Finite Group Databasebeta

← C22×#19104704 #35227752

Subgroups — C22×#19104704 #35227752

Order 20864.

4544 conjugacy classes of subgroups; only the 805 normal subgroups are shown.

normal subgroups only

This group has 4544 conjugacy classes of subgroups — too many to draw. Only its 805 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×C22⋊D163C213yesmaximal, complemented
2C2×#19104704C2124yesmaximal, direct product factor
3C22×C22⋊Dic163C213yesmaximal, complemented
4C26×C163C211yesmaximal, characteristic, complemented
5C24×D163C2211yescharacteristic, complemented
6C22×C22⋊D163—224no—
7C22×C22⋊D163C22136yescomplemented
8C23×Dic163C2213yescomplemented
9#19104704C22164yesdirect product factor
10C2×C22⋊Dic163C22136yescomplemented
11C25×C163—224no—
12C25×C163C2213yes—
13C25×C163C22112yescomplemented
14C23×D163—212no—
15C23×D163C2316yescomplemented
16C2×C22⋊D163—2144no—
17C2×C22⋊D163C23148yescomplemented
18C22×Dic163—212no—
19C22×Dic163C23118yescomplemented
20C22⋊Dic163C23148yescomplemented
21C24×C163—2128no—
22C24×C163—224no—
23C24×C163—2144no—
24C24×C163D4124yescomplemented
25C24×C163C23118yes—
26C24×C163C2311yescharacteristic
27C24×C163C23116yescomplemented
28C22×D163—236no—
29C22×D163—416no—
30C22×D163C2414yescomplemented
31C22⋊D163—296no—
32C2×Dic163—236no—
33C2×Dic163C24112yescomplemented
34C23×C163—2384no—
35C23×C163—296no—
36C23×C163—272no—
37C23×C163—296no—
38C23×C163C2×D4112yes—
39C23×C163C2×D4172yescomplemented
40C23×C163C24112yes—
41C23×C163C2413yes—
42C22×C163—2128no—
43C22×C163—2144no—
44C22×C163—224no—
45C22×C163#120116yescomplemented
46C22×C163C22×D4118yes—
47C22×C163C22×D4124yescomplemented
48C22×C163C2511yescharacteristic
49C2×D163—212no—
50C2×D163—424no—
51Dic163—212no—
52C2×C163—224no—
53C2×C163C2×#120112yescomplemented
54C2×C163C23×D413yes—
55D163—44no—
56C163C22×#12011yescharacteristic, complemented, Sylow
57C22×#120—1631nomaximal, Sylow
58C26D16311yescharacteristic, complemented
59C23×D4—1633no—
60C2×#120—16324no—
61C22×C22⋊C4—1633no—
62C25—224no—
63C25—1631no—
64C25C2×D16313yes—
65C25C2×D163112yescomplemented
66C22×D4—32624no—
67C22×D4—16336no—
68C23×C4—1633no—
69#120—16364no—
70C2×C22⋊C4—16336no—
71C24—2128no—
72C24—224no—
73C24—2144no—
74C24—32612no—
75C24—1636no—
76C24C22⋊D163124yescomplemented
77C24C22×D163118yes—
78C24C22×D16311yescharacteristic
79C24C22×D163116yescomplemented
80C2×D4—326144no—
81C2×D4—16348no—
82C22×C4—32612no—
83C22×C4—16318no—
84C22⋊C4—16348no—
85C23—2384no—
86C23—272no—
87C23—296no—
88C23—296no—
89C23—32636no—
90C23—65216no—
91C23—1634no—
92C23C2×C22⋊D163112yes—
93C23C2×C22⋊D163172yescomplemented
94C23C23×D163112yes—
95C23C23×D16313yes—
96D4—32696no—
97C2×C4—32636no—
98C2×C4—16312no—
99C22—2128no—
100C22—2144no—
101C22—224no—
102C22—32612no—
103C22—65224no—
104C22#19104704116yesdirect product factor
105C22C22×C22⋊D163118yes—
106C22C22×C22⋊D163124yescomplemented
107C22C24×D16311yescharacteristic
108C4—32612no—
109C2—224no—
110C2—6524no—
111C2C2×#19104704112yesdirect product factor
112C2C23×C22⋊D16313yes—
subgroupG/H
centerC24C22×D163
commutator, cosocleC22×C163C25
FrattiniC22C24×D163
FittingC26×C163C2
socleC24×C163C23
hypercenterC26D163
hypocenterC163C22×#120
LCSG ▷ C22×C163 ▷ C163
UCSC1 ◁ C24 ◁ C26

Complements

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

NH ≅ G/Ntype
1. C23×C22⋊D163C2semidirect
2. C2×#19104704C2direct (N × H)
3. C22×C22⋊Dic163C2semidirect
4. C26×C163C2semidirect
5. C24×D163C22semidirect
7. C22×C22⋊D163C22semidirect
8. C23×Dic163C22semidirect
9. #19104704C22direct (N × H)
10. C2×C22⋊Dic163C22semidirect
13. C25×C163C22semidirect
15. C23×D163C23semidirect
17. C2×C22⋊D163C23semidirect
19. C22×Dic163C23semidirect
20. C22⋊Dic163C23semidirect
24. C24×C163D4semidirect
27. C24×C163C23semidirect
30. C22×D163C24semidirect
33. C2×Dic163C24semidirect
39. C23×C163C2×D4semidirect
45. C22×C163#120semidirect
47. C22×C163C22×D4semidirect
53. C2×C163C2×#120semidirect
56. C163C22×#120semidirect
58. C26D163semidirect
65. C25C2×D163semidirect
76. C24C22⋊D163semidirect
79. C24C22×D163semidirect
93. C23C2×C22⋊D163semidirect
104. C22#19104704direct (N × H)
106. C22C22×C22⋊D163semidirect
111. C2C2×#19104704direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C22⋊D1632
C2×#191047042
C22×C22⋊Dic1632
C26×C1632
C22×#120163