Finite Group Databasebeta

← C23×C4×D3×C7 #13062349

Subgroups — C23×C4×D3×C7 #13062349

Order 1344.

2724 conjugacy classes of subgroups; only the 1598 normal subgroups are shown.

normal subgroups only

This group has 2724 conjugacy classes of subgroups — too many to draw. Only its 1598 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
1C24×D3×C7C211yesmaximal, characteristic
2C23×C3×C4×C7C211yesmaximal, characteristic, complemented
3C23×C7×Dic3C211yesmaximal, characteristic, complemented
4C22×C4×D3×C7C2128yesmaximal, direct product factor
5C24×C4×C7—31nomaximal, Hall
6C24×C3×C7C2211yescharacteristic
7C23×D3×C7C4116yesdirect product factor
8C23×D3×C7C22114yes—
9C22×C3×C4×C7C22114yescomplemented
10C22×C7×Dic3C22114yescomplemented
11C2×C4×D3×C7C221112yesdirect product factor
12C25×C7—31no—
13C23×C4×C7—31no—
14C23×C4×C7—328no—
15C23×C4×C7D311yescharacteristic, direct product factor
16C23×C4×D3C711yesmaximal, characteristic, direct product factor, Hall
17C23×C3×C7C2×C418yescomplemented
18C23×C3×C7C2317yes—
19C22×D3×C7C2×C41112yesdirect product factor
20C22×D3×C7C23128yes—
21C2×C3×C4×C7C23128yescomplemented
22C2×C7×Dic3C23128yescomplemented
23C4×D3×C7C23164yesdirect product factor
24C24×C7—316no—
25C24×C7—314no—
26C24×C7C2×D311yescharacteristic
27C22×C4×C7—314no—
28C22×C4×C7—3112no—
29C22×C4×C7C2×D3114yesdirect product factor
30C24×D3C2×C711yescharacteristic
31C23×C3×C4C2×C711yescharacteristic, complemented
32C23×Dic3C2×C711yescharacteristic, complemented
33C22×C4×D3C2×C7128yesdirect product factor
34C22×C3×C7C22×C4128yescomplemented
35C22×C3×C7C2417yes—
36C2×D3×C7C22×C41112yesdirect product factor
37C2×D3×C7C2418yes—
38C3×C4×C7C2418yescomplemented
39C7×Dic3C2418yescomplemented
40C24×C4—31noSylow
41C23×C7—3112no—
42C23×C7—328no—
43C23×C7C4×D318yesdirect product factor
44C23×C7C22×D317yes—
45C2×C4×C7—328no—
46C2×C4×C7—364no—
47C2×C4×C7C22×D3128yesdirect product factor
48C24×C3C22×C711yescharacteristic
49C23×D3C4×C7116yesdirect product factor
50C23×D3C22×C7114yes—
51C22×C3×C4C22×C7114yescomplemented
52C22×Dic3C22×C7114yescomplemented
53C2×C4×D3C22×C71112yesdirect product factor
54C2×C3×C7C23×C4114yescomplemented
55C2×C3×C7C2511yescharacteristic
56D3×C7C23×C4116yesdirect product factor
57C25—31no—
58C23×C4—31no—
59C23×C4—328no—
60C23×C4D3×C711yescharacteristic, direct product factor
61C22×C7—3112no—
62C22×C7—38no—
63C22×C7C2×C4×D3128yesdirect product factor
64C22×C7C23×D317yes—
65C4×C7—38no—
66C4×C7C23×D318yesdirect product factor
67C23×C3C2×C4×C718yescomplemented
68C23×C3C23×C717yes—
69C22×D3C2×C4×C71112yesdirect product factor
70C22×D3C23×C7128yes—
71C2×C3×C4C23×C7128yescomplemented
72C2×Dic3C23×C7128yescomplemented
73C4×D3C23×C7164yesdirect product factor
74C3×C7C24×C411yescharacteristic, complemented, Hall
75C24—316no—
76C24—314no—
77C24C2×D3×C711yescharacteristic
78C22×C4—314no—
79C22×C4—3112no—
80C22×C4C2×D3×C7114yesdirect product factor
81C2×C7—316no—
82C2×C7C22×C4×D3114yesdirect product factor
83C2×C7C24×D311yescharacteristic
84C22×C3C22×C4×C7128yescomplemented
85C22×C3C24×C717yes—
86C2×D3C22×C4×C71112yesdirect product factor
87C2×D3C24×C718yes—
88C3×C4C24×C718yescomplemented
89Dic3C24×C718yescomplemented
90C23—3112no—
91C23—328no—
92C23C4×D3×C718yesdirect product factor
93C23C22×D3×C717yes—
94C2×C4—328no—
95C2×C4—364no—
96C2×C4C22×D3×C7128yesdirect product factor
97C7C23×C4×D311yescharacteristic, direct product factor, Sylow
98C2×C3C23×C4×C7114yescomplemented
99C2×C3C25×C711yescharacteristic
100D3C23×C4×C7116yesdirect product factor
101C22—38no—
102C22—3112no—
103C22C2×C4×D3×C7128yesdirect product factor
104C22C23×D3×C717yes—
105C4—38no—
106C4C23×D3×C718yesdirect product factor
107C3C24×C4×C711yescharacteristic, complemented, Sylow
108C2—316no—
109C2C22×C4×D3×C7114yesdirect product factor
110C2C24×D3×C711yescharacteristic
subgroupG/H
center, hypercenterC23×C4×C7D3
commutator, hypocenterC3C24×C4×C7
FrattiniC2C24×D3×C7
FittingC23×C3×C4×C7C2
socleC24×C3×C7C22
prime radicalC24×D3×C7C2
cosocleC2×C3C25×C7
LCSG ▷ C3
UCSC1 ◁ C23×C4×C7

