Finite Group Databasebeta

← ΓL(1,𝔽8)×(C5⋊D5)⋊D7 #28413332

Subgroups — ΓL(1,𝔽8)×(C5⋊D5)⋊D7 #28413332

Order 14700.

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

normal subgroups only

This group has 208 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
1C7×ΓL(1,𝔽8)×C5⋊D5C211yesmaximal, characteristic, complemented
2C5×ΓL(1,𝔽8)×D35C212yesmaximal, complemented
3C7×(C5⋊D5)⋊D7C311yesmaximal, characteristic, complemented, Hall
4C52×C7×ΓL(1,𝔽8)C2211yescharacteristic, complemented, Hall
5D5×D7×ΓL(1,𝔽8)—52nomaximal
6C72×C5⋊D5C2×C311yescharacteristic, complemented
7C5×C7×D35C2×C312yescomplemented
8C3×(C5⋊D5)⋊D7—71nomaximal
9D52×ΓL(1,𝔽8)—71nomaximal
10ΓL(1,𝔽8)×D35D512yescomplemented
11C7×D5×ΓL(1,𝔽8)—52no—
12C7×D5×ΓL(1,𝔽8)—102no—
13C5×D7×ΓL(1,𝔽8)—52no—
14C52×C72C22×C311yescharacteristic, complemented, Hall
15C3×C7×C5⋊D5—71no—
16C3×C5×D35—72no—
17ΓL(1,𝔽8)×C5⋊D5D711yescharacteristic, complemented
18C5×D5×ΓL(1,𝔽8)—72no—
19C7×D5×D7—52no—
20C5×C7×ΓL(1,𝔽8)—22no—
21C5×C7×ΓL(1,𝔽8)C2×D512yescomplemented
22(C5⋊D5)⋊D7ΓL(1,𝔽8)11yescharacteristic, direct product factor
23C7×D52—71no—
24C2×D7×ΓL(1,𝔽8)—251noHall
25C3×C52×C7—71no—
26C52×ΓL(1,𝔽8)C2×D711yescharacteristic, complemented
27C7×D35C3×D512yescomplemented
28C72×D5—52no—
29C72×D5—102no—
30C5×C7×D7—52no—
31C3×D5×D7—352no—
32C2×D5×ΓL(1,𝔽8)—352no—
33C7×C5⋊D5—61no—
34C7×C5⋊D5C2×ΓL(1,𝔽8)11yescharacteristic, complemented
35C7×C5⋊D5C3×D711yescharacteristic, complemented
36C5×D35C2×ΓL(1,𝔽8)12yescomplemented
37C5×C7×D5—72no—
38C3×D52—491noHall
39C2×C7×ΓL(1,𝔽8)—251no—
40D7×ΓL(1,𝔽8)—52no—
41C5×C72—22no—
42C5×C72C2×C3×D512yescomplemented
43C3×D35—72no—
44C3×C7×D5—352no—
45C3×C7×D5—702no—
46C3×C5×D7—352no—
47C2×C5×ΓL(1,𝔽8)—352no—
48D5×ΓL(1,𝔽8)—52no—
49D5×ΓL(1,𝔽8)—72no—
50D5×ΓL(1,𝔽8)—102no—
51C2×C7×D7—251noHall
52C52×C7—61no—
53C52×C7C22×ΓL(1,𝔽8)11yescharacteristic, complemented
54C52×C7C2×C3×D711yescharacteristic, complemented
55C3×C5⋊D5—71no—
56C3×C5×D5—492no—
57C7×ΓL(1,𝔽8)D5211yescharacteristic, complemented, Hall
58C2×C7×D5—352no—
59D5×D7—52no—
60C3×C5×C7—72no—
61C3×C5×C7—142no—
62C5×ΓL(1,𝔽8)—22no—
63C5×ΓL(1,𝔽8)D5×D712yescomplemented
64D52—71noHall
65C2×C72—251no—
66C7×D7—52no—
67C2×C3×D7—1751no—
68C22×ΓL(1,𝔽8)—1751no—
69C3×C52—71noHall
70C2×C5×C7—352no—
71D35D5×ΓL(1,𝔽8)12yescomplemented
72C5×D7—52no—
73C7×D5—52no—
74C7×D5—52no—
75C7×D5—72no—
76C7×D5—102no—
77C7×D5—102no—
78C7×D5—302no—
79C7×D5—304no—
80C2×C3×D5—2452no—
81C5⋊D5D7×ΓL(1,𝔽8)11yescharacteristic, complemented
82C5×D5—72no—
83C72C3×D5211yescharacteristic, complemented, Sylow
84C2×C3×C7—1751no—
85C3×D7—352no—
86C2×ΓL(1,𝔽8)—251no—
87C2×ΓL(1,𝔽8)—352no—
88C5×C7—22no—
89C5×C7—22no—
90C5×C7—64no—
91C5×C7—62no—
92C5×C7C2×D5×ΓL(1,𝔽8)12yescomplemented
93C5×C7C3×D5×D712yescomplemented
94C2×C3×C5—2452no—
95C3×D5—352no—
96C3×D5—492no—
97C3×D5—702no—
98C22×C7—1751no—
99C2×D7—251no—
100C52C2×D7×ΓL(1,𝔽8)11yescharacteristic, complemented, Sylow
101C3×C7—71no—
102ΓL(1,𝔽8)(C5⋊D5)⋊D711yescharacteristic, direct product factor
103C2×D5—352no—
104C3×C5—72no—
105C3×C5—142no—
106C2×C7—251no—
107C2×C7—251no—
108C2×C7—352no—
109C2×C7—1501no—
110D7—52no—
111C22×C3—12251noHall
112C2×C5—352no—
113D5—52no—
114D5—72no—
115D5—102no—
116C7—61no—
117C7D52×ΓL(1,𝔽8)11yescharacteristic, complemented
118C7C3×(C5⋊D5)⋊D711yescharacteristic, complemented
119C2×C3—1751no—
120C2×C3—2452no—
121C5—22no—
122C5D5×D7×ΓL(1,𝔽8)12yescomplemented
123C22—1751noSylow
124C3—71noSylow
125C2—251no—
126C2—352no—
subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC52×C72C22×C3
LCSG ▷ C52×C72
UCSC1

