Finite Group Databasebeta

← C3×C4×C7×D4 #10621441

Subgroups — C3×C4×C7×D4 #10621441

Order 672.

188 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 188 conjugacy classes of subgroups — too many to draw. Only its 128 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
1C2×C3×C7×D4C211yesmaximal, characteristic
2C22×C3×C4×C7C212yesmaximal, complemented
3C3×C7×C4⋊C4C211yesmaximal, characteristic, complemented
4C3×C7×C22⋊C4C212yesmaximal, complemented
5C3×C42×C7C211yesmaximal, characteristic, complemented
6C4×C7×D4C311yesmaximal, characteristic, direct product factor, Hall
7C23×C3×C7C2212yes—
8C3×C7×D4C414yesdirect product factor
9C2×C3×C4×C7—24no—
10C2×C3×C4×C7C2211yescharacteristic
11C2×C3×C4×C7C2212yes—
12C2×C3×C4×C7C2211yescharacteristic
13C2×C3×C4×C7C2211yescharacteristic
14C2×C7×D4C2×C311yescharacteristic
15C22×C4×C7C2×C312yescomplemented
16C7×C4⋊C4C2×C311yescharacteristic, complemented
17C7×C22⋊C4C2×C312yescomplemented
18C42×C7C2×C311yescharacteristic, complemented
19C3×C4×D4C711yesmaximal, characteristic, direct product factor, Hall
20C22×C3×C7—22no—
21C22×C3×C7—22no—
22C22×C3×C7C2×C414yescomplemented
23C22×C3×C7C2311yescharacteristic
24C3×C4×C7—22no—
25C3×C4×C7—21no—
26C3×C4×C7C2×C412yescomplemented
27C3×C4×C7D412yesdirect product factor
28C23×C7C22×C312yes—
29C7×D4C3×C414yesdirect product factor
30C2×C4×C7—24no—
31C2×C4×C7C22×C311yescharacteristic
32C2×C4×C7C22×C312yes—
33C2×C4×C7C22×C311yescharacteristic
34C2×C4×C7C22×C311yescharacteristic
35C2×C3×D4C2×C711yescharacteristic
36C22×C3×C4C2×C712yescomplemented
37C3×C4⋊C4C2×C711yescharacteristic, complemented
38C3×C22⋊C4C2×C712yescomplemented
39C3×C42C2×C711yescharacteristic, complemented
40C2×C3×C7—24no—
41C2×C3×C7C22×C411yescharacteristic
42C2×C3×C7C2×D411yescharacteristic
43C2×C3×C7#4111yescharacteristic
44C4×D4C3×C711yescharacteristic, direct product factor, Sylow
45C22×C7—22no—
46C22×C7—22no—
47C22×C7C2×C3×C414yescomplemented
48C22×C7C23×C311yescharacteristic
49C4×C7—22no—
50C4×C7—21no—
51C4×C7C2×C3×C412yescomplemented
52C4×C7C3×D412yesdirect product factor
53C23×C3C22×C712yes—
54C3×D4C4×C714yesdirect product factor
55C2×C3×C4—24no—
56C2×C3×C4C22×C711yescharacteristic
57C2×C3×C4C22×C712yes—
58C2×C3×C4C22×C711yescharacteristic
59C2×C3×C4C22×C711yescharacteristic
60C3×C7C4×D411yescharacteristic, direct product factor, Hall
61C2×D4C2×C3×C711yescharacteristic
62C22×C4C2×C3×C712yescomplemented
63C4⋊C4C2×C3×C711yescharacteristic, complemented
64C22⋊C4C2×C3×C712yescomplemented
65C42C2×C3×C711yescharacteristic, complemented
66C2×C7—24no—
67C2×C7C22×C3×C411yescharacteristic
68C2×C7C2×C3×D411yescharacteristic
69C2×C7C3×#4111yescharacteristic
70C22×C3—22no—
71C22×C3—22no—
72C22×C3C2×C4×C714yescomplemented
73C22×C3C23×C711yescharacteristic
74C3×C4—22no—
75C3×C4—21no—
76C3×C4C2×C4×C712yescomplemented
77C3×C4C7×D412yesdirect product factor
78C23C22×C3×C712yes—
79D4C3×C4×C714yesdirect product factor
80C2×C4—24no—
81C2×C4C22×C3×C711yescharacteristic
82C2×C4C22×C3×C712yes—
83C2×C4C22×C3×C711yescharacteristic
84C2×C4C22×C3×C711yescharacteristic
85C7C3×C4×D411yescharacteristic, direct product factor, Sylow
86C2×C3—24no—
87C2×C3C22×C4×C711yescharacteristic
88C2×C3C2×C7×D411yescharacteristic
89C2×C3C7×#4111yescharacteristic
90C22—22no—
91C22—22no—
92C22C2×C3×C4×C714yescomplemented
93C22C23×C3×C711yescharacteristic
94C4—22no—
95C4—21no—
96C4C2×C3×C4×C712yescomplemented
97C4C3×C7×D412yesdirect product factor
98C3C4×C7×D411yescharacteristic, direct product factor, Sylow
99C2—24no—
100C2C22×C3×C4×C711yescharacteristic
101C2C2×C3×C7×D411yescharacteristic
102C2C3×C7×#4111yescharacteristic
subgroupG/H
centerC2×C3×C4×C7C22
commutatorC2C22×C3×C4×C7
Frattini, cosocleC22C23×C3×C7
socleC22×C3×C7C23
prime radicalC2×C3×C7×D4C2
LCSG ▷ C2 ▷ C1
UCSC1 ◁ C2×C3×C4×C7 ◁ G

