Subgroups — C₃×C₇²×D₅₁ #28418438

Order 14994.

180 conjugacy classes of subgroups.

Conjugacy classes of subgroups

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

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

Complements

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

N	H ≅ G/N	type
1. C₃²×C₇²×C₁₇	C₂	semidirect
2. C₇²×D₅₁	C₃	direct (N × H)
5. C₃×C₇²×C₁₇	D₃	semidirect
6. C₃×C₇²×C₁₇	C₂×C₃	semidirect
7. C₃×C₇×D₅₁	C₇	direct (N × H)
9. C₃²×C₇×C₁₇	C₂×C₇	semidirect
11. C₇²×C₁₇	C₃×D₃	semidirect
12. C₇×D₅₁	C₃×C₇	direct (N × H)
14. C₃²×C₇²	D₁₇	semidirect
16. C₃×C₇×C₁₇	D₃×C₇	semidirect
17. C₃×C₇×C₁₇	C₂×C₃×C₇	semidirect
18. C₃×D₅₁	C₇²	direct (N × H)
22. C₃²×C₁₇	C₂×C₇²	semidirect
24. C₃×C₇²	C₃×D₁₇	semidirect
25. C₃×C₇²	D₅₁	direct (N × H)
27. C₇×C₁₇	C₃×D₃×C₇	semidirect
28. D₅₁	C₃×C₇²	direct (N × H)
31. C₃²×C₇	C₇×D₁₇	semidirect
33. C₃×C₁₇	D₃×C₇²	semidirect
34. C₃×C₁₇	C₂×C₃×C₇²	semidirect
35. C₇²	C₃×D₅₁	direct (N × H)
40. C₃×C₇	C₃×C₇×D₁₇	semidirect
41. C₃×C₇	C₇×D₅₁	direct (N × H)
43. C₁₇	C₃×D₃×C₇²	semidirect
45. C₃²	C₇²×D₁₇	semidirect
46. C₇	C₃×C₇×D₅₁	direct (N × H)
50. C₃	C₃×C₇²×D₁₇	semidirect
51. C₃	C₇²×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
C₃²×C₇²×C₁₇	2
C₇²×D₅₁	3
C₃×C₇²×D₁₇	3
C₃×C₇×D₅₁	7
C₃×D₃×C₇²	17

Subgroups — C3×C72×D51 #28418438

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₇²×D₅₁ #28418438