Subgroups — C₃×C₁₇×C₃⋊(C₇⋊D₇) #28418435

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	C₁₇×C₃⋊(C₇⋊D₇)	C₃	1	1	yes	maximal, characteristic, direct product factor
3	C₃×C₁₇×C₇⋊D₇	—	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×C₁₇×D₂₁	—	7	8	no	maximal
8	C₁₇×C₇⋊D₇	—	3	1	no	Hall
9	C₃²×C₇×C₁₇	D₇	1	8	yes	complemented
10	C₃×C₃⋊(C₇⋊D₇)	C₁₇	1	1	yes	maximal, characteristic, direct product factor, Hall
11	C₇²×C₁₇	C₃×D₃	1	1	yes	characteristic, complemented, Hall
12	C₁₇×D₂₁	—	7	8	no	—
13	C₃×D₇×C₁₇	—	21	8	no	—
14	C₃²×C₇²	C₂×C₁₇	1	1	yes	characteristic, complemented, Hall
15	C₃×C₇×C₁₇	—	2	8	no	—
16	C₃×C₇×C₁₇	C₃×D₇	1	8	yes	complemented
17	C₃×C₇×C₁₇	D₂₁	1	8	yes	complemented
18	C₃×D₃×C₁₇	—	49	1	no	Hall
19	C₃⋊(C₇⋊D₇)	C₃×C₁₇	1	1	yes	characteristic, direct product factor
20	C₃×C₇⋊D₇	—	3	1	no	—
21	D₇×C₁₇	—	21	8	no	—
22	C₃²×C₁₇	C₇⋊D₇	1	1	yes	characteristic, complemented, Hall
23	C₃×C₇²	—	2	1	no	—
24	C₃×C₇²	D₃×C₁₇	1	1	yes	characteristic, complemented
25	C₃×C₇²	C₂×C₃×C₁₇	1	1	yes	characteristic, complemented
26	C₃×D₂₁	—	7	8	no	—
27	C₇×C₁₇	C₃×D₂₁	1	8	yes	complemented
28	C₂×C₃×C₁₇	—	147	1	no	—
29	D₃×C₁₇	—	49	1	no	—
30	C₇⋊D₇	—	3	1	no	Hall
31	C₃²×C₇	D₇×C₁₇	1	8	yes	complemented
32	C₃×C₁₇	—	2	1	no	—
33	C₃×C₁₇	C₃×C₇⋊D₇	1	1	yes	characteristic, complemented
34	C₃×C₁₇	C₃⋊(C₇⋊D₇)	1	1	yes	characteristic, direct product factor
35	C₇²	C₃×D₃×C₁₇	1	1	yes	characteristic, complemented, Sylow
36	D₂₁	—	7	8	no	—
37	C₃×D₇	—	21	8	no	—
38	C₂×C₁₇	—	147	1	no	Hall
39	C₃×C₇	—	2	8	no	—
40	C₃×C₇	C₃×D₇×C₁₇	1	8	yes	complemented
41	C₃×C₇	C₁₇×D₂₁	1	8	yes	complemented
42	C₃×D₃	—	49	1	no	Hall
43	C₁₇	C₃×C₃⋊(C₇⋊D₇)	1	1	yes	characteristic, direct product factor, Sylow
44	D₇	—	21	8	no	—
45	C₃²	C₁₇×C₇⋊D₇	1	1	yes	characteristic, complemented, Sylow
46	C₇	C₃×C₁₇×D₂₁	1	8	yes	complemented
47	C₂×C₃	—	147	1	no	—
48	D₃	—	49	1	no	—
49	C₃	—	2	1	no	—
50	C₃	C₃×C₁₇×C₇⋊D₇	1	1	yes	characteristic, complemented
51	C₃	C₁₇×C₃⋊(C₇⋊D₇)	1	1	yes	characteristic, direct product factor
52	C₂	—	147	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₃×C₁₇	C₃⋊(C₇⋊D₇)
commutator, cosocle, hypocenter	C₃×C₇²	C₂×C₃×C₁₇
Frattini	C₁	G
Fitting, socle	C₃²×C₇²×C₁₇	C₂
LCS	G ▷ C₃×C₇²
UCS	C₁ ◁ C₃×C₁₇

Complements

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

N	H ≅ G/N	type
1. C₃²×C₇²×C₁₇	C₂	semidirect
2. C₁₇×C₃⋊(C₇⋊D₇)	C₃	direct (N × H)
5. C₃×C₇²×C₁₇	D₃	semidirect
6. C₃×C₇²×C₁₇	C₂×C₃	semidirect
9. C₃²×C₇×C₁₇	D₇	semidirect
10. C₃×C₃⋊(C₇⋊D₇)	C₁₇	direct (N × H)
11. C₇²×C₁₇	C₃×D₃	semidirect
14. C₃²×C₇²	C₂×C₁₇	semidirect
16. C₃×C₇×C₁₇	C₃×D₇	semidirect
17. C₃×C₇×C₁₇	D₂₁	semidirect
19. C₃⋊(C₇⋊D₇)	C₃×C₁₇	direct (N × H)
22. C₃²×C₁₇	C₇⋊D₇	semidirect
24. C₃×C₇²	D₃×C₁₇	semidirect
25. C₃×C₇²	C₂×C₃×C₁₇	semidirect
27. C₇×C₁₇	C₃×D₂₁	semidirect
31. C₃²×C₇	D₇×C₁₇	semidirect
33. C₃×C₁₇	C₃×C₇⋊D₇	semidirect
34. C₃×C₁₇	C₃⋊(C₇⋊D₇)	direct (N × H)
35. C₇²	C₃×D₃×C₁₇	semidirect
40. C₃×C₇	C₃×D₇×C₁₇	semidirect
41. C₃×C₇	C₁₇×D₂₁	semidirect
43. C₁₇	C₃×C₃⋊(C₇⋊D₇)	direct (N × H)
45. C₃²	C₁₇×C₇⋊D₇	semidirect
46. C₇	C₃×C₁₇×D₂₁	semidirect
50. C₃	C₃×C₁₇×C₇⋊D₇	semidirect
51. C₃	C₁₇×C₃⋊(C₇⋊D₇)	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
C₁₇×C₃⋊(C₇⋊D₇)	3
C₃×C₁₇×C₇⋊D₇	3
C₃×C₁₇×D₂₁	7
C₃×C₃⋊(C₇⋊D₇)	17

Subgroups — C3×C17×C3⋊(C7⋊D7) #28418435

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₁₇×C₃⋊(C₇⋊D₇) #28418435