Subgroups — C₂×C₇²×D₅₁ #23756276

Order 9996.

200 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇²×D₅₁	C₂	1	2	yes	maximal, direct product factor
2	C₂×C₃×C₇²×C₁₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₇²×D₁₇	—	3	1	no	maximal, Hall
4	C₃×C₇²×C₁₇	C₂²	1	1	yes	characteristic, complemented, Hall
5	C₂×C₇²×C₁₇	D₃	1	1	yes	characteristic, complemented
6	C₇²×D₁₇	—	3	2	no	—
7	C₂×C₇×D₅₁	C₇	1	8	yes	maximal, direct product factor
8	C₇²×C₁₇	C₂×D₃	1	1	yes	characteristic, complemented, Hall
9	C₂×C₃×C₇×C₁₇	C₂×C₇	1	8	yes	complemented
10	C₇×D₅₁	C₂×C₇	1	16	yes	direct product factor
11	C₂×D₃×C₇²	—	17	1	no	maximal, Hall
12	C₂×C₇×D₁₇	—	3	8	no	—
13	C₃×C₇×C₁₇	C₂²×C₇	1	8	yes	complemented
14	C₂×C₃×C₇²	D₁₇	1	1	yes	characteristic, complemented
15	D₃×C₇²	—	17	2	no	—
16	C₂×C₇×C₁₇	D₃×C₇	1	8	yes	complemented
17	C₇×D₁₇	—	3	16	no	—
18	C₂×D₅₁	C₇²	1	1	yes	characteristic, direct product factor, Hall
19	C₂²×C₇²	—	51	1	no	Hall
20	C₃×C₇²	C₂×D₁₇	1	1	yes	characteristic, complemented, Hall
21	C₇×C₁₇	C₂×D₃×C₇	1	8	yes	complemented
22	C₂×C₃×C₁₇	C₂×C₇²	1	1	yes	characteristic, complemented
23	D₅₁	C₂×C₇²	1	2	yes	direct product factor
24	C₂×C₇²	—	51	2	no	—
25	C₂×C₇²	D₅₁	1	1	yes	characteristic, direct product factor
26	C₂×D₃×C₇	—	17	8	no	—
27	C₂×D₁₇	—	3	1	no	Hall
28	C₃×C₁₇	C₂²×C₇²	1	1	yes	characteristic, complemented, Hall
29	C₇²	C₂×D₅₁	1	1	yes	characteristic, direct product factor, Sylow
30	C₂×C₃×C₇	C₇×D₁₇	1	8	yes	complemented
31	D₃×C₇	—	17	16	no	—
32	C₂×C₁₇	D₃×C₇²	1	1	yes	characteristic, complemented
33	D₁₇	—	3	2	no	—
34	C₂²×C₇	—	51	8	no	—
35	C₃×C₇	C₂×C₇×D₁₇	1	8	yes	complemented
36	C₁₇	C₂×D₃×C₇²	1	1	yes	characteristic, complemented, Sylow
37	C₂×C₇	—	51	16	no	—
38	C₂×C₇	C₇×D₅₁	1	8	yes	direct product factor
39	C₂×D₃	—	17	1	no	Hall
40	C₇	C₂×C₇×D₅₁	1	8	yes	direct product factor
41	C₂×C₃	C₇²×D₁₇	1	1	yes	characteristic, complemented
42	D₃	—	17	2	no	—
43	C₂²	—	51	1	no	Sylow
44	C₃	C₂×C₇²×D₁₇	1	1	yes	characteristic, complemented, Sylow
45	C₂	—	51	2	no	—
46	C₂	C₇²×D₅₁	1	1	yes	characteristic, direct product factor

	subgroup	G/H
center, hypercenter	C₂×C₇²	D₅₁
commutator, cosocle, hypocenter	C₃×C₁₇	C₂²×C₇²
Frattini	C₁	G
Fitting, socle	C₂×C₃×C₇²×C₁₇	C₂
LCS	G ▷ C₃×C₁₇
UCS	C₁ ◁ C₂×C₇²

Complements

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

N	H ≅ G/N	type
1. C₇²×D₅₁	C₂	direct (N × H)
2. C₂×C₃×C₇²×C₁₇	C₂	semidirect
4. C₃×C₇²×C₁₇	C₂²	semidirect
5. C₂×C₇²×C₁₇	D₃	semidirect
7. C₂×C₇×D₅₁	C₇	direct (N × H)
8. C₇²×C₁₇	C₂×D₃	semidirect
9. C₂×C₃×C₇×C₁₇	C₂×C₇	semidirect
10. C₇×D₅₁	C₂×C₇	direct (N × H)
13. C₃×C₇×C₁₇	C₂²×C₇	semidirect
14. C₂×C₃×C₇²	D₁₇	semidirect
16. C₂×C₇×C₁₇	D₃×C₇	semidirect
18. C₂×D₅₁	C₇²	direct (N × H)
20. C₃×C₇²	C₂×D₁₇	semidirect
21. C₇×C₁₇	C₂×D₃×C₇	semidirect
22. C₂×C₃×C₁₇	C₂×C₇²	semidirect
23. D₅₁	C₂×C₇²	direct (N × H)
25. C₂×C₇²	D₅₁	direct (N × H)
28. C₃×C₁₇	C₂²×C₇²	semidirect
29. C₇²	C₂×D₅₁	direct (N × H)
30. C₂×C₃×C₇	C₇×D₁₇	semidirect
32. C₂×C₁₇	D₃×C₇²	semidirect
35. C₃×C₇	C₂×C₇×D₁₇	semidirect
36. C₁₇	C₂×D₃×C₇²	semidirect
38. C₂×C₇	C₇×D₅₁	direct (N × H)
40. C₇	C₂×C₇×D₅₁	direct (N × H)
41. C₂×C₃	C₇²×D₁₇	semidirect
44. C₃	C₂×C₇²×D₁₇	semidirect
46. C₂	C₇²×D₅₁	direct (N × H)

Subgroups — C2×C72×D51 #23756276

Conjugacy classes of subgroups

Complements

Subgroups — C₂×C₇²×D₅₁ #23756276