Finite Group Databasebeta

← C7×C11⋊(ΓL(1,𝔽8)⋊AGL(1,𝔽5)) #44568652

Subgroups — C7×C11⋊(ΓL(1,𝔽8)⋊AGL(1,𝔽5)) #44568652

Order 32340.

104 conjugacy classes of subgroups; only the 40 normal subgroups are shown.

normal subgroups only

This group has 104 conjugacy classes of subgroups — too many to draw. Only its 40 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×D5×C11×ΓL(1,𝔽8)C211yesmaximal, characteristic
2C7×C7⋊(C11⋊AGL(1,𝔽5))C311yesmaximal, characteristic, complemented, Hall
3C5×C7×C11×ΓL(1,𝔽8)C411yescharacteristic, complemented, Hall
4C7×C11⋊(ΓL(1,𝔽8)⋊C4)—51nomaximal, Hall
5C72×D5×C11C2×C311yescharacteristic
6C3×C7×C11⋊AGL(1,𝔽5)—71nomaximal
7C11⋊(ΓL(1,𝔽8)⋊AGL(1,𝔽5))C711yesmaximal, characteristic, direct product factor
8C2×C7×C11×ΓL(1,𝔽8)—51no—
9C7×ΓL(1,𝔽8)⋊AGL(1,𝔽5)—111nomaximal, Hall
10C5×C72×C11C3×C411yescharacteristic, complemented, Hall
11D5×C11×ΓL(1,𝔽8)C2×C711yescharacteristic
12C3×C7×D5×C11—71no—
13C7×Dic77—51noHall
14C7×C11×ΓL(1,𝔽8)AGL(1,𝔽5)11yescharacteristic, complemented, Hall
15C7⋊(C11⋊AGL(1,𝔽5))C3×C711yescharacteristic, complemented
16C7×C11⋊AGL(1,𝔽5)—71no—
17C7×D5×ΓL(1,𝔽8)D1111yescharacteristic
18C3×C5×C7×C11—71no—
19C5×C11×ΓL(1,𝔽8)C4×C711yescharacteristic, complemented
20C2×C72×C11—51no—
21C7×C7⋊AGL(1,𝔽5)—111noHall
22C3×C7×Dic11—351no—
23C11⋊(ΓL(1,𝔽8)⋊C4)—51no—
24C7×D5×C11—61no—
25C7×D5×C11AGL(1,𝔽7)11yescharacteristic
26C7×D5×C11C2×C3×C711yescharacteristic
27C5×C7×ΓL(1,𝔽8)Dic1111yescharacteristic, complemented, Hall
28C3×C11⋊AGL(1,𝔽5)—71noHall
29C7×ΓL(1,𝔽8)⋊C4—551noHall
30C72×C11C3×AGL(1,𝔽5)11yescharacteristic, complemented, Hall
31C72×D5C3×D1111yescharacteristic
32C2×C3×C7×C11—701no—
33C2×C11×ΓL(1,𝔽8)—51no—
34C3×C7×AGL(1,𝔽5)—771no—
35ΓL(1,𝔽8)⋊AGL(1,𝔽5)—111no—
36C5×C7×C11—61no—
37C5×C7×C11ΓL(1,𝔽8)⋊C411yescharacteristic, complemented
38C5×C7×C11C3×C4×C711yescharacteristic, complemented
39C3×D5×C11—141no—
40Dic77—51no—
41C7×Dic11—351no—
42C2×C7×ΓL(1,𝔽8)—51no—
43C5×C72C3×Dic1111yescharacteristic, complemented, Hall
44C3×C7×C11—141no—
45C11×ΓL(1,𝔽8)C7×AGL(1,𝔽5)11yescharacteristic, complemented
46C11⋊AGL(1,𝔽5)—71noHall
47C3×C7×D5—141no—
48D5×ΓL(1,𝔽8)C7×D1111yescharacteristic
49C7×Dic7—551noHall
50C3×C5×C11—141noHall
51C2×C7×C11—51no—
52C2×C7×C11—51no—
53C2×C7×C11—301no—
54C7×ΓL(1,𝔽8)C11⋊AGL(1,𝔽5)11yescharacteristic, complemented, Hall
55C7⋊AGL(1,𝔽5)—111no—
56C7×AGL(1,𝔽5)—771no—
57C3×Dic11—351noHall
58D5×C11C7×AGL(1,𝔽7)11yescharacteristic
59C3×C5×C7—141no—
60C5×ΓL(1,𝔽8)C7×Dic1111yescharacteristic, complemented
61C2×C72—51no—
62C3×C4×C7—3851no—
63ΓL(1,𝔽8)⋊C4—551no—
64C7×C11—61no—
65C7×C11ΓL(1,𝔽8)⋊AGL(1,𝔽5)11yescharacteristic, complemented
66C7×C11C3×C7×AGL(1,𝔽5)11yescharacteristic, complemented
67C7×D5—61no—
68C7×D5C11⋊AGL(1,𝔽7)11yescharacteristic
69C7×D5C3×C7×D1111yescharacteristic
70C2×C3×C11—701no—
71C3×AGL(1,𝔽5)—771noHall
72C5×C11C7×ΓL(1,𝔽8)⋊C411yescharacteristic, complemented, Hall
73C72C3×C11⋊AGL(1,𝔽5)11yescharacteristic, complemented, Sylow
74Dic11—351noHall
75C2×C3×C7—701no—
76C2×ΓL(1,𝔽8)—51no—
77C5×C7—61no—
78C5×C7C11⋊(ΓL(1,𝔽8)⋊C4)11yescharacteristic, complemented
79C5×C7C3×C7×Dic1111yescharacteristic, complemented
80C3×C11—141noHall
81C3×D5—141no—
82C4×C7—3851no—
83Dic7—551no—
84C2×C11—51no—
85C3×C7—141no—
86ΓL(1,𝔽8)C7×C11⋊AGL(1,𝔽5)11yescharacteristic, complemented
87AGL(1,𝔽5)—771noHall
88C3×C5—141noHall
89C2×C7—51no—
90C2×C7—51no—
91C2×C7—301no—
92C3×C4—3851noHall
93C11C7×ΓL(1,𝔽8)⋊AGL(1,𝔽5)11yescharacteristic, complemented, Sylow
94D5C7×C11⋊AGL(1,𝔽7)11yescharacteristic
95C7—61no—
96C7C11⋊(ΓL(1,𝔽8)⋊AGL(1,𝔽5))11yescharacteristic, direct product factor
97C7C3×C7×C11⋊AGL(1,𝔽5)11yescharacteristic, complemented
98C2×C3—701no—
99C5C7×C11⋊(ΓL(1,𝔽8)⋊C4)11yescharacteristic, complemented, Sylow
100C4—3851noSylow
101C3—141noSylow
102C2—51no—
subgroupG/H
center, hypercenterC7C11⋊(ΓL(1,𝔽8)⋊AGL(1,𝔽5))
commutator, hypocenterC5×C7×C11C3×C4×C7
FrattiniC1G
Fitting, socleC5×C72×C11C3×C4
prime radicalC7×D5×C11×ΓL(1,𝔽8)C2
cosocleC7×D5×C11C2×C3×C7
LCSG ▷ C5×C7×C11
UCSC1 ◁ C7

