Subgroups — C₂×C₅²×ΓL(1,𝔽₈)×D₁₇ #49087853

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×ΓL(1,𝔽₈)×D₁₇	C₂	1	2	yes	maximal, direct product factor
3	C₂×C₅²×C₇×D₁₇	C₃	1	1	yes	maximal, characteristic, complemented, Hall
4	C₅²×C₁₇×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic, complemented, Hall
5	C₂×C₅×ΓL(1,𝔽₈)×D₁₇	C₅	1	6	yes	maximal, direct product factor
6	C₅²×C₇×D₁₇	C₂×C₃	1	2	yes	complemented
7	C₂×C₅²×C₇×C₁₇	C₂×C₃	1	1	yes	characteristic, complemented
8	C₂×C₃×C₅²×D₁₇	—	7	1	no	maximal, Hall
9	C₅×ΓL(1,𝔽₈)×D₁₇	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	C₃×C₅²×D₁₇	—	7	2	no	—
13	C₂×C₃×C₅²×C₁₇	—	7	1	no	—
14	C₂×C₅×C₇×D₁₇	C₃×C₅	1	6	yes	complemented
15	C₂²×C₅²×ΓL(1,𝔽₈)	—	17	1	no	maximal, Hall
16	C₅×C₁₇×ΓL(1,𝔽₈)	C₂²×C₅	1	6	yes	complemented
17	C₂×C₅²×D₁₇	ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Hall
18	C₂×ΓL(1,𝔽₈)×D₁₇	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×C₇×D₁₇	C₂×C₃×C₅	1	12	yes	complemented
22	C₂×C₅²×ΓL(1,𝔽₈)	—	17	2	no	—
23	C₂×C₅²×ΓL(1,𝔽₈)	D₁₇	1	1	yes	characteristic, direct product factor
24	C₂×C₃×C₅×D₁₇	—	7	6	no	—
25	C₂×C₅²×C₁₇	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
26	C₅²×D₁₇	C₂×ΓL(1,𝔽₈)	1	2	yes	direct product factor
27	C₂×C₁₇×ΓL(1,𝔽₈)	C₂×C₅²	1	1	yes	characteristic, complemented
28	ΓL(1,𝔽₈)×D₁₇	C₂×C₅²	1	2	yes	direct product factor
29	C₂²×C₅²×C₇	—	17	1	no	Hall
30	C₅×C₇×C₁₇	C₂²×C₃×C₅	1	6	yes	complemented
31	C₅²×ΓL(1,𝔽₈)	C₂×D₁₇	1	1	yes	characteristic, direct product factor, Hall
32	C₂×C₃×C₅×C₁₇	—	7	6	no	—
33	C₃×C₅×D₁₇	—	7	12	no	—
34	C₂×C₇×D₁₇	C₃×C₅²	1	1	yes	characteristic, complemented, Hall
35	C₅²×C₁₇	C₂²×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Hall
36	C₂²×C₅×ΓL(1,𝔽₈)	—	17	6	no	—
37	C₁₇×ΓL(1,𝔽₈)	C₂²×C₅²	1	1	yes	characteristic, complemented, Hall
38	C₂×C₅²×C₇	—	17	2	no	—
39	C₂×C₅²×C₇	C₃×D₁₇	1	1	yes	characteristic, complemented
40	C₂×C₅×D₁₇	C₅×ΓL(1,𝔽₈)	1	6	yes	direct product factor
41	C₂²×C₃×C₅²	—	119	1	no	Hall
42	C₃×C₅×C₁₇	—	7	6	no	—
43	C₂×C₇×C₁₇	C₂×C₃×C₅²	1	1	yes	characteristic, complemented
44	C₇×D₁₇	C₂×C₃×C₅²	1	2	yes	complemented
