Subgroups — C₂×C₅²×C₁₇×AGL(1,𝔽₇) #49087851

Order 35700.

320 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅²×C₁₇×ΓL(1,𝔽₈)	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅²×C₁₇×AGL(1,𝔽₇)	C₂	1	2	yes	maximal, direct product factor
3	C₂×C₅²×D₇×C₁₇	C₃	1	1	yes	maximal, characteristic, complemented, Hall
4	C₅²×C₁₇×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic, complemented, Hall
5	C₂×C₅×C₁₇×AGL(1,𝔽₇)	C₅	1	6	yes	maximal, direct product factor
6	C₅²×D₇×C₁₇	C₂×C₃	1	2	yes	complemented
7	C₂×C₅²×C₇×C₁₇	C₂×C₃	1	1	yes	characteristic, complemented
8	C₂²×C₃×C₅²×C₁₇	—	7	1	no	maximal, Hall
9	C₅×C₁₇×AGL(1,𝔽₇)	C₂×C₅	1	12	yes	direct product factor
10	C₂×C₅×C₁₇×ΓL(1,𝔽₈)	C₂×C₅	1	6	yes	complemented
11	C₅²×C₇×C₁₇	C₂²×C₃	1	1	yes	characteristic, complemented, Hall
12	C₂×C₃×C₅²×C₁₇	—	7	2	no	—
13	C₂×C₃×C₅²×C₁₇	—	7	1	no	—
14	C₂×C₅×D₇×C₁₇	C₃×C₅	1	6	yes	complemented
15	C₂×C₅²×AGL(1,𝔽₇)	C₁₇	1	1	yes	maximal, characteristic, direct product factor, Hall
16	C₅×C₁₇×ΓL(1,𝔽₈)	C₂²×C₅	1	6	yes	complemented
17	C₂²×C₅²×C₁₇	—	7	1	no	Hall
18	C₂×C₁₇×AGL(1,𝔽₇)	C₅²	1	1	yes	characteristic, direct product factor, Hall
19	C₃×C₅²×C₁₇	—	7	1	no	Hall
20	C₂×C₅×C₇×C₁₇	C₂×C₃×C₅	1	6	yes	complemented
21	C₅×D₇×C₁₇	C₂×C₃×C₅	1	12	yes	complemented
22	C₂×C₅²×ΓL(1,𝔽₈)	C₂×C₁₇	1	1	yes	characteristic, complemented
23	C₅²×AGL(1,𝔽₇)	C₂×C₁₇	1	2	yes	direct product factor
24	C₂²×C₃×C₅×C₁₇	—	7	6	no	—
25	C₂×C₅²×C₁₇	—	7	2	no	—
26	C₂×C₅²×C₁₇	AGL(1,𝔽₇)	1	1	yes	characteristic, direct product factor
27	C₂×C₁₇×ΓL(1,𝔽₈)	C₂×C₅²	1	1	yes	characteristic, complemented
28	C₁₇×AGL(1,𝔽₇)	C₂×C₅²	1	2	yes	direct product factor
29	C₂×C₅²×D₇	C₃×C₁₇	1	1	yes	characteristic, complemented, Hall
30	C₅×C₇×C₁₇	C₂²×C₃×C₅	1	6	yes	complemented
31	C₅²×ΓL(1,𝔽₈)	C₂²×C₁₇	1	1	yes	characteristic, complemented, Hall
32	C₂×C₃×C₅×C₁₇	—	7	12	no	—
33	C₂×C₃×C₅×C₁₇	—	7	6	no	—
34	C₂×D₇×C₁₇	C₃×C₅²	1	1	yes	characteristic, complemented, Hall
35	C₅²×C₁₇	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic, direct product factor, Hall
36	C₂×C₅×AGL(1,𝔽₇)	C₅×C₁₇	1	6	yes	direct product factor
37	C₁₇×ΓL(1,𝔽₈)	C₂²×C₅²	1	1	yes	characteristic, complemented, Hall
38	C₂×C₅²×C₇	C₂×C₃×C₁₇	1	1	yes	characteristic, complemented
39	C₅²×D₇	C₂×C₃×C₁₇	1	2	yes	complemented
40	C₂²×C₅×C₁₇	—	7	6	no	—
