Finite Group Databasebeta

← C2×D3×C7×Dic7 #72246331

Subgroups — C2×D3×C7×Dic7 #72246331

Order 2352.

312 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 112 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
1C22×D3×C72C211yesmaximal, characteristic
2C2×C7×Dic21C211yesmaximal, characteristic, complemented
3C2×C3×C7×Dic7C211yesmaximal, characteristic, complemented
4D3×C7×Dic7C214yesmaximal, direct product factor
5C22×C7×Dic7—31nomaximal, Hall
6C22×C3×C72C2211yescharacteristic
7C2×D3×C72C414yescomplemented
8C2×D3×C72C2212yes—
9C7×Dic21C2212yescomplemented
10C3×C7×Dic7C2212yescomplemented
11C23×C72—31no—
12C2×C7×Dic7—31no—
13C2×C7×Dic7—34no—
14C2×C7×Dic7D311yescharacteristic, direct product factor
15C2×C4×D3×C7—71nomaximal
16C2×D3×Dic7C711yesmaximal, characteristic, direct product factor
17C2×C3×C72C2×C412yescomplemented
18C2×C3×C72C2311yescharacteristic
19D3×C72C2×C414yescomplemented
20C22×C72—34no—
21C22×C72—32no—
22C22×C72C2×D311yescharacteristic
23C7×Dic7—32no—
24C7×Dic7C2×D312yesdirect product factor
25C22×D3×C7—23no—
26C22×D3×C7D711yescharacteristic
27C22×D3×C7C2×C711yescharacteristic
28C2×C3×C4×C7—71no—
29C2×Dic21C2×C711yescharacteristic, complemented
30C2×C7×Dic3—71no—
31C4×D3×C7—74no—
32C2×C3×Dic7C2×C711yescharacteristic, complemented
33D3×Dic7C2×C714yesdirect product factor
34C3×C72C22×C411yescharacteristic, complemented, Hall
35C22×C4×C7—211no—
36C22×Dic7—31no—
37C2×C72—34no—
38C2×C72C4×D312yescomplemented
39C2×C72C22×D311yescharacteristic
40C22×C3×C7—23no—
41C22×C3×C7C2×D711yescharacteristic
42C22×C3×C7C22×C711yescharacteristic
43C2×D3×C7—212no—
44C2×D3×C7—26no—
45C2×D3×C7Dic714yesdirect product factor
46C2×D3×C7C4×C714yescomplemented
47C2×D3×C7C2×D712yes—
48C2×D3×C7C22×C712yes—
49C3×C4×C7—72no—
50Dic21C22×C712yescomplemented
51C3×Dic7C22×C712yescomplemented
52C7×Dic3—72no—
53C23×C7—31no—
54C23×C7—31no—
55C23×C7—63no—
56C2×C4×C7—71no—
57C2×C4×C7—214no—
58C2×C4×C7—211no—
59C2×Dic7—31no—
60C2×Dic7—34no—
61C2×Dic7D3×C711yescharacteristic, direct product factor
62C72C2×C4×D311yescharacteristic, complemented, Sylow
63C2×C4×D3—71noHall
64C2×C3×C7—26no—
65C2×C3×C7—23no—
66C2×C3×C7C2×Dic712yescomplemented
67C2×C3×C7C2×C4×C712yescomplemented
68C2×C3×C7C22×D711yescharacteristic
69C2×C3×C7C23×C711yescharacteristic
70D3×C7—212no—
71D3×C7C2×Dic714yesdirect product factor
72D3×C7C2×C4×C714yescomplemented
73C22×C7—23no—
74C22×C7—34no—
75C22×C7—34no—
76C22×C7—32no—
77C22×C7—32no—
78C22×C7—612no—
79C22×C7—66no—
80C22×C7D3×D711yescharacteristic
81C22×C7C2×D3×C711yescharacteristic
82C4×C7—72no—
83C4×C7—212no—
84Dic7—32no—
85Dic7C2×D3×C712yesdirect product factor
86C22×D3C7×D711yescharacteristic
87C2×C3×C4—71no—
88C2×Dic3—71no—
89C4×D3—74no—
90C3×C7—23no—
91C3×C7C22×Dic711yescharacteristic, complemented
92C3×C7C22×C4×C711yescharacteristic, complemented
93C22×C4—211noSylow
94C2×C7—26no—
95C2×C7—23no—
96C2×C7—34no—
97C2×C7—34no—
98C2×C7—612no—
99C2×C7D3×Dic712yesdirect product factor
100C2×C7C4×D3×C712yescomplemented
101C2×C7C2×D3×D711yescharacteristic
102C2×C7C22×D3×C711yescharacteristic
103C22×C3C2×C7×D711yescharacteristic
104C2×D3C7×Dic714yesdirect product factor
105C2×D3C2×C7×D712yes—
106C3×C4—72no—
107Dic3—72no—
108C23—31no—
109C2×C4—71no—
110C2×C4—214no—
111C2×C4—211no—
112C7—23no—
113C7C2×D3×Dic711yescharacteristic, direct product factor
114C7C2×C4×D3×C711yescharacteristic, complemented
115C2×C3C2×C7×Dic712yescomplemented
116C2×C3C22×C7×D711yescharacteristic
117D3C2×C7×Dic714yesdirect product factor
118C22—34no—
119C22—32no—
120C22D3×C7×D711yescharacteristic
121C4—72no—
122C4—212no—
123C3C22×C7×Dic711yescharacteristic, complemented, Sylow
124C2—34no—
125C2D3×C7×Dic712yesdirect product factor
126C2C2×D3×C7×D711yescharacteristic
subgroupG/H
center, hypercenterC22×C7D3×D7
commutator, hypocenterC3×C7C22×C4×C7
FrattiniC2C2×D3×C7×D7
Fitting, socleC22×C3×C72C22
prime radicalC22×D3×C72C2
cosocleC2×C3×C7C23×C7
LCSG ▷ C3×C7
UCSC1 ◁ C22×C7