Complements

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

NH ≅ G/Ntype
2. C23×C3×C4×C7C2semidirect
3. C23×C7×Dic3C2semidirect
4. C22×C4×D3×C7C2direct (N × H)
7. C23×D3×C7C4direct (N × H)
9. C22×C3×C4×C7C22semidirect
10. C22×C7×Dic3C22semidirect
11. C2×C4×D3×C7C22direct (N × H)
15. C23×C4×C7D3direct (N × H)
16. C23×C4×D3C7direct (N × H)
17. C23×C3×C7C2×C4semidirect
19. C22×D3×C7C2×C4direct (N × H)
21. C2×C3×C4×C7C23semidirect
22. C2×C7×Dic3C23semidirect
23. C4×D3×C7C23direct (N × H)
29. C22×C4×C7C2×D3direct (N × H)
31. C23×C3×C4C2×C7semidirect
32. C23×Dic3C2×C7semidirect
33. C22×C4×D3C2×C7direct (N × H)
34. C22×C3×C7C22×C4semidirect
36. C2×D3×C7C22×C4direct (N × H)
38. C3×C4×C7C24semidirect
39. C7×Dic3C24semidirect
43. C23×C7C4×D3direct (N × H)
47. C2×C4×C7C22×D3direct (N × H)
49. C23×D3C4×C7direct (N × H)
51. C22×C3×C4C22×C7semidirect
52. C22×Dic3C22×C7semidirect
53. C2×C4×D3C22×C7direct (N × H)
54. C2×C3×C7C23×C4semidirect
56. D3×C7C23×C4direct (N × H)
60. C23×C4D3×C7direct (N × H)
63. C22×C7C2×C4×D3direct (N × H)
66. C4×C7C23×D3direct (N × H)
67. C23×C3C2×C4×C7semidirect
69. C22×D3C2×C4×C7direct (N × H)
71. C2×C3×C4C23×C7semidirect
72. C2×Dic3C23×C7semidirect
73. C4×D3C23×C7direct (N × H)
74. C3×C7C24×C4semidirect
80. C22×C4C2×D3×C7direct (N × H)
82. C2×C7C22×C4×D3direct (N × H)
84. C22×C3C22×C4×C7semidirect
86. C2×D3C22×C4×C7direct (N × H)
88. C3×C4C24×C7semidirect
89. Dic3C24×C7semidirect
92. C23C4×D3×C7direct (N × H)
96. C2×C4C22×D3×C7direct (N × H)
97. C7C23×C4×D3direct (N × H)
98. C2×C3C23×C4×C7semidirect
100. D3C23×C4×C7direct (N × H)
103. C22C2×C4×D3×C7direct (N × H)
106. C4C23×D3×C7direct (N × H)
107. C3C24×C4×C7semidirect
109. C2C22×C4×D3×C7direct (N × H)

Maximal subgroups

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

subgroupindex
C24×D3×C72
C23×C3×C4×C72
C23×C7×Dic32
C22×C4×D3×C72
C24×C4×C73
C23×C4×D37