Finite Group Databasebeta

← ΓL(1,𝔽8)×(C5⋊D5)⋊D11 #37528539

Subgroups — ΓL(1,𝔽8)×(C5⋊D5)⋊D11 #37528539

Order 23100.

192 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 192 conjugacy classes of subgroups — too many to draw. Only its 45 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
1C11×ΓL(1,𝔽8)×C5⋊D5C211yesmaximal, characteristic, complemented
2C5×ΓL(1,𝔽8)×D55C212yesmaximal, complemented
3C7×(C5⋊D5)⋊D11C311yesmaximal, characteristic, complemented, Hall
4C52×C11×ΓL(1,𝔽8)C2211yescharacteristic, complemented, Hall
5D5×ΓL(1,𝔽8)×D11—52nomaximal
6C7×C11×C5⋊D5C2×C311yescharacteristic, complemented
7C5×C7×D55C2×C312yescomplemented
8C3×(C5⋊D5)⋊D11—71nomaximal, Hall
9ΓL(1,𝔽8)×D55D512yescomplemented
10D5×C11×ΓL(1,𝔽8)—52no—
11D5×C11×ΓL(1,𝔽8)—102no—
12C5×ΓL(1,𝔽8)×D11—52no—
13D52×ΓL(1,𝔽8)—111nomaximal, Hall
14C52×C7×C11C22×C311yescharacteristic, complemented, Hall
15C3×C11×C5⋊D5—71no—
16C3×C5×D55—72no—
17C7×D5×D11—52no—
18C5×C11×ΓL(1,𝔽8)—22no—
19C5×C11×ΓL(1,𝔽8)C2×D512yescomplemented
20(C5⋊D5)⋊D11ΓL(1,𝔽8)11yescharacteristic, direct product factor, Hall
21ΓL(1,𝔽8)×C5⋊D5D1111yescharacteristic, complemented
22C5×D5×ΓL(1,𝔽8)—112no—
23C2×ΓL(1,𝔽8)×D11—251noHall
24C3×C52×C11—71noHall
25C7×D55C3×D512yescomplemented
26C7×D5×C11—52no—
27C7×D5×C11—102no—
28C5×C7×D11—52no—
29C7×D52—111noHall
30C3×D5×D11—352no—
31C11×C5⋊D5C2×ΓL(1,𝔽8)11yescharacteristic, complemented
32C5×D55C2×ΓL(1,𝔽8)12yescomplemented
33C52×ΓL(1,𝔽8)C2×D1111yescharacteristic, complemented, Hall
34C2×C11×ΓL(1,𝔽8)—251no—
35ΓL(1,𝔽8)×D11—52no—
36C2×D5×ΓL(1,𝔽8)—552no—
37C5×C7×C11—22no—
38C5×C7×C11C2×C3×D512yescomplemented
39C7×C5⋊D5C3×D1111yescharacteristic, complemented
40C5×C7×D5—112no—
41C3×D55—72no—
42C3×D5×C11—352no—
43C3×D5×C11—702no—
44C3×C5×D11—352no—
45C2×C7×D11—251noHall
46C3×D52—771noHall
47C52×C11C22×ΓL(1,𝔽8)11yescharacteristic, complemented, Hall
48C11×ΓL(1,𝔽8)D5211yescharacteristic, complemented, Hall
49D5×D11—52no—
50C2×C5×ΓL(1,𝔽8)—552no—
51D5×ΓL(1,𝔽8)—52no—
52D5×ΓL(1,𝔽8)—102no—
53D5×ΓL(1,𝔽8)—112no—
54C52×C7C2×C3×D1111yescharacteristic, complemented, Hall
55C3×C5×C11—72no—
56C3×C5×C11—142no—
57C2×C7×C11—251no—
58C7×D11—52no—
59C3×C5⋊D5—71no—
60C3×C5×D5—772no—
61C2×C7×D5—552no—
62C2×C3×D11—1751noHall
63D55D5×ΓL(1,𝔽8)12yescomplemented
64C5×D11—52no—
65D5×C11—52no—
66D5×C11—102no—
67C5×ΓL(1,𝔽8)—22no—
