Finite Group Databasebeta

← C22×C3×Dic7⋊C4 #13061125

Subgroups — C22×C3×Dic7⋊C4 #13061125

Order 1344.

836 conjugacy classes of subgroups; only the 494 normal subgroups are shown.

normal subgroups only

This group has 836 conjugacy classes of subgroups — too many to draw. Only its 494 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×Dic7C211yesmaximal, characteristic
3C23×C3×Dic7C211yesmaximal, characteristic
4C2×C3×Dic7⋊C4C2112yesmaximal, direct product factor
5C22×Dic7⋊C4C311yesmaximal, characteristic, direct product factor, Hall
6C24×C3×C7C2211yescharacteristic
7C22×C3×C4×C7—24no—
8C22×C3×C4×C7C2216yes—
9C22×C3×Dic7—24no—
10C22×C3×Dic7C418yescomplemented
11C22×C3×Dic7C2216yes—
12C22×C3×Dic7C2216yes—
13C3×Dic7⋊C4C22116yesdirect product factor
14C23×C4×C7C2×C311yescharacteristic
15C23×Dic7C2×C311yescharacteristic
16C23×Dic7C2×C311yescharacteristic
17C2×Dic7⋊C4C2×C3112yesdirect product factor
18C22×C3×C4⋊C4—71nomaximal, Hall
19C23×C3×C7C2×C414yes—
20C23×C3×C7D414yes—
21C23×C3×C7Q814yes—
22C23×C3×C7C2313yes—
23C2×C3×C4×C7—212no—
24C2×C3×C4×C7C2314yes—
25C2×C3×Dic7—212no—
26C2×C3×Dic7C2×C4124yescomplemented
27C2×C3×Dic7C2314yes—
28C2×C3×Dic7C2314yes—
29C24×C7C22×C311yescharacteristic
30C22×C4×C7—24no—
31C22×C4×C7C22×C316yes—
32C22×Dic7—24no—
33C22×Dic7C3×C418yescomplemented
34C22×Dic7C22×C316yes—
35C22×Dic7C22×C316yes—
36Dic7⋊C4C22×C3116yesdirect product factor
37C23×C3×C4—71no—
38C23×C3×C4—71no—
39C23×C3×C4D711yescharacteristic
40C2×C3×C4⋊C4—712no—
41C22×C3×C7C4⋊C4116yescomplemented
42C22×C3×C7C22×C416yes—
43C22×C3×C7C2×D416yes—
44C22×C3×C7C2×Q816yes—
45C22×C3×C7C2411yescharacteristic
46C3×C4×C7—24no—
47C3×Dic7—24no—
48C3×Dic7C22×C418yescomplemented
49C22×C4⋊C4—71noSylow
50C23×C7C2×C3×C414yes—
51C23×C7C3×D414yes—
52C23×C7C3×Q814yes—
53C23×C7C23×C313yes—
54C2×C4×C7—212no—
55C2×C4×C7C23×C314yes—
56C2×Dic7—212no—
57C2×Dic7C2×C3×C4124yescomplemented
58C2×Dic7C23×C314yes—
59C2×Dic7C23×C314yes—
60C24×C3C2×D711yescharacteristic
61C22×C3×C4—24no—
62C22×C3×C4—78no—
63C22×C3×C4—76no—
64C22×C3×C4—76no—
65C22×C3×C4—144no—
66C22×C3×C4C2×D716yes—
67C3×C4⋊C4—716no—
68C2×C3×C7C2×C4⋊C4112yescomplemented
69C2×C3×C7C23×C411yescharacteristic
70C2×C3×C7C22×D411yescharacteristic
71C2×C3×C7C22×Q811yescharacteristic
72C23×C4—71no—
73C23×C4—71no—
74C23×C4C3×D711yescharacteristic
75C2×C4⋊C4—712no—
76C22×C7C3×C4⋊C4116yescomplemented
77C22×C7C22×C3×C416yes—
78C22×C7C2×C3×D416yes—
79C22×C7C2×C3×Q816yes—
80C22×C7C24×C311yescharacteristic
