Finite Group Databasebeta

← C17×(C5⋊AGL(1,𝔽7))⋊D5 #49087805

Subgroups — C17×(C5⋊AGL(1,𝔽7))⋊D5 #49087805

Order 35700.

192 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 192 conjugacy classes of subgroups — too many to draw. Only its 50 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C17×ΓL(1,𝔽8)×C5⋊D5C211yesmaximal, characteristic, complemented
2C5×C17×C5⋊AGL(1,𝔽7)C212yesmaximal, complemented
3C17×(C5⋊D5)⋊D7C311yesmaximal, characteristic, complemented, Hall
4C52×C17×ΓL(1,𝔽8)C2211yescharacteristic, complemented, Hall
5D5×C17×AGL(1,𝔽7)—52nomaximal
6C7×C17×C5⋊D5C2×C311yescharacteristic, complemented
7C5×C17×D35C2×C312yescomplemented
8C3×D52×C17—71nomaximal, Hall
9C17×C5⋊AGL(1,𝔽7)D512yescomplemented
10D5×C17×ΓL(1,𝔽8)—52no—
11D5×C17×ΓL(1,𝔽8)—102no—
12C5×C17×AGL(1,𝔽7)—52no—
13C52×C7×C17C22×C311yescharacteristic, complemented, Hall
14C3×C17×C5⋊D5—71no—
15C3×C5×D5×C17—72no—
16D5×D7×C17—52no—
17(C5⋊AGL(1,𝔽7))⋊D5C1711yesmaximal, characteristic, direct product factor, Hall
18C5×C17×ΓL(1,𝔽8)—22no—
19C5×C17×ΓL(1,𝔽8)C2×D512yescomplemented
20D52×C17—71noHall
21C2×C17×AGL(1,𝔽7)—251noHall
22C3×C52×C17—71noHall
23C17×D35C3×D512yescomplemented
24C7×D5×C17—52no—
25C7×D5×C17—102no—
26C5×D7×C17—52no—
27ΓL(1,𝔽8)×C5⋊D5C2×C1711yescharacteristic, complemented
28C5×C5⋊AGL(1,𝔽7)C2×C1712yescomplemented
29C2×C3×D5×C17—352no—
30C17×C5⋊D5AGL(1,𝔽7)11yescharacteristic, complemented
31C5×D5×C17—72no—
32C2×C17×ΓL(1,𝔽8)—251no—
33C17×AGL(1,𝔽7)—52no—
34(C5⋊D5)⋊D7C3×C1711yescharacteristic, complemented, Hall
35C5×C7×C17—22no—
36C5×C7×C17C2×C3×D512yescomplemented
37C52×ΓL(1,𝔽8)C22×C1711yescharacteristic, complemented, Hall
38C2×C3×C5×C17—352no—
39C3×D5×C17—72no—
40C3×D5×C17—352no—
41C3×D5×C17—702no—
42C2×D7×C17—251noHall
43C52×C17C2×AGL(1,𝔽7)11yescharacteristic, complemented, Hall
44D5×AGL(1,𝔽7)—52no—
45C17×ΓL(1,𝔽8)D5211yescharacteristic, complemented, Hall
46C7×C5⋊D5C2×C3×C1711yescharacteristic, complemented
47C5×D35C2×C3×C1712yescomplemented
48C2×D5×C17—352no—
49C3×D52—71noHall
50C3×C5×C17—72no—
51C3×C5×C17—142no—
52C2×C7×C17—251no—
53D7×C17—52no—
54C5⋊AGL(1,𝔽7)D5×C1712yescomplemented
55D5×ΓL(1,𝔽8)—52no—
56D5×ΓL(1,𝔽8)—102no—
57C5×AGL(1,𝔽7)—52no—
58C22×C3×C17—1751noHall
59C52×C7C22×C3×C1711yescharacteristic, complemented, Hall
60C2×C5×C17—352no—
61D5×C17—52no—
62D5×C17—72no—
63D5×C17—102no—
64C3×C5⋊D5—71no—
65C3×C5×D5—72no—
66D5×D7—52no—
67C7×C17C3×D5211yescharacteristic, complemented, Hall
68C5×ΓL(1,𝔽8)—22no—
69C5×ΓL(1,𝔽8)C2×D5×C1712yescomplemented
70C2×C3×C17—352no—
71C2×C3×C17—1751no—
72D52—71noHall
73C5×C17—22no—
74C5×C17D5×AGL(1,𝔽7)12yescomplemented
75C2×AGL(1,𝔽7)—251noHall
76C3×C52—71noHall
77D35C3×D5×C1712yescomplemented
78C5×D7—52no—
79C7×D5—52no—
80C7×D5—102no—
81C22×C17—1751noHall
82C2×C3×D5—352no—
83C3×C17—71noHall
84C5⋊D5C17×AGL(1,𝔽7)11yescharacteristic, complemented
85C5×D5—72no—
86C2×ΓL(1,𝔽8)—251no—
87AGL(1,𝔽7)—52no—
88C5×C7—22no—
89C5×C7C2×C3×D5×C1712yescomplemented
90C2×C17—251no—
91C2×C17—352no—
92C2×C3×C5—352no—
93C3×D5—72no—
94C3×D5—352no—
95C3×D5—702no—
96C2×D7—251noHall
97C52C2×C17×AGL(1,𝔽7)11yescharacteristic, complemented, Sylow
98ΓL(1,𝔽8)D52×C1711yescharacteristic, complemented, Hall
99C2×D5—352no—
100C17(C5⋊AGL(1,𝔽7))⋊D511yescharacteristic, direct product factor, Sylow
101C3×C5—72no—
102C3×C5—142no—
103C2×C7—251no—
104D7—52no—
105C22×C3—1751noHall
106C2×C5—352no—
107D5—52no—
108D5—72no—
109D5—102no—
110C7C3×D52×C1711yescharacteristic, complemented, Sylow
111C2×C3—352no—
112C2×C3—1751no—
113C5—22no—
114C5D5×C17×AGL(1,𝔽7)12yescomplemented
115C22—1751noSylow
116C3—71noSylow
117C2—251no—
118C2—352no—
subgroupG/H
center, hypercenterC17(C5⋊AGL(1,𝔽7))⋊D5
commutator, cosocle, hypocenterC52×C7C22×C3×C17
FrattiniC1G
Fitting, socleC52×C7×C17C22×C3
LCSG ▷ C52×C7
UCSC1 ◁ C17

