Finite Group Databasebeta

← C7×C11⋊C5×ΓL(1,𝔽8)⋊C4 #44568439

Subgroups — C7×C11⋊C5×ΓL(1,𝔽8)⋊C4 #44568439

Order 32340.

104 conjugacy classes of subgroups; only the 48 normal subgroups are shown.

normal subgroups only

This group has 104 conjugacy classes of subgroups — too many to draw. Only its 48 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
1C2×C7×ΓL(1,𝔽8)×C11⋊C5C211yesmaximal, characteristic
2C7×Dic7×C11⋊C5C311yesmaximal, characteristic, complemented, Hall
3C7×ΓL(1,𝔽8)×C11⋊C5C411yescharacteristic, complemented, Hall
4C7×C11×ΓL(1,𝔽8)⋊C4C511yesmaximal, characteristic, complemented, Hall
5C2×C72×C11⋊C5C2×C311yescharacteristic
6C11⋊C5×ΓL(1,𝔽8)⋊C4C711yesmaximal, characteristic, direct product factor
7C3×C4×C7×C11⋊C5—71nomaximal
8C2×C7×C11×ΓL(1,𝔽8)C2×C511yescharacteristic
9C5×C7×ΓL(1,𝔽8)⋊C4—111nomaximal, Hall
10C72×C11⋊C5C3×C411yescharacteristic, complemented, Hall
11C2×ΓL(1,𝔽8)×C11⋊C5C2×C711yescharacteristic
12C2×C3×C7×C11⋊C5—71no—
13C7×C11×Dic7C3×C511yescharacteristic, complemented, Hall
14C7×C11×ΓL(1,𝔽8)C4×C511yescharacteristic, complemented, Hall
15C4×C7×C11⋊C5—71no—
16Dic7×C11⋊C5C3×C711yescharacteristic, complemented
17C2×C5×C7×ΓL(1,𝔽8)—111no—
18C3×C7×C11⋊C5—71no—
19ΓL(1,𝔽8)×C11⋊C5C4×C711yescharacteristic, complemented
20C2×C72×C11C2×C3×C511yescharacteristic
21C5×C7×Dic7—111noHall
22C3×C4×C7×C11—71no—
23C11×ΓL(1,𝔽8)⋊C4C5×C711yescharacteristic, complemented
24C2×C7×C11⋊C5—61no—
25C2×C7×C11⋊C5AGL(1,𝔽7)11yescharacteristic
26C2×C7×C11⋊C5C2×C3×C711yescharacteristic
27C5×C7×ΓL(1,𝔽8)—111noHall
28C3×C4×C11⋊C5—71noHall
29C7×ΓL(1,𝔽8)⋊C4C11⋊C511yescharacteristic, direct product factor, Hall
30C72×C11C3×C4×C511yescharacteristic, complemented, Hall
31C2×C5×C72—111no—
32C2×C3×C7×C11—71no—
33C2×C11×ΓL(1,𝔽8)C2×C5×C711yescharacteristic
34C3×C4×C5×C7—771no—
35C5×ΓL(1,𝔽8)⋊C4—111no—
36C7×C11⋊C5—61no—
37C7×C11⋊C5ΓL(1,𝔽8)⋊C411yescharacteristic, direct product factor
38C7×C11⋊C5C3×C4×C711yescharacteristic, complemented
39C2×C3×C11⋊C5—71no—
40C4×C7×C11—71no—
41C11×Dic7C3×C5×C711yescharacteristic, complemented
42C2×C7×ΓL(1,𝔽8)C2×C11⋊C511yescharacteristic
43C5×C72—111noHall
44C3×C7×C11—71no—
45C11×ΓL(1,𝔽8)C4×C5×C711yescharacteristic, complemented
46C4×C11⋊C5—71noHall
47C2×C3×C5×C7—771no—
48C2×C5×ΓL(1,𝔽8)—111no—
49C7×Dic7C3×C11⋊C511yescharacteristic, complemented, Hall
50C3×C11⋊C5—71noHall
51C2×C7×C11—61no—
52C2×C7×C11C5×AGL(1,𝔽7)11yescharacteristic
53C2×C7×C11C2×C3×C5×C711yescharacteristic
54C7×ΓL(1,𝔽8)C4×C11⋊C511yescharacteristic, complemented, Hall
55C4×C5×C7—771no—
56C5×Dic7—111no—
57C3×C4×C11—71noHall
58C2×C11⋊C5C7×AGL(1,𝔽7)11yescharacteristic
59C3×C5×C7—771no—
60C5×ΓL(1,𝔽8)—111no—
61C2×C72C2×C3×C11⋊C511yescharacteristic
62C3×C4×C7—71no—
63ΓL(1,𝔽8)⋊C4C7×C11⋊C511yescharacteristic, direct product factor
64C7×C11—61no—
65C7×C11C5×ΓL(1,𝔽8)⋊C411yescharacteristic, complemented
66C7×C11C3×C4×C5×C711yescharacteristic, complemented
67C2×C5×C7—111no—
68C2×C5×C7—111no—
69C2×C5×C7—661no—
70C2×C3×C11—71no—
71C3×C4×C5—771noHall
72C11⋊C5C7×ΓL(1,𝔽8)⋊C411yescharacteristic, direct product factor, Hall
73C72C3×C4×C11⋊C511yescharacteristic, complemented, Sylow
74C4×C11—71noHall
75C2×C3×C7—71no—
76C2×ΓL(1,𝔽8)C2×C7×C11⋊C511yescharacteristic
77C5×C7—111no—
78C5×C7—111no—
79C5×C7—661no—
80C3×C11—71noHall
81C2×C3×C5—771no—
82C4×C7—71no—
83Dic7C3×C7×C11⋊C511yescharacteristic, complemented
84C2×C11C5×C7×AGL(1,𝔽7)11yescharacteristic
85C3×C7—71no—
86ΓL(1,𝔽8)C4×C7×C11⋊C511yescharacteristic, complemented
87C4×C5—771noHall
88C3×C5—771noHall
89C2×C7—61no—
90C2×C7C2×C3×C7×C11⋊C511yescharacteristic
91C2×C7AGL(1,𝔽7)×C11⋊C511yescharacteristic
92C3×C4—71noHall
93C11C5×C7×ΓL(1,𝔽8)⋊C411yescharacteristic, complemented, Sylow
94C2×C5—111no—
95C7—61no—
96C7C3×C4×C7×C11⋊C511yescharacteristic, complemented
97C7C11⋊C5×ΓL(1,𝔽8)⋊C411yescharacteristic, direct product factor
98C2×C3—71no—
99C5—111noSylow
100C4—71noSylow
101C3—71noSylow
102C2C7×AGL(1,𝔽7)×C11⋊C511yescharacteristic
subgroupG/H
center, hypercenterC2×C7AGL(1,𝔽7)×C11⋊C5
commutator, hypocenterC7×C11C3×C4×C5×C7
FrattiniC2C7×AGL(1,𝔽7)×C11⋊C5
Fitting, socleC2×C72×C11C2×C3×C5
prime radicalC2×C7×ΓL(1,𝔽8)×C11⋊C5C2
cosocleC2×C7×C11C2×C3×C5×C7
LCSG ▷ C7×C11
UCSC1 ◁ C2×C7

