Subgroups — C₂×C₇²×C₃⋊D₅₁ #44473021

Order 29988.

600 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇²×C₃⋊D₅₁	C₂	1	2	yes	maximal, direct product factor
2	C₂×C₃²×C₇²×C₁₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₇²×D₅₁	—	3	4	no	maximal
4	C₃²×C₇²×C₁₇	C₂²	1	1	yes	characteristic, complemented, Hall
5	C₇²×D₅₁	—	3	8	no	—
6	C₂×C₃×C₇²×C₁₇	D₃	1	4	yes	complemented
7	C₂×C₇×C₃⋊D₅₁	C₇	1	8	yes	maximal, direct product factor
8	C₂×C₇²×D₁₇	—	9	1	no	Hall
9	C₃×C₇²×C₁₇	C₂×D₃	1	4	yes	complemented
10	C₇×C₃⋊D₅₁	C₂×C₇	1	16	yes	direct product factor
11	C₂×C₃²×C₇×C₁₇	C₂×C₇	1	8	yes	complemented
12	C₂×C₇²×C₃⋊D₃	—	17	1	no	maximal, Hall
13	C₂×C₇²×C₁₇	C₃⋊D₃	1	1	yes	characteristic, complemented
14	C₇²×D₁₇	—	9	2	no	—
15	C₂×C₇×D₅₁	—	3	32	no	—
16	C₃²×C₇×C₁₇	C₂²×C₇	1	8	yes	complemented
17	C₂×C₃²×C₇²	D₁₇	1	1	yes	characteristic, complemented
18	C₇²×C₃⋊D₃	—	17	2	no	—
19	C₇²×C₁₇	C₂×C₃⋊D₃	1	1	yes	characteristic, complemented, Hall
20	C₂×C₃×C₇×C₁₇	D₃×C₇	1	32	yes	complemented
21	C₇×D₅₁	—	3	64	no	—
22	C₂×C₃⋊D₅₁	C₇²	1	1	yes	characteristic, direct product factor, Hall
23	C₂×D₃×C₇²	—	51	4	no	—
24	C₂×C₇×D₁₇	—	9	8	no	—
25	C₃²×C₇²	C₂×D₁₇	1	1	yes	characteristic, complemented, Hall
26	C₃×C₇×C₁₇	C₂×D₃×C₇	1	32	yes	complemented
27	C₂×C₃²×C₁₇	C₂×C₇²	1	1	yes	characteristic, complemented
28	C₃⋊D₅₁	C₂×C₇²	1	2	yes	direct product factor
29	C₂×C₃×C₇²	D₅₁	1	4	yes	complemented
30	D₃×C₇²	—	51	8	no	—
31	C₂×C₇×C₃⋊D₃	—	17	8	no	—
32	C₂×C₇×C₁₇	C₇×C₃⋊D₃	1	8	yes	complemented
33	C₇×D₁₇	—	9	16	no	—
34	C₂×D₅₁	—	3	4	no	—
35	C₂²×C₇²	—	153	1	no	Hall
36	C₃²×C₁₇	C₂²×C₇²	1	1	yes	characteristic, complemented, Hall
37	C₃×C₇²	C₂×D₅₁	1	4	yes	complemented
38	C₂×C₃²×C₇	C₇×D₁₇	1	8	yes	complemented
39	C₇×C₃⋊D₃	—	17	16	no	—
40	C₇×C₁₇	C₂×C₇×C₃⋊D₃	1	8	yes	complemented
41	C₂×C₃×C₁₇	D₃×C₇²	1	4	yes	complemented
42	D₅₁	—	3	8	no	—
43	C₂×C₇²	—	153	2	no	—
44	C₂×C₇²	C₃⋊D₅₁	1	1	yes	characteristic, direct product factor
45	C₂×D₃×C₇	—	51	32	no	—
46	C₂×D₁₇	—	9	1	no	Hall
47	C₃²×C₇	C₂×C₇×D₁₇	1	8	yes	complemented
48	C₃×C₁₇	C₂×D₃×C₇²	1	4	yes	complemented
49	C₇²	C₂×C₃⋊D₅₁	1	1	yes	characteristic, direct product factor, Sylow
50	C₂×C₃×C₇	C₇×D₅₁	1	32	yes	complemented
51	D₃×C₇	—	51	64	no	—
52	C₂×C₃⋊D₃	—	17	1	no	Hall
53	C₂×C₁₇	C₇²×C₃⋊D₃	1	1	yes	characteristic, complemented
54	D₁₇	—	9	2	no	—
55	C₂²×C₇	—	153	8	no	—
56	C₃×C₇	C₂×C₇×D₅₁	1	32	yes	complemented
57	C₂×C₃²	C₇²×D₁₇	1	1	yes	characteristic, complemented
58	C₃⋊D₃	—	17	2	no	—
59	C₁₇	C₂×C₇²×C₃⋊D₃	1	1	yes	characteristic, complemented, Sylow
60	C₂×C₇	—	153	16	no	—
61	C₂×C₇	C₇×C₃⋊D₅₁	1	8	yes	direct product factor
62	C₂×D₃	—	51	4	no	—
63	C₃²	C₂×C₇²×D₁₇	1	1	yes	characteristic, complemented, Sylow
64	C₇	C₂×C₇×C₃⋊D₅₁	1	8	yes	direct product factor
65	C₂×C₃	C₇²×D₅₁	1	4	yes	complemented
66	D₃	—	51	8	no	—
67	C₂²	—	153	1	no	Sylow
68	C₃	C₂×C₇²×D₅₁	1	4	yes	complemented
69	C₂	—	153	2	no	—
70	C₂	C₇²×C₃⋊D₅₁	1	1	yes	characteristic, direct product factor