41	C₂²×C₃×C₅²	—	7	1	no	Hall
42	C₃×C₅×C₁₇	—	7	6	no	—
43	C₂×C₇×C₁₇	C₂×C₃×C₅²	1	1	yes	characteristic, complemented
44	D₇×C₁₇	C₂×C₃×C₅²	1	2	yes	complemented
45	C₂×C₅×ΓL(1,𝔽₈)	C₂×C₅×C₁₇	1	6	yes	complemented
46	C₅×AGL(1,𝔽₇)	C₂×C₅×C₁₇	1	12	yes	direct product factor
47	C₂²×C₃×C₁₇	—	7	1	no	Hall
48	C₅²×C₇	C₂²×C₃×C₁₇	1	1	yes	characteristic, complemented, Hall
49	C₂×C₅×C₁₇	—	7	12	no	—
50	C₂×C₅×C₁₇	C₅×AGL(1,𝔽₇)	1	6	yes	direct product factor
51	C₂×C₃×C₅²	—	7	2	no	—
52	C₂×C₃×C₅²	—	7	1	no	—
53	C₂×C₅×D₇	C₃×C₅×C₁₇	1	6	yes	complemented
54	C₇×C₁₇	C₂²×C₃×C₅²	1	1	yes	characteristic, complemented, Hall
55	C₅×ΓL(1,𝔽₈)	C₂²×C₅×C₁₇	1	6	yes	complemented
56	C₂×C₃×C₁₇	—	7	2	no	—
57	C₂×C₃×C₁₇	—	7	1	no	—
58	C₂²×C₅²	—	7	1	no	Hall
59	C₅×C₁₇	C₂×C₅×AGL(1,𝔽₇)	1	6	yes	direct product factor
60	C₂×AGL(1,𝔽₇)	C₅²×C₁₇	1	1	yes	characteristic, direct product factor, Hall
61	C₃×C₅²	—	7	1	no	Hall
62	C₂×C₅×C₇	C₂×C₃×C₅×C₁₇	1	6	yes	complemented
63	C₅×D₇	C₂×C₃×C₅×C₁₇	1	12	yes	complemented
64	C₂²×C₁₇	—	7	1	no	Hall
65	C₂²×C₃×C₅	—	7	6	no	—
66	C₃×C₁₇	—	7	1	no	Hall
67	C₂×C₅²	—	7	2	no	—
68	C₂×C₅²	C₁₇×AGL(1,𝔽₇)	1	1	yes	characteristic, direct product factor
69	C₂×ΓL(1,𝔽₈)	C₂×C₅²×C₁₇	1	1	yes	characteristic, complemented
70	AGL(1,𝔽₇)	C₂×C₅²×C₁₇	1	2	yes	direct product factor
71	C₅×C₇	C₂²×C₃×C₅×C₁₇	1	6	yes	complemented
72	C₂×C₁₇	—	7	2	no	—
73	C₂×C₁₇	C₅²×AGL(1,𝔽₇)	1	1	yes	characteristic, direct product factor
74	C₂×C₃×C₅	—	7	12	no	—
75	C₂×C₃×C₅	—	7	6	no	—
76	C₂×D₇	C₃×C₅²×C₁₇	1	1	yes	characteristic, complemented, Hall
77	C₅²	C₂×C₁₇×AGL(1,𝔽₇)	1	1	yes	characteristic, direct product factor, Sylow
78	ΓL(1,𝔽₈)	C₂²×C₅²×C₁₇	1	1	yes	characteristic, complemented, Hall
79	C₂²×C₅	—	7	6	no	—
80	C₁₇	C₂×C₅²×AGL(1,𝔽₇)	1	1	yes	characteristic, direct product factor, Sylow
81	C₃×C₅	—	7	6	no	—
82	C₂×C₇	C₂×C₃×C₅²×C₁₇	1	1	yes	characteristic, complemented
83	D₇	C₂×C₃×C₅²×C₁₇	1	2	yes	complemented
84	C₂²×C₃	—	7	1	no	Hall
85	C₂×C₅	—	7	12	no	—
86	C₂×C₅	C₅×C₁₇×AGL(1,𝔽₇)	1	6	yes	direct product factor
87	C₇	C₂²×C₃×C₅²×C₁₇	1	1	yes	characteristic, complemented, Sylow
88	C₂×C₃	—	7	2	no	—
89	C₂×C₃	—	7	1	no	—
90	C₅	C₂×C₅×C₁₇×AGL(1,𝔽₇)	1	6	yes	direct product factor
91	C₂²	—	7	1	no	Sylow
92	C₃	—	7	1	no	Sylow
93	C₂	—	7	2	no	—
94	C₂	C₅²×C₁₇×AGL(1,𝔽₇)	1	1	yes	characteristic, direct product factor

