Finite Group Databasebeta

← ΓL(1,𝔽8)×D11×C5⋊D5 #37528545

Subgroups — ΓL(1,𝔽8)×D11×C5⋊D5 #37528545

Order 23100.

320 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 320 conjugacy classes of subgroups — too many to draw. Only its 84 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
2C11×ΓL(1,𝔽8)×C5⋊D5C211yesmaximal, characteristic, complemented
3C52×ΓL(1,𝔽8)×D11C211yesmaximal, characteristic, complemented
4C7×D11×C5⋊D5C311yesmaximal, characteristic, complemented, Hall
5C52×C11×ΓL(1,𝔽8)C2211yescharacteristic, complemented, Hall
6D5×ΓL(1,𝔽8)×D11—56nomaximal
7C7×C5⋊D55C2×C311yescharacteristic, complemented
8C7×C11×C5⋊D5C2×C311yescharacteristic, complemented
9C52×C7×D11C2×C311yescharacteristic, complemented
10C3×D11×C5⋊D5—71nomaximal, Hall
11ΓL(1,𝔽8)×D55—56no—
12D5×C11×ΓL(1,𝔽8)—56no—
13C5×ΓL(1,𝔽8)×D11D516yescomplemented
14C2×ΓL(1,𝔽8)×C5⋊D5—111nomaximal, Hall
15C52×C7×C11C22×C311yescharacteristic, complemented, Hall
16C3×C5⋊D55—71no—
17C3×C11×C5⋊D5—71no—
18C3×C52×D11—71no—
19C7×D5×D11—56no—
20C5×C11×ΓL(1,𝔽8)C2×D516yescomplemented
21D11×C5⋊D5ΓL(1,𝔽8)11yescharacteristic, direct product factor, Hall
22C2×C52×ΓL(1,𝔽8)—111no—
23ΓL(1,𝔽8)×C5⋊D5—111no—
24ΓL(1,𝔽8)×C5⋊D5D1111yescharacteristic, direct product factor
25C2×ΓL(1,𝔽8)×D11—251noHall
26C3×C52×C11—71noHall
27C7×D55—56no—
28C7×D5×C11—56no—
29C5×C7×D11C3×D516yescomplemented
30C2×C7×C5⋊D5—111noHall
31C3×D5×D11—356no—
32C5⋊D55C2×ΓL(1,𝔽8)11yescharacteristic, complemented
33C11×C5⋊D5C2×ΓL(1,𝔽8)11yescharacteristic, complemented
34C52×D11C2×ΓL(1,𝔽8)11yescharacteristic, complemented
35C52×ΓL(1,𝔽8)C2×D1111yescharacteristic, complemented, Hall
36C2×C11×ΓL(1,𝔽8)—251no—
37ΓL(1,𝔽8)×D11—251no—
38ΓL(1,𝔽8)×D11C5⋊D511yescharacteristic, direct product factor
39C2×D5×ΓL(1,𝔽8)—556no—
40C5×C7×C11C2×C3×D516yescomplemented
41C2×C52×C7—111no—
42C7×C5⋊D5—111no—
43C7×C5⋊D5C3×D1111yescharacteristic, complemented
44C3×D55—356no—
45C3×D5×C11—356no—
46C3×C5×D11—76no—
47C2×C7×D11—251noHall
48C2×C3×C5⋊D5—771noHall
49C52×C11C22×ΓL(1,𝔽8)11yescharacteristic, complemented, Hall
50C11×ΓL(1,𝔽8)C2×C5⋊D511yescharacteristic, complemented, Hall
51D5×D11—56no—
52C2×C5×ΓL(1,𝔽8)—116no—
53D5×ΓL(1,𝔽8)—56no—
54D5×ΓL(1,𝔽8)—556no—
55C52×C7C2×C3×D1111yescharacteristic, complemented, Hall
56C3×C5×C11—76no—
57C2×C7×C11—251no—
58C7×D11—251no—
59C7×D11C3×C5⋊D511yescharacteristic, complemented
60C2×C3×C52—771no—
61C3×C5⋊D5—71no—
62C3×C5⋊D5—771no—
63C2×C7×D5—556no—
64C2×C3×D11—1751noHall
65D55—56no—
66C5×D11D5×ΓL(1,𝔽8)16yescomplemented
67D5×C11—56no—
68C5×ΓL(1,𝔽8)D5×D1116yescomplemented
69C2×C5⋊D5—111noHall
70C22×ΓL(1,𝔽8)—2751noHall
