Finite Group Databasebeta

← C5×ΓL(1,𝔽8)×Dic77 #44568487

Subgroups — C5×ΓL(1,𝔽8)×Dic77 #44568487

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
2C5×C7×Dic77C311yesmaximal, characteristic, complemented, Hall
3C5×C7×C11×ΓL(1,𝔽8)C411yescharacteristic, complemented, Hall
4ΓL(1,𝔽8)×Dic77C511yesmaximal, characteristic, direct product factor, Hall
5C2×C5×C72×C11C2×C311yescharacteristic
6C5×ΓL(1,𝔽8)×Dic11—71nomaximal
7C3×C5×Dic77—71nomaximal
8C2×C7×C11×ΓL(1,𝔽8)C2×C511yescharacteristic
9C5×ΓL(1,𝔽8)×Dic7—111nomaximal, Hall
10C5×C72×C11C3×C411yescharacteristic, complemented, Hall
11C2×C5×C11×ΓL(1,𝔽8)D711yescharacteristic
12C2×C3×C5×C7×C11—71no—
13C7×Dic77C3×C511yescharacteristic, complemented, Hall
14C7×C11×ΓL(1,𝔽8)C4×C511yescharacteristic, complemented, Hall
15C5×Dic77ΓL(1,𝔽8)11yescharacteristic, direct product factor
16C5×C7×Dic11—71no—
17C2×C5×C7×ΓL(1,𝔽8)D1111yescharacteristic
18C3×C5×C7×C11—71no—
19C5×C11×ΓL(1,𝔽8)Dic711yescharacteristic, complemented
20C2×C72×C11C2×C3×C511yescharacteristic
21C5×C7×Dic7—111noHall
22C3×Dic77—71no—
23ΓL(1,𝔽8)×Dic11—71no—
24C2×C5×C7×C11—61no—
25C2×C5×C7×C11C2×ΓL(1,𝔽8)11yescharacteristic
26C2×C5×C7×C11C3×D711yescharacteristic
27C5×C7×ΓL(1,𝔽8)Dic1111yescharacteristic, complemented, Hall
28C3×C5×Dic11—491noHall
29ΓL(1,𝔽8)×Dic7—111noHall
30C72×C11C3×C4×C511yescharacteristic, complemented, Hall
31C2×C5×C72C3×D1111yescharacteristic
32C2×C3×C7×C11—71no—
33C2×C11×ΓL(1,𝔽8)C5×D711yescharacteristic
34C3×C5×Dic7—771no—
35C4×C5×ΓL(1,𝔽8)—771no—
36C5×C7×C11—61no—
37C5×C7×C11C4×ΓL(1,𝔽8)11yescharacteristic, complemented
38C5×C7×C11C3×Dic711yescharacteristic, complemented
39C2×C3×C5×C11—71no—
40Dic77C5×ΓL(1,𝔽8)11yescharacteristic, direct product factor
41C7×Dic11—71no—
42C2×C7×ΓL(1,𝔽8)C5×D1111yescharacteristic
43C5×C72C3×Dic1111yescharacteristic, complemented, Hall
44C3×C7×C11—71no—
45C11×ΓL(1,𝔽8)C5×Dic711yescharacteristic, complemented
46C5×Dic11—71noHall
47C2×C3×C5×C7—71no—
48C2×C5×ΓL(1,𝔽8)D7711yescharacteristic
49C7×Dic7—111noHall
50C3×C5×C11—71noHall
51C2×C7×C11—61no—
52C2×C7×C11C2×C5×ΓL(1,𝔽8)11yescharacteristic
53C2×C7×C11C3×C5×D711yescharacteristic
54C7×ΓL(1,𝔽8)C5×Dic1111yescharacteristic, complemented, Hall
55C4×C5×C7—771no—
56C5×Dic7—111no—
57C3×Dic11—491noHall
58C2×C5×C11D7×ΓL(1,𝔽8)11yescharacteristic
59C3×C5×C7—71no—
60C5×ΓL(1,𝔽8)Dic7711yescharacteristic, direct product factor
61C2×C72C3×C5×D1111yescharacteristic
62C3×Dic7—771no—
63C4×ΓL(1,𝔽8)—771no—
64C7×C11—61no—
65C7×C11C4×C5×ΓL(1,𝔽8)11yescharacteristic, complemented
66C7×C11C3×C5×Dic711yescharacteristic, complemented
67C2×C5×C7—61no—
68C2×C5×C7ΓL(1,𝔽8)×D1111yescharacteristic
69C2×C5×C7C3×D7711yescharacteristic
70C2×C3×C11—71no—
71C3×C4×C5—5391noHall
72C5×C11ΓL(1,𝔽8)×Dic711yescharacteristic, complemented, Hall
73C72C3×C5×Dic1111yescharacteristic, complemented, Sylow
74Dic11—71noHall
75C2×C3×C7—71no—
76C2×ΓL(1,𝔽8)C5×D7711yescharacteristic
77C5×C7—61no—
78C5×C7ΓL(1,𝔽8)×Dic1111yescharacteristic, complemented
79C5×C7C3×Dic7711yescharacteristic, complemented
80C3×C11—71noHall
81C2×C3×C5—71no—
82C4×C7—771no—
83Dic7—111no—
84C2×C11C5×D7×ΓL(1,𝔽8)11yescharacteristic
85C3×C7—71no—
86ΓL(1,𝔽8)C5×Dic7711yescharacteristic, direct product factor
87C4×C5—771noHall
88C3×C5—71noHall
89C2×C7—61no—
90C2×C7C3×C5×D7711yescharacteristic
91C2×C7C5×ΓL(1,𝔽8)×D1111yescharacteristic
92C3×C4—5391noHall
93C11C5×ΓL(1,𝔽8)×Dic711yescharacteristic, complemented, Sylow
94C2×C5ΓL(1,𝔽8)×D7711yescharacteristic
95C7—61no—
96C7C3×C5×Dic7711yescharacteristic, complemented
97C7C5×ΓL(1,𝔽8)×Dic1111yescharacteristic, complemented
98C2×C3—71no—
99C5ΓL(1,𝔽8)×Dic7711yescharacteristic, direct product factor, Sylow
100C4—771noSylow
101C3—71noSylow
102C2C5×ΓL(1,𝔽8)×D7711yescharacteristic
subgroupG/H
center, hypercenterC2×C5ΓL(1,𝔽8)×D77
commutator, hypocenterC72×C11C3×C4×C5
FrattiniC2C5×ΓL(1,𝔽8)×D77
Fitting, socleC2×C5×C72×C11C2×C3
prime radicalC2×C5×C7×C11×ΓL(1,𝔽8)C2
cosocleC2×C72×C11C2×C3×C5
LCSG ▷ C72×C11
UCSC1 ◁ C2×C5