	subgroup	G/H
center, hypercenter	C₂×C₅²×C₁₇	AGL(1,𝔽₇)
commutator, cosocle, hypocenter	C₇	C₂²×C₃×C₅²×C₁₇
Frattini	C₁	G
Fitting, socle	C₂×C₅²×C₇×C₁₇	C₂×C₃
LCS	G ▷ C₇
UCS	C₁ ◁ C₂×C₅²×C₁₇

Complements

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

N	H ≅ G/N	type
1. C₂×C₅²×C₁₇×ΓL(1,𝔽₈)	C₂	semidirect
2. C₅²×C₁₇×AGL(1,𝔽₇)	C₂	direct (N × H)
3. C₂×C₅²×D₇×C₁₇	C₃	semidirect
4. C₅²×C₁₇×ΓL(1,𝔽₈)	C₂²	semidirect
5. C₂×C₅×C₁₇×AGL(1,𝔽₇)	C₅	direct (N × H)
6. C₅²×D₇×C₁₇	C₂×C₃	semidirect
7. C₂×C₅²×C₇×C₁₇	C₂×C₃	semidirect
9. C₅×C₁₇×AGL(1,𝔽₇)	C₂×C₅	direct (N × H)
10. C₂×C₅×C₁₇×ΓL(1,𝔽₈)	C₂×C₅	semidirect
11. C₅²×C₇×C₁₇	C₂²×C₃	semidirect
14. C₂×C₅×D₇×C₁₇	C₃×C₅	semidirect
15. C₂×C₅²×AGL(1,𝔽₇)	C₁₇	direct (N × H)
16. C₅×C₁₇×ΓL(1,𝔽₈)	C₂²×C₅	semidirect
18. C₂×C₁₇×AGL(1,𝔽₇)	C₅²	direct (N × H)
20. C₂×C₅×C₇×C₁₇	C₂×C₃×C₅	semidirect
21. C₅×D₇×C₁₇	C₂×C₃×C₅	semidirect
22. C₂×C₅²×ΓL(1,𝔽₈)	C₂×C₁₇	semidirect
23. C₅²×AGL(1,𝔽₇)	C₂×C₁₇	direct (N × H)
26. C₂×C₅²×C₁₇	AGL(1,𝔽₇)	direct (N × H)
27. C₂×C₁₇×ΓL(1,𝔽₈)	C₂×C₅²	semidirect
28. C₁₇×AGL(1,𝔽₇)	C₂×C₅²	direct (N × H)
29. C₂×C₅²×D₇	C₃×C₁₇	semidirect
30. C₅×C₇×C₁₇	C₂²×C₃×C₅	semidirect
31. C₅²×ΓL(1,𝔽₈)	C₂²×C₁₇	semidirect
34. C₂×D₇×C₁₇	C₃×C₅²	semidirect
35. C₅²×C₁₇	C₂×AGL(1,𝔽₇)	direct (N × H)
36. C₂×C₅×AGL(1,𝔽₇)	C₅×C₁₇	direct (N × H)
37. C₁₇×ΓL(1,𝔽₈)	C₂²×C₅²	semidirect
38. C₂×C₅²×C₇	C₂×C₃×C₁₇	semidirect
39. C₅²×D₇	C₂×C₃×C₁₇	semidirect
43. C₂×C₇×C₁₇	C₂×C₃×C₅²	semidirect
44. D₇×C₁₇	C₂×C₃×C₅²	semidirect
45. C₂×C₅×ΓL(1,𝔽₈)	C₂×C₅×C₁₇	semidirect
46. C₅×AGL(1,𝔽₇)	C₂×C₅×C₁₇	direct (N × H)
48. C₅²×C₇	C₂²×C₃×C₁₇	semidirect
50. C₂×C₅×C₁₇	C₅×AGL(1,𝔽₇)	direct (N × H)
53. C₂×C₅×D₇	C₃×C₅×C₁₇	semidirect
54. C₇×C₁₇	C₂²×C₃×C₅²	semidirect
55. C₅×ΓL(1,𝔽₈)	C₂²×C₅×C₁₇	semidirect
59. C₅×C₁₇	C₂×C₅×AGL(1,𝔽₇)	direct (N × H)
60. C₂×AGL(1,𝔽₇)	C₅²×C₁₇	direct (N × H)
62. C₂×C₅×C₇	C₂×C₃×C₅×C₁₇	semidirect
63. C₅×D₇	C₂×C₃×C₅×C₁₇	semidirect
68. C₂×C₅²	C₁₇×AGL(1,𝔽₇)	direct (N × H)
69. C₂×ΓL(1,𝔽₈)	C₂×C₅²×C₁₇	semidirect
70. AGL(1,𝔽₇)	C₂×C₅²×C₁₇	direct (N × H)
71. C₅×C₇	C₂²×C₃×C₅×C₁₇	semidirect
73. C₂×C₁₇	C₅²×AGL(1,𝔽₇)	direct (N × H)
76. C₂×D₇	C₃×C₅²×C₁₇	semidirect
77. C₅²	C₂×C₁₇×AGL(1,𝔽₇)	direct (N × H)
78. ΓL(1,𝔽₈)	C₂²×C₅²×C₁₇	semidirect
80. C₁₇	C₂×C₅²×AGL(1,𝔽₇)	direct (N × H)
82. C₂×C₇	C₂×C₃×C₅²×C₁₇	semidirect
83. D₇	C₂×C₃×C₅²×C₁₇	semidirect
86. C₂×C₅	C₅×C₁₇×AGL(1,𝔽₇)	direct (N × H)
87. C₇	C₂²×C₃×C₅²×C₁₇	semidirect
90. C₅	C₂×C₅×C₁₇×AGL(1,𝔽₇)	direct (N × H)
94. C₂	C₅²×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×C₅²×C₁₇×ΓL(1,𝔽₈)	2
C₅²×C₁₇×AGL(1,𝔽₇)	2
C₂×C₅²×D₇×C₁₇	3
C₂×C₅×C₁₇×AGL(1,𝔽₇)	5
C₂²×C₃×C₅²×C₁₇	7
C₂×C₅²×AGL(1,𝔽₇)	17

Subgroups — C2×C52×C17×AGL(1,𝔽7) #49087851

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅²×C₁₇×AGL(1,𝔽₇) #49087851