Subgroups — C₂×C₃×D₃×C₇ #7003

Order 252.

44 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×C₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×D₃×C₇	C₂	1	2	yes	maximal, direct product factor
3	C₂²×C₃×C₇	—	3	1	no	maximal
4	C₂×D₃×C₇	C₃	1	1	yes	maximal, characteristic, direct product factor
5	C₃²×C₇	C₂²	1	1	yes	characteristic, complemented, Hall
6	C₂×C₃×C₇	—	2	1	no	—
7	C₂×C₃×C₇	—	3	2	no	—
8	C₂×C₃×C₇	D₃	1	1	yes	characteristic, direct product factor
9	C₂×C₃×C₇	C₂×C₃	1	1	yes	characteristic, complemented
10	D₃×C₇	C₂×C₃	1	2	yes	direct product factor
11	C₂×C₃×D₃	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
12	C₂²×C₇	—	3	1	no	Hall
13	C₃×C₇	—	2	1	no	—
14	C₃×C₇	C₂×D₃	1	1	yes	characteristic, direct product factor
15	C₃×C₇	C₂²×C₃	1	1	yes	characteristic, complemented
16	C₂×C₃²	C₂×C₇	1	1	yes	characteristic, complemented
17	C₃×D₃	C₂×C₇	1	2	yes	direct product factor
18	C₂×C₇	—	3	2	no	—
19	C₂×C₇	C₃×D₃	1	1	yes	characteristic, direct product factor
20	C₂²×C₃	—	3	1	no	—
21	C₂×D₃	C₃×C₇	1	1	yes	characteristic, direct product factor
22	C₃²	C₂²×C₇	1	1	yes	characteristic, complemented, Sylow
23	C₇	C₂×C₃×D₃	1	1	yes	characteristic, direct product factor, Sylow
24	C₂×C₃	—	2	1	no	—
25	C₂×C₃	—	3	2	no	—
26	C₂×C₃	D₃×C₇	1	1	yes	characteristic, direct product factor
27	C₂×C₃	C₂×C₃×C₇	1	1	yes	characteristic, complemented
28	D₃	C₂×C₃×C₇	1	2	yes	direct product factor
29	C₂²	—	3	1	no	Sylow
30	C₃	—	2	1	no	—
31	C₃	C₂×D₃×C₇	1	1	yes	characteristic, direct product factor
32	C₃	C₂²×C₃×C₇	1	1	yes	characteristic, complemented
33	C₂	—	3	2	no	—
34	C₂	C₃×D₃×C₇	1	1	yes	characteristic, direct product factor

	subgroup	G/H
center, hypercenter	C₂×C₃×C₇	D₃
commutator, cosocle, hypocenter	C₃	C₂²×C₃×C₇
Frattini	C₁	G
Fitting, socle	C₂×C₃²×C₇	C₂
LCS	G ▷ C₃
UCS	C₁ ◁ C₂×C₃×C₇

Complements

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

N	H ≅ G/N	type
1. C₂×C₃²×C₇	C₂	semidirect
2. C₃×D₃×C₇	C₂	direct (N × H)
4. C₂×D₃×C₇	C₃	direct (N × H)
5. C₃²×C₇	C₂²	semidirect
8. C₂×C₃×C₇	D₃	direct (N × H)
9. C₂×C₃×C₇	C₂×C₃	semidirect
10. D₃×C₇	C₂×C₃	direct (N × H)
11. C₂×C₃×D₃	C₇	direct (N × H)
14. C₃×C₇	C₂×D₃	direct (N × H)
15. C₃×C₇	C₂²×C₃	semidirect
16. C₂×C₃²	C₂×C₇	semidirect
17. C₃×D₃	C₂×C₇	direct (N × H)
19. C₂×C₇	C₃×D₃	direct (N × H)
21. C₂×D₃	C₃×C₇	direct (N × H)
22. C₃²	C₂²×C₇	semidirect
23. C₇	C₂×C₃×D₃	direct (N × H)
26. C₂×C₃	D₃×C₇	direct (N × H)
27. C₂×C₃	C₂×C₃×C₇	semidirect
28. D₃	C₂×C₃×C₇	direct (N × H)
31. C₃	C₂×D₃×C₇	direct (N × H)
32. C₃	C₂²×C₃×C₇	semidirect
34. C₂	C₃×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×C₃²×C₇	2
C₃×D₃×C₇	2
C₂²×C₃×C₇	3
C₂×D₃×C₇	3
C₂×C₃×D₃	7

Subgroups — C2×C3×D3×C7 #7003

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×D₃×C₇ #7003