Complements

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

NH ≅ G/Ntype
2. C2×C7×Dic21C2semidirect
3. C2×C3×C7×Dic7C2semidirect
4. D3×C7×Dic7C2direct (N × H)
7. C2×D3×C72C4semidirect
9. C7×Dic21C22semidirect
10. C3×C7×Dic7C22semidirect
14. C2×C7×Dic7D3direct (N × H)
16. C2×D3×Dic7C7direct (N × H)
17. C2×C3×C72C2×C4semidirect
19. D3×C72C2×C4semidirect
24. C7×Dic7C2×D3direct (N × H)
29. C2×Dic21C2×C7semidirect
32. C2×C3×Dic7C2×C7semidirect
33. D3×Dic7C2×C7direct (N × H)
34. C3×C72C22×C4semidirect
38. C2×C72C4×D3semidirect
45. C2×D3×C7Dic7direct (N × H)
46. C2×D3×C7C4×C7semidirect
50. Dic21C22×C7semidirect
51. C3×Dic7C22×C7semidirect
61. C2×Dic7D3×C7direct (N × H)
62. C72C2×C4×D3semidirect
66. C2×C3×C7C2×Dic7semidirect
67. C2×C3×C7C2×C4×C7semidirect
71. D3×C7C2×Dic7direct (N × H)
72. D3×C7C2×C4×C7semidirect
85. Dic7C2×D3×C7direct (N × H)
91. C3×C7C22×Dic7semidirect
92. C3×C7C22×C4×C7semidirect
99. C2×C7D3×Dic7direct (N × H)
100. C2×C7C4×D3×C7semidirect
104. C2×D3C7×Dic7direct (N × H)
113. C7C2×D3×Dic7direct (N × H)
114. C7C2×C4×D3×C7semidirect
115. C2×C3C2×C7×Dic7semidirect
117. D3C2×C7×Dic7direct (N × H)
123. C3C22×C7×Dic7semidirect
125. C2D3×C7×Dic7direct (N × H)

Maximal subgroups

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

subgroupindex
C22×D3×C722
C2×C7×Dic212
C2×C3×C7×Dic72
D3×C7×Dic72
C22×C7×Dic73
C2×C4×D3×C77
C2×D3×Dic77