Finite Group Databasebeta

← C2×C3×C4×C5×D7 #13132831

Subgroups — C2×C3×C4×C5×D7 #13132831

Order 1680.

216 conjugacy classes of subgroups; only the 140 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 140 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×C4×C5×C7C211yesmaximal, characteristic, complemented
3C2×C3×C5×Dic7C211yesmaximal, characteristic, complemented
4C3×C4×C5×D7C214yesmaximal, direct product factor
5C2×C4×C5×D7C311yesmaximal, characteristic, direct product factor, Hall
6C22×C3×C5×C7C2211yescharacteristic
7C2×C3×C5×D7C414yesdirect product factor
8C2×C3×C5×D7C2212yes—
9C3×C4×C5×C7C2212yescomplemented
10C3×C5×Dic7C2212yescomplemented
11C2×C3×C4×D7C511yesmaximal, characteristic, direct product factor, Hall
12C22×C5×D7C2×C311yescharacteristic
13C2×C4×C5×C7C2×C311yescharacteristic, complemented
14C2×C5×Dic7C2×C311yescharacteristic, complemented
15C4×C5×D7C2×C314yesdirect product factor
16C22×C3×C4×C5—71nomaximal, Hall
17C2×C3×C5×C7C2×C412yescomplemented
18C2×C3×C5×C7C2311yescharacteristic
19C3×C5×D7C2×C414yesdirect product factor
20C22×C3×D7C2×C511yescharacteristic
21C2×C3×C4×C7C2×C511yescharacteristic, complemented
22C2×C3×Dic7C2×C511yescharacteristic, complemented
23C3×C4×D7C2×C514yesdirect product factor
24C22×C5×C7C22×C311yescharacteristic
25C2×C5×D7C3×C414yesdirect product factor
26C2×C5×D7C22×C312yes—
27C4×C5×C7C22×C312yescomplemented
28C5×Dic7C22×C312yescomplemented
29C23×C3×C5—71no—
30C2×C3×C4×C5—71no—
31C2×C3×C4×C5—74no—
32C2×C3×C4×C5D711yescharacteristic, direct product factor
33C2×C4×D7C3×C511yescharacteristic, direct product factor, Hall
34C3×C5×C7C22×C411yescharacteristic, complemented, Hall
35C22×C3×C7C22×C511yescharacteristic
36C2×C3×D7C4×C514yesdirect product factor
37C2×C3×D7C22×C512yes—
38C3×C4×C7C22×C512yescomplemented
39C3×Dic7C22×C512yescomplemented
40C22×C4×C5—71noHall
41C2×C5×C7C2×C3×C412yescomplemented
42C2×C5×C7C23×C311yescharacteristic
43C5×D7C2×C3×C414yesdirect product factor
44C22×C3×C5—74no—
45C22×C3×C5—72no—
46C22×C3×C5C2×D711yescharacteristic
47C3×C4×C5—72no—
48C3×C4×C5C2×D712yesdirect product factor
49C22×D7C2×C3×C511yescharacteristic
50C2×C4×C7C2×C3×C511yescharacteristic, complemented
51C2×Dic7C2×C3×C511yescharacteristic, complemented
52C4×D7C2×C3×C514yesdirect product factor
53C22×C3×C4—71noHall
54C2×C3×C7C2×C4×C512yescomplemented
55C2×C3×C7C23×C511yescharacteristic
56C3×D7C2×C4×C514yesdirect product factor
57C23×C5—71no—
58C2×C4×C5—71no—
59C2×C4×C5—74no—
60C2×C4×C5C3×D711yescharacteristic, direct product factor
61C5×C7C22×C3×C411yescharacteristic, complemented, Hall
62C2×C3×C5—74no—
63C2×C3×C5C4×D712yesdirect product factor
64C2×C3×C5C22×D711yescharacteristic
65C22×C7C22×C3×C511yescharacteristic
66C2×D7C3×C4×C514yesdirect product factor
67C2×D7C22×C3×C512yes—
68C4×C7C22×C3×C512yescomplemented
69Dic7C22×C3×C512yescomplemented
70C23×C3—71no—
71C2×C3×C4—71no—
72C2×C3×C4—74no—
73C2×C3×C4C5×D711yescharacteristic, direct product factor
74C3×C7C22×C4×C511yescharacteristic, complemented, Hall
75C22×C5—74no—
76C22×C5—72no—
77C22×C5C2×C3×D711yescharacteristic
78C4×C5—72no—
79C4×C5C2×C3×D712yesdirect product factor
80C22×C4—71noSylow
81C3×C5C2×C4×D711yescharacteristic, direct product factor, Hall
82C2×C7C2×C3×C4×C512yescomplemented
83C2×C7C23×C3×C511yescharacteristic
84D7C2×C3×C4×C514yesdirect product factor
85C22×C3—74no—
86C22×C3—72no—
87C22×C3C2×C5×D711yescharacteristic
88C3×C4—72no—
89C3×C4C2×C5×D712yesdirect product factor
90C2×C5—74no—
91C2×C5C3×C4×D712yesdirect product factor
92C2×C5C22×C3×D711yescharacteristic
93C23—71no—
94C2×C4—71no—
95C2×C4—74no—
96C2×C4C3×C5×D711yescharacteristic, direct product factor
97C7C22×C3×C4×C511yescharacteristic, complemented, Sylow
98C2×C3—74no—
99C2×C3C4×C5×D712yesdirect product factor
100C2×C3C22×C5×D711yescharacteristic
101C5C2×C3×C4×D711yescharacteristic, direct product factor, Sylow
102C22—74no—
103C22—72no—
104C22C2×C3×C5×D711yescharacteristic
105C4—72no—
106C4C2×C3×C5×D712yesdirect product factor
107C3C2×C4×C5×D711yescharacteristic, direct product factor, Sylow
108C2—74no—
109C2C3×C4×C5×D712yesdirect product factor
110C2C22×C3×C5×D711yescharacteristic
subgroupG/H
center, hypercenterC2×C3×C4×C5D7
commutator, hypocenterC7C22×C3×C4×C5
FrattiniC2C22×C3×C5×D7
FittingC2×C3×C4×C5×C7C2
socleC22×C3×C5×C7C22
prime radicalC22×C3×C5×D7C2
cosocleC2×C7C23×C3×C5
LCSG ▷ C7
UCSC1 ◁ C2×C3×C4×C5

