Finite Group Databasebeta

← AGL(1,𝔽7)×C5⋊D55 #37528542

Subgroups — AGL(1,𝔽7)×C5⋊D55 #37528542

Order 23100.

320 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 320 conjugacy classes of subgroups — too many to draw. Only its 87 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
1ΓL(1,𝔽8)×C5⋊D55C211yesmaximal, characteristic, complemented
2C5⋊(C5⋊(C11⋊AGL(1,𝔽7)))C211yesmaximal, characteristic, complemented
3C52×C11×AGL(1,𝔽7)C211yesmaximal, characteristic, complemented
4D7×C5⋊D55C311yesmaximal, characteristic, complemented, Hall
5C52×C11×ΓL(1,𝔽8)C2211yescharacteristic, complemented, Hall
6AGL(1,𝔽7)×D55—56nomaximal
7C5⋊D385C2×C311yescharacteristic, complemented
8C7×C5⋊D55C2×C311yescharacteristic, complemented
9C52×C11×D7C2×C311yescharacteristic, complemented
10C2×C3×C5⋊D55—71nomaximal, Hall
11C5⋊(C11⋊AGL(1,𝔽7))—56no—
12ΓL(1,𝔽8)×D55—56no—
13C5×C11×AGL(1,𝔽7)D516yescomplemented
14AGL(1,𝔽7)×C5⋊D5—111nomaximal, Hall
15C52×C7×C11C22×C311yescharacteristic, complemented, Hall
16C2×C3×C52×C11—71no—
17C3×C5⋊D55—71no—
18C3×C5⋊D55—141no—
19D7×D55—56no—
20C5×C11×ΓL(1,𝔽8)C2×D516yescomplemented
21C2×C5⋊D55—71noHall
22C5⋊(C5⋊AGL(1,𝔽7))—111no—
23ΓL(1,𝔽8)×C5⋊D5—111no—
24C52×AGL(1,𝔽7)D1111yescharacteristic, complemented
25D11×AGL(1,𝔽7)—251noHall
26C3×C52×C11—141noHall
27D385—56no—
28C7×D55—56no—
29C5×C11×D7C3×D516yescomplemented
30D7×C5⋊D5—111noHall
31C2×C3×D55—356no—
32C2×C52×C11—71no—
33C5⋊D55—71no—
34C5⋊D55AGL(1,𝔽7)11yescharacteristic, direct product factor
35C52×ΓL(1,𝔽8)C2×D1111yescharacteristic, complemented, Hall
36C11⋊AGL(1,𝔽7)—251no—
37C11×AGL(1,𝔽7)C5⋊D511yescharacteristic, complemented
38ΓL(1,𝔽8)×D11—251no—
39D5×AGL(1,𝔽7)—556no—
40C5×C7×C11C2×C3×D516yescomplemented
41C5⋊D35—111no—
42C7×C5⋊D5—111no—
43C52×D7C3×D1111yescharacteristic, complemented
44C2×C3×C5×C11—76no—
45C3×D55—356no—
46C3×D55—706no—
47D7×D11—251noHall
48C2×C3×C5⋊D5—771noHall
49C52×C11C2×AGL(1,𝔽7)11yescharacteristic, complemented, Hall
50C11×ΓL(1,𝔽8)C2×C5⋊D511yescharacteristic, complemented, Hall
51C2×D55—356no—
52C5⋊AGL(1,𝔽7)—556no—
53D5×ΓL(1,𝔽8)—556no—
54C5×AGL(1,𝔽7)D5516yescomplemented
55C52×C7C2×C3×D1111yescharacteristic, complemented, Hall
56C3×C5×C11—146no—
57D77—251no—
58C7×D11—251no—
59C11×D7C3×C5⋊D511yescharacteristic, complemented
60C2×C3×C52—71no—
61C3×C5⋊D5—771no—
62C3×C5⋊D5—1541no—
63D5×D7—556no—
64C2×C3×D11—1751noHall
65C2×C5×C11—76no—
66D55—56no—
67D55—356no—
68C5×ΓL(1,𝔽8)C2×D5516yescomplemented
69C2×C5⋊D5—771noHall
70C2×AGL(1,𝔽7)—2751noHall
71C7×C11C2×C3×C5⋊D511yescharacteristic, complemented, Hall
72C3×C52—141noHall
73D35—556no—
74C5×D7C3×D5516yescomplemented
75C7×D5—556no—
76C2×C3×C11—71no—
77C3×D11—1751no—
78C3×D11—1751no—
79C2×C3×D5—3856no—
80C5×C11D5×AGL(1,𝔽7)16yescomplemented
81C2×C52—71no—
82C5⋊D5—111no—
83C5⋊D5—771no—
84C2×D11—1751noHall
85C2×ΓL(1,𝔽8)—2751no—
86AGL(1,𝔽7)—2751no—
87AGL(1,𝔽7)C5⋊D5511yescharacteristic, direct product factor
88C5×C7C2×C3×D5516yescomplemented
89C3×C11—71noHall
90C2×C3×C5—76no—
91C3×D5—7706no—
92C3×D5—3856no—
93C2×D7—2751noHall
94C52D11×AGL(1,𝔽7)11yescharacteristic, complemented, Sylow
95C2×C11—71no—
96D11—251no—
97D11—1751no—
98ΓL(1,𝔽8)C2×C5⋊D5511yescharacteristic, complemented, Hall
99C2×D5—3856no—
100C3×C5—146no—
101C2×C7—2751no—
102D7—2751no—
103D7C3×C5⋊D5511yescharacteristic, complemented
104C22×C3—19251noHall
105C11AGL(1,𝔽7)×C5⋊D511yescharacteristic, complemented, Sylow
106C2×C5—76no—
107D5—556no—
108D5—3856no—
109C7C2×C3×C5⋊D5511yescharacteristic, complemented, Sylow
110C2×C3—71no—
111C2×C3—38501no—
112C2×C3—19251no—
113C5AGL(1,𝔽7)×D5516yescomplemented
114C22—19251noSylow
115C3—141noSylow
116C2—71no—
117C2—2751no—
118C2—19251no—
subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC52×C7×C11C22×C3
LCSG ▷ C52×C7×C11
UCSC1

