Subgroups — C₂×C₁₇⋊(C₁₇⋊AGL(1,𝔽₇)) #37566248

Order 24276.

400 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×C₇⋊(C₁₇⋊D₁₇)	C₃	1	1	yes	maximal, characteristic, complemented, Hall
4	C₁₇²×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic, complemented, Hall
5	C₇⋊(C₁₇⋊D₁₇)	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)	—	17	2	no	maximal
12	C₂×C₁₇⋊AGL(1,𝔽₇)	—	17	16	no	maximal
13	C₂×C₁₇⋊D₁₇	—	7	1	no	Hall
14	C₃×C₁₇²	—	14	1	no	Hall
15	C₂×C₁₇×ΓL(1,𝔽₈)	D₁₇	1	2	yes	complemented
16	C₂×C₁₇×ΓL(1,𝔽₈)	D₁₇	1	16	yes	complemented
17	C₁₇⋊AGL(1,𝔽₇)	—	17	4	no	—
18	C₁₇⋊AGL(1,𝔽₇)	—	17	32	no	—
19	C₂×C₁₇²	AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
20	C₁₇⋊D₁₇	—	7	2	no	—
21	C₂×D₁₁₉	—	17	2	no	—
22	C₂×D₁₁₉	—	17	16	no	—
23	C₁₇×ΓL(1,𝔽₈)	C₂×D₁₇	1	2	yes	complemented
24	C₁₇×ΓL(1,𝔽₈)	C₂×D₁₇	1	16	yes	complemented
25	C₁₇²	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic, complemented, Sylow
26	C₂×C₇×C₁₇	C₃×D₁₇	1	2	yes	complemented
27	C₂×C₇×C₁₇	C₃×D₁₇	1	16	yes	complemented
28	D₁₁₉	—	17	4	no	—
29	D₁₁₉	—	17	32	no	—
30	C₂×C₃×D₁₇	—	119	2	no	—
31	C₂×C₃×D₁₇	—	119	16	no	—
32	C₇×C₁₇	C₂×C₃×D₁₇	1	2	yes	complemented
33	C₇×C₁₇	C₂×C₃×D₁₇	1	16	yes	complemented
34	C₂×C₃×C₁₇	—	7	2	no	—
35	C₂×C₃×C₁₇	—	14	16	no	—
36	C₃×D₁₇	—	119	4	no	—
37	C₃×D₁₇	—	119	32	no	—
38	C₂×AGL(1,𝔽₇)	—	289	1	no	Hall
39	C₂×D₁₇	—	119	2	no	—
40	C₂×D₁₇	—	119	16	no	—
41	C₃×C₁₇	—	7	2	no	—
42	C₃×C₁₇	—	14	16	no	—
43	C₂×ΓL(1,𝔽₈)	C₁₇⋊D₁₇	1	1	yes	characteristic, complemented
44	AGL(1,𝔽₇)	—	289	2	no	—
45	C₂×C₁₇	C₁₇⋊AGL(1,𝔽₇)	1	2	yes	complemented
46	C₂×C₁₇	C₁₇⋊AGL(1,𝔽₇)	1	16	yes	complemented
47	D₁₇	—	119	4	no	—
48	D₁₇	—	119	32	no	—
49	C₂×D₇	—	289	1	no	Hall
50	ΓL(1,𝔽₈)	C₂×C₁₇⋊D₁₇	1	1	yes	characteristic, complemented, Hall
51	C₁₇	C₂×C₁₇⋊AGL(1,𝔽₇)	1	2	yes	complemented
52	C₁₇	C₂×C₁₇⋊AGL(1,𝔽₇)	1	16	yes	complemented
53	C₂×C₇	C₃×C₁₇⋊D₁₇	1	1	yes	characteristic, complemented
54	D₇	—	289	2	no	—
55	C₂²×C₃	—	2023	1	no	Hall
56	C₇	C₂×C₃×C₁₇⋊D₁₇	1	1	yes	characteristic, complemented, Sylow
57	C₂×C₃	—	14	1	no	—
58	C₂×C₃	—	2023	2	no	—
59	C₂²	—	2023	1	no	Sylow
60	C₃	—	14	1	no	Sylow
61	C₂	—	2023	2	no	—
62	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×C₁₇²	C₂²×C₃
Frattini	C₁	G
Fitting, socle	C₂×C₇×C₁₇²	C₂×C₃
LCS	G ▷ C₇×C₁₇²
UCS	C₁ ◁ C₂

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×C₇⋊(C₁₇⋊D₁₇)	C₃	semidirect
4. C₁₇²×ΓL(1,𝔽₈)	C₂²	semidirect
5. C₇⋊(C₁₇⋊D₁₇)	C₂×C₃	semidirect
6. C₂×C₇×C₁₇²	C₂×C₃	semidirect
8. C₇×C₁₇²	C₂²×C₃	semidirect
15. C₂×C₁₇×ΓL(1,𝔽₈)	D₁₇	semidirect
16. C₂×C₁₇×ΓL(1,𝔽₈)	D₁₇	semidirect
19. C₂×C₁₇²	AGL(1,𝔽₇)	semidirect
23. C₁₇×ΓL(1,𝔽₈)	C₂×D₁₇	semidirect
24. C₁₇×ΓL(1,𝔽₈)	C₂×D₁₇	semidirect
25. C₁₇²	C₂×AGL(1,𝔽₇)	semidirect
26. C₂×C₇×C₁₇	C₃×D₁₇	semidirect
27. C₂×C₇×C₁₇	C₃×D₁₇	semidirect
32. C₇×C₁₇	C₂×C₃×D₁₇	semidirect
33. C₇×C₁₇	C₂×C₃×D₁₇	semidirect
43. C₂×ΓL(1,𝔽₈)	C₁₇⋊D₁₇	semidirect
45. C₂×C₁₇	C₁₇⋊AGL(1,𝔽₇)	semidirect
46. C₂×C₁₇	C₁₇⋊AGL(1,𝔽₇)	semidirect
50. ΓL(1,𝔽₈)	C₂×C₁₇⋊D₁₇	semidirect
51. C₁₇	C₂×C₁₇⋊AGL(1,𝔽₇)	semidirect
52. C₁₇	C₂×C₁₇⋊AGL(1,𝔽₇)	semidirect
53. C₂×C₇	C₃×C₁₇⋊D₁₇	semidirect
56. C₇	C₂×C₃×C₁₇⋊D₁₇	semidirect
62. 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×C₇⋊(C₁₇⋊D₁₇)	3
C₂×C₃×C₁₇⋊D₁₇	7
C₂×C₁₇⋊AGL(1,𝔽₇)	17

Subgroups — C2×C17⋊(C17⋊AGL(1,𝔽7)) #37566248

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₇⋊(C₁₇⋊AGL(1,𝔽₇)) #37566248