Finite Group Databasebeta

← C22×C3×C4⋊Dic7 #13061127

Subgroups — C22×C3×C4⋊Dic7 #13061127

Order 1344.

836 conjugacy classes of subgroups; only the 574 normal subgroups are shown.

normal subgroups only

This group has 836 conjugacy classes of subgroups — too many to draw. Only its 574 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
1C23×C3×C4×C7C211yesmaximal, characteristic
2C23×C3×Dic7C212yesmaximal
3C2×C3×C4⋊Dic7C2112yesmaximal, direct product factor
4C22×C4⋊Dic7C311yesmaximal, characteristic, direct product factor, Hall
5C24×C3×C7C2211yescharacteristic
6C22×C3×C4×C7C418yescomplemented
7C22×C3×C4×C7C2216yes—
8C22×C3×Dic7—28no—
9C22×C3×Dic7C22112yes—
10C3×C4⋊Dic7C22116yesdirect product factor
11C23×C4×C7C2×C311yescharacteristic
12C23×Dic7C2×C312yes—
13C2×C4⋊Dic7C2×C3112yesdirect product factor
14C22×C3×C4⋊C4—71nomaximal, Hall
15C23×C3×C7C2×C414yes—
16C23×C3×C7D414yes—
17C23×C3×C7Q814yes—
18C23×C3×C7C2313yes—
19C2×C3×C4×C7C2×C4124yescomplemented
20C2×C3×C4×C7C2314yes—
21C2×C3×Dic7—224no—
22C2×C3×Dic7C2318yes—
23C24×C7C22×C311yescharacteristic
24C22×C4×C7C3×C418yescomplemented
25C22×C4×C7C22×C316yes—
26C22×Dic7—28no—
27C22×Dic7C22×C3112yes—
28C4⋊Dic7C22×C3116yesdirect product factor
29C23×C3×C4—72no—
30C23×C3×C4D711yescharacteristic
31C2×C3×C4⋊C4—712no—
32C22×C3×C7C4⋊C4116yescomplemented
33C22×C3×C7C22×C416yes—
34C22×C3×C7C2×D416yes—
35C22×C3×C7C2×Q816yes—
36C22×C3×C7C2411yescharacteristic
37C3×C4×C7C22×C418yescomplemented
38C3×Dic7—28no—
39C22×C4⋊C4—71noSylow
40C23×C7C2×C3×C414yes—
41C23×C7C3×D414yes—
42C23×C7C3×Q814yes—
43C23×C7C23×C313yes—
44C2×C4×C7C2×C3×C4124yescomplemented
45C2×C4×C7C23×C314yes—
46C2×Dic7—224no—
47C2×Dic7C23×C318yes—
48C24×C3C2×D711yescharacteristic
49C22×C3×C4—712no—
50C22×C3×C4—148no—
51C22×C3×C4Dic718yescomplemented
52C22×C3×C4C2×D716yes—
53C3×C4⋊C4—716no—
54C2×C3×C7C2×C4⋊C4112yescomplemented
55C2×C3×C7C23×C411yescharacteristic
56C2×C3×C7C22×D411yescharacteristic
57C2×C3×C7C22×Q811yescharacteristic
58C23×C4—72no—
59C23×C4C3×D711yescharacteristic
60C2×C4⋊C4—712no—
61C22×C7C3×C4⋊C4116yescomplemented
62C22×C7C22×C3×C416yes—
63C22×C7C2×C3×D416yes—
64C22×C7C2×C3×Q816yes—
65C22×C7C24×C311yescharacteristic
66C4×C7C22×C3×C418yescomplemented
67Dic7—28no—
68C23×C3Dic1414yes—
69C23×C3D2814yes—
70C23×C3C2×Dic714yes—
71C23×C3C22×D713yes—
72C2×C3×C4—78no—
73C2×C3×C4—1424no—
74C2×C3×C4C2×Dic7124yescomplemented
75C2×C3×C4C22×D714yes—
76C3×C7C22×C4⋊C411yescharacteristic, complemented, Hall
77C24C2×C3×D711yescharacteristic
78C22×C4—712no—
79C22×C4—148no—
80C22×C4C3×Dic718yescomplemented
81C22×C4C2×C3×D716yes—
82C4⋊C4—716no—
83C2×C7C2×C3×C4⋊C4112yescomplemented
84C2×C7C23×C3×C411yescharacteristic
85C2×C7C22×C3×D411yescharacteristic
86C2×C7C22×C3×Q811yescharacteristic
87C22×C3C4⋊Dic7116yesdirect product factor
88C22×C3C2×Dic1416yes—
89C22×C3C2×D2816yes—
90C22×C3C22×Dic716yes—
91C22×C3C23×D711yescharacteristic
92C3×C4—148no—
93C3×C4C22×Dic718yescomplemented
94C23C3×Dic1414yes—
95C23C3×D2814yes—
96C23C2×C3×Dic714yes—
97C23C22×C3×D713yes—
98C2×C4—78no—
99C2×C4—1424no—
100C2×C4C2×C3×Dic7124yescomplemented
101C2×C4C22×C3×D714yes—
102C7C22×C3×C4⋊C411yescharacteristic, complemented, Sylow
103C2×C3C2×C4⋊Dic7112yesdirect product factor
104C2×C3C22×Dic1411yescharacteristic
105C2×C3C22×D2811yescharacteristic
106C2×C3C23×Dic711yescharacteristic
107C22C3×C4⋊Dic7116yesdirect product factor
108C22C2×C3×Dic1416yes—
109C22C2×C3×D2816yes—
110C22C22×C3×Dic716yes—
111C22C23×C3×D711yescharacteristic
112C4—148no—
113C4C22×C3×Dic718yescomplemented
114C3C22×C4⋊Dic711yescharacteristic, direct product factor, Sylow
115C2C2×C3×C4⋊Dic7112yesdirect product factor
116C2C22×C3×Dic1411yescharacteristic
117C2C22×C3×D2811yescharacteristic
118C2C23×C3×Dic711yescharacteristic
subgroupG/H
centerC24×C3C2×D7
commutatorC2×C7C23×C3×C4
FrattiniC22C23×C3×D7
FittingC23×C3×C4×C7C2
socle, prime radicalC24×C3×C7C22
cosocleC22×C7C24×C3
hypercenterC23×C3×C4D7
hypocenterC7C22×C3×C4⋊C4
LCSG ▷ C2×C7 ▷ C7
UCSC1 ◁ C24×C3 ◁ C23×C3×C4

