Subgroups — D₅₁⋊(C₇⋊D₇) #23756254

Order 9996.

200 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₇×C₃⋊(C₇⋊D₇)	C₂	1	1	yes	maximal, characteristic, complemented
2	C₇²×D₅₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₇⋊D₁₁₉	C₂	1	1	yes	maximal, characteristic, complemented
4	D₁₇×C₇⋊D₇	—	3	1	no	maximal, Hall
5	C₃×C₇²×C₁₇	C₂²	1	1	yes	characteristic, complemented, Hall
6	C₇⋊D₁₁₉	D₃	1	1	yes	characteristic, complemented
7	C₁₇×C₇⋊D₇	—	3	1	no	—
8	C₇²×D₁₇	—	3	1	no	—
9	D₂₁⋊D₁₇	—	7	8	no	maximal
10	C₇²×C₁₇	C₂×D₃	1	1	yes	characteristic, complemented, Hall
11	C₁₇×D₂₁	—	7	8	no	—
12	C₇×D₅₁	D₇	1	8	yes	complemented
13	C₃×D₁₁₉	—	7	8	no	—
14	D₃×C₇⋊D₇	—	17	1	no	maximal, Hall
15	D₇×D₁₇	—	21	8	no	—
16	C₃×C₇×C₁₇	C₂×D₇	1	8	yes	complemented
17	C₃⋊(C₇⋊D₇)	D₁₇	1	1	yes	characteristic, complemented
18	D₃×C₇²	—	17	1	no	—
19	C₃×C₇⋊D₇	—	17	1	no	—
20	D₁₁₉	—	7	8	no	—
21	C₇×D₁₇	—	3	8	no	—
22	D₇×C₁₇	—	21	8	no	—
23	D₃×D₁₇	—	49	1	no	Hall
24	C₂×C₇⋊D₇	—	51	1	no	Hall
25	C₃×C₇²	C₂×D₁₇	1	1	yes	characteristic, complemented, Hall
26	C₇×C₁₇	D₃×D₇	1	8	yes	complemented
27	D₅₁	C₇⋊D₇	1	1	yes	characteristic, complemented
28	C₃×D₁₇	—	49	1	no	—
29	D₃×C₁₇	—	49	1	no	—
30	C₂×C₇²	—	51	1	no	—
31	C₇⋊D₇	—	3	1	no	—
32	C₇⋊D₇	—	17	1	no	—
33	D₃×D₇	—	119	8	no	—
34	C₂×D₁₇	—	147	1	no	Hall
35	C₃×C₁₇	C₂×C₇⋊D₇	1	1	yes	characteristic, complemented, Hall
36	C₇²	D₃×D₁₇	1	1	yes	characteristic, complemented, Sylow
37	D₂₁	—	7	8	no	—
38	C₃×D₇	—	119	8	no	—
39	D₃×C₇	—	17	8	no	—
40	C₂×C₁₇	—	147	1	no	—
41	D₁₇	—	3	1	no	—
42	D₁₇	—	49	1	no	—
43	C₂×D₇	—	357	8	no	—
44	C₃×C₇	D₇×D₁₇	1	8	yes	complemented
45	C₁₇	D₃×C₇⋊D₇	1	1	yes	characteristic, complemented, Sylow
46	C₂×C₇	—	51	8	no	—
47	D₇	—	21	8	no	—
48	D₇	—	119	8	no	—
49	C₂×D₃	—	833	1	no	Hall
50	C₇	D₂₁⋊D₁₇	1	8	yes	complemented
51	C₂×C₃	—	833	1	no	—
52	D₃	—	17	1	no	—
53	D₃	—	49	1	no	—
54	C₂²	—	2499	1	no	Sylow
55	C₃	D₁₇×C₇⋊D₇	1	1	yes	characteristic, complemented, Sylow
56	C₂	—	51	1	no	—
57	C₂	—	833	1	no	—
58	C₂	—	147	1	no	—

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

Complements

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

N	H ≅ G/N	type
1. C₁₇×C₃⋊(C₇⋊D₇)	C₂	semidirect
2. C₇²×D₅₁	C₂	semidirect
3. C₃×C₇⋊D₁₁₉	C₂	semidirect
5. C₃×C₇²×C₁₇	C₂²	semidirect
6. C₇⋊D₁₁₉	D₃	semidirect
10. C₇²×C₁₇	C₂×D₃	semidirect
12. C₇×D₅₁	D₇	semidirect
16. C₃×C₇×C₁₇	C₂×D₇	semidirect
17. C₃⋊(C₇⋊D₇)	D₁₇	semidirect
25. C₃×C₇²	C₂×D₁₇	semidirect
26. C₇×C₁₇	D₃×D₇	semidirect
27. D₅₁	C₇⋊D₇	semidirect
35. C₃×C₁₇	C₂×C₇⋊D₇	semidirect
36. C₇²	D₃×D₁₇	semidirect
44. C₃×C₇	D₇×D₁₇	semidirect
45. C₁₇	D₃×C₇⋊D₇	semidirect
50. C₇	D₂₁⋊D₁₇	semidirect
55. C₃	D₁₇×C₇⋊D₇	semidirect

Subgroups — D51⋊(C7⋊D7) #23756254

Conjugacy classes of subgroups

Complements

Subgroups — D₅₁⋊(C₇⋊D₇) #23756254