Complements

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

NH ≅ G/Ntype
2. C5×C7×Dic77C3semidirect
3. C5×C7×C11×ΓL(1,𝔽8)C4semidirect
4. ΓL(1,𝔽8)×Dic77C5direct (N × H)
10. C5×C72×C11C3×C4semidirect
13. C7×Dic77C3×C5semidirect
14. C7×C11×ΓL(1,𝔽8)C4×C5semidirect
15. C5×Dic77ΓL(1,𝔽8)direct (N × H)
19. C5×C11×ΓL(1,𝔽8)Dic7semidirect
27. C5×C7×ΓL(1,𝔽8)Dic11semidirect
30. C72×C11C3×C4×C5semidirect
37. C5×C7×C11C4×ΓL(1,𝔽8)semidirect
38. C5×C7×C11C3×Dic7semidirect
40. Dic77C5×ΓL(1,𝔽8)direct (N × H)
43. C5×C72C3×Dic11semidirect
45. C11×ΓL(1,𝔽8)C5×Dic7semidirect
54. C7×ΓL(1,𝔽8)C5×Dic11semidirect
60. C5×ΓL(1,𝔽8)Dic77direct (N × H)
65. C7×C11C4×C5×ΓL(1,𝔽8)semidirect
66. C7×C11C3×C5×Dic7semidirect
72. C5×C11ΓL(1,𝔽8)×Dic7semidirect
73. C72C3×C5×Dic11semidirect
78. C5×C7ΓL(1,𝔽8)×Dic11semidirect
79. C5×C7C3×Dic77semidirect
86. ΓL(1,𝔽8)C5×Dic77direct (N × H)
93. C11C5×ΓL(1,𝔽8)×Dic7semidirect
96. C7C3×C5×Dic77semidirect
97. C7C5×ΓL(1,𝔽8)×Dic11semidirect
99. C5ΓL(1,𝔽8)×Dic77direct (N × H)

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
C5×C7×Dic773
ΓL(1,𝔽8)×Dic775
C5×ΓL(1,𝔽8)×Dic117
C3×C5×Dic777
C5×ΓL(1,𝔽8)×Dic711