Subgroups — C₂×C₁₇×C₁₇⋊AGL(1,𝔽₇) #37566247

Order 24276.

144 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₇²×ΓL(1,𝔽₈)	C₂	1	1	yes	maximal, characteristic, complemented
2	C₁₇×C₁₇⋊AGL(1,𝔽₇)	C₂	1	2	yes	maximal, direct product factor
3	C₂×C₁₇×D₁₁₉	C₃	1	1	yes	maximal, characteristic, complemented, Hall
4	C₁₇²×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic, complemented, Hall
5	C₁₇×D₁₁₉	C₂×C₃	1	2	yes	complemented
6	C₂×C₇×C₁₇²	C₂×C₃	1	1	yes	characteristic, complemented
7	C₂×C₃×C₁₇×D₁₇	—	7	1	no	maximal, Hall
8	C₇×C₁₇²	C₂²×C₃	1	1	yes	characteristic, complemented, Hall
9	C₂×C₃×C₁₇²	—	14	1	no	—
10	C₃×C₁₇×D₁₇	—	7	2	no	—
11	C₂×C₁₇⋊AGL(1,𝔽₇)	C₁₇	1	1	yes	maximal, characteristic, direct product factor
12	C₂×C₁₇×AGL(1,𝔽₇)	—	17	1	no	maximal
13	C₂×C₁₇×D₁₇	—	7	1	no	Hall
14	C₃×C₁₇²	—	14	1	no	Hall
15	C₂×C₁₇×ΓL(1,𝔽₈)	—	2	8	no	—
16	C₂×C₁₇×ΓL(1,𝔽₈)	D₁₇	1	1	yes	characteristic, complemented
17	C₂×C₁₇×ΓL(1,𝔽₈)	C₂×C₁₇	1	1	yes	characteristic, complemented
18	C₁₇⋊AGL(1,𝔽₇)	C₂×C₁₇	1	2	yes	direct product factor
19	C₁₇×AGL(1,𝔽₇)	—	17	2	no	—
20	C₂×C₁₇²	AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
21	C₁₇×D₁₇	—	7	2	no	—
22	C₂×D₁₁₉	C₃×C₁₇	1	1	yes	characteristic, complemented
23	C₂×D₇×C₁₇	—	17	1	no	—
24	C₁₇×ΓL(1,𝔽₈)	—	2	8	no	—
25	C₁₇×ΓL(1,𝔽₈)	C₂×D₁₇	1	1	yes	characteristic, complemented
26	C₁₇×ΓL(1,𝔽₈)	C₂²×C₁₇	1	1	yes	characteristic, complemented
27	C₁₇²	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic, complemented, Sylow
28	C₂×C₇×C₁₇	—	2	8	no	—
29	C₂×C₇×C₁₇	C₃×D₁₇	1	1	yes	characteristic, complemented
30	C₂×C₇×C₁₇	C₂×C₃×C₁₇	1	1	yes	characteristic, complemented
31	D₁₁₉	C₂×C₃×C₁₇	1	2	yes	complemented
32	D₇×C₁₇	—	17	2	no	—
33	C₂²×C₃×C₁₇	—	119	1	no	—
34	C₂×C₃×D₁₇	—	7	1	no	—
35	C₇×C₁₇	—	2	8	no	—
36	C₇×C₁₇	C₂×C₃×D₁₇	1	1	yes	characteristic, complemented
37	C₇×C₁₇	C₂²×C₃×C₁₇	1	1	yes	characteristic, complemented
38	C₂×C₃×C₁₇	—	14	8	no	—
39	C₂×C₃×C₁₇	—	14	1	no	—
40	C₂×C₃×C₁₇	—	14	1	no	—
41	C₂×C₃×C₁₇	—	119	2	no	—
42	C₃×D₁₇	—	7	2	no	—
43	C₂×AGL(1,𝔽₇)	—	17	1	no	Hall
44	C₂²×C₁₇	—	119	1	no	—
45	C₂×D₁₇	—	7	1	no	—
46	C₃×C₁₇	—	14	8	no	—
47	C₃×C₁₇	—	14	1	no	—
48	C₃×C₁₇	—	14	1	no	—
49	C₂×ΓL(1,𝔽₈)	C₁₇×D₁₇	1	1	yes	characteristic, complemented
50	AGL(1,𝔽₇)	—	17	2	no	—
51	C₂×C₁₇	—	2	8	no	—
52	C₂×C₁₇	—	119	2	no	—
53	C₂×C₁₇	C₁₇×AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
54	C₂×C₁₇	C₁₇⋊AGL(1,𝔽₇)	1	1	yes	characteristic, direct product factor
55	D₁₇	—	7	2	no	—
56	C₂×D₇	—	17	1	no	Hall
57	ΓL(1,𝔽₈)	C₂×C₁₇×D₁₇	1	1	yes	characteristic, complemented, Hall
58	C₁₇	—	2	8	no	—
59	C₁₇	C₂×C₁₇×AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
60	C₁₇	C₂×C₁₇⋊AGL(1,𝔽₇)	1	1	yes	characteristic, direct product factor
61	C₂×C₇	C₃×C₁₇×D₁₇	1	1	yes	characteristic, complemented
62	D₇	—	17	2	no	—
63	C₂²×C₃	—	119	1	no	Hall
64	C₇	C₂×C₃×C₁₇×D₁₇	1	1	yes	characteristic, complemented, Sylow
65	C₂×C₃	—	14	1	no	—
66	C₂×C₃	—	119	2	no	—
67	C₂²	—	119	1	no	Sylow
68	C₃	—	14	1	no	Sylow
69	C₂	—	119	2	no	—
70	C₂	C₁₇×C₁₇⋊AGL(1,𝔽₇)	1	1	yes	characteristic, direct product factor

