Finite Group Databasebeta

← C11×ΓL(1,𝔽8)×C7⋊AGL(1,𝔽5) #44568655

Subgroups — C11×ΓL(1,𝔽8)×C7⋊AGL(1,𝔽5) #44568655

Order 32340.

104 conjugacy classes of subgroups; only the 42 normal subgroups are shown.

normal subgroups only

This group has 104 conjugacy classes of subgroups — too many to draw. Only its 42 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×C11×C7⋊AGL(1,𝔽5)C311yesmaximal, characteristic, complemented, Hall
3C5×C7×C11×ΓL(1,𝔽8)C411yescharacteristic, complemented, Hall
4C11×ΓL(1,𝔽8)×Dic7—51nomaximal, Hall
5C72×D5×C11C2×C311yescharacteristic
6C3×C11×C7⋊AGL(1,𝔽5)—71nomaximal
7C11×AGL(1,𝔽5)×ΓL(1,𝔽8)—71nomaximal
8C2×C7×C11×ΓL(1,𝔽8)—51no—
9ΓL(1,𝔽8)×C7⋊AGL(1,𝔽5)C1111yesmaximal, characteristic, direct product factor, Hall
10C5×C72×C11C3×C411yescharacteristic, complemented, Hall
11D5×C11×ΓL(1,𝔽8)D711yescharacteristic
12C3×C7×D5×C11—71no—
13C7×C11×Dic7—51noHall
14C7×C11×ΓL(1,𝔽8)AGL(1,𝔽5)11yescharacteristic, complemented, Hall
15C11×C7⋊AGL(1,𝔽5)ΓL(1,𝔽8)11yescharacteristic, direct product factor
16C7×C11×AGL(1,𝔽5)—71no—
17C7×D5×ΓL(1,𝔽8)C2×C1111yescharacteristic
18C3×C5×C7×C11—71no—
19C5×C11×ΓL(1,𝔽8)Dic711yescharacteristic, complemented
20C2×C72×C11—51no—
21C7×C7⋊AGL(1,𝔽5)C3×C1111yescharacteristic, complemented, Hall
22C3×C11×Dic7—351no—
23C4×C11×ΓL(1,𝔽8)—351no—
24C7×D5×C11—61no—
25C7×D5×C11C2×ΓL(1,𝔽8)11yescharacteristic
26C7×D5×C11C3×D711yescharacteristic
27C5×C7×ΓL(1,𝔽8)C4×C1111yescharacteristic, complemented, Hall
28C3×C11×AGL(1,𝔽5)—491noHall
29ΓL(1,𝔽8)×Dic7—51noHall
30C72×C11C3×AGL(1,𝔽5)11yescharacteristic, complemented, Hall
31C72×D5C2×C3×C1111yescharacteristic
32C2×C3×C7×C11—351no—
33C2×C11×ΓL(1,𝔽8)—51no—
34C3×C7⋊AGL(1,𝔽5)—71no—
35AGL(1,𝔽5)×ΓL(1,𝔽8)—71no—
36C5×C7×C11—61no—
37C5×C7×C11C4×ΓL(1,𝔽8)11yescharacteristic, complemented
38C5×C7×C11C3×Dic711yescharacteristic, complemented
39C3×D5×C11—71no—
40C4×C7×C11—351no—
41C11×Dic7—51no—
42C2×C7×ΓL(1,𝔽8)—51no—
43C5×C72C3×C4×C1111yescharacteristic, complemented, Hall
44C3×C7×C11—71no—
45C11×ΓL(1,𝔽8)C7⋊AGL(1,𝔽5)11yescharacteristic, direct product factor
46C11×AGL(1,𝔽5)—71noHall
47C3×C7×D5—71no—
48D5×ΓL(1,𝔽8)C11×D711yescharacteristic
49C7×Dic7—51noHall
50C3×C5×C11—71noHall
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)C11×ΓL(1,𝔽8)11yescharacteristic, direct product factor
56C7×AGL(1,𝔽5)—71no—
57C3×C4×C11—2451noHall
58D5×C11D7×ΓL(1,𝔽8)11yescharacteristic
59C3×C5×C7—71no—
60C5×ΓL(1,𝔽8)C11×Dic711yescharacteristic, complemented
61C2×C72—51no—
62C3×Dic7—351no—
63C4×ΓL(1,𝔽8)—351no—
64C7×C11—61no—
65C7×C11AGL(1,𝔽5)×ΓL(1,𝔽8)11yescharacteristic, complemented
66C7×C11C3×C7⋊AGL(1,𝔽5)11yescharacteristic, complemented
67C7×D5—61no—
68C7×D5C2×C11×ΓL(1,𝔽8)11yescharacteristic
69C7×D5C3×C11×D711yescharacteristic
70C2×C3×C11—351no—
71C3×AGL(1,𝔽5)—491noHall
72C5×C11ΓL(1,𝔽8)×Dic711yescharacteristic, complemented, Hall
73C72C3×C11×AGL(1,𝔽5)11yescharacteristic, complemented, Sylow
74C4×C11—351noHall
75C2×C3×C7—351no—
76C2×ΓL(1,𝔽8)—51no—
77C5×C7—61no—
78C5×C7C4×C11×ΓL(1,𝔽8)11yescharacteristic, complemented
79C5×C7C3×C11×Dic711yescharacteristic, complemented
80C3×C11—71noHall
81C3×D5—71no—
82C4×C7—351no—
83Dic7—51no—
84C2×C11—51no—
85C3×C7—71no—
86ΓL(1,𝔽8)C11×C7⋊AGL(1,𝔽5)11yescharacteristic, direct product factor
87AGL(1,𝔽5)—71noHall
88C3×C5—71noHall
89C2×C7—51no—
90C2×C7—51no—
91C2×C7—301no—
92C3×C4—2451noHall
93C11ΓL(1,𝔽8)×C7⋊AGL(1,𝔽5)11yescharacteristic, direct product factor, Sylow
94D5C11×D7×ΓL(1,𝔽8)11yescharacteristic
95C7—61no—
96C7C11×AGL(1,𝔽5)×ΓL(1,𝔽8)11yescharacteristic, complemented
97C7C3×C11×C7⋊AGL(1,𝔽5)11yescharacteristic, complemented
98C2×C3—351no—
99C5C11×ΓL(1,𝔽8)×Dic711yescharacteristic, complemented, Sylow
100C4—351noSylow
101C3—71noSylow
102C2—51no—
subgroupG/H
center, hypercenterC11ΓL(1,𝔽8)×C7⋊AGL(1,𝔽5)
commutator, hypocenterC5×C72C3×C4×C11
FrattiniC1G
Fitting, socleC5×C72×C11C3×C4
prime radicalC7×D5×C11×ΓL(1,𝔽8)C2
cosocleC72×D5C2×C3×C11
LCSG ▷ C5×C72
UCSC1 ◁ C11

