H)
27. C₂×C₃	C₅×D₅	direct (N × H)
29. C₅	C₂×C₃×D₅	direct (N × H)
30. C₅	C₂²×C₃×C₅	semidirect
32. C₃	C₂×C₅×D₅	direct (N × H)
34. C₂	C₃×C₅×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
C₂×C₃×C₅²	2
C₃×C₅×D₅	2
C₂×C₅×D₅	3
C₂²×C₃×C₅	5
C₂×C₃×D₅	5

Subgroups — C2×C3×C5×D5 #64501

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×C₅×D₅ #64501