Finite Group Databasebeta

← C5×C11×ΓL(1,𝔽8)⋊Dic7 #44568482

Subgroups — C5×C11×ΓL(1,𝔽8)⋊Dic7 #44568482

Order 32340.

120 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 120 conjugacy classes of subgroups — too many to draw. Only its 56 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×C11×C7⋊Dic7C311yesmaximal, characteristic, complemented, Hall
3C5×C7×C11×ΓL(1,𝔽8)C411yescharacteristic, complemented, Hall
4C11×ΓL(1,𝔽8)⋊Dic7C511yesmaximal, characteristic, direct product factor, Hall
5C2×C5×C72×C11C2×C311yescharacteristic
6C5×C11×ΓL(1,𝔽8)⋊C4—71nomaximal
7C3×C5×C11×Dic7—71nomaximal
8C2×C7×C11×ΓL(1,𝔽8)C2×C511yescharacteristic
9C5×ΓL(1,𝔽8)⋊Dic7C1111yesmaximal, characteristic, direct product factor, Hall
10C5×C72×C11C3×C411yescharacteristic, complemented, Hall
11C2×C5×C11×ΓL(1,𝔽8)D711yescharacteristic
12C2×C3×C5×C7×C11—71no—
13C11×C7⋊Dic7C3×C511yescharacteristic, complemented, Hall
14C7×C11×ΓL(1,𝔽8)C4×C511yescharacteristic, complemented, Hall
15C5×C11×Dic7—71no—
16C5×C11×Dic7—71no—
17C5×C11×Dic7—212no—
18C2×C5×C7×ΓL(1,𝔽8)C2×C1111yescharacteristic
19C3×C5×C7×C11—71no—
20C5×C11×ΓL(1,𝔽8)Dic711yescharacteristic, complemented
21C2×C72×C11C2×C3×C511yescharacteristic
22C5×C7⋊Dic7C3×C1111yescharacteristic, complemented, Hall
23C3×C11×Dic7—71no—
24C11×ΓL(1,𝔽8)⋊C4—71no—
25C2×C5×C7×C11—32no—
26C2×C5×C7×C11AGL(1,𝔽7)11yescharacteristic
27C2×C5×C7×C11C3×D711yescharacteristic
28C5×C7×ΓL(1,𝔽8)C4×C1111yescharacteristic, complemented, Hall
29C3×C4×C5×C11—491noHall
30ΓL(1,𝔽8)⋊Dic7C5×C1111yescharacteristic, direct product factor, Hall
31C72×C11C3×C4×C511yescharacteristic, complemented, Hall
32C2×C5×C72C2×C3×C1111yescharacteristic
33C2×C3×C7×C11—71no—
34C2×C11×ΓL(1,𝔽8)C5×D711yescharacteristic
35C3×C5×Dic7—71no—
36C5×ΓL(1,𝔽8)⋊C4—71no—
37C5×C7×C11—32no—
38C5×C7×C11ΓL(1,𝔽8)⋊C411yescharacteristic, complemented
39C5×C7×C11C3×Dic711yescharacteristic, complemented
40C2×C3×C5×C11—71no—
41C11×Dic7—71no—
42C11×Dic7—71no—
43C11×Dic7—212no—
44C2×C7×ΓL(1,𝔽8)C2×C5×C1111yescharacteristic
45C5×C72C3×C4×C1111yescharacteristic, complemented, Hall
46C3×C7×C11—71no—
47C11×ΓL(1,𝔽8)C5×Dic711yescharacteristic, complemented
48C4×C5×C11—491noHall
49C2×C3×C5×C7—71no—
50C2×C5×ΓL(1,𝔽8)C11×D711yescharacteristic
51C7⋊Dic7C3×C5×C1111yescharacteristic, complemented, Hall
52C3×C5×C11—71noHall
53C2×C7×C11—32no—
54C2×C7×C11C5×AGL(1,𝔽7)11yescharacteristic
55C2×C7×C11C3×C5×D711yescharacteristic
56C7×ΓL(1,𝔽8)C4×C5×C1111yescharacteristic, complemented, Hall
57C5×Dic7—71no—
58C5×Dic7—71no—
59C5×Dic7—212no—
60C3×C4×C11—491noHall
61C2×C5×C11ΓL(1,𝔽8)⋊D711yescharacteristic
62C3×C5×C7—71no—
63C5×ΓL(1,𝔽8)C11×Dic711yescharacteristic, complemented
64C2×C72C2×C3×C5×C1111yescharacteristic
65C3×Dic7—71no—
66ΓL(1,𝔽8)⋊C4—71no—
67C7×C11—32no—
68C7×C11C5×ΓL(1,𝔽8)⋊C411yescharacteristic, complemented
69C7×C11C3×C5×Dic711yescharacteristic, complemented
70C2×C5×C7—32no—
71C2×C5×C7C11×AGL(1,𝔽7)11yescharacteristic
72C2×C5×C7C3×C11×D711yescharacteristic
73C2×C3×C11—71no—
74C3×C4×C5—491noHall
75C5×C11ΓL(1,𝔽8)⋊Dic711yescharacteristic, direct product factor, Hall
76C72C3×C4×C5×C1111yescharacteristic, complemented, Sylow
77C4×C11—491noHall
78C2×C3×C7—71no—
79C2×ΓL(1,𝔽8)C5×C11×D711yescharacteristic
80C5×C7—32no—
81C5×C7C11×ΓL(1,𝔽8)⋊C411yescharacteristic, complemented
82C5×C7C3×C11×Dic711yescharacteristic, complemented
83C3×C11—71noHall
84C2×C3×C5—71no—
85Dic7—71no—
86Dic7—71no—
87Dic7—212no—
88C2×C11C5×ΓL(1,𝔽8)⋊D711yescharacteristic
89C3×C7—71no—
90ΓL(1,𝔽8)C5×C11×Dic711yescharacteristic, complemented
91C4×C5—491noHall
92C3×C5—71noHall
93C2×C7—32no—
94C2×C7C3×C5×C11×D711yescharacteristic
95C2×C7C5×C11×AGL(1,𝔽7)11yescharacteristic
96C3×C4—491noHall
97C11C5×ΓL(1,𝔽8)⋊Dic711yescharacteristic, direct product factor, Sylow
98C2×C5C11×ΓL(1,𝔽8)⋊D711yescharacteristic
99C7—32no—
100C7C3×C5×C11×Dic711yescharacteristic, complemented
101C7C5×C11×ΓL(1,𝔽8)⋊C411yescharacteristic, complemented
102C2×C3—71no—
103C5C11×ΓL(1,𝔽8)⋊Dic711yescharacteristic, direct product factor, Sylow
104C4—491noSylow
105C3—71noSylow
106C2C5×C11×ΓL(1,𝔽8)⋊D711yescharacteristic
subgroupG/H
center, hypercenterC2×C5×C11ΓL(1,𝔽8)⋊D7
commutator, hypocenterC72C3×C4×C5×C11
FrattiniC2C5×C11×ΓL(1,𝔽8)⋊D7
Fitting, socleC2×C5×C72×C11C2×C3
prime radicalC2×C5×C7×C11×ΓL(1,𝔽8)C2
cosocleC2×C72C2×C3×C5×C11
LCSG ▷ C72
UCSC1 ◁ C2×C5×C11