Complements

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

NH ≅ G/Ntype
1. C7×ΓL(1,𝔽8)×C5⋊D5C2semidirect
2. C5×ΓL(1,𝔽8)×D35C2semidirect
3. C7×(C5⋊D5)⋊D7C3semidirect
4. C52×C7×ΓL(1,𝔽8)C22semidirect
6. C72×C5⋊D5C2×C3semidirect
7. C5×C7×D35C2×C3semidirect
10. ΓL(1,𝔽8)×D35D5semidirect
14. C52×C72C22×C3semidirect
17. ΓL(1,𝔽8)×C5⋊D5D7semidirect
21. C5×C7×ΓL(1,𝔽8)C2×D5semidirect
22. (C5⋊D5)⋊D7ΓL(1,𝔽8)direct (N × H)
26. C52×ΓL(1,𝔽8)C2×D7semidirect
27. C7×D35C3×D5semidirect
34. C7×C5⋊D5C2×ΓL(1,𝔽8)semidirect
35. C7×C5⋊D5C3×D7semidirect
36. C5×D35C2×ΓL(1,𝔽8)semidirect
42. C5×C72C2×C3×D5semidirect
53. C52×C7C22×ΓL(1,𝔽8)semidirect
54. C52×C7C2×C3×D7semidirect
57. C7×ΓL(1,𝔽8)D52semidirect
63. C5×ΓL(1,𝔽8)D5×D7semidirect
71. D35D5×ΓL(1,𝔽8)semidirect
81. C5⋊D5D7×ΓL(1,𝔽8)semidirect
83. C72C3×D52semidirect
92. C5×C7C2×D5×ΓL(1,𝔽8)semidirect
93. C5×C7C3×D5×D7semidirect
100. C52C2×D7×ΓL(1,𝔽8)semidirect
102. ΓL(1,𝔽8)(C5⋊D5)⋊D7direct (N × H)
117. C7D52×ΓL(1,𝔽8)semidirect
118. C7C3×(C5⋊D5)⋊D7semidirect
122. C5D5×D7×ΓL(1,𝔽8)semidirect

Maximal subgroups

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

subgroupindex
C7×ΓL(1,𝔽8)×C5⋊D52
C5×ΓL(1,𝔽8)×D352
C7×(C5⋊D5)⋊D73
D5×D7×ΓL(1,𝔽8)5
C3×(C5⋊D5)⋊D77
D52×ΓL(1,𝔽8)7