Complements

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

NH ≅ G/Ntype
1. ΓL(1,𝔽8)×C5⋊D55C2semidirect
2. C5⋊(C5⋊(C11⋊AGL(1,𝔽7)))C2semidirect
3. C52×C11×AGL(1,𝔽7)C2semidirect
4. D7×C5⋊D55C3semidirect
5. C52×C11×ΓL(1,𝔽8)C22semidirect
7. C5⋊D385C2×C3semidirect
8. C7×C5⋊D55C2×C3semidirect
9. C52×C11×D7C2×C3semidirect
13. C5×C11×AGL(1,𝔽7)D5semidirect
15. C52×C7×C11C22×C3semidirect
20. C5×C11×ΓL(1,𝔽8)C2×D5semidirect
24. C52×AGL(1,𝔽7)D11semidirect
29. C5×C11×D7C3×D5semidirect
34. C5⋊D55AGL(1,𝔽7)direct (N × H)
35. C52×ΓL(1,𝔽8)C2×D11semidirect
37. C11×AGL(1,𝔽7)C5⋊D5semidirect
40. C5×C7×C11C2×C3×D5semidirect
43. C52×D7C3×D11semidirect
49. C52×C11C2×AGL(1,𝔽7)semidirect
50. C11×ΓL(1,𝔽8)C2×C5⋊D5semidirect
54. C5×AGL(1,𝔽7)D55semidirect
55. C52×C7C2×C3×D11semidirect
59. C11×D7C3×C5⋊D5semidirect
68. C5×ΓL(1,𝔽8)C2×D55semidirect
71. C7×C11C2×C3×C5⋊D5semidirect
74. C5×D7C3×D55semidirect
80. C5×C11D5×AGL(1,𝔽7)semidirect
87. AGL(1,𝔽7)C5⋊D55direct (N × H)
88. C5×C7C2×C3×D55semidirect
94. C52D11×AGL(1,𝔽7)semidirect
98. ΓL(1,𝔽8)C2×C5⋊D55semidirect
103. D7C3×C5⋊D55semidirect
105. C11AGL(1,𝔽7)×C5⋊D5semidirect
109. C7C2×C3×C5⋊D55semidirect
113. C5AGL(1,𝔽7)×D55semidirect

Maximal subgroups

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

subgroupindex
ΓL(1,𝔽8)×C5⋊D552
C5⋊(C5⋊(C11⋊AGL(1,𝔽7)))2
C52×C11×AGL(1,𝔽7)2
D7×C5⋊D553
AGL(1,𝔽7)×D555
C2×C3×C5⋊D557
AGL(1,𝔽7)×C5⋊D511