Finite Group Databasebeta

← C11×ΓL(1,𝔽8)×Dic35 #44568508

Subgroups — C11×ΓL(1,𝔽8)×Dic35 #44568508

Order 32340.

104 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 104 conjugacy classes of subgroups — too many to draw. Only its 54 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×C5×C7×C11×ΓL(1,𝔽8)C211yesmaximal, characteristic
2C7×C11×Dic35C311yesmaximal, characteristic, complemented, Hall
3C5×C7×C11×ΓL(1,𝔽8)C411yescharacteristic, complemented, Hall
4C11×ΓL(1,𝔽8)×Dic7—51nomaximal, Hall
5C2×C5×C72×C11C2×C311yescharacteristic
6C11×Dic5×ΓL(1,𝔽8)—71nomaximal
7C3×C11×Dic35—71nomaximal
8C2×C7×C11×ΓL(1,𝔽8)D511yescharacteristic
9ΓL(1,𝔽8)×Dic35C1111yesmaximal, characteristic, direct product factor, Hall
10C5×C72×C11C3×C411yescharacteristic, complemented, Hall
11C2×C5×C11×ΓL(1,𝔽8)D711yescharacteristic
12C2×C3×C5×C7×C11—71no—
13C7×C11×Dic7—51noHall
14C7×C11×ΓL(1,𝔽8)Dic511yescharacteristic, complemented, Hall
15C11×Dic35ΓL(1,𝔽8)11yescharacteristic, direct product factor
16C7×C11×Dic5—71no—
17C2×C5×C7×ΓL(1,𝔽8)C2×C1111yescharacteristic
18C3×C5×C7×C11—71no—
19C5×C11×ΓL(1,𝔽8)Dic711yescharacteristic, complemented
20C2×C72×C11C3×D511yescharacteristic
21C7×Dic35C3×C1111yescharacteristic, complemented, Hall
22C3×C11×Dic7—351no—
23C4×C11×ΓL(1,𝔽8)—351no—
24C2×C5×C7×C11—61no—
25C2×C5×C7×C11C2×ΓL(1,𝔽8)11yescharacteristic
26C2×C5×C7×C11C3×D711yescharacteristic
27C5×C7×ΓL(1,𝔽8)C4×C1111yescharacteristic, complemented, Hall
28C3×C11×Dic5—491noHall
29ΓL(1,𝔽8)×Dic7—51noHall
30C72×C11C3×Dic511yescharacteristic, complemented, Hall
31C2×C5×C72C2×C3×C1111yescharacteristic
32C2×C3×C7×C11—71no—
33C2×C11×ΓL(1,𝔽8)D3511yescharacteristic
34C3×Dic35—71no—
35Dic5×ΓL(1,𝔽8)—71no—
36C5×C7×C11—61no—
37C5×C7×C11C4×ΓL(1,𝔽8)11yescharacteristic, complemented
38C5×C7×C11C3×Dic711yescharacteristic, complemented
39C2×C3×C5×C11—71no—
40C4×C7×C11—351no—
41C11×Dic7—51no—
42C2×C7×ΓL(1,𝔽8)D5×C1111yescharacteristic
43C5×C72C3×C4×C1111yescharacteristic, complemented, Hall
44C3×C7×C11—71no—
45C11×ΓL(1,𝔽8)Dic3511yescharacteristic, direct product factor
46C11×Dic5—71noHall
47C2×C3×C5×C7—71no—
48C2×C5×ΓL(1,𝔽8)C11×D711yescharacteristic
49C7×Dic7—51noHall
50C3×C5×C11—71noHall
51C2×C7×C11—61no—
52C2×C7×C11D5×ΓL(1,𝔽8)11yescharacteristic
53C2×C7×C11C3×D3511yescharacteristic
54C7×ΓL(1,𝔽8)C11×Dic511yescharacteristic, complemented, Hall
55Dic35C11×ΓL(1,𝔽8)11yescharacteristic, direct product factor
56C7×Dic5—71no—
57C3×C4×C11—2451noHall
58C2×C5×C11D7×ΓL(1,𝔽8)11yescharacteristic
59C3×C5×C7—71no—
60C5×ΓL(1,𝔽8)C11×Dic711yescharacteristic, complemented
61C2×C72C3×D5×C1111yescharacteristic
62C3×Dic7—351no—
63C4×ΓL(1,𝔽8)—351no—
64C7×C11—61no—
65C7×C11Dic5×ΓL(1,𝔽8)11yescharacteristic, complemented
66C7×C11C3×Dic3511yescharacteristic, complemented
67C2×C5×C7—61no—
68C2×C5×C7C2×C11×ΓL(1,𝔽8)11yescharacteristic
69C2×C5×C7C3×C11×D711yescharacteristic
70C2×C3×C11—71no—
71C3×Dic5—491noHall
72C5×C11ΓL(1,𝔽8)×Dic711yescharacteristic, complemented, Hall
73C72C3×C11×Dic511yescharacteristic, complemented, Sylow
74C4×C11—351noHall
75C2×C3×C7—71no—
76C2×ΓL(1,𝔽8)C11×D3511yescharacteristic
77C5×C7—61no—
78C5×C7C4×C11×ΓL(1,𝔽8)11yescharacteristic, complemented
79C5×C7C3×C11×Dic711yescharacteristic, complemented
80C3×C11—71noHall
81C2×C3×C5—71no—
82C4×C7—351no—
83Dic7—51no—
84C2×C11ΓL(1,𝔽8)×D3511yescharacteristic
85C3×C7—71no—
86ΓL(1,𝔽8)C11×Dic3511yescharacteristic, direct product factor
87Dic5—71noHall
88C3×C5—71noHall
89C2×C7—61no—
90C2×C7C3×C11×D3511yescharacteristic
91C2×C7D5×C11×ΓL(1,𝔽8)11yescharacteristic
92C3×C4—2451noHall
93C11ΓL(1,𝔽8)×Dic3511yescharacteristic, direct product factor, Sylow
94C2×C5C11×D7×ΓL(1,𝔽8)11yescharacteristic
95C7—61no—
96C7C3×C11×Dic3511yescharacteristic, complemented
97C7C11×Dic5×ΓL(1,𝔽8)11yescharacteristic, complemented
98C2×C3—71no—
99C5C11×ΓL(1,𝔽8)×Dic711yescharacteristic, complemented, Sylow
100C4—351noSylow
101C3—71noSylow
102C2C11×ΓL(1,𝔽8)×D3511yescharacteristic
subgroupG/H
center, hypercenterC2×C11ΓL(1,𝔽8)×D35
commutator, hypocenterC5×C72C3×C4×C11
FrattiniC2C11×ΓL(1,𝔽8)×D35
Fitting, socleC2×C5×C72×C11C2×C3
prime radicalC2×C5×C7×C11×ΓL(1,𝔽8)C2
cosocleC2×C5×C72C2×C3×C11
LCSG ▷ C5×C72
UCSC1 ◁ C2×C11