68C5×ΓL(1,𝔽8)D5×D1112yescomplemented
69D52—111noHall
70C22×ΓL(1,𝔽8)—2751noHall
71C7×C11C3×D5211yescharacteristic, complemented, Hall
72C3×C52—71noHall
73C2×C5×C7—552no—
74C7×D5—52no—
75C7×D5—102no—
76C7×D5—112no—
77C2×C3×C11—1751no—
78C3×D11—352no—
79C2×C3×D5—3852no—
80C5×C11—22no—
81C5×C11C2×D5×ΓL(1,𝔽8)12yescomplemented
82C5⋊D5ΓL(1,𝔽8)×D1111yescharacteristic, complemented
83C5×D5—112no—
84C2×D11—251noHall
85C2×ΓL(1,𝔽8)—251no—
86C2×ΓL(1,𝔽8)—552no—
87C5×C7—22no—
88C5×C7C3×D5×D1112yescomplemented
89C3×C11—71noHall
90C2×C3×C5—3852no—
91C3×D5—352no—
92C3×D5—702no—
93C3×D5—772no—
94C22×C7—2751noHall
95C52C2×ΓL(1,𝔽8)×D1111yescharacteristic, complemented, Sylow
96C2×C11—251no—
97D11—52no—
98ΓL(1,𝔽8)(C5⋊D5)⋊D1111yescharacteristic, direct product factor, Hall
99C2×D5—552no—
100C3×C5—72no—
101C3×C5—142no—
102C2×C7—251no—
103C2×C7—552no—
104C22×C3—19251noHall
105C11D52×ΓL(1,𝔽8)11yescharacteristic, complemented, Sylow
106C2×C5—552no—
107D5—52no—
108D5—102no—
109D5—112no—
110C7C3×(C5⋊D5)⋊D1111yescharacteristic, complemented, Sylow
111C2×C3—3852no—
112C2×C3—1751no—
113C5—22no—
114C5D5×ΓL(1,𝔽8)×D1112yescomplemented
115C22—2751noSylow
116C3—71noSylow
117C2—251no—
118C2—552no—
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. C11×ΓL(1,𝔽8)×C5⋊D5C2semidirect
2. C5×ΓL(1,𝔽8)×D55C2semidirect
3. C7×(C5⋊D5)⋊D11C3semidirect
4. C52×C11×ΓL(1,𝔽8)C22semidirect
6. C7×C11×C5⋊D5C2×C3semidirect
7. C5×C7×D55C2×C3semidirect
9. ΓL(1,𝔽8)×D55D5semidirect
14. C52×C7×C11C22×C3semidirect
19. C5×C11×ΓL(1,𝔽8)C2×D5semidirect
20. (C5⋊D5)⋊D11ΓL(1,𝔽8)direct (N × H)
21. ΓL(1,𝔽8)×C5⋊D5D11semidirect
25. C7×D55C3×D5semidirect
31. C11×C5⋊D5C2×ΓL(1,𝔽8)semidirect
32. C5×D55C2×ΓL(1,𝔽8)semidirect
33. C52×ΓL(1,𝔽8)C2×D11semidirect
38. C5×C7×C11C2×C3×D5semidirect
39. C7×C5⋊D5C3×D11semidirect
47. C52×C11C22×ΓL(1,𝔽8)semidirect
48. C11×ΓL(1,𝔽8)D52semidirect
54. C52×C7C2×C3×D11semidirect
63. D55D5×ΓL(1,𝔽8)semidirect
68. C5×ΓL(1,𝔽8)D5×D11semidirect
71. C7×C11C3×D52semidirect
81. C5×C11C2×D5×ΓL(1,𝔽8)semidirect
82. C5⋊D5ΓL(1,𝔽8)×D11semidirect
88. C5×C7C3×D5×D11semidirect
95. C52C2×ΓL(1,𝔽8)×D11semidirect
98. ΓL(1,𝔽8)(C5⋊D5)⋊D11direct (N × H)
105. C11D52×ΓL(1,𝔽8)semidirect
110. C7C3×(C5⋊D5)⋊D11semidirect
114. 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
C11×ΓL(1,𝔽8)×C5⋊D52
C5×ΓL(1,𝔽8)×D552
C7×(C5⋊D5)⋊D113
D5×ΓL(1,𝔽8)×D115
C3×(C5⋊D5)⋊D117
D52×ΓL(1,𝔽8)11