Subgroups — C₂×C₅×D₃×C₇ #88312

Order 420.

40 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×C₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅×D₃×C₇	C₂	1	2	yes	maximal, direct product factor
3	C₂²×C₅×C₇	—	3	1	no	maximal, Hall
4	C₃×C₅×C₇	C₂²	1	1	yes	characteristic, complemented, Hall
5	C₂×D₃×C₇	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₂×C₅×C₇	—	3	2	no	—
7	C₂×C₅×C₇	D₃	1	1	yes	characteristic, direct product factor
8	C₂×C₅×D₃	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₂×C₃×C₇	C₂×C₅	1	1	yes	characteristic, complemented
10	D₃×C₇	C₂×C₅	1	2	yes	direct product factor
11	C₅×C₇	C₂×D₃	1	1	yes	characteristic, direct product factor, Hall
12	C₂×C₃×C₅	C₂×C₇	1	1	yes	characteristic, complemented
13	C₅×D₃	C₂×C₇	1	2	yes	direct product factor
14	C₂²×C₇	—	3	1	no	Hall
15	C₃×C₇	C₂²×C₅	1	1	yes	characteristic, complemented, Hall
16	C₂²×C₅	—	3	1	no	Hall
17	C₃×C₅	C₂²×C₇	1	1	yes	characteristic, complemented, Hall
18	C₂×C₇	—	3	2	no	—
19	C₂×C₇	C₅×D₃	1	1	yes	characteristic, direct product factor
20	C₂×D₃	C₅×C₇	1	1	yes	characteristic, direct product factor, Hall
21	C₂×C₅	—	3	2	no	—
22	C₂×C₅	D₃×C₇	1	1	yes	characteristic, direct product factor
23	C₇	C₂×C₅×D₃	1	1	yes	characteristic, direct product factor, Sylow
24	C₂×C₃	C₂×C₅×C₇	1	1	yes	characteristic, complemented
25	D₃	C₂×C₅×C₇	1	2	yes	direct product factor
26	C₅	C₂×D₃×C₇	1	1	yes	characteristic, direct product factor, Sylow
27	C₂²	—	3	1	no	Sylow
28	C₃	C₂²×C₅×C₇	1	1	yes	characteristic, complemented, Sylow
29	C₂	—	3	2	no	—
30	C₂	C₅×D₃×C₇	1	1	yes	characteristic, direct product factor

	subgroup	G/H
center, hypercenter	C₂×C₅×C₇	D₃
commutator, cosocle, hypocenter	C₃	C₂²×C₅×C₇
Frattini	C₁	G
Fitting, socle	C₂×C₃×C₅×C₇	C₂
LCS	G ▷ C₃
UCS	C₁ ◁ C₂×C₅×C₇

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×C₅×C₇	C₂	semidirect
2. C₅×D₃×C₇	C₂	direct (N × H)
4. C₃×C₅×C₇	C₂²	semidirect
5. C₂×D₃×C₇	C₅	direct (N × H)
7. C₂×C₅×C₇	D₃	direct (N × H)
8. C₂×C₅×D₃	C₇	direct (N × H)
9. C₂×C₃×C₇	C₂×C₅	semidirect
10. D₃×C₇	C₂×C₅	direct (N × H)
11. C₅×C₇	C₂×D₃	direct (N × H)
12. C₂×C₃×C₅	C₂×C₇	semidirect
13. C₅×D₃	C₂×C₇	direct (N × H)
15. C₃×C₇	C₂²×C₅	semidirect
17. C₃×C₅	C₂²×C₇	semidirect
19. C₂×C₇	C₅×D₃	direct (N × H)
20. C₂×D₃	C₅×C₇	direct (N × H)
22. C₂×C₅	D₃×C₇	direct (N × H)
23. C₇	C₂×C₅×D₃	direct (N × H)
24. C₂×C₃	C₂×C₅×C₇	semidirect
25. D₃	C₂×C₅×C₇	direct (N × H)
26. C₅	C₂×D₃×C₇	direct (N × H)
28. C₃	C₂²×C₅×C₇	semidirect
30. C₂	C₅×D₃×C₇	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×C₅×C₇	2
C₅×D₃×C₇	2
C₂²×C₅×C₇	3
C₂×D₃×C₇	5
C₂×C₅×D₃	7

Subgroups — C2×C5×D3×C7 #88312

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×D₃×C₇ #88312