81C4×C7—24no—
82Dic7—24no—
83Dic7C22×C3×C418yescomplemented
84C23×C3Dic1414yes—
85C23×C3C4×D714yes—
86C23×C3C22⋊D714yes—
87C23×C3C22×D713yes—
88C2×C3×C4—212no—
89C2×C3×C4—724no—
90C2×C3×C4—74no—
91C2×C3×C4—74no—
92C2×C3×C4—1412no—
93C2×C3×C4C22×D714yes—
94C3×C7C22×C4⋊C411yescharacteristic, complemented, Hall
95C24C2×C3×D711yescharacteristic
96C22×C4—24no—
97C22×C4—78no—
98C22×C4—76no—
99C22×C4—76no—
100C22×C4—144no—
101C22×C4C2×C3×D716yes—
102C4⋊C4—716no—
103C2×C7C2×C3×C4⋊C4112yescomplemented
104C2×C7C23×C3×C411yescharacteristic
105C2×C7C22×C3×D411yescharacteristic
106C2×C7C22×C3×Q811yescharacteristic
107C22×C3Dic7⋊C4116yesdirect product factor
108C22×C3C2×Dic1416yes—
109C22×C3C2×C4×D716yes—
110C22×C3C2×C22⋊D716yes—
111C22×C3C23×D711yescharacteristic
112C3×C4—24no—
113C3×C4—78no—
114C3×C4—144no—
115C23C3×Dic1414yes—
116C23C3×C4×D714yes—
117C23C3×C22⋊D714yes—
118C23C22×C3×D713yes—
119C2×C4—212no—
120C2×C4—724no—
121C2×C4—74no—
122C2×C4—74no—
123C2×C4—1412no—
124C2×C4C22×C3×D714yes—
125C7C22×C3×C4⋊C411yescharacteristic, complemented, Sylow
126C2×C3C2×Dic7⋊C4112yesdirect product factor
127C2×C3C22×Dic1411yescharacteristic
128C2×C3C22×C4×D711yescharacteristic
129C2×C3C22×C22⋊D711yescharacteristic
130C22C3×Dic7⋊C4116yesdirect product factor
131C22C2×C3×Dic1416yes—
132C22C2×C3×C4×D716yes—
133C22C2×C3×C22⋊D716yes—
134C22C23×C3×D711yescharacteristic
135C4—24no—
136C4—78no—
137C4—144no—
138C3C22×Dic7⋊C411yescharacteristic, direct product factor, Sylow
139C2C2×C3×Dic7⋊C4112yesdirect product factor
140C2C22×C3×Dic1411yescharacteristic
141C2C22×C3×C4×D711yescharacteristic
142C2C22×C3×C22⋊D711yescharacteristic
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
4. C2×C3×Dic7⋊C4C2direct (N × H)
5. C22×Dic7⋊C4C3direct (N × H)
10. C22×C3×Dic7C4semidirect
13. C3×Dic7⋊C4C22direct (N × H)
17. C2×Dic7⋊C4C2×C3direct (N × H)
26. C2×C3×Dic7C2×C4semidirect
33. C22×Dic7C3×C4semidirect
36. Dic7⋊C4C22×C3direct (N × H)
41. C22×C3×C7C4⋊C4semidirect
48. C3×Dic7C22×C4semidirect
57. C2×Dic7C2×C3×C4semidirect
68. C2×C3×C7C2×C4⋊C4semidirect
76. C22×C7C3×C4⋊C4semidirect
83. Dic7C22×C3×C4semidirect
94. C3×C7C22×C4⋊C4semidirect
103. C2×C7C2×C3×C4⋊C4semidirect
107. C22×C3Dic7⋊C4direct (N × H)
125. C7C22×C3×C4⋊C4semidirect
126. C2×C3C2×Dic7⋊C4direct (N × H)
130. C22C3×Dic7⋊C4direct (N × H)
138. C3C22×Dic7⋊C4direct (N × H)
139. C2C2×C3×Dic7⋊C4direct (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×Dic7⋊C42
C22×Dic7⋊C43
C22×C3×C4⋊C47