Finite Group Databasebeta

← C23×C7×#67 #13062367

Subgroups — C23×C7×#67 #13062367

Order 1344.

2724 conjugacy classes of subgroups; only the 1086 normal subgroups are shown.

normal subgroups only

This group has 2724 conjugacy classes of subgroups — too many to draw. Only its 1086 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
1C25×C3×C7C211yesmaximal, characteristic, complemented
2C24×D3×C7C211yesmaximal, characteristic, complemented
3C22×C7×#67C2128yesmaximal, direct product factor
4C23×C7×Dic3C211yesmaximal, characteristic, complemented
5C23×C7×D4—31nomaximal, Hall
6C24×C3×C7—28no—
7C24×C3×C7C2211yescharacteristic
8C24×C3×C7C22114yescomplemented
9C23×D3×C7—28no—
10C23×D3×C7C22114yescomplemented
11C2×C7×#67C221112yesdirect product factor
12C22×C7×Dic3C22114yescomplemented
13C25×C7—31no—
14C25×C7D311yescharacteristic, complemented
15C22×C7×D4—328no—
16C23×C4×C7—31no—
17C23×#67C711yesmaximal, characteristic, direct product factor, Hall
18C23×C3×C7—256no—
19C23×C3×C7D418yescomplemented
20C23×C3×C7C2317yes—
21C23×C3×C7C23128yescomplemented
22C22×D3×C7—256no—
23C22×D3×C7C23128yescomplemented
24C7×#67C23164yesdirect product factor
25C2×C7×Dic3C23128yescomplemented
26C24×C7—28no—
27C24×C7—314no—
28C24×C7—68no—
29C24×C7C2×D311yescharacteristic
30C24×C7C2×D3114yescomplemented
31C2×C7×D4—3112no—
32C22×C4×C7—314no—
33C25×C3C2×C711yescharacteristic, complemented
34C24×D3C2×C711yescharacteristic, complemented
35C22×#67C2×C7128yesdirect product factor
36C23×Dic3C2×C711yescharacteristic, complemented
37C22×C3×C7—256no—
38C22×C3×C7C2×D4128yescomplemented
39C22×C3×C7C2417yes—
40C22×C3×C7C2418yescomplemented
41C2×D3×C7—256no—
42C2×D3×C7C2418yescomplemented
43C7×Dic3C2418yescomplemented
44C23×D4—31noSylow
45C23×C7—256no—
46C23×C7—328no—
47C23×C7—656no—
48C23×C7#6718yesdirect product factor
49C23×C7C22×D317yes—
50C23×C7C22×D3128yescomplemented
51C7×D4—364no—
52C2×C4×C7—328no—
53C24×C3—28no—
54C24×C3C22×C711yescharacteristic
55C24×C3C22×C7114yescomplemented
56C23×D3—28no—
57C23×D3C22×C7114yescomplemented
58C2×#67C22×C71112yesdirect product factor
59C22×Dic3C22×C7114yescomplemented
60C2×C3×C7—28no—
61C2×C3×C7C22×D4114yescomplemented
62C2×C3×C7C2511yescharacteristic
63D3×C7—28no—
64C25—31no—
65C25D3×C711yescharacteristic, complemented
66C22×D4—328no—
67C23×C4—31no—
68C22×C7—256no—
69C22×C7—38no—
70C22×C7—656no—
71C22×C7C2×#67128yesdirect product factor
72C22×C7C23×D317yes—
73C22×C7C23×D318yescomplemented
74C4×C7—38no—
75C23×C3—256no—
76C23×C3C7×D418yescomplemented
77C23×C3C23×C717yes—
78C23×C3C23×C7128yescomplemented
79C22×D3—256no—
80C22×D3C23×C7128yescomplemented
81#67C23×C7164yesdirect product factor
82C2×Dic3C23×C7128yescomplemented
83C3×C7C23×D411yescharacteristic, complemented, Hall
84C24—28no—
85C24—314no—
86C24—68no—
87C24C2×D3×C711yescharacteristic
88C24C2×D3×C7114yescomplemented
89C2×D4—3112no—
90C22×C4—314no—
91C2×C7—28no—
92C2×C7—68no—
93C2×C7C22×#67114yesdirect product factor
94C2×C7C24×D311yescharacteristic
95C22×C3—256no—
96C22×C3C2×C7×D4128yescomplemented
97C22×C3C24×C717yes—
98C22×C3C24×C718yescomplemented
99C2×D3—256no—
100C2×D3C24×C718yescomplemented
101Dic3C24×C718yescomplemented
102C23—256no—
103C23—328no—
104C23—656no—
105C23C7×#6718yesdirect product factor
106C23C22×D3×C717yes—
107C23C22×D3×C7128yescomplemented
108D4—364no—
109C2×C4—328no—
110C7C23×#6711yescharacteristic, direct product factor, Sylow
111C2×C3—28no—
112C2×C3C22×C7×D4114yescomplemented
113C2×C3C25×C711yescharacteristic
114D3—28no—
115C22—256no—
116C22—38no—
117C22—656no—
118C22C2×C7×#67128yesdirect product factor
119C22C23×D3×C717yes—
120C22C23×D3×C718yescomplemented
121C4—38no—
122C3C23×C7×D411yescharacteristic, complemented, Sylow
123C2—28no—
124C2—68no—
125C2C22×C7×#67114yesdirect product factor
126C2C24×D3×C711yescharacteristic
subgroupG/H
centerC24×C7C2×D3
commutator, cosocleC2×C3C25×C7
FrattiniC2C24×D3×C7
FittingC25×C3×C7C2
socleC24×C3×C7C22
hypercenterC25×C7D3
hypocenterC3C23×C7×D4
LCSG ▷ C2×C3 ▷ C3
UCSC1 ◁ C24×C7 ◁ C25×C7