	subgroup	G/H
center, hypercenter	C₂×C₇²	C₃⋊D₅₁
commutator, cosocle, hypocenter	C₃²×C₁₇	C₂²×C₇²
Frattini	C₁	G
Fitting, socle	C₂×C₃²×C₇²×C₁₇	C₂
LCS	G ▷ C₃²×C₁₇
UCS	C₁ ◁ C₂×C₇²

Complements

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

N	H ≅ G/N	type
1. C₇²×C₃⋊D₅₁	C₂	direct (N × H)
2. C₂×C₃²×C₇²×C₁₇	C₂	semidirect
4. C₃²×C₇²×C₁₇	C₂²	semidirect
6. C₂×C₃×C₇²×C₁₇	D₃	semidirect
7. C₂×C₇×C₃⋊D₅₁	C₇	direct (N × H)
9. C₃×C₇²×C₁₇	C₂×D₃	semidirect
10. C₇×C₃⋊D₅₁	C₂×C₇	direct (N × H)
11. C₂×C₃²×C₇×C₁₇	C₂×C₇	semidirect
13. C₂×C₇²×C₁₇	C₃⋊D₃	semidirect
16. C₃²×C₇×C₁₇	C₂²×C₇	semidirect
17. C₂×C₃²×C₇²	D₁₇	semidirect
19. C₇²×C₁₇	C₂×C₃⋊D₃	semidirect
20. C₂×C₃×C₇×C₁₇	D₃×C₇	semidirect
22. C₂×C₃⋊D₅₁	C₇²	direct (N × H)
25. C₃²×C₇²	C₂×D₁₇	semidirect
26. C₃×C₇×C₁₇	C₂×D₃×C₇	semidirect
27. C₂×C₃²×C₁₇	C₂×C₇²	semidirect
28. C₃⋊D₅₁	C₂×C₇²	direct (N × H)
29. C₂×C₃×C₇²	D₅₁	semidirect
32. C₂×C₇×C₁₇	C₇×C₃⋊D₃	semidirect
36. C₃²×C₁₇	C₂²×C₇²	semidirect
37. C₃×C₇²	C₂×D₅₁	semidirect
38. C₂×C₃²×C₇	C₇×D₁₇	semidirect
40. C₇×C₁₇	C₂×C₇×C₃⋊D₃	semidirect
41. C₂×C₃×C₁₇	D₃×C₇²	semidirect
44. C₂×C₇²	C₃⋊D₅₁	direct (N × H)
47. C₃²×C₇	C₂×C₇×D₁₇	semidirect
48. C₃×C₁₇	C₂×D₃×C₇²	semidirect
49. C₇²	C₂×C₃⋊D₅₁	direct (N × H)
50. C₂×C₃×C₇	C₇×D₅₁	semidirect
53. C₂×C₁₇	C₇²×C₃⋊D₃	semidirect
56. C₃×C₇	C₂×C₇×D₅₁	semidirect
57. C₂×C₃²	C₇²×D₁₇	semidirect
59. C₁₇	C₂×C₇²×C₃⋊D₃	semidirect
61. C₂×C₇	C₇×C₃⋊D₅₁	direct (N × H)
63. C₃²	C₂×C₇²×D₁₇	semidirect
64. C₇	C₂×C₇×C₃⋊D₅₁	direct (N × H)
65. C₂×C₃	C₇²×D₅₁	semidirect
68. C₃	C₂×C₇²×D₅₁	semidirect
70. C₂	C₇²×C₃⋊D₅₁	direct (N × H)

Maximal subgroups

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

subgroup	index
C₇²×C₃⋊D₅₁	2
C₂×C₃²×C₇²×C₁₇	2
C₂×C₇²×D₅₁	3
C₂×C₇×C₃⋊D₅₁	7
C₂×C₇²×C₃⋊D₃	17

Subgroups — C2×C72×C3⋊D51 #44473021

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₇²×C₃⋊D₅₁ #44473021