Finite Group Databasebeta

← C2×C3×#63307 #13132653

Subgroups — C2×C3×#63307 #13132653

Order 1680.

216 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 80 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×D35C211yesmaximal, characteristic, complemented
2C22×C3×C7×D5C211yesmaximal, characteristic, complemented
3C2×C3×C5×Dic7C211yesmaximal, characteristic, complemented
4C3×#63307C214yesmaximal, direct product factor
5C2×#63307C311yesmaximal, characteristic, direct product factor, Hall
6C22×C3×C5×C7C2211yescharacteristic
7C2×C3×D35—22no—
8C2×C3×D35C2212yescomplemented
9C2×C3×C7×D5—22no—
10C2×C3×C7×D5C2212yescomplemented
11C3×C5×Dic7C2212yescomplemented
12C2×C3×C22⋊D7—51nomaximal, Hall
13C22×D35C2×C311yescharacteristic, complemented
14C22×C7×D5C2×C311yescharacteristic, complemented
15C2×C5×Dic7C2×C311yescharacteristic, complemented
16#63307C2×C314yesdirect product factor
17C2×C3×D20—71nomaximal, Hall
18C2×C3×C5×C7D412yescomplemented
19C2×C3×C5×C7C2311yescharacteristic
20C3×D35—22no—
21C3×C7×D5—22no—
22C23×C3×C7—51no—
23C22×C3×D7—51no—
24C3×C22⋊D7—54no—
25C2×C3×Dic7D511yescharacteristic, complemented
26C22×C5×C7C22×C311yescharacteristic
27C2×D35—22no—
28C2×D35C22×C312yescomplemented
29C2×C7×D5—22no—
30C2×C7×D5C22×C312yescomplemented
31C5×Dic7C22×C312yescomplemented
32C22×C3×D5—71no—
33C22×C3×D5D711yescharacteristic, complemented
34C2×C3×C4×C5—71no—
35C3×D20—74no—
36C2×C22⋊D7—51noHall
37C3×C5×C7C2×D411yescharacteristic, complemented, Hall
38C22×C3×C7—52no—
39C22×C3×C7—102no—
40C22×C3×C7C2×D511yescharacteristic
41C2×C3×D7—52no—
42C2×C3×D7—102no—
43C3×Dic7C2×D512yescomplemented
44C2×D20—71noHall
45C2×C5×C7C3×D412yescomplemented
46C2×C5×C7C23×C311yescharacteristic
47D35—22no—
48C7×D5—22no—
49C22×C3×C5C2×D711yescharacteristic
50C2×C3×D5—22no—
51C2×C3×D5—72no—
52C2×C3×D5—142no—
53C2×C3×D5C2×D712yescomplemented
54C3×C4×C5—72no—
55C23×C7—51no—
56C22×D7—51no—
57C22⋊D7—54no—
58C2×Dic7C3×D511yescharacteristic, complemented
59C2×C3×D4—351noHall
60C2×C3×C7—102no—
61C2×C3×C7D2012yescomplemented
62C2×C3×C7C22×D511yescharacteristic
63C3×D7—102no—
64C22×D5—71no—
65C22×D5C3×D711yescharacteristic, complemented
66C2×C4×C5—71no—
67D20—74no—
68C5×C7C2×C3×D411yescharacteristic, complemented, Hall
69C2×C3×C5C22⋊D712yescomplemented
70C2×C3×C5C22×D711yescharacteristic
71C3×D5—22no—
72C3×D5—142no—
73C22×C7—52no—
74C22×C7—102no—
75C22×C7C2×C3×D511yescharacteristic
76C2×D7—52no—
77C2×D7—102no—
78Dic7C2×C3×D512yescomplemented
79C23×C3—51no—
80C23×C3—351no—
81C3×D4—354no—
82C2×C3×C4—71no—
83C3×C7C2×D2011yescharacteristic, complemented, Hall
84C22×C5C2×C3×D711yescharacteristic
85C2×D5—22no—
86C2×D5—72no—
87C2×D5—142no—
88C2×D5C2×C3×D712yescomplemented
89C4×C5—72no—
90C2×D4—351noSylow
91C3×C5C2×C22⋊D711yescharacteristic, complemented, Hall
92C2×C7—102no—
93C2×C7C3×D2012yescomplemented
94C2×C7C22×C3×D511yescharacteristic
95D7—102no—
96C22×C3—52no—
97C22×C3—102no—
98C22×C3—352no—
99C22×C3—702no—
100C22×C3D5×D711yescharacteristic
101C3×C4—72no—
102C2×C5C3×C22⋊D712yescomplemented
103C2×C5C22×C3×D711yescharacteristic
104D5—22no—
105D5—142no—
106C23—51no—
107C23—351no—
108D4—354no—
109C2×C4—71no—
110C7C2×C3×D2011yescharacteristic, complemented, Sylow
111C2×C3—102no—
112C2×C3—702no—
113C2×C3#6330712yesdirect product factor
114C2×C3C2×D5×D711yescharacteristic
115C5C2×C3×C22⋊D711yescharacteristic, complemented, Sylow
116C22—52no—
117C22—102no—
118C22—352no—
119C22—702no—
120C22C3×D5×D711yescharacteristic
121C4—72no—
122C3C2×#6330711yescharacteristic, direct product factor, Sylow
123C2—102no—
124C2—702no—
125C2C3×#6330712yesdirect product factor
126C2C2×C3×D5×D711yescharacteristic
subgroupG/H
center, hypercenterC22×C3D5×D7
commutator, cosocleC2×C5×C7C23×C3
FrattiniC2C2×C3×D5×D7
Fitting, socleC22×C3×C5×C7C22
hypocenterC5×C7C2×C3×D4
LCSG ▷ C2×C5×C7 ▷ C5×C7
UCSC1 ◁ C22×C3

