Subgroups — C₁₇×C₃⋊#64422 #28418413

Order 14994.

76 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₁₇×#3862	C₂	1	1	yes	maximal, characteristic, complemented, Hall
2	C₇×C₁₇×D₂₁	C₃	1	1	yes	maximal, characteristic, complemented
3	C₁₇×#64422	—	3	1	no	maximal
4	C₁₇×#3862	—	2	1	no	—
5	C₁₇×#3862	D₃	1	1	yes	characteristic, complemented
6	C₃×C₇²×C₁₇	C₂×C₃	1	1	yes	characteristic, complemented
7	D₃×C₁₇×ΓL(1,𝔽₈)	—	7	1	no	maximal
8	C₁₇×C₃⋊AGL(1,𝔽₇)	—	7	1	no	maximal
9	C₇×D₇×C₁₇	—	3	1	no	Hall
10	C₃×C₁₇×ΓL(1,𝔽₈)	—	7	1	no	—
11	C₃×C₁₇×ΓL(1,𝔽₈)	—	7	1	no	—
12	C₃⋊#64422	C₁₇	1	1	yes	maximal, characteristic, direct product factor, Hall
13	C₇²×C₁₇	C₃×D₃	1	1	yes	characteristic, complemented, Hall
14	C₁₇×D₂₁	ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
15	D₃×C₇×C₁₇	—	7	1	no	—
16	C₂×C₁₇×ΓL(1,𝔽₈)	—	21	1	no	—
17	C₁₇×AGL(1,𝔽₇)	—	21	1	no	—
18	C₃×#3862	C₂×C₁₇	1	1	yes	characteristic, complemented, Hall
19	C₃×C₇×C₁₇	—	6	1	no	—
20	C₃×C₇×C₁₇	AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
21	C₃×C₇×C₁₇	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
22	C₁₇×ΓL(1,𝔽₈)	—	7	1	no	—
23	C₁₇×ΓL(1,𝔽₈)	—	7	1	no	—
24	C₁₇×ΓL(1,𝔽₈)	—	14	1	no	—
25	C₁₇×ΓL(1,𝔽₈)	—	14	1	no	—
26	C₃×D₃×C₁₇	—	49	1	no	Hall
27	C₇×D₂₁	C₃×C₁₇	1	1	yes	characteristic, complemented
28	#64422	—	3	1	no	—
29	C₂×C₇×C₁₇	—	21	1	no	—
30	D₇×C₁₇	—	3	1	no	—
31	C₃²×C₁₇	—	49	1	no	Hall
32	C₃×C₇²	C₂×C₃×C₁₇	1	1	yes	characteristic, complemented
33	#3862	—	2	1	no	—
34	#3862	D₃×C₁₇	1	1	yes	characteristic, complemented
35	C₃⋊AGL(1,𝔽₇)	—	7	1	no	—
36	D₃×ΓL(1,𝔽₈)	—	7	1	no	—
37	C₇×C₁₇	—	6	1	no	—
38	C₇×C₁₇	D₃×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
39	C₇×C₁₇	C₃⋊AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
40	C₂×C₃×C₁₇	—	147	1	no	—
41	D₃×C₁₇	—	7	1	no	—
42	C₇×D₇	—	3	1	no	Hall
43	C₃×ΓL(1,𝔽₈)	—	7	1	no	—
44	C₃×ΓL(1,𝔽₈)	—	7	1	no	—
45	C₃×C₁₇	—	49	1	no	—
46	C₃×C₁₇	—	98	1	no	—
47	C₃×C₁₇	#64422	1	1	yes	characteristic, complemented
48	C₇²	C₃×D₃×C₁₇	1	1	yes	characteristic, complemented, Sylow
49	D₂₁	C₁₇×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
50	D₃×C₇	—	7	1	no	—
51	C₂×ΓL(1,𝔽₈)	—	21	1	no	—
52	AGL(1,𝔽₇)	—	21	1	no	—
53	C₂×C₁₇	—	21	1	no	Hall
54	C₃×C₇	—	6	1	no	—
55	C₃×C₇	C₁₇×AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
56	C₃×C₇	C₂×C₁₇×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
57	ΓL(1,𝔽₈)	—	7	1	no	—
58	ΓL(1,𝔽₈)	—	7	1	no	—
59	ΓL(1,𝔽₈)	—	14	1	no	—
60	ΓL(1,𝔽₈)	—	14	1	no	—
61	C₃×D₃	—	49	1	no	Hall
62	C₁₇	C₃⋊#64422	1	1	yes	characteristic, direct product factor, Sylow
63	C₂×C₇	—	21	1	no	—
64	D₇	—	3	1	no	—
65	C₃²	—	49	1	no	Sylow
66	C₇	—	6	1	no	—
67	C₇	C₁₇×C₃⋊AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
68	C₇	D₃×C₁₇×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
69	C₂×C₃	—	147	1	no	—
70	D₃	—	7	1	no	—
71	C₃	—	49	1	no	—
72	C₃	—	98	1	no	—
73	C₃	C₁₇×#64422	1	1	yes	characteristic, complemented
74	C₂	—	21	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₁₇	C₃⋊#64422
commutator, cosocle, hypocenter	C₃×C₇²	C₂×C₃×C₁₇
Frattini	C₁	G
Fitting, socle	C₃×C₇²×C₁₇	C₂×C₃
LCS	G ▷ C₃×C₇²
UCS	C₁ ◁ C₁₇

Complements

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

N	H ≅ G/N	type
1. C₃×C₁₇×#3862	C₂	semidirect
2. C₇×C₁₇×D₂₁	C₃	semidirect
5. C₁₇×#3862	D₃	semidirect
6. C₃×C₇²×C₁₇	C₂×C₃	semidirect
12. C₃⋊#64422	C₁₇	direct (N × H)
13. C₇²×C₁₇	C₃×D₃	semidirect
14. C₁₇×D₂₁	ΓL(1,𝔽₈)	semidirect
18. C₃×#3862	C₂×C₁₇	semidirect
20. C₃×C₇×C₁₇	AGL(1,𝔽₇)	semidirect
21. C₃×C₇×C₁₇	C₂×ΓL(1,𝔽₈)	semidirect
27. C₇×D₂₁	C₃×C₁₇	semidirect
32. C₃×C₇²	C₂×C₃×C₁₇	semidirect
34. #3862	D₃×C₁₇	semidirect
38. C₇×C₁₇	D₃×ΓL(1,𝔽₈)	semidirect
39. C₇×C₁₇	C₃⋊AGL(1,𝔽₇)	semidirect
47. C₃×C₁₇	#64422	semidirect
48. C₇²	C₃×D₃×C₁₇	semidirect
49. D₂₁	C₁₇×ΓL(1,𝔽₈)	semidirect
55. C₃×C₇	C₁₇×AGL(1,𝔽₇)	semidirect
56. C₃×C₇	C₂×C₁₇×ΓL(1,𝔽₈)	semidirect
62. C₁₇	C₃⋊#64422	direct (N × H)
67. C₇	C₁₇×C₃⋊AGL(1,𝔽₇)	semidirect
68. C₇	D₃×C₁₇×ΓL(1,𝔽₈)	semidirect
73. C₃	C₁₇×#64422	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₁₇×#3862	2
C₇×C₁₇×D₂₁	3
C₁₇×#64422	3
D₃×C₁₇×ΓL(1,𝔽₈)	7
C₁₇×C₃⋊AGL(1,𝔽₇)	7
C₃⋊#64422	17

Subgroups — C17×C3⋊#64422 #28418413

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₇×C₃⋊#64422 #28418413