Finite Group Databasebeta

← C3×C7×ΓL(1,𝔽8)⋊Dic13 #37524701

Subgroups — C3×C7×ΓL(1,𝔽8)⋊Dic13 #37524701

Order 22932.

152 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 152 conjugacy classes of subgroups — too many to draw. Only its 76 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×C3×C7×C13×ΓL(1,𝔽8)C211yesmaximal, characteristic
2C3×C7×Dic91C311yesmaximal, characteristic, complemented
3C7×ΓL(1,𝔽8)⋊Dic13C313yesmaximal, direct product factor
4C3×C7×C13×ΓL(1,𝔽8)C411yescharacteristic, complemented, Hall
5C2×C7×C13×ΓL(1,𝔽8)C2×C313yes—
6C2×C3×C72×C13C2×C311yescharacteristic
7C32×C7×Dic13—71nomaximal
8C3×ΓL(1,𝔽8)⋊Dic13C711yesmaximal, characteristic, direct product factor
9C7×Dic91C3211yescharacteristic, complemented, Hall
10C3×C72×C13C3×C411yescharacteristic, complemented
11C7×C13×ΓL(1,𝔽8)C3×C413yescomplemented
12C3×C7×ΓL(1,𝔽8)⋊C4—131nomaximal, Hall
13C2×C32×C7×C13—71no—
14C2×C3×C13×ΓL(1,𝔽8)C2×C711yescharacteristic
15C2×C72×C13C2×C3211yescharacteristic
16C3×Dic91C3×C711yescharacteristic, complemented
17C3×C7×Dic13—73no—
18C3×C7×Dic13—211no—
19ΓL(1,𝔽8)⋊Dic13C3×C713yesdirect product factor
20C2×C3×C7×ΓL(1,𝔽8)D1311yescharacteristic
21C32×C7×C13—71no—
22C3×C13×ΓL(1,𝔽8)C4×C711yescharacteristic, complemented
23C72×C13C32×C411yescharacteristic, complemented, Hall
24C3×C7×Dic7—131no—
25C7×ΓL(1,𝔽8)⋊C4—133no—
26C2×C3×C7×C13—61no—
27C2×C3×C7×C13—73no—
28C2×C3×C7×C13AGL(1,𝔽7)11yescharacteristic
29C2×C3×C7×C13C2×C3×C711yescharacteristic
30C2×C13×ΓL(1,𝔽8)C2×C3×C713yes—
31C32×Dic13—71noHall
32C3×C7×ΓL(1,𝔽8)Dic1311yescharacteristic, complemented, Hall
33Dic91C32×C711yescharacteristic, complemented
34C7×Dic13—211no—
35C2×C3×C72C3×D1311yescharacteristic
36C2×C7×ΓL(1,𝔽8)C3×D1313yes—
37C3×C7×C13—61no—
38C3×C7×C13—73no—
39C3×C7×C13ΓL(1,𝔽8)⋊C411yescharacteristic, complemented
40C3×C7×C13C3×C4×C711yescharacteristic, complemented
41C13×ΓL(1,𝔽8)C3×C4×C713yescomplemented
42C32×C4×C7—911no—
43C3×ΓL(1,𝔽8)⋊C4—131no—
44C2×C32×C13—71no—
45C7×Dic7—131noHall
46C2×C7×C13—61no—
47C2×C7×C13C3×AGL(1,𝔽7)11yescharacteristic
48C2×C7×C13C2×C32×C711yescharacteristic
49C3×Dic13—73no—
50C3×Dic13—71no—
51C3×C72C3×Dic1311yescharacteristic, complemented
52C7×ΓL(1,𝔽8)C3×Dic1313yescomplemented
53C2×C32×C7—71no—
54C2×C3×ΓL(1,𝔽8)C7×D1311yescharacteristic
55C32×C13—71noHall
56C2×C72C32×D1311yescharacteristic
57C7×C13—61no—
58C7×C13C3×ΓL(1,𝔽8)⋊C411yescharacteristic, complemented
59C7×C13C32×C4×C711yescharacteristic, complemented
60C3×C4×C7—2731no—
61C3×C4×C7—913no—
62C3×Dic7—131no—
63ΓL(1,𝔽8)⋊C4—133no—
64C2×C3×C13—73no—
65C2×C3×C13C7×AGL(1,𝔽7)11yescharacteristic
66C32×C7—71no—
67C3×ΓL(1,𝔽8)C7×Dic1311yescharacteristic, complemented
68Dic13—71noHall
69C72C32×Dic1311yescharacteristic, complemented, Sylow
70C2×C3×C7—61no—
71C2×C3×C7—73no—
72C2×C3×C7ΓL(1,𝔽8)⋊D1311yescharacteristic
73C2×C3×C7C3×C7×D1311yescharacteristic
74C2×ΓL(1,𝔽8)C3×C7×D1313yes—
75C3×C13—73no—
76C3×C13C7×ΓL(1,𝔽8)⋊C411yescharacteristic, complemented
77C32×C4—911noHall
78C4×C7—2731no—
79Dic7—131no—
80C2×C13C3×C7×AGL(1,𝔽7)11yescharacteristic
81C3×C7—61no—
82C3×C7—73no—
83C3×C7ΓL(1,𝔽8)⋊Dic1311yescharacteristic, direct product factor
84C3×C7C3×C7×Dic1311yescharacteristic, complemented
85ΓL(1,𝔽8)C3×C7×Dic1313yescomplemented
86C2×C32—71no—
87C2×C7—61no—
88C2×C7C3×ΓL(1,𝔽8)⋊D1311yescharacteristic
89C2×C7C32×C7×D1311yescharacteristic
90C13C3×C7×ΓL(1,𝔽8)⋊C411yescharacteristic, complemented, Sylow
91C3×C4—913no—
92C3×C4—911no—
93C32—71noSylow
94C7—61no—
95C7C3×ΓL(1,𝔽8)⋊Dic1311yescharacteristic, direct product factor
96C7C32×C7×Dic1311yescharacteristic, complemented
97C2×C3—73no—
98C2×C3C7×ΓL(1,𝔽8)⋊D1311yescharacteristic
99C4—911noSylow
100C3—73no—
101C3C7×ΓL(1,𝔽8)⋊Dic1311yescharacteristic, direct product factor
102C2C3×C7×ΓL(1,𝔽8)⋊D1311yescharacteristic
subgroupG/H
center, hypercenterC2×C3×C7ΓL(1,𝔽8)⋊D13
commutator, hypocenterC7×C13C32×C4×C7
FrattiniC2C3×C7×ΓL(1,𝔽8)⋊D13
Fitting, socleC2×C3×C72×C13C2×C3
prime radicalC2×C3×C7×C13×ΓL(1,𝔽8)C2
cosocleC2×C7×C13C2×C32×C7
LCSG ▷ C7×C13
UCSC1 ◁ C2×C3×C7