Complements

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

NH ≅ G/Ntype
1. C25×C3×C7C2semidirect
2. C24×D3×C7C2semidirect
3. C22×C7×#67C2direct (N × H)
4. C23×C7×Dic3C2semidirect
8. C24×C3×C7C22semidirect
10. C23×D3×C7C22semidirect
11. C2×C7×#67C22direct (N × H)
12. C22×C7×Dic3C22semidirect
14. C25×C7D3semidirect
17. C23×#67C7direct (N × H)
19. C23×C3×C7D4semidirect
21. C23×C3×C7C23semidirect
23. C22×D3×C7C23semidirect
24. C7×#67C23direct (N × H)
25. C2×C7×Dic3C23semidirect
30. C24×C7C2×D3semidirect
33. C25×C3C2×C7semidirect
34. C24×D3C2×C7semidirect
35. C22×#67C2×C7direct (N × H)
36. C23×Dic3C2×C7semidirect
38. C22×C3×C7C2×D4semidirect
40. C22×C3×C7C24semidirect
42. C2×D3×C7C24semidirect
43. C7×Dic3C24semidirect
48. C23×C7#67direct (N × H)
50. C23×C7C22×D3semidirect
55. C24×C3C22×C7semidirect
57. C23×D3C22×C7semidirect
58. C2×#67C22×C7direct (N × H)
59. C22×Dic3C22×C7semidirect
61. C2×C3×C7C22×D4semidirect
65. C25D3×C7semidirect
71. C22×C7C2×#67direct (N × H)
73. C22×C7C23×D3semidirect
76. C23×C3C7×D4semidirect
78. C23×C3C23×C7semidirect
80. C22×D3C23×C7semidirect
81. #67C23×C7direct (N × H)
82. C2×Dic3C23×C7semidirect
83. C3×C7C23×D4semidirect
88. C24C2×D3×C7semidirect
93. C2×C7C22×#67direct (N × H)
96. C22×C3C2×C7×D4semidirect
98. C22×C3C24×C7semidirect
100. C2×D3C24×C7semidirect
101. Dic3C24×C7semidirect
105. C23C7×#67direct (N × H)
107. C23C22×D3×C7semidirect
110. C7C23×#67direct (N × H)
112. C2×C3C22×C7×D4semidirect
118. C22C2×C7×#67direct (N × H)
120. C22C23×D3×C7semidirect
122. C3C23×C7×D4semidirect
125. C2C22×C7×#67direct (N × H)

Maximal subgroups

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

subgroupindex
C25×C3×C72
C24×D3×C72
C22×C7×#672
C23×C7×Dic32
C23×C7×D43
C23×#677