71C7×C11C2×C3×C5⋊D511yescharacteristic, complemented, Hall
72C3×C52—71noHall
73C2×C5×C7—116no—
74C7×D5—56no—
75C7×D5—556no—
76C2×C3×C11—1751no—
77C3×D11—71no—
78C3×D11—1751no—
79C2×C3×D5—3856no—
80C5×C11C2×D5×ΓL(1,𝔽8)16yescomplemented
81C2×C52—111no—
82C5⋊D5—111no—
83C5⋊D5ΓL(1,𝔽8)×D1111yescharacteristic, direct product factor
84C2×D11—251noHall
85C2×ΓL(1,𝔽8)—111no—
86C2×ΓL(1,𝔽8)—2751no—
87C2×ΓL(1,𝔽8)—251no—
88C5×C7C3×D5×D1116yescomplemented
89C3×C11—71noHall
90C2×C3×C5—776no—
91C3×D5—356no—
92C3×D5—3856no—
93C22×C7—2751noHall
94C52C2×ΓL(1,𝔽8)×D1111yescharacteristic, complemented, Sylow
95C2×C11—251no—
96D11—251no—
97D11ΓL(1,𝔽8)×C5⋊D511yescharacteristic, direct product factor
98ΓL(1,𝔽8)D11×C5⋊D511yescharacteristic, direct product factor, Hall
99C2×D5—556no—
100C3×C5—76no—
101C2×C7—111no—
102C2×C7—2751no—
103C2×C7—251no—
104C22×C3—19251noHall
105C11C2×ΓL(1,𝔽8)×C5⋊D511yescharacteristic, complemented, Sylow
106C2×C5—116no—
107D5—56no—
108D5—556no—
109C7C3×D11×C5⋊D511yescharacteristic, complemented, Sylow
110C2×C3—771no—
111C2×C3—19251no—
112C2×C3—1751no—
113C5D5×ΓL(1,𝔽8)×D1116yescomplemented
114C22—2751noSylow
115C3—71noSylow
116C2—111no—
117C2—2751no—
118C2—251no—
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. C11×ΓL(1,𝔽8)×C5⋊D5C2semidirect
3. C52×ΓL(1,𝔽8)×D11C2semidirect
4. C7×D11×C5⋊D5C3semidirect
5. C52×C11×ΓL(1,𝔽8)C22semidirect
7. C7×C5⋊D55C2×C3semidirect
8. C7×C11×C5⋊D5C2×C3semidirect
9. C52×C7×D11C2×C3semidirect
13. C5×ΓL(1,𝔽8)×D11D5semidirect
15. C52×C7×C11C22×C3semidirect
20. C5×C11×ΓL(1,𝔽8)C2×D5semidirect
21. D11×C5⋊D5ΓL(1,𝔽8)direct (N × H)
24. ΓL(1,𝔽8)×C5⋊D5D11direct (N × H)
29. C5×C7×D11C3×D5semidirect
32. C5⋊D55C2×ΓL(1,𝔽8)semidirect
33. C11×C5⋊D5C2×ΓL(1,𝔽8)semidirect
34. C52×D11C2×ΓL(1,𝔽8)semidirect
35. C52×ΓL(1,𝔽8)C2×D11semidirect
38. ΓL(1,𝔽8)×D11C5⋊D5direct (N × H)
40. C5×C7×C11C2×C3×D5semidirect
43. C7×C5⋊D5C3×D11semidirect
49. C52×C11C22×ΓL(1,𝔽8)semidirect
50. C11×ΓL(1,𝔽8)C2×C5⋊D5semidirect
55. C52×C7C2×C3×D11semidirect
59. C7×D11C3×C5⋊D5semidirect
66. C5×D11D5×ΓL(1,𝔽8)semidirect
68. C5×ΓL(1,𝔽8)D5×D11semidirect
71. C7×C11C2×C3×C5⋊D5semidirect
80. C5×C11C2×D5×ΓL(1,𝔽8)semidirect
83. C5⋊D5ΓL(1,𝔽8)×D11direct (N × H)
88. C5×C7C3×D5×D11semidirect
94. C52C2×ΓL(1,𝔽8)×D11semidirect
97. D11ΓL(1,𝔽8)×C5⋊D5direct (N × H)
98. ΓL(1,𝔽8)D11×C5⋊D5direct (N × H)
105. C11C2×ΓL(1,𝔽8)×C5⋊D5semidirect
109. C7C3×D11×C5⋊D5semidirect
113. C5D5×ΓL(1,𝔽8)×D11semidirect

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
C11×ΓL(1,𝔽8)×C5⋊D52
C52×ΓL(1,𝔽8)×D112
C7×D11×C5⋊D53
D5×ΓL(1,𝔽8)×D115
C3×D11×C5⋊D57
C2×ΓL(1,𝔽8)×C5⋊D511