Complements

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

NH ≅ G/Ntype
2. C5×C11×C7⋊Dic7C3semidirect
3. C5×C7×C11×ΓL(1,𝔽8)C4semidirect
4. C11×ΓL(1,𝔽8)⋊Dic7C5direct (N × H)
9. C5×ΓL(1,𝔽8)⋊Dic7C11direct (N × H)
10. C5×C72×C11C3×C4semidirect
13. C11×C7⋊Dic7C3×C5semidirect
14. C7×C11×ΓL(1,𝔽8)C4×C5semidirect
20. C5×C11×ΓL(1,𝔽8)Dic7semidirect
22. C5×C7⋊Dic7C3×C11semidirect
28. C5×C7×ΓL(1,𝔽8)C4×C11semidirect
30. ΓL(1,𝔽8)⋊Dic7C5×C11direct (N × H)
31. C72×C11C3×C4×C5semidirect
38. C5×C7×C11ΓL(1,𝔽8)⋊C4semidirect
39. C5×C7×C11C3×Dic7semidirect
45. C5×C72C3×C4×C11semidirect
47. C11×ΓL(1,𝔽8)C5×Dic7semidirect
51. C7⋊Dic7C3×C5×C11semidirect
56. C7×ΓL(1,𝔽8)C4×C5×C11semidirect
63. C5×ΓL(1,𝔽8)C11×Dic7semidirect
68. C7×C11C5×ΓL(1,𝔽8)⋊C4semidirect
69. C7×C11C3×C5×Dic7semidirect
75. C5×C11ΓL(1,𝔽8)⋊Dic7direct (N × H)
76. C72C3×C4×C5×C11semidirect
81. C5×C7C11×ΓL(1,𝔽8)⋊C4semidirect
82. C5×C7C3×C11×Dic7semidirect
90. ΓL(1,𝔽8)C5×C11×Dic7semidirect
97. C11C5×ΓL(1,𝔽8)⋊Dic7direct (N × H)
100. C7C3×C5×C11×Dic7semidirect
101. C7C5×C11×ΓL(1,𝔽8)⋊C4semidirect
103. C5C11×ΓL(1,𝔽8)⋊Dic7direct (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×C11×C7⋊Dic73
C11×ΓL(1,𝔽8)⋊Dic75
C5×C11×ΓL(1,𝔽8)⋊C47
C3×C5×C11×Dic77
C5×ΓL(1,𝔽8)⋊Dic711