Finite Group Databasebeta

← C2×C3×D5×Dic7 #13132648

Subgroups — C2×C3×D5×Dic7 #13132648

Order 1680.

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

normal subgroups only

This group has 216 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×C3×C7×D5C211yesmaximal, characteristic
2C2×C3×Dic35C211yesmaximal, characteristic, complemented
3C2×C3×C5×Dic7C211yesmaximal, characteristic, complemented
4C3×D5×Dic7C214yesmaximal, direct product factor
5C2×D5×Dic7C311yesmaximal, characteristic, direct product factor, Hall
6C22×C3×C5×C7C2211yescharacteristic
7C2×C3×C7×D5C414yescomplemented
8C2×C3×C7×D5C2212yes—
9C3×Dic35C2212yescomplemented
10C3×C5×Dic7C2212yescomplemented
11C22×C3×Dic7—51nomaximal, Hall
12C22×C7×D5C2×C311yescharacteristic
13C2×Dic35C2×C311yescharacteristic, complemented
14C2×C5×Dic7C2×C311yescharacteristic, complemented
15D5×Dic7C2×C314yesdirect product factor
16C2×C3×C4×D5—71nomaximal, Hall
17C2×C3×C5×C7C2×C412yescomplemented
18C2×C3×C5×C7C2311yescharacteristic
19C3×C7×D5C2×C414yescomplemented
20C23×C3×C7—51no—
21C2×C3×Dic7—51no—
22C2×C3×Dic7—54no—
23C2×C3×Dic7D511yescharacteristic, direct product factor
24C22×C5×C7C22×C311yescharacteristic
25C2×C7×D5C3×C414yescomplemented
26C2×C7×D5C22×C312yes—
27Dic35C22×C312yescomplemented
28C5×Dic7C22×C312yescomplemented
29C22×C3×D5D711yescharacteristic
30C2×C3×C4×C5—71no—
31C2×C3×Dic5—71no—
32C3×C4×D5—74no—
33C22×Dic7—51noHall
34C3×C5×C7C22×C411yescharacteristic, complemented, Hall
35C22×C3×C7—54no—
36C22×C3×C7—52no—
37C22×C3×C7C2×D511yescharacteristic
38C3×Dic7—52no—
39C3×Dic7C2×D512yesdirect product factor
40C2×C4×D5—71noHall
41C2×C5×C7C2×C3×C412yescomplemented
42C2×C5×C7C23×C311yescharacteristic
43C7×D5C2×C3×C414yescomplemented
44C22×C3×C5C2×D711yescharacteristic
45C2×C3×D5Dic714yesdirect product factor
46C2×C3×D5C2×D712yes—
47C3×C4×C5—72no—
48C3×Dic5—72no—
49C23×C7—51no—
50C2×Dic7—51no—
51C2×Dic7—54no—
52C2×Dic7C3×D511yescharacteristic, direct product factor
53C22×C3×C4—351noHall
54C2×C3×C7—54no—
55C2×C3×C7C4×D512yescomplemented
56C2×C3×C7C22×D511yescharacteristic
57C22×D5C3×D711yescharacteristic
58C2×C4×C5—71no—
59C2×Dic5—71no—
60C4×D5—74no—
61C5×C7C22×C3×C411yescharacteristic, complemented, Hall
62C2×C3×C5C2×Dic712yescomplemented
63C2×C3×C5C22×D711yescharacteristic
64C3×D5C2×Dic714yesdirect product factor
65C22×C7—54no—
66C22×C7—52no—
67C22×C7C2×C3×D511yescharacteristic
68Dic7—52no—
69Dic7C2×C3×D512yesdirect product factor
70C23×C3—51no—
71C2×C3×C4—71no—
72C2×C3×C4—354no—
73C2×C3×C4—351no—
74C3×C7C2×C4×D511yescharacteristic, complemented, Hall
75C22×C5C2×C3×D711yescharacteristic
76C2×D5C3×Dic714yesdirect product factor
77C2×D5C2×C3×D712yes—
78C4×C5—72no—
79Dic5—72no—
80C22×C4—351noSylow
81C3×C5C22×Dic711yescharacteristic, complemented, Hall
82C2×C7—54no—
83C2×C7C3×C4×D512yescomplemented
84C2×C7C22×C3×D511yescharacteristic
85C22×C3—54no—
86C22×C3—52no—
87C22×C3D5×D711yescharacteristic
88C3×C4—72no—
89C3×C4—352no—
90C2×C5C2×C3×Dic712yescomplemented
91C2×C5C22×C3×D711yescharacteristic
92D5C2×C3×Dic714yesdirect product factor
93C23—51no—
94C2×C4—71no—
95C2×C4—354no—
96C2×C4—351no—
97C7C2×C3×C4×D511yescharacteristic, complemented, Sylow
98C2×C3—54no—
99C2×C3D5×Dic712yesdirect product factor
100C2×C3C2×D5×D711yescharacteristic
101C5C22×C3×Dic711yescharacteristic, complemented, Sylow
102C22—54no—
103C22—52no—
104C22C3×D5×D711yescharacteristic
105C4—72no—
106C4—352no—
107C3C2×D5×Dic711yescharacteristic, direct product factor, Sylow
108C2—54no—
109C2C3×D5×Dic712yesdirect product factor
110C2C2×C3×D5×D711yescharacteristic
subgroupG/H
center, hypercenterC22×C3D5×D7
commutator, hypocenterC5×C7C22×C3×C4
FrattiniC2C2×C3×D5×D7
Fitting, socleC22×C3×C5×C7C22
prime radicalC22×C3×C7×D5C2
cosocleC2×C5×C7C23×C3
LCSG ▷ C5×C7
UCSC1 ◁ C22×C3

