Subgroups — C₃×C₅×D₃×C₇² #17984404

Order 4410.

180 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₅×C₇²	C₂	1	1	yes	maximal, characteristic, complemented, Hall
2	C₂×C₃×C₅×C₇²	—	3	1	no	maximal
3	C₅×D₃×C₇²	C₃	1	1	yes	maximal, characteristic, direct product factor
4	C₃×D₃×C₇²	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
5	C₃×C₅×C₇²	—	2	1	no	—
6	C₃×C₅×C₇²	D₃	1	1	yes	characteristic, direct product factor
7	C₃×C₅×C₇²	C₂×C₃	1	1	yes	characteristic, complemented
8	C₃×C₅×D₃×C₇	C₇	1	8	yes	maximal, direct product factor
9	C₂×C₅×C₇²	—	3	1	no	Hall
10	C₃²×C₇²	C₂×C₅	1	1	yes	characteristic, complemented, Hall
11	C₃²×C₅×C₇	C₂×C₇	1	8	yes	complemented
12	C₂×C₃×C₇²	—	3	1	no	—
13	D₃×C₇²	C₃×C₅	1	1	yes	characteristic, direct product factor
14	C₅×C₇²	C₃×D₃	1	1	yes	characteristic, direct product factor, Hall
15	C₂×C₃×C₅×C₇	—	3	8	no	—
16	C₅×D₃×C₇	C₃×C₇	1	8	yes	direct product factor
17	C₃×C₇²	—	2	1	no	—
18	C₃×C₇²	C₅×D₃	1	1	yes	characteristic, direct product factor
19	C₃×C₇²	C₂×C₃×C₅	1	1	yes	characteristic, complemented
20	C₃×D₃×C₇	C₅×C₇	1	8	yes	direct product factor
21	C₃×C₅×C₇	—	2	8	no	—
22	C₃×C₅×C₇	D₃×C₇	1	8	yes	direct product factor
23	C₃×C₅×C₇	C₂×C₃×C₇	1	8	yes	complemented
24	C₂×C₇²	—	3	1	no	Hall
25	C₃×C₅×D₃	C₇²	1	1	yes	characteristic, direct product factor, Hall
26	C₂×C₅×C₇	—	3	8	no	—
27	C₃²×C₇	C₂×C₅×C₇	1	8	yes	complemented
28	C₇²	C₃×C₅×D₃	1	1	yes	characteristic, direct product factor, Sylow
29	C₃²×C₅	C₂×C₇²	1	1	yes	characteristic, complemented, Hall
30	C₂×C₃×C₇	—	3	8	no	—
31	D₃×C₇	C₃×C₅×C₇	1	8	yes	direct product factor
32	C₅×C₇	C₃×D₃×C₇	1	8	yes	direct product factor
33	C₂×C₃×C₅	—	3	1	no	—
34	C₅×D₃	C₃×C₇²	1	1	yes	characteristic, direct product factor
35	C₃×C₇	—	2	8	no	—
36	C₃×C₇	C₅×D₃×C₇	1	8	yes	direct product factor
37	C₃×C₇	C₂×C₃×C₅×C₇	1	8	yes	complemented
38	C₃×D₃	C₅×C₇²	1	1	yes	characteristic, direct product factor, Hall
39	C₃×C₅	—	2	1	no	—
40	C₃×C₅	D₃×C₇²	1	1	yes	characteristic, direct product factor
41	C₃×C₅	C₂×C₃×C₇²	1	1	yes	characteristic, complemented
42	C₂×C₇	—	3	8	no	—
43	C₂×C₅	—	3	1	no	Hall
44	C₃²	C₂×C₅×C₇²	1	1	yes	characteristic, complemented, Sylow
45	C₇	C₃×C₅×D₃×C₇	1	8	yes	direct product factor
46	C₂×C₃	—	3	1	no	—
47	D₃	C₃×C₅×C₇²	1	1	yes	characteristic, direct product factor
48	C₅	C₃×D₃×C₇²	1	1	yes	characteristic, direct product factor, Sylow
49	C₃	—	2	1	no	—
50	C₃	C₅×D₃×C₇²	1	1	yes	characteristic, direct product factor
51	C₃	C₂×C₃×C₅×C₇²	1	1	yes	characteristic, complemented
52	C₂	—	3	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₃×C₅×C₇²	D₃
commutator, cosocle, hypocenter	C₃	C₂×C₃×C₅×C₇²
Frattini	C₁	G
Fitting, socle	C₃²×C₅×C₇²	C₂
LCS	G ▷ C₃
UCS	C₁ ◁ C₃×C₅×C₇²

Complements

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

N	H ≅ G/N	type
1. C₃²×C₅×C₇²	C₂	semidirect
3. C₅×D₃×C₇²	C₃	direct (N × H)
4. C₃×D₃×C₇²	C₅	direct (N × H)
6. C₃×C₅×C₇²	D₃	direct (N × H)
7. C₃×C₅×C₇²	C₂×C₃	semidirect
8. C₃×C₅×D₃×C₇	C₇	direct (N × H)
10. C₃²×C₇²	C₂×C₅	semidirect
11. C₃²×C₅×C₇	C₂×C₇	semidirect
13. D₃×C₇²	C₃×C₅	direct (N × H)
14. C₅×C₇²	C₃×D₃	direct (N × H)
16. C₅×D₃×C₇	C₃×C₇	direct (N × H)
18. C₃×C₇²	C₅×D₃	direct (N × H)
19. C₃×C₇²	C₂×C₃×C₅	semidirect
20. C₃×D₃×C₇	C₅×C₇	direct (N × H)
22. C₃×C₅×C₇	D₃×C₇	direct (N × H)
23. C₃×C₅×C₇	C₂×C₃×C₇	semidirect
25. C₃×C₅×D₃	C₇²	direct (N × H)
27. C₃²×C₇	C₂×C₅×C₇	semidirect
28. C₇²	C₃×C₅×D₃	direct (N × H)
29. C₃²×C₅	C₂×C₇²	semidirect
31. D₃×C₇	C₃×C₅×C₇	direct (N × H)
32. C₅×C₇	C₃×D₃×C₇	direct (N × H)
34. C₅×D₃	C₃×C₇²	direct (N × H)
36. C₃×C₇	C₅×D₃×C₇	direct (N × H)
37. C₃×C₇	C₂×C₃×C₅×C₇	semidirect
38. C₃×D₃	C₅×C₇²	direct (N × H)
40. C₃×C₅	D₃×C₇²	direct (N × H)
41. C₃×C₅	C₂×C₃×C₇²	semidirect
44. C₃²	C₂×C₅×C₇²	semidirect
45. C₇	C₃×C₅×D₃×C₇	direct (N × H)
47. D₃	C₃×C₅×C₇²	direct (N × H)
48. C₅	C₃×D₃×C₇²	direct (N × H)
50. C₃	C₅×D₃×C₇²	direct (N × H)
51. C₃	C₂×C₃×C₅×C₇²	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₃²×C₅×C₇²	2
C₂×C₃×C₅×C₇²	3
C₅×D₃×C₇²	3
C₃×D₃×C₇²	5
C₃×C₅×D₃×C₇	7

Subgroups — C3×C5×D3×C72 #17984404

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×D₃×C₇² #17984404