Complements

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

NH ≅ G/Ntype
2. C3×C7×Dic91C3semidirect
3. C7×ΓL(1,𝔽8)⋊Dic13C3direct (N × H)
4. C3×C7×C13×ΓL(1,𝔽8)C4semidirect
8. C3×ΓL(1,𝔽8)⋊Dic13C7direct (N × H)
9. C7×Dic91C32semidirect
10. C3×C72×C13C3×C4semidirect
11. C7×C13×ΓL(1,𝔽8)C3×C4semidirect
16. C3×Dic91C3×C7semidirect
19. ΓL(1,𝔽8)⋊Dic13C3×C7direct (N × H)
22. C3×C13×ΓL(1,𝔽8)C4×C7semidirect
23. C72×C13C32×C4semidirect
32. C3×C7×ΓL(1,𝔽8)Dic13semidirect
33. Dic91C32×C7semidirect
39. C3×C7×C13ΓL(1,𝔽8)⋊C4semidirect
40. C3×C7×C13C3×C4×C7semidirect
41. C13×ΓL(1,𝔽8)C3×C4×C7semidirect
51. C3×C72C3×Dic13semidirect
52. C7×ΓL(1,𝔽8)C3×Dic13semidirect
58. C7×C13C3×ΓL(1,𝔽8)⋊C4semidirect
59. C7×C13C32×C4×C7semidirect
67. C3×ΓL(1,𝔽8)C7×Dic13semidirect
69. C72C32×Dic13semidirect
76. C3×C13C7×ΓL(1,𝔽8)⋊C4semidirect
83. C3×C7ΓL(1,𝔽8)⋊Dic13direct (N × H)
84. C3×C7C3×C7×Dic13semidirect
85. ΓL(1,𝔽8)C3×C7×Dic13semidirect
90. C13C3×C7×ΓL(1,𝔽8)⋊C4semidirect
95. C7C3×ΓL(1,𝔽8)⋊Dic13direct (N × H)
96. C7C32×C7×Dic13semidirect
101. C3C7×ΓL(1,𝔽8)⋊Dic13direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×C7×C13×ΓL(1,𝔽8)2
C3×C7×Dic913
C7×ΓL(1,𝔽8)⋊Dic133
C32×C7×Dic137
C3×ΓL(1,𝔽8)⋊Dic137
C3×C7×ΓL(1,𝔽8)⋊C413