	subgroup	G/H
center, hypercenter	C₂×C₁₇	C₁₇⋊AGL(1,𝔽₇)
commutator, cosocle, hypocenter	C₇×C₁₇	C₂²×C₃×C₁₇
Frattini	C₁	G
Fitting, socle	C₂×C₇×C₁₇²	C₂×C₃
LCS	G ▷ C₇×C₁₇
UCS	C₁ ◁ C₂×C₁₇

Complements

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

N	H ≅ G/N	type
1. C₂×C₁₇²×ΓL(1,𝔽₈)	C₂	semidirect
2. C₁₇×C₁₇⋊AGL(1,𝔽₇)	C₂	direct (N × H)
3. C₂×C₁₇×D₁₁₉	C₃	semidirect
4. C₁₇²×ΓL(1,𝔽₈)	C₂²	semidirect
5. C₁₇×D₁₁₉	C₂×C₃	semidirect
6. C₂×C₇×C₁₇²	C₂×C₃	semidirect
8. C₇×C₁₇²	C₂²×C₃	semidirect
11. C₂×C₁₇⋊AGL(1,𝔽₇)	C₁₇	direct (N × H)
16. C₂×C₁₇×ΓL(1,𝔽₈)	D₁₇	semidirect
17. C₂×C₁₇×ΓL(1,𝔽₈)	C₂×C₁₇	semidirect
18. C₁₇⋊AGL(1,𝔽₇)	C₂×C₁₇	direct (N × H)
20. C₂×C₁₇²	AGL(1,𝔽₇)	semidirect
22. C₂×D₁₁₉	C₃×C₁₇	semidirect
25. C₁₇×ΓL(1,𝔽₈)	C₂×D₁₇	semidirect
26. C₁₇×ΓL(1,𝔽₈)	C₂²×C₁₇	semidirect
27. C₁₇²	C₂×AGL(1,𝔽₇)	semidirect
29. C₂×C₇×C₁₇	C₃×D₁₇	semidirect
30. C₂×C₇×C₁₇	C₂×C₃×C₁₇	semidirect
31. D₁₁₉	C₂×C₃×C₁₇	semidirect
36. C₇×C₁₇	C₂×C₃×D₁₇	semidirect
37. C₇×C₁₇	C₂²×C₃×C₁₇	semidirect
49. C₂×ΓL(1,𝔽₈)	C₁₇×D₁₇	semidirect
53. C₂×C₁₇	C₁₇×AGL(1,𝔽₇)	semidirect
54. C₂×C₁₇	C₁₇⋊AGL(1,𝔽₇)	direct (N × H)
57. ΓL(1,𝔽₈)	C₂×C₁₇×D₁₇	semidirect
59. C₁₇	C₂×C₁₇×AGL(1,𝔽₇)	semidirect
60. C₁₇	C₂×C₁₇⋊AGL(1,𝔽₇)	direct (N × H)
61. C₂×C₇	C₃×C₁₇×D₁₇	semidirect
64. C₇	C₂×C₃×C₁₇×D₁₇	semidirect
70. C₂	C₁₇×C₁₇⋊AGL(1,𝔽₇)	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₁₇²×ΓL(1,𝔽₈)	2
C₁₇×C₁₇⋊AGL(1,𝔽₇)	2
C₂×C₁₇×D₁₁₉	3
C₂×C₃×C₁₇×D₁₇	7
C₂×C₁₇⋊AGL(1,𝔽₇)	17
C₂×C₁₇×AGL(1,𝔽₇)	17

Subgroups — C2×C17×C17⋊AGL(1,𝔽7) #37566247

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₇×C₁₇⋊AGL(1,𝔽₇) #37566247