Complements

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

NH ≅ G/Ntype
2. C22×C3×C4×C7C2semidirect
3. C3×C7×C4⋊C4C2semidirect
4. C3×C7×C22⋊C4C2semidirect
5. C3×C42×C7C2semidirect
6. C4×C7×D4C3direct (N × H)
8. C3×C7×D4C4direct (N × H)
15. C22×C4×C7C2×C3semidirect
16. C7×C4⋊C4C2×C3semidirect
17. C7×C22⋊C4C2×C3semidirect
18. C42×C7C2×C3semidirect
19. C3×C4×D4C7direct (N × H)
22. C22×C3×C7C2×C4semidirect
26. C3×C4×C7C2×C4semidirect
27. C3×C4×C7D4direct (N × H)
29. C7×D4C3×C4direct (N × H)
36. C22×C3×C4C2×C7semidirect
37. C3×C4⋊C4C2×C7semidirect
38. C3×C22⋊C4C2×C7semidirect
39. C3×C42C2×C7semidirect
44. C4×D4C3×C7direct (N × H)
47. C22×C7C2×C3×C4semidirect
51. C4×C7C2×C3×C4semidirect
52. C4×C7C3×D4direct (N × H)
54. C3×D4C4×C7direct (N × H)
60. C3×C7C4×D4direct (N × H)
62. C22×C4C2×C3×C7semidirect
63. C4⋊C4C2×C3×C7semidirect
64. C22⋊C4C2×C3×C7semidirect
65. C42C2×C3×C7semidirect
72. C22×C3C2×C4×C7semidirect
76. C3×C4C2×C4×C7semidirect
77. C3×C4C7×D4direct (N × H)
79. D4C3×C4×C7direct (N × H)
85. C7C3×C4×D4direct (N × H)
92. C22C2×C3×C4×C7semidirect
96. C4C2×C3×C4×C7semidirect
97. C4C3×C7×D4direct (N × H)
98. C3C4×C7×D4direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×C7×D42
C22×C3×C4×C72
C3×C7×C4⋊C42
C3×C7×C22⋊C42
C3×C42×C72
C4×C7×D43
C3×C4×D47