Complements

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

NH ≅ G/Ntype
2. C7×C11×C7⋊AGL(1,𝔽5)C3semidirect
3. C5×C7×C11×ΓL(1,𝔽8)C4semidirect
9. ΓL(1,𝔽8)×C7⋊AGL(1,𝔽5)C11direct (N × H)
10. C5×C72×C11C3×C4semidirect
14. C7×C11×ΓL(1,𝔽8)AGL(1,𝔽5)semidirect
15. C11×C7⋊AGL(1,𝔽5)ΓL(1,𝔽8)direct (N × H)
19. C5×C11×ΓL(1,𝔽8)Dic7semidirect
21. C7×C7⋊AGL(1,𝔽5)C3×C11semidirect
27. C5×C7×ΓL(1,𝔽8)C4×C11semidirect
30. C72×C11C3×AGL(1,𝔽5)semidirect
37. C5×C7×C11C4×ΓL(1,𝔽8)semidirect
38. C5×C7×C11C3×Dic7semidirect
43. C5×C72C3×C4×C11semidirect
45. C11×ΓL(1,𝔽8)C7⋊AGL(1,𝔽5)direct (N × H)
54. C7×ΓL(1,𝔽8)C11×AGL(1,𝔽5)semidirect
55. C7⋊AGL(1,𝔽5)C11×ΓL(1,𝔽8)direct (N × H)
60. C5×ΓL(1,𝔽8)C11×Dic7semidirect
65. C7×C11AGL(1,𝔽5)×ΓL(1,𝔽8)semidirect
66. C7×C11C3×C7⋊AGL(1,𝔽5)semidirect
72. C5×C11ΓL(1,𝔽8)×Dic7semidirect
73. C72C3×C11×AGL(1,𝔽5)semidirect
78. C5×C7C4×C11×ΓL(1,𝔽8)semidirect
79. C5×C7C3×C11×Dic7semidirect
86. ΓL(1,𝔽8)C11×C7⋊AGL(1,𝔽5)direct (N × H)
93. C11ΓL(1,𝔽8)×C7⋊AGL(1,𝔽5)direct (N × H)
96. C7C11×AGL(1,𝔽5)×ΓL(1,𝔽8)semidirect
97. C7C3×C11×C7⋊AGL(1,𝔽5)semidirect
99. C5C11×ΓL(1,𝔽8)×Dic7semidirect

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×C11×C7⋊AGL(1,𝔽5)3
C11×ΓL(1,𝔽8)×Dic75
C3×C11×C7⋊AGL(1,𝔽5)7
C11×AGL(1,𝔽5)×ΓL(1,𝔽8)7
ΓL(1,𝔽8)×C7⋊AGL(1,𝔽5)11