Complements

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

NH ≅ G/Ntype
2. C2×C3×Dic35C2semidirect
3. C2×C3×C5×Dic7C2semidirect
4. C3×D5×Dic7C2direct (N × H)
5. C2×D5×Dic7C3direct (N × H)
7. C2×C3×C7×D5C4semidirect
9. C3×Dic35C22semidirect
10. C3×C5×Dic7C22semidirect
13. C2×Dic35C2×C3semidirect
14. C2×C5×Dic7C2×C3semidirect
15. D5×Dic7C2×C3direct (N × H)
17. C2×C3×C5×C7C2×C4semidirect
19. C3×C7×D5C2×C4semidirect
23. C2×C3×Dic7D5direct (N × H)
25. C2×C7×D5C3×C4semidirect
27. Dic35C22×C3semidirect
28. C5×Dic7C22×C3semidirect
34. C3×C5×C7C22×C4semidirect
39. C3×Dic7C2×D5direct (N × H)
41. C2×C5×C7C2×C3×C4semidirect
43. C7×D5C2×C3×C4semidirect
45. C2×C3×D5Dic7direct (N × H)
52. C2×Dic7C3×D5direct (N × H)
55. C2×C3×C7C4×D5semidirect
61. C5×C7C22×C3×C4semidirect
62. C2×C3×C5C2×Dic7semidirect
64. C3×D5C2×Dic7direct (N × H)
69. Dic7C2×C3×D5direct (N × H)
74. C3×C7C2×C4×D5semidirect
76. C2×D5C3×Dic7direct (N × H)
81. C3×C5C22×Dic7semidirect
83. C2×C7C3×C4×D5semidirect
90. C2×C5C2×C3×Dic7semidirect
92. D5C2×C3×Dic7direct (N × H)
97. C7C2×C3×C4×D5semidirect
99. C2×C3D5×Dic7direct (N × H)
101. C5C22×C3×Dic7semidirect
107. C3C2×D5×Dic7direct (N × H)
109. C2C3×D5×Dic7direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×C7×D52
C2×C3×Dic352
C2×C3×C5×Dic72
C3×D5×Dic72
C2×D5×Dic73
C22×C3×Dic75
C2×C3×C4×D57