Complements

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

NH ≅ G/Ntype
2. C7×Dic7×C11⋊C5C3semidirect
3. C7×ΓL(1,𝔽8)×C11⋊C5C4semidirect
4. C7×C11×ΓL(1,𝔽8)⋊C4C5semidirect
6. C11⋊C5×ΓL(1,𝔽8)⋊C4C7direct (N × H)
10. C72×C11⋊C5C3×C4semidirect
13. C7×C11×Dic7C3×C5semidirect
14. C7×C11×ΓL(1,𝔽8)C4×C5semidirect
16. Dic7×C11⋊C5C3×C7semidirect
19. ΓL(1,𝔽8)×C11⋊C5C4×C7semidirect
23. C11×ΓL(1,𝔽8)⋊C4C5×C7semidirect
29. C7×ΓL(1,𝔽8)⋊C4C11⋊C5direct (N × H)
30. C72×C11C3×C4×C5semidirect
37. C7×C11⋊C5ΓL(1,𝔽8)⋊C4direct (N × H)
38. C7×C11⋊C5C3×C4×C7semidirect
41. C11×Dic7C3×C5×C7semidirect
45. C11×ΓL(1,𝔽8)C4×C5×C7semidirect
49. C7×Dic7C3×C11⋊C5semidirect
54. C7×ΓL(1,𝔽8)C4×C11⋊C5semidirect
63. ΓL(1,𝔽8)⋊C4C7×C11⋊C5direct (N × H)
65. C7×C11C5×ΓL(1,𝔽8)⋊C4semidirect
66. C7×C11C3×C4×C5×C7semidirect
72. C11⋊C5C7×ΓL(1,𝔽8)⋊C4direct (N × H)
73. C72C3×C4×C11⋊C5semidirect
83. Dic7C3×C7×C11⋊C5semidirect
86. ΓL(1,𝔽8)C4×C7×C11⋊C5semidirect
93. C11C5×C7×ΓL(1,𝔽8)⋊C4semidirect
96. C7C3×C4×C7×C11⋊C5semidirect
97. C7C11⋊C5×ΓL(1,𝔽8)⋊C4direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×ΓL(1,𝔽8)×C11⋊C52
C7×Dic7×C11⋊C53
C7×C11×ΓL(1,𝔽8)⋊C45
C11⋊C5×ΓL(1,𝔽8)⋊C47
C3×C4×C7×C11⋊C57
C5×C7×ΓL(1,𝔽8)⋊C411