Complements

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

NH ≅ G/Ntype
3. C2×C3×C4⋊Dic7C2direct (N × H)
4. C22×C4⋊Dic7C3direct (N × H)
6. C22×C3×C4×C7C4semidirect
10. C3×C4⋊Dic7C22direct (N × H)
13. C2×C4⋊Dic7C2×C3direct (N × H)
19. C2×C3×C4×C7C2×C4semidirect
24. C22×C4×C7C3×C4semidirect
28. C4⋊Dic7C22×C3direct (N × H)
32. C22×C3×C7C4⋊C4semidirect
37. C3×C4×C7C22×C4semidirect
44. C2×C4×C7C2×C3×C4semidirect
51. C22×C3×C4Dic7semidirect
54. C2×C3×C7C2×C4⋊C4semidirect
61. C22×C7C3×C4⋊C4semidirect
66. C4×C7C22×C3×C4semidirect
74. C2×C3×C4C2×Dic7semidirect
76. C3×C7C22×C4⋊C4semidirect
80. C22×C4C3×Dic7semidirect
83. C2×C7C2×C3×C4⋊C4semidirect
87. C22×C3C4⋊Dic7direct (N × H)
93. C3×C4C22×Dic7semidirect
100. C2×C4C2×C3×Dic7semidirect
102. C7C22×C3×C4⋊C4semidirect
103. C2×C3C2×C4⋊Dic7direct (N × H)
107. C22C3×C4⋊Dic7direct (N × H)
113. C4C22×C3×Dic7semidirect
114. C3C22×C4⋊Dic7direct (N × H)
115. C2C2×C3×C4⋊Dic7direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C3×C4×C72
C23×C3×Dic72
C2×C3×C4⋊Dic72
C22×C4⋊Dic73
C22×C3×C4⋊C47