Finite Group Databasebeta

← C2×C3×D7×Dic5 #13132645

Subgroups — C2×C3×D7×Dic5 #13132645

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×C5×D7C211yesmaximal, characteristic
2C2×C3×Dic35C211yesmaximal, characteristic, complemented
3C2×C3×C7×Dic5C211yesmaximal, characteristic, complemented
4C3×D7×Dic5C214yesmaximal, direct product factor
5C2×D7×Dic5C311yesmaximal, characteristic, direct product factor, Hall
6C22×C3×C5×C7C2211yescharacteristic
7C2×C3×C5×D7C414yescomplemented
8C2×C3×C5×D7C2212yes—
9C3×Dic35C2212yescomplemented
10C3×C7×Dic5C2212yescomplemented
11C2×C3×C4×D7—51nomaximal, Hall
12C22×C5×D7C2×C311yescharacteristic
13C2×Dic35C2×C311yescharacteristic, complemented
14C2×C7×Dic5C2×C311yescharacteristic, complemented
15D7×Dic5C2×C314yesdirect product factor
16C22×C3×Dic5—71nomaximal, Hall
17C2×C3×C5×C7C2×C412yescomplemented
18C2×C3×C5×C7C2311yescharacteristic
19C3×C5×D7C2×C414yescomplemented
20C22×C3×D7D511yescharacteristic
21C2×C3×C4×C7—51no—
22C2×C3×Dic7—51no—
23C3×C4×D7—54no—
24C22×C5×C7C22×C311yescharacteristic
25C2×C5×D7C3×C414yescomplemented
26C2×C5×D7C22×C312yes—
27Dic35C22×C312yescomplemented
28C7×Dic5C22×C312yescomplemented
29C23×C3×C5—71no—
30C2×C3×Dic5—71no—
31C2×C3×Dic5—74no—
32C2×C3×Dic5D711yescharacteristic, direct product factor
33C2×C4×D7—51noHall
34C3×C5×C7C22×C411yescharacteristic, complemented, Hall
35C22×C3×C7C2×D511yescharacteristic
36C2×C3×D7Dic514yesdirect product factor
37C2×C3×D7C2×D512yes—
38C3×C4×C7—52no—
39C3×Dic7—52no—
40C22×Dic5—71noHall
41C2×C5×C7C2×C3×C412yescomplemented
42C2×C5×C7C23×C311yescharacteristic
43C5×D7C2×C3×C414yescomplemented
44C22×C3×C5—74no—
45C22×C3×C5—72no—
46C22×C3×C5C2×D711yescharacteristic
47C3×Dic5—72no—
48C3×Dic5C2×D712yesdirect product factor
49C22×D7C3×D511yescharacteristic
50C2×C4×C7—51no—
51C2×Dic7—51no—
52C4×D7—54no—
53C22×C3×C4—351noHall
54C2×C3×C7C2×Dic512yescomplemented
55C2×C3×C7C22×D511yescharacteristic
56C3×D7C2×Dic514yesdirect product factor
57C23×C5—71no—
58C2×Dic5—71no—
59C2×Dic5—74no—
60C2×Dic5C3×D711yescharacteristic, direct product factor
61C5×C7C22×C3×C411yescharacteristic, complemented, Hall
62C2×C3×C5—74no—
63C2×C3×C5C4×D712yescomplemented
64C2×C3×C5C22×D711yescharacteristic
65C22×C7C2×C3×D511yescharacteristic
66C2×D7C3×Dic514yesdirect product factor
67C2×D7C2×C3×D512yes—
68C4×C7—52no—
69Dic7—52no—
70C23×C3—71no—
71C2×C3×C4—51no—
72C2×C3×C4—354no—
73C2×C3×C4—351no—
74C3×C7C22×Dic511yescharacteristic, complemented, Hall
75C22×C5—74no—
76C22×C5—72no—
77C22×C5C2×C3×D711yescharacteristic
78Dic5—72no—
79Dic5C2×C3×D712yesdirect product factor
80C22×C4—351noSylow
81C3×C5C2×C4×D711yescharacteristic, complemented, Hall
82C2×C7C2×C3×Dic512yescomplemented
83C2×C7C22×C3×D511yescharacteristic
84D7C2×C3×Dic514yesdirect product factor
85C22×C3—74no—
86C22×C3—72no—
87C22×C3D5×D711yescharacteristic
88C3×C4—52no—
89C3×C4—352no—
90C2×C5—74no—
91C2×C5C3×C4×D712yescomplemented
92C2×C5C22×C3×D711yescharacteristic
93C23—71no—
94C2×C4—51no—
95C2×C4—354no—
96C2×C4—351no—
97C7C22×C3×Dic511yescharacteristic, complemented, Sylow
98C2×C3—74no—
99C2×C3D7×Dic512yesdirect product factor
100C2×C3C2×D5×D711yescharacteristic
101C5C2×C3×C4×D711yescharacteristic, complemented, Sylow
102C22—74no—
103C22—72no—
104C22C3×D5×D711yescharacteristic
105C4—52no—
106C4—352no—
107C3C2×D7×Dic511yescharacteristic, direct product factor, Sylow
108C2—74no—
109C2C3×D7×Dic512yesdirect 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×C5×D7C2
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×C7×Dic5C2semidirect
4. C3×D7×Dic5C2direct (N × H)
5. C2×D7×Dic5C3direct (N × H)
7. C2×C3×C5×D7C4semidirect
9. C3×Dic35C22semidirect
10. C3×C7×Dic5C22semidirect
13. C2×Dic35C2×C3semidirect
14. C2×C7×Dic5C2×C3semidirect
15. D7×Dic5C2×C3direct (N × H)
17. C2×C3×C5×C7C2×C4semidirect
19. C3×C5×D7C2×C4semidirect
25. C2×C5×D7C3×C4semidirect
27. Dic35C22×C3semidirect
28. C7×Dic5C22×C3semidirect
32. C2×C3×Dic5D7direct (N × H)
34. C3×C5×C7C22×C4semidirect
36. C2×C3×D7Dic5direct (N × H)
41. C2×C5×C7C2×C3×C4semidirect
43. C5×D7C2×C3×C4semidirect
48. C3×Dic5C2×D7direct (N × H)
54. C2×C3×C7C2×Dic5semidirect
56. C3×D7C2×Dic5direct (N × H)
60. C2×Dic5C3×D7direct (N × H)
61. C5×C7C22×C3×C4semidirect
63. C2×C3×C5C4×D7semidirect
66. C2×D7C3×Dic5direct (N × H)
74. C3×C7C22×Dic5semidirect
79. Dic5C2×C3×D7direct (N × H)
81. C3×C5C2×C4×D7semidirect
82. C2×C7C2×C3×Dic5semidirect
84. D7C2×C3×Dic5direct (N × H)
91. C2×C5C3×C4×D7semidirect
97. C7C22×C3×Dic5semidirect
99. C2×C3D7×Dic5direct (N × H)
101. C5C2×C3×C4×D7semidirect
107. C3C2×D7×Dic5direct (N × H)
109. C2C3×D7×Dic5direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×C5×D72
C2×C3×Dic352
C2×C3×C7×Dic52
C3×D7×Dic52
C2×D7×Dic53
C2×C3×C4×D75
C22×C3×Dic57