Finite Group Databasebeta

← C23×D72 #11918534

Subgroups — C23×D72 #11918534

Order 1152.

2964 conjugacy classes of subgroups; only the 814 normal subgroups are shown.

normal subgroups only

This group has 2964 conjugacy classes of subgroups — too many to draw. Only its 814 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
1C23×D36C212yesmaximal, complemented
2C23×C8×C9C211yesmaximal, characteristic, complemented
3C22×D72C2128yesmaximal, direct product factor
4C23×D24—31nomaximal
5C24×D9—22no—
6C23×C4×C9C2211yescharacteristic
7C22×D36—214no—
8C22×D36C22128yescomplemented
9C22×C8×C9C22114yescomplemented
10C2×D72C221112yesdirect product factor
11C23×D12—32no—
12C23×C3×C8D311yescharacteristic
13C22×D24—328no—
14C24×C9D411yescharacteristic
15C23×D9—228no—
16C23×D9—416no—
17C22×C4×C9D417yes—
18C22×C4×C9C2317yes—
19C2×D36—284no—
20C2×D36C23156yescomplemented
21C2×C8×C9C23128yescomplemented
22D72C23164yesdirect product factor
23C23×D8—91noSylow
24C24×D3—62no—
25C23×C3×C4C2×D311yescharacteristic
26C22×D12—328no—
27C22×D12—614no—
28C22×C3×C8C2×D3114yes—
29C2×D24—3112no—
30C23×C9D818yescomplemented
31C23×C9C2×D417yes—
32C22×D9—256no—
33C22×D9—4112no—
34C2×C4×C9C2×D4121yes—
35C2×C4×C9C2417yes—
36D36—256no—
37D36C24116yescomplemented
38C8×C9C2418yescomplemented
39C23×D4—92no—
40C22×D8—928no—
41C23×C8D911yescharacteristic, complemented
42C24×C3D1211yescharacteristic
43C23×D3—628no—
44C23×D3—1216no—
45C22×C3×C4D1217yes—
46C22×C3×C4C22×D317yes—
47C2×D12—356no—
48C2×D12—684no—
49C2×C3×C8C22×D3128yes—
50D24—364no—
51C22×C9C2×D8128yescomplemented
52C22×C9C22×D417yes—
53C2×D9—216no—
54C2×D9—4112no—
55C4×C9C22×D417yes—
56C4×C9C2511yescharacteristic
57C25—182no—
58C22×D4—928no—
59C22×D4—1814no—
60C23×C4C2×D911yescharacteristic
61C2×D8—9112no—
62C22×C8C2×D9114yescomplemented
63C23×C3D2418yes—
64C23×C3C2×D1217yes—
65C22×D3—656no—
66C22×D3—12112no—
67C2×C3×C4C2×D12121yes—
68C2×C3×C4C23×D317yes—
69D12—316no—
70D12—656no—
71C3×C8C23×D318yes—
72C2×C9C22×D8114yescomplemented
73C2×C9C23×D411yescharacteristic
74D9—416no—
75C24—1828no—
76C24—3616no—
77C24D3611yescharacteristic
78C2×D4—956no—
79C2×D4—1884no—
80C22×C4D3617yes—
81C22×C4C22×D917yes—
82D8—964no—
83C2×C8C22×D9128yescomplemented
84C22×C3C2×D24128yes—
85C22×C3C22×D1217yes—
86C2×D3—616no—
87C2×D3—12112no—
88C3×C4C22×D1217yes—
89C3×C4C24×D311yescharacteristic
90C9C23×D811yescharacteristic, complemented, Sylow
91C23—1856no—
92C23—36112no—
93C23D7218yesdirect product factor
94C23C2×D3617yes—
95D4—916no—
96D4—1856no—
97C2×C4C2×D36121yes—
98C2×C4C23×D917yes—
99C8C23×D918yescomplemented
100C2×C3C22×D24114yes—
101C2×C3C23×D1211yescharacteristic
102D3—1216no—
103C22—1816no—
104C22—36112no—
105C22C2×D72128yesdirect product factor
106C22C22×D3617yes—
107C4C22×D3617yes—
108C4C24×D911yescharacteristic
109C3C23×D2411yescharacteristic
110C2—3616no—
111C2C22×D72114yesdirect product factor
112C2C23×D3611yescharacteristic
subgroupG/H
centerC24D36
commutator, cosocleC4×C9C25
FrattiniC3×C4C24×D3
FittingC23×C8×C9C2
socleC24×C3D12
hypercenterC23×C8D9
hypocenterC9C23×D8
LCSG ▷ C4×C9 ▷ C2×C9 ▷ C9
UCSC1 ◁ C24 ◁ C23×C4 ◁ C23×C8

Complements

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

NH ≅ G/Ntype
1. C23×D36C2semidirect
2. C23×C8×C9C2semidirect
3. C22×D72C2direct (N × H)
8. C22×D36C22semidirect
9. C22×C8×C9C22semidirect
10. C2×D72C22direct (N × H)
20. C2×D36C23semidirect
21. C2×C8×C9C23semidirect
22. D72C23direct (N × H)
30. C23×C9D8semidirect
37. D36C24semidirect
38. C8×C9C24semidirect
41. C23×C8D9semidirect
51. C22×C9C2×D8semidirect
62. C22×C8C2×D9semidirect
72. C2×C9C22×D8semidirect
83. C2×C8C22×D9semidirect
90. C9C23×D8semidirect
93. C23D72direct (N × H)
99. C8C23×D9semidirect
105. C22C2×D72direct (N × H)
111. C2C22×D72direct (N × H)

Maximal subgroups

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

subgroupindex
C23×D362
C23×C8×C92
C22×D722
C23×D243