Subgroups — C₁₇×C₃⋊#64423 #28418411

Order 14994.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₁₇×#3861	C₂	1	1	yes	maximal, characteristic, complemented, Hall
2	C₇×C₁₇×D₂₁	C₃	1	1	yes	maximal, characteristic, complemented
3	C₁₇×#64423	—	3	1	no	maximal
4	C₁₇×#3861	—	2	1	no	—
5	C₁₇×#3861	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×C₁₇×ΓL(1,𝔽₈)	—	14	3	no	—
13	C₃⋊#64423	C₁₇	1	1	yes	maximal, characteristic, direct product factor, Hall
14	C₇²×C₁₇	C₃×D₃	1	1	yes	characteristic, complemented, Hall
15	C₁₇×D₂₁	ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
16	D₃×C₇×C₁₇	—	7	1	no	—
17	C₂×C₁₇×ΓL(1,𝔽₈)	—	21	1	no	—
18	C₁₇×AGL(1,𝔽₇)	—	21	1	no	—
19	C₃×#3861	C₂×C₁₇	1	1	yes	characteristic, complemented, Hall
20	C₃×C₇×C₁₇	—	2	3	no	—
21	C₃×C₇×C₁₇	AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
22	C₃×C₇×C₁₇	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
23	C₁₇×ΓL(1,𝔽₈)	—	7	1	no	—
24	C₁₇×ΓL(1,𝔽₈)	—	7	1	no	—
25	C₁₇×ΓL(1,𝔽₈)	—	14	3	no	—
26	C₁₇×ΓL(1,𝔽₈)	—	14	6	no	—
27	C₁₇×ΓL(1,𝔽₈)	—	14	1	no	—
28	C₁₇×ΓL(1,𝔽₈)	—	14	1	no	—
29	C₃×D₃×C₁₇	—	49	1	no	Hall
30	C₇×D₂₁	C₃×C₁₇	1	1	yes	characteristic, complemented
31	#64423	—	3	1	no	—
32	C₂×C₇×C₁₇	—	21	1	no	—
33	D₇×C₁₇	—	3	1	no	—
34	C₃²×C₁₇	—	49	1	no	Hall
35	C₃×C₇²	C₂×C₃×C₁₇	1	1	yes	characteristic, complemented
36	#3861	—	2	1	no	—
37	#3861	D₃×C₁₇	1	1	yes	characteristic, complemented
38	C₃⋊AGL(1,𝔽₇)	—	7	1	no	—
39	D₃×ΓL(1,𝔽₈)	—	7	1	no	—
40	C₇×C₁₇	—	2	3	no	—
41	C₇×C₁₇	D₃×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
42	C₇×C₁₇	C₃⋊AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
43	C₂×C₃×C₁₇	—	147	1	no	—
44	D₃×C₁₇	—	7	1	no	—
45	C₇×D₇	—	3	1	no	Hall
46	C₃×ΓL(1,𝔽₈)	—	7	1	no	—
47	C₃×ΓL(1,𝔽₈)	—	7	1	no	—
48	C₃×ΓL(1,𝔽₈)	—	14	3	no	—
49	C₃×C₁₇	—	49	1	no	—
50	C₃×C₁₇	—	98	1	no	—
51	C₃×C₁₇	#64423	1	1	yes	characteristic, complemented
52	C₇²	C₃×D₃×C₁₇	1	1	yes	characteristic, complemented, Sylow
53	D₂₁	C₁₇×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
54	D₃×C₇	—	7	1	no	—
55	C₂×ΓL(1,𝔽₈)	—	21	1	no	—
56	AGL(1,𝔽₇)	—	21	1	no	—
57	C₂×C₁₇	—	21	1	no	Hall
58	C₃×C₇	—	2	3	no	—
59	C₃×C₇	C₁₇×AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
60	C₃×C₇	C₂×C₁₇×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
61	ΓL(1,𝔽₈)	—	7	1	no	—
62	ΓL(1,𝔽₈)	—	7	1	no	—
63	ΓL(1,𝔽₈)	—	14	3	no	—
64	ΓL(1,𝔽₈)	—	14	6	no	—
65	ΓL(1,𝔽₈)	—	14	1	no	—
66	ΓL(1,𝔽₈)	—	14	1	no	—
67	C₃×D₃	—	49	1	no	Hall
68	C₁₇	C₃⋊#64423	1	1	yes	characteristic, direct product factor, Sylow
69	C₂×C₇	—	21	1	no	—
70	D₇	—	3	1	no	—
71	C₃²	—	49	1	no	Sylow
72	C₇	—	2	3	no	—
73	C₇	C₁₇×C₃⋊AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
74	C₇	D₃×C₁₇×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
75	C₂×C₃	—	147	1	no	—
76	D₃	—	7	1	no	—
77	C₃	—	49	1	no	—
78	C₃	—	98	1	no	—
79	C₃	C₁₇×#64423	1	1	yes	characteristic, complemented
80	C₂	—	21	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₁₇	C₃⋊#64423
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×#3861	C₂	semidirect
2. C₇×C₁₇×D₂₁	C₃	semidirect
5. C₁₇×#3861	D₃	semidirect
6. C₃×C₇²×C₁₇	C₂×C₃	semidirect
13. C₃⋊#64423	C₁₇	direct (N × H)
14. C₇²×C₁₇	C₃×D₃	semidirect
15. C₁₇×D₂₁	ΓL(1,𝔽₈)	semidirect
19. C₃×#3861	C₂×C₁₇	semidirect
21. C₃×C₇×C₁₇	AGL(1,𝔽₇)	semidirect
22. C₃×C₇×C₁₇	C₂×ΓL(1,𝔽₈)	semidirect
30. C₇×D₂₁	C₃×C₁₇	semidirect
35. C₃×C₇²	C₂×C₃×C₁₇	semidirect
37. #3861	D₃×C₁₇	semidirect
41. C₇×C₁₇	D₃×ΓL(1,𝔽₈)	semidirect
42. C₇×C₁₇	C₃⋊AGL(1,𝔽₇)	semidirect
51. C₃×C₁₇	#64423	semidirect
52. C₇²	C₃×D₃×C₁₇	semidirect
53. D₂₁	C₁₇×ΓL(1,𝔽₈)	semidirect
59. C₃×C₇	C₁₇×AGL(1,𝔽₇)	semidirect
60. C₃×C₇	C₂×C₁₇×ΓL(1,𝔽₈)	semidirect
68. C₁₇	C₃⋊#64423	direct (N × H)
73. C₇	C₁₇×C₃⋊AGL(1,𝔽₇)	semidirect
74. C₇	D₃×C₁₇×ΓL(1,𝔽₈)	semidirect
79. C₃	C₁₇×#64423	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₁₇×#3861	2
C₇×C₁₇×D₂₁	3
C₁₇×#64423	3
D₃×C₁₇×ΓL(1,𝔽₈)	7
C₁₇×C₃⋊AGL(1,𝔽₇)	7
C₃⋊#64423	17

Subgroups — C17×C3⋊#64423 #28418411

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₇×C₃⋊#64423 #28418411