Complements

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

NH ≅ G/Ntype
2. C2×C3×C4×C5×C7C2semidirect
3. C2×C3×C5×Dic7C2semidirect
4. C3×C4×C5×D7C2direct (N × H)
5. C2×C4×C5×D7C3direct (N × H)
7. C2×C3×C5×D7C4direct (N × H)
9. C3×C4×C5×C7C22semidirect
10. C3×C5×Dic7C22semidirect
11. C2×C3×C4×D7C5direct (N × H)
13. C2×C4×C5×C7C2×C3semidirect
14. C2×C5×Dic7C2×C3semidirect
15. C4×C5×D7C2×C3direct (N × H)
17. C2×C3×C5×C7C2×C4semidirect
19. C3×C5×D7C2×C4direct (N × H)
21. C2×C3×C4×C7C2×C5semidirect
22. C2×C3×Dic7C2×C5semidirect
23. C3×C4×D7C2×C5direct (N × H)
25. C2×C5×D7C3×C4direct (N × H)
27. C4×C5×C7C22×C3semidirect
28. C5×Dic7C22×C3semidirect
32. C2×C3×C4×C5D7direct (N × H)
33. C2×C4×D7C3×C5direct (N × H)
34. C3×C5×C7C22×C4semidirect
36. C2×C3×D7C4×C5direct (N × H)
38. C3×C4×C7C22×C5semidirect
39. C3×Dic7C22×C5semidirect
41. C2×C5×C7C2×C3×C4semidirect
43. C5×D7C2×C3×C4direct (N × H)
48. C3×C4×C5C2×D7direct (N × H)
50. C2×C4×C7C2×C3×C5semidirect
51. C2×Dic7C2×C3×C5semidirect
52. C4×D7C2×C3×C5direct (N × H)
54. C2×C3×C7C2×C4×C5semidirect
56. C3×D7C2×C4×C5direct (N × H)
60. C2×C4×C5C3×D7direct (N × H)
61. C5×C7C22×C3×C4semidirect
63. C2×C3×C5C4×D7direct (N × H)
66. C2×D7C3×C4×C5direct (N × H)
68. C4×C7C22×C3×C5semidirect
69. Dic7C22×C3×C5semidirect
73. C2×C3×C4C5×D7direct (N × H)
74. C3×C7C22×C4×C5semidirect
79. C4×C5C2×C3×D7direct (N × H)
81. C3×C5C2×C4×D7direct (N × H)
82. C2×C7C2×C3×C4×C5semidirect
84. D7C2×C3×C4×C5direct (N × H)
89. C3×C4C2×C5×D7direct (N × H)
91. C2×C5C3×C4×D7direct (N × H)
96. C2×C4C3×C5×D7direct (N × H)
97. C7C22×C3×C4×C5semidirect
99. C2×C3C4×C5×D7direct (N × H)
101. C5C2×C3×C4×D7direct (N × H)
106. C4C2×C3×C5×D7direct (N × H)
107. C3C2×C4×C5×D7direct (N × H)
109. C2C3×C4×C5×D7direct (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×C4×C5×C72
C2×C3×C5×Dic72
C3×C4×C5×D72
C2×C4×C5×D73
C2×C3×C4×D75
C22×C3×C4×C57