Complements

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

NH ≅ G/Ntype
2. C7×C11×Dic35C3semidirect
3. C5×C7×C11×ΓL(1,𝔽8)C4semidirect
9. ΓL(1,𝔽8)×Dic35C11direct (N × H)
10. C5×C72×C11C3×C4semidirect
14. C7×C11×ΓL(1,𝔽8)Dic5semidirect
15. C11×Dic35ΓL(1,𝔽8)direct (N × H)
19. C5×C11×ΓL(1,𝔽8)Dic7semidirect
21. C7×Dic35C3×C11semidirect
27. C5×C7×ΓL(1,𝔽8)C4×C11semidirect
30. C72×C11C3×Dic5semidirect
37. C5×C7×C11C4×ΓL(1,𝔽8)semidirect
38. C5×C7×C11C3×Dic7semidirect
43. C5×C72C3×C4×C11semidirect
45. C11×ΓL(1,𝔽8)Dic35direct (N × H)
54. C7×ΓL(1,𝔽8)C11×Dic5semidirect
55. Dic35C11×ΓL(1,𝔽8)direct (N × H)
60. C5×ΓL(1,𝔽8)C11×Dic7semidirect
65. C7×C11Dic5×ΓL(1,𝔽8)semidirect
66. C7×C11C3×Dic35semidirect
72. C5×C11ΓL(1,𝔽8)×Dic7semidirect
73. C72C3×C11×Dic5semidirect
78. C5×C7C4×C11×ΓL(1,𝔽8)semidirect
79. C5×C7C3×C11×Dic7semidirect
86. ΓL(1,𝔽8)C11×Dic35direct (N × H)
93. C11ΓL(1,𝔽8)×Dic35direct (N × H)
96. C7C3×C11×Dic35semidirect
97. C7C11×Dic5×ΓL(1,𝔽8)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
C2×C5×C7×C11×ΓL(1,𝔽8)2
C7×C11×Dic353
C11×ΓL(1,𝔽8)×Dic75
C11×Dic5×ΓL(1,𝔽8)7
C3×C11×Dic357
ΓL(1,𝔽8)×Dic3511