Complements

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

NH ≅ G/Ntype
1. C22×C3×D35C2semidirect
2. C22×C3×C7×D5C2semidirect
3. C2×C3×C5×Dic7C2semidirect
4. C3×#63307C2direct (N × H)
5. C2×#63307C3direct (N × H)
8. C2×C3×D35C22semidirect
10. C2×C3×C7×D5C22semidirect
11. C3×C5×Dic7C22semidirect
13. C22×D35C2×C3semidirect
14. C22×C7×D5C2×C3semidirect
15. C2×C5×Dic7C2×C3semidirect
16. #63307C2×C3direct (N × H)
18. C2×C3×C5×C7D4semidirect
25. C2×C3×Dic7D5semidirect
28. C2×D35C22×C3semidirect
30. C2×C7×D5C22×C3semidirect
31. C5×Dic7C22×C3semidirect
33. C22×C3×D5D7semidirect
37. C3×C5×C7C2×D4semidirect
43. C3×Dic7C2×D5semidirect
45. C2×C5×C7C3×D4semidirect
53. C2×C3×D5C2×D7semidirect
58. C2×Dic7C3×D5semidirect
61. C2×C3×C7D20semidirect
65. C22×D5C3×D7semidirect
68. C5×C7C2×C3×D4semidirect
69. C2×C3×C5C22⋊D7semidirect
78. Dic7C2×C3×D5semidirect
83. C3×C7C2×D20semidirect
88. C2×D5C2×C3×D7semidirect
91. C3×C5C2×C22⋊D7semidirect
93. C2×C7C3×D20semidirect
102. C2×C5C3×C22⋊D7semidirect
110. C7C2×C3×D20semidirect
113. C2×C3#63307direct (N × H)
115. C5C2×C3×C22⋊D7semidirect
122. C3C2×#63307direct (N × H)
125. C2C3×#63307direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×D352
C22×C3×C7×D52
C2×C3×C5×Dic72
C3×#633072
C2×#633073
C2×C3×C22⋊D75
C2×C3×D207