Finite Group Databasebeta

← C23×#4245 #13062285

Subgroups — C23×#4245 #13062285

Order 1344.

2724 conjugacy classes of subgroups; only the 728 normal subgroups are shown.

normal subgroups only

This group has 2724 conjugacy classes of subgroups — too many to draw. Only its 728 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, complemented
2C24×C3×D7C211yesmaximal, characteristic, complemented
3C23×Dic21C211yesmaximal, characteristic, complemented
4C22×#4245C2128yesmaximal, direct product factor
5C23×C22⋊D7—31nomaximal, Hall
6C24×C3×C7C2211yescharacteristic
7C23×D3×C7—28no—
8C23×D3×C7C22114yescomplemented
9C23×C3×D7—28no—
10C23×C3×D7C22114yescomplemented
11C22×Dic21C22114yescomplemented
12C2×#4245C221112yesdirect product factor
13C25×C7—31no—
14C24×D7D311yescharacteristic, complemented
15C22×C22⋊D7—328no—
16C23×Dic7—31no—
17C23×#67—71nomaximal, Hall
18C23×C3×C7D418yescomplemented
19C23×C3×C7C2317yes—
20C22×D3×C7—256no—
21C22×D3×C7C23128yescomplemented
22C22×C3×D7—256no—
23C22×C3×D7C23128yescomplemented
24C2×Dic21C23128yescomplemented
25#4245C23164yesdirect product factor
26C24×C7—314no—
27C24×C7—68no—
28C24×C7C2×D311yescharacteristic
29C23×D7—28no—
30C23×D7C2×D3114yescomplemented
31C2×C22⋊D7—3112no—
32C22×Dic7—314no—
33C25×C3—71no—
34C24×D3D711yescharacteristic, complemented
35C22×#67—728no—
36C23×Dic3—71no—
37C22×C3×C7C2×D4128yescomplemented
38C22×C3×C7C2417yes—
39C2×D3×C7—256no—
40C2×D3×C7C2418yescomplemented
41C2×C3×D7—256no—
42C2×C3×D7C2418yescomplemented
43Dic21C2418yescomplemented
44C23×D4—211noSylow
45C23×C7—328no—
46C23×C7—656no—
47C23×C7#6718yescomplemented
48C23×C7C22×D317yes—
49C22×D7—256no—
50C22×D7C22×D3128yescomplemented
51C22⋊D7—364no—
52C2×Dic7—328no—
53C24×C3—714no—
54C24×C3—148no—
55C24×C3C2×D711yescharacteristic
56C23×D3—28no—
57C23×D3C2×D7114yescomplemented
58C2×#67—7112no—
59C22×Dic3—714no—
60C2×C3×C7C22×D4114yescomplemented
61C2×C3×C7C2511yescharacteristic
62C3×D7—28no—
63D3×C7—28no—
64C25—31no—
65C25—71no—
66C22×D4—2128no—
67C23×C4—211no—
68C22×C7—38no—
69C22×C7—656no—
70C22×C7C2×#67128yescomplemented
71C22×C7C23×D317yes—
72C2×D7—256no—
73C2×D7C23×D318yescomplemented
74Dic7—38no—
75C23×C3—728no—
76C23×C3—1456no—
77C23×C3C22⋊D718yescomplemented
78C23×C3C22×D717yes—
79C22×D3—256no—
80C22×D3C22×D7128yescomplemented
81#67—764no—
82C2×Dic3—728no—
83C3×C7C23×D411yescharacteristic, complemented, Hall
84C24—314no—
85C24—68no—
86C24—714no—
87C24—148no—
88C24D3×D711yescharacteristic
89C2×D4—21112no—
90C22×C4—2114no—
91C2×C7—68no—
92C2×C7C22×#67114yescomplemented
93C2×C7C24×D311yescharacteristic
94D7—28no—
95C22×C3—78no—
96C22×C3—1456no—
97C22×C3C2×C22⋊D7128yescomplemented
98C22×C3C23×D717yes—
99C2×D3—256no—
100C2×D3C23×D718yescomplemented
101Dic3—78no—
102C23—328no—
103C23—656no—
104C23—728no—
105C23—1456no—
106C23#424518yesdirect product factor
107C23C2×D3×D717yes—
108D4—2164no—
109C2×C4—2128no—
110C7C23×#6711yescharacteristic, complemented, Sylow
111C2×C3—148no—
112C2×C3C22×C22⋊D7114yescomplemented
113C2×C3C24×D711yescharacteristic
114D3—28no—
115C22—38no—
116C22—656no—
117C22—78no—
118C22—1456no—
119C22C2×#4245128yesdirect product factor
120C22C22×D3×D717yes—
121C4—218no—
122C3C23×C22⋊D711yescharacteristic, complemented, Sylow
123C2—68no—
124C2—148no—
125C2C22×#4245114yesdirect product factor
126C2C23×D3×D711yescharacteristic
subgroupG/H
center, hypercenterC24D3×D7
commutator, cosocleC2×C3×C7C25
FrattiniC2C23×D3×D7
Fitting, socleC24×C3×C7C22
hypocenterC3×C7C23×D4
LCSG ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C24

Complements

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

NH ≅ G/Ntype
1. C24×D3×C7C2semidirect
2. C24×C3×D7C2semidirect
3. C23×Dic21C2semidirect
4. C22×#4245C2direct (N × H)
8. C23×D3×C7C22semidirect
10. C23×C3×D7C22semidirect
11. C22×Dic21C22semidirect
12. C2×#4245C22direct (N × H)
14. C24×D7D3semidirect
18. C23×C3×C7D4semidirect
21. C22×D3×C7C23semidirect
23. C22×C3×D7C23semidirect
24. C2×Dic21C23semidirect
25. #4245C23direct (N × H)
30. C23×D7C2×D3semidirect
34. C24×D3D7semidirect
37. C22×C3×C7C2×D4semidirect
40. C2×D3×C7C24semidirect
42. C2×C3×D7C24semidirect
43. Dic21C24semidirect
47. C23×C7#67semidirect
50. C22×D7C22×D3semidirect
57. C23×D3C2×D7semidirect
60. C2×C3×C7C22×D4semidirect
70. C22×C7C2×#67semidirect
73. C2×D7C23×D3semidirect
77. C23×C3C22⋊D7semidirect
80. C22×D3C22×D7semidirect
83. C3×C7C23×D4semidirect
92. C2×C7C22×#67semidirect
97. C22×C3C2×C22⋊D7semidirect
100. C2×D3C23×D7semidirect
106. C23#4245direct (N × H)
110. C7C23×#67semidirect
112. C2×C3C22×C22⋊D7semidirect
119. C22C2×#4245direct (N × H)
122. C3C23×C22⋊D7semidirect
125. C2C22×#4245direct (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
C24×C3×D72
C23×Dic212
C22×#42452
C23×C22⋊D73
C23×#677