Complements

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

NH ≅ G/Ntype
2. C7×C7⋊(C11⋊AGL(1,𝔽5))C3semidirect
3. C5×C7×C11×ΓL(1,𝔽8)C4semidirect
7. C11⋊(ΓL(1,𝔽8)⋊AGL(1,𝔽5))C7direct (N × H)
10. C5×C72×C11C3×C4semidirect
14. C7×C11×ΓL(1,𝔽8)AGL(1,𝔽5)semidirect
15. C7⋊(C11⋊AGL(1,𝔽5))C3×C7semidirect
19. C5×C11×ΓL(1,𝔽8)C4×C7semidirect
27. C5×C7×ΓL(1,𝔽8)Dic11semidirect
30. C72×C11C3×AGL(1,𝔽5)semidirect
37. C5×C7×C11ΓL(1,𝔽8)⋊C4semidirect
38. C5×C7×C11C3×C4×C7semidirect
43. C5×C72C3×Dic11semidirect
45. C11×ΓL(1,𝔽8)C7×AGL(1,𝔽5)semidirect
54. C7×ΓL(1,𝔽8)C11⋊AGL(1,𝔽5)semidirect
60. C5×ΓL(1,𝔽8)C7×Dic11semidirect
65. C7×C11ΓL(1,𝔽8)⋊AGL(1,𝔽5)semidirect
66. C7×C11C3×C7×AGL(1,𝔽5)semidirect
72. C5×C11C7×ΓL(1,𝔽8)⋊C4semidirect
73. C72C3×C11⋊AGL(1,𝔽5)semidirect
78. C5×C7C11⋊(ΓL(1,𝔽8)⋊C4)semidirect
79. C5×C7C3×C7×Dic11semidirect
86. ΓL(1,𝔽8)C7×C11⋊AGL(1,𝔽5)semidirect
93. C11C7×ΓL(1,𝔽8)⋊AGL(1,𝔽5)semidirect
96. C7C11⋊(ΓL(1,𝔽8)⋊AGL(1,𝔽5))direct (N × H)
97. C7C3×C7×C11⋊AGL(1,𝔽5)semidirect
99. C5C7×C11⋊(ΓL(1,𝔽8)⋊C4)semidirect

Maximal subgroups

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

subgroupindex
C7×D5×C11×ΓL(1,𝔽8)2
C7×C7⋊(C11⋊AGL(1,𝔽5))3
C7×C11⋊(ΓL(1,𝔽8)⋊C4)5
C3×C7×C11⋊AGL(1,𝔽5)7
C11⋊(ΓL(1,𝔽8)⋊AGL(1,𝔽5))7
C7×ΓL(1,𝔽8)⋊AGL(1,𝔽5)11