45	C₂×C₅×ΓL(1,𝔽₈)	—	17	12	no	—
46	C₂×C₅×ΓL(1,𝔽₈)	C₅×D₁₇	1	6	yes	direct product factor
47	C₂×C₃×D₁₇	—	7	1	no	Hall
48	C₅²×C₇	C₂×C₃×D₁₇	1	1	yes	characteristic, complemented, Hall
49	C₂×C₅×C₁₇	C₂×C₅×ΓL(1,𝔽₈)	1	6	yes	complemented
50	C₅×D₁₇	C₂×C₅×ΓL(1,𝔽₈)	1	12	yes	direct product factor
51	C₂×C₃×C₅²	—	7	1	no	—
52	C₂×C₃×C₅²	—	119	2	no	—
53	C₂²×C₅×C₇	—	17	6	no	—
54	C₇×C₁₇	C₂²×C₃×C₅²	1	1	yes	characteristic, complemented, Hall
55	C₅×ΓL(1,𝔽₈)	C₂×C₅×D₁₇	1	6	yes	direct product factor
56	C₂×C₃×C₁₇	—	7	1	no	—
57	C₃×D₁₇	—	7	2	no	—
58	C₂²×C₅²	—	17	1	no	Hall
59	C₅×C₁₇	C₂²×C₅×ΓL(1,𝔽₈)	1	6	yes	complemented
60	C₂²×ΓL(1,𝔽₈)	—	17	1	no	Hall
61	C₃×C₅²	—	7	1	no	Hall
62	C₂×C₅×C₇	—	17	12	no	—
63	C₂×C₅×C₇	C₃×C₅×D₁₇	1	6	yes	complemented
64	C₂×D₁₇	C₅²×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Hall
65	C₂²×C₃×C₅	—	119	6	no	—
66	C₃×C₁₇	—	7	1	no	Hall
67	C₂×C₅²	—	17	2	no	—
68	C₂×C₅²	ΓL(1,𝔽₈)×D₁₇	1	1	yes	characteristic, direct product factor
69	C₂×ΓL(1,𝔽₈)	—	17	2	no	—
70	C₂×ΓL(1,𝔽₈)	C₅²×D₁₇	1	1	yes	characteristic, direct product factor
71	C₅×C₇	C₂×C₃×C₅×D₁₇	1	6	yes	complemented
72	C₂×C₁₇	C₂×C₅²×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
73	D₁₇	C₂×C₅²×ΓL(1,𝔽₈)	1	2	yes	direct product factor
74	C₂×C₃×C₅	—	7	6	no	—
75	C₂×C₃×C₅	—	119	12	no	—
76	C₂²×C₇	—	17	1	no	Hall
77	C₅²	C₂×ΓL(1,𝔽₈)×D₁₇	1	1	yes	characteristic, direct product factor, Sylow
78	ΓL(1,𝔽₈)	C₂×C₅²×D₁₇	1	1	yes	characteristic, direct product factor, Hall
79	C₂²×C₅	—	17	6	no	—
80	C₁₇	C₂²×C₅²×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Sylow
81	C₃×C₅	—	7	6	no	—
82	C₂×C₇	—	17	2	no	—
83	C₂×C₇	C₃×C₅²×D₁₇	1	1	yes	characteristic, complemented
84	C₂²×C₃	—	119	1	no	Hall
85	C₂×C₅	—	17	12	no	—
86	C₂×C₅	C₅×ΓL(1,𝔽₈)×D₁₇	1	6	yes	direct product factor
87	C₇	C₂×C₃×C₅²×D₁₇	1	1	yes	characteristic, complemented, Sylow
88	C₂×C₃	—	7	1	no	—
89	C₂×C₃	—	119	2	no	—
90	C₅	C₂×C₅×ΓL(1,𝔽₈)×D₁₇	1	6	yes	direct product factor
91	C₂²	—	17	1	no	Sylow
92	C₃	—	7	1	no	Sylow
93	C₂	—	17	2	no	—
94	C₂	C₅²×ΓL(1,𝔽₈)×D₁₇	1	1	yes	characteristic, direct product factor