Complements

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

NH ≅ G/Ntype
1. C17×ΓL(1,𝔽8)×C5⋊D5C2semidirect
2. C5×C17×C5⋊AGL(1,𝔽7)C2semidirect
3. C17×(C5⋊D5)⋊D7C3semidirect
4. C52×C17×ΓL(1,𝔽8)C22semidirect
6. C7×C17×C5⋊D5C2×C3semidirect
7. C5×C17×D35C2×C3semidirect
9. C17×C5⋊AGL(1,𝔽7)D5semidirect
13. C52×C7×C17C22×C3semidirect
17. (C5⋊AGL(1,𝔽7))⋊D5C17direct (N × H)
19. C5×C17×ΓL(1,𝔽8)C2×D5semidirect
23. C17×D35C3×D5semidirect
27. ΓL(1,𝔽8)×C5⋊D5C2×C17semidirect
28. C5×C5⋊AGL(1,𝔽7)C2×C17semidirect
30. C17×C5⋊D5AGL(1,𝔽7)semidirect
34. (C5⋊D5)⋊D7C3×C17semidirect
36. C5×C7×C17C2×C3×D5semidirect
37. C52×ΓL(1,𝔽8)C22×C17semidirect
43. C52×C17C2×AGL(1,𝔽7)semidirect
45. C17×ΓL(1,𝔽8)D52semidirect
46. C7×C5⋊D5C2×C3×C17semidirect
47. C5×D35C2×C3×C17semidirect
54. C5⋊AGL(1,𝔽7)D5×C17semidirect
59. C52×C7C22×C3×C17semidirect
67. C7×C17C3×D52semidirect
69. C5×ΓL(1,𝔽8)C2×D5×C17semidirect
74. C5×C17D5×AGL(1,𝔽7)semidirect
77. D35C3×D5×C17semidirect
84. C5⋊D5C17×AGL(1,𝔽7)semidirect
89. C5×C7C2×C3×D5×C17semidirect
97. C52C2×C17×AGL(1,𝔽7)semidirect
98. ΓL(1,𝔽8)D52×C17semidirect
100. C17(C5⋊AGL(1,𝔽7))⋊D5direct (N × H)
110. C7C3×D52×C17semidirect
114. C5D5×C17×AGL(1,𝔽7)semidirect

Maximal subgroups

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

subgroupindex
C17×ΓL(1,𝔽8)×C5⋊D52
C5×C17×C5⋊AGL(1,𝔽7)2
C17×(C5⋊D5)⋊D73
D5×C17×AGL(1,𝔽7)5
C3×D52×C177
(C5⋊AGL(1,𝔽7))⋊D517