Finite Group Databasebeta

← C2×C3×#63304 #13132650

Subgroups — C2×C3×#63304 #13132650

Order 1680.

216 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 96 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
2C2×C3×C7×Dic5C211yesmaximal, characteristic, complemented
3C2×C3×C5×Dic7C211yesmaximal, characteristic, complemented
4C3×#63304C214yesmaximal, direct product factor
5C2×#63304C311yesmaximal, characteristic, direct product factor, Hall
6C22×C3×C5×C7C2211yescharacteristic
7C2×C3×D35C414yescomplemented
8C2×C3×D35C2212yes—
9C3×C7×Dic5C2212yescomplemented
10C3×C5×Dic7C2212yescomplemented
11C2×C3×C4×D7—51nomaximal, Hall
12C22×D35C2×C311yescharacteristic
13C2×C7×Dic5C2×C311yescharacteristic, complemented
14C2×C5×Dic7C2×C311yescharacteristic, complemented
15#63304C2×C314yesdirect product factor
16C2×C3×C4×D5—71nomaximal, Hall
17C2×C3×C5×C7C2×C412yescomplemented
18C2×C3×C5×C7C2311yescharacteristic
19C3×D35C2×C414yescomplemented
20C22×C3×D7—51no—
21C2×C3×C4×C7—51no—
22C2×C3×Dic7D511yescharacteristic, complemented
23C3×C4×D7—54no—
24C22×C5×C7C22×C311yescharacteristic
25C2×D35C3×C414yescomplemented
26C2×D35C22×C312yes—
27C5×Dic7C22×C312yescomplemented
28C7×Dic5C22×C312yescomplemented
29C22×C3×D5—71no—
30C2×C3×C4×C5—71no—
31C2×C3×Dic5D711yescharacteristic, complemented
32C3×C4×D5—74no—
33C2×C4×D7—51noHall
34C3×C5×C7C22×C411yescharacteristic, complemented, Hall
35C22×C3×C7C2×D511yescharacteristic
36C2×C3×D7—54no—
37C2×C3×D7—52no—
38C3×C4×C7—52no—
39C3×Dic7C2×D512yescomplemented
40C2×C4×D5—71noHall
41C2×C5×C7C2×C3×C412yescomplemented
42C2×C5×C7C23×C311yescharacteristic
43D35C2×C3×C414yescomplemented
44C22×C3×C5C2×D711yescharacteristic
45C2×C3×D5—74no—
46C2×C3×D5—72no—
47C3×C4×C5—72no—
48C3×Dic5C2×D712yescomplemented
49C22×D7—51no—
50C2×C4×C7—51no—
51C2×Dic7C3×D511yescharacteristic, complemented
52C4×D7—54no—
53C22×C3×C4—351noHall
54C2×C3×C7C4×D512yescomplemented
55C2×C3×C7C22×D511yescharacteristic
56C3×D7—54no—
57C22×D5—71no—
58C2×C4×C5—71no—
59C2×Dic5C3×D711yescharacteristic, complemented
60C4×D5—74no—
61C5×C7C22×C3×C411yescharacteristic, complemented, Hall
62C2×C3×C5C4×D712yescomplemented
63C2×C3×C5C22×D711yescharacteristic
64C3×D5—74no—
65C22×C7C2×C3×D511yescharacteristic
66C2×D7—54no—
67C2×D7—52no—
68C4×C7—52no—
69Dic7C2×C3×D512yescomplemented
70C23×C3—351no—
71C2×C3×C4—51no—
72C2×C3×C4—71no—
73C2×C3×C4—354no—
74C3×C7C2×C4×D511yescharacteristic, complemented, Hall
75C22×C5C2×C3×D711yescharacteristic
76C2×D5—74no—
77C2×D5—72no—
78C4×C5—72no—
79Dic5C2×C3×D712yescomplemented
80C22×C4—351noSylow
81C3×C5C2×C4×D711yescharacteristic, complemented, Hall
82C2×C7C3×C4×D512yescomplemented
83C2×C7C22×C3×D511yescharacteristic
84D7—54no—
85C22×C3—354no—
86C22×C3—352no—
87C22×C3D5×D711yescharacteristic
88C3×C4—52no—
89C3×C4—72no—
90C2×C5C3×C4×D712yescomplemented
91C2×C5C22×C3×D711yescharacteristic
92D5—74no—
93C23—351no—
94C2×C4—51no—
95C2×C4—71no—
96C2×C4—354no—
97C7C2×C3×C4×D511yescharacteristic, complemented, Sylow
98C2×C3—354no—
99C2×C3#6330412yesdirect product factor
100C2×C3C2×D5×D711yescharacteristic
101C5C2×C3×C4×D711yescharacteristic, complemented, Sylow
102C22—354no—
103C22—352no—
104C22C3×D5×D711yescharacteristic
105C4—52no—
106C4—72no—
107C3C2×#6330411yescharacteristic, direct product factor, Sylow
108C2—354no—
109C2C3×#6330412yesdirect 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×D35C2
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×C7×Dic5C2semidirect
3. C2×C3×C5×Dic7C2semidirect
4. C3×#63304C2direct (N × H)
5. C2×#63304C3direct (N × H)
7. C2×C3×D35C4semidirect
9. C3×C7×Dic5C22semidirect
10. C3×C5×Dic7C22semidirect
13. C2×C7×Dic5C2×C3semidirect
14. C2×C5×Dic7C2×C3semidirect
15. #63304C2×C3direct (N × H)
17. C2×C3×C5×C7C2×C4semidirect
19. C3×D35C2×C4semidirect
22. C2×C3×Dic7D5semidirect
25. C2×D35C3×C4semidirect
27. C5×Dic7C22×C3semidirect
28. C7×Dic5C22×C3semidirect
31. C2×C3×Dic5D7semidirect
34. C3×C5×C7C22×C4semidirect
39. C3×Dic7C2×D5semidirect
41. C2×C5×C7C2×C3×C4semidirect
43. D35C2×C3×C4semidirect
48. C3×Dic5C2×D7semidirect
51. C2×Dic7C3×D5semidirect
54. C2×C3×C7C4×D5semidirect
59. C2×Dic5C3×D7semidirect
61. C5×C7C22×C3×C4semidirect
62. C2×C3×C5C4×D7semidirect
69. Dic7C2×C3×D5semidirect
74. C3×C7C2×C4×D5semidirect
79. Dic5C2×C3×D7semidirect
81. C3×C5C2×C4×D7semidirect
82. C2×C7C3×C4×D5semidirect
90. C2×C5C3×C4×D7semidirect
97. C7C2×C3×C4×D5semidirect
99. C2×C3#63304direct (N × H)
101. C5C2×C3×C4×D7semidirect
107. C3C2×#63304direct (N × H)
109. C2C3×#63304direct (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
C2×C3×C7×Dic52
C2×C3×C5×Dic72
C3×#633042
C2×#633043
C2×C3×C4×D75
C2×C3×C4×D57