	subgroup	G/H
center, hypercenter	C₂×C₅²	ΓL(1,𝔽₈)×D₁₇
commutator, cosocle, hypocenter	C₇×C₁₇	C₂²×C₃×C₅²
Frattini	C₁	G
Fitting, socle	C₂×C₅²×C₇×C₁₇	C₂×C₃
LCS	G ▷ C₇×C₁₇
UCS	C₁ ◁ C₂×C₅²

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×ΓL(1,𝔽₈)×D₁₇	C₂	direct (N × H)
3. C₂×C₅²×C₇×D₁₇	C₃	semidirect
4. C₅²×C₁₇×ΓL(1,𝔽₈)	C₂²	semidirect
5. C₂×C₅×ΓL(1,𝔽₈)×D₁₇	C₅	direct (N × H)
6. C₅²×C₇×D₁₇	C₂×C₃	semidirect
7. C₂×C₅²×C₇×C₁₇	C₂×C₃	semidirect
9. C₅×ΓL(1,𝔽₈)×D₁₇	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×C₇×D₁₇	C₃×C₅	semidirect
16. C₅×C₁₇×ΓL(1,𝔽₈)	C₂²×C₅	semidirect
17. C₂×C₅²×D₁₇	ΓL(1,𝔽₈)	direct (N × H)
18. C₂×ΓL(1,𝔽₈)×D₁₇	C₅²	direct (N × H)
20. C₂×C₅×C₇×C₁₇	C₂×C₃×C₅	semidirect
21. C₅×C₇×D₁₇	C₂×C₃×C₅	semidirect
23. C₂×C₅²×ΓL(1,𝔽₈)	D₁₇	direct (N × H)
25. C₂×C₅²×C₁₇	C₂×ΓL(1,𝔽₈)	semidirect
26. C₅²×D₁₇	C₂×ΓL(1,𝔽₈)	direct (N × H)
27. C₂×C₁₇×ΓL(1,𝔽₈)	C₂×C₅²	semidirect
28. ΓL(1,𝔽₈)×D₁₇	C₂×C₅²	direct (N × H)
30. C₅×C₇×C₁₇	C₂²×C₃×C₅	semidirect
31. C₅²×ΓL(1,𝔽₈)	C₂×D₁₇	direct (N × H)
34. C₂×C₇×D₁₇	C₃×C₅²	semidirect
35. C₅²×C₁₇	C₂²×ΓL(1,𝔽₈)	semidirect
37. C₁₇×ΓL(1,𝔽₈)	C₂²×C₅²	semidirect
39. C₂×C₅²×C₇	C₃×D₁₇	semidirect
40. C₂×C₅×D₁₇	C₅×ΓL(1,𝔽₈)	direct (N × H)
43. C₂×C₇×C₁₇	C₂×C₃×C₅²	semidirect
44. C₇×D₁₇	C₂×C₃×C₅²	semidirect
46. C₂×C₅×ΓL(1,𝔽₈)	C₅×D₁₇	direct (N × H)
48. C₅²×C₇	C₂×C₃×D₁₇	semidirect
49. C₂×C₅×C₁₇	C₂×C₅×ΓL(1,𝔽₈)	semidirect
50. C₅×D₁₇	C₂×C₅×ΓL(1,𝔽₈)	direct (N × H)
54. C₇×C₁₇	C₂²×C₃×C₅²	semidirect
55. C₅×ΓL(1,𝔽₈)	C₂×C₅×D₁₇	direct (N × H)
59. C₅×C₁₇	C₂²×C₅×ΓL(1,𝔽₈)	semidirect
63. C₂×C₅×C₇	C₃×C₅×D₁₇	semidirect
64. C₂×D₁₇	C₅²×ΓL(1,𝔽₈)	direct (N × H)
68. C₂×C₅²	ΓL(1,𝔽₈)×D₁₇	direct (N × H)
70. C₂×ΓL(1,𝔽₈)	C₅²×D₁₇	direct (N × H)
71. C₅×C₇	C₂×C₃×C₅×D₁₇	semidirect
72. C₂×C₁₇	C₂×C₅²×ΓL(1,𝔽₈)	semidirect
73. D₁₇	C₂×C₅²×ΓL(1,𝔽₈)	direct (N × H)
77. C₅²	C₂×ΓL(1,𝔽₈)×D₁₇	direct (N × H)
78. ΓL(1,𝔽₈)	C₂×C₅²×D₁₇	direct (N × H)
80. C₁₇	C₂²×C₅²×ΓL(1,𝔽₈)	semidirect
83. C₂×C₇	C₃×C₅²×D₁₇	semidirect
86. C₂×C₅	C₅×ΓL(1,𝔽₈)×D₁₇	direct (N × H)
87. C₇	C₂×C₃×C₅²×D₁₇	semidirect
90. C₅	C₂×C₅×ΓL(1,𝔽₈)×D₁₇	direct (N × H)
94. C₂	C₅²×ΓL(1,𝔽₈)×D₁₇	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×ΓL(1,𝔽₈)×D₁₇	2
C₂×C₅²×C₇×D₁₇	3
C₂×C₅×ΓL(1,𝔽₈)×D₁₇	5
C₂×C₃×C₅²×D₁₇	7
C₂²×C₅²×ΓL(1,𝔽₈)	17

Subgroups — C2×C52×ΓL(1,𝔽8)×D17 #49087853

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅²×ΓL(1,𝔽₈)×D₁₇ #49087853