Finite Group Databasebeta

← C22×#16812078 #28349677

Subgroups — C22×#16812078 #28349677

Order 13696.

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⋊D107C213yesmaximal, complemented
2C2×#16812078C2124yesmaximal, direct product factor
3C22×C22⋊Dic107C213yesmaximal, complemented
4C26×C107C211yesmaximal, characteristic, complemented
5C24×D107C2211yescharacteristic, complemented
6C22×C22⋊D107—224no—
7C22×C22⋊D107C22136yescomplemented
8C23×Dic107C2213yescomplemented
9#16812078C22164yesdirect product factor
10C2×C22⋊Dic107C22136yescomplemented
11C25×C107—224no—
12C25×C107C2213yes—
13C25×C107C22112yescomplemented
14C23×D107—212no—
15C23×D107C2316yescomplemented
16C2×C22⋊D107—2144no—
17C2×C22⋊D107C23148yescomplemented
18C22×Dic107—212no—
19C22×Dic107C23118yescomplemented
20C22⋊Dic107C23148yescomplemented
21C24×C107—2128no—
22C24×C107—224no—
23C24×C107—2144no—
24C24×C107D4124yescomplemented
25C24×C107C23118yes—
26C24×C107C2311yescharacteristic
27C24×C107C23116yescomplemented
28C23×C107—2384no—
29C23×C107—296no—
30C23×C107—272no—
31C23×C107—296no—
32C23×C107C2×D4112yes—
33C23×C107C2×D4172yescomplemented
34C23×C107C24112yes—
35C23×C107C2413yes—
36C22×D107—236no—
37C22×D107—416no—
38C22×D107C2414yescomplemented
39C22⋊D107—296no—
40C2×Dic107—236no—
41C2×Dic107C24112yescomplemented
42C22×C107—2128no—
43C22×C107—2144no—
44C22×C107—224no—
45C22×C107#120116yescomplemented
46C22×C107C22×D4118yes—
47C22×C107C22×D4124yescomplemented
48C22×C107C2511yescharacteristic
49C2×D107—212no—
50C2×D107—424no—
51Dic107—212no—
52C2×C107—224no—
53C2×C107C2×#120112yescomplemented
54C2×C107C23×D413yes—
55D107—44no—
56C22×#120—1071nomaximal, Sylow
57C107C22×#12011yescharacteristic, complemented, Sylow
58C26D10711yescharacteristic, complemented
59C23×D4—1073no—
60C2×#120—10724no—
61C22×C22⋊C4—1073no—
62C25—224no—
63C25—1071no—
64C25C2×D10713yes—
65C25C2×D107112yescomplemented
66C22×D4—10736no—
67C22×D4—21424no—
68C23×C4—1073no—
69#120—10764no—
70C2×C22⋊C4—10736no—
71C24—2128no—
72C24—224no—
73C24—2144no—
74C24—1076no—
75C24—21412no—
76C24C22⋊D107124yescomplemented
77C24C22×D107118yes—
78C24C22×D10711yescharacteristic
79C24C22×D107116yescomplemented
80C2×D4—10748no—
81C2×D4—214144no—
82C22×C4—10718no—
83C22×C4—21412no—
84C22⋊C4—10748no—
85C23—2384no—
86C23—296no—
87C23—272no—
88C23—296no—
89C23—1074no—
90C23—42816no—
91C23—21436no—
92C23C2×C22⋊D107112yes—
93C23C2×C22⋊D107172yescomplemented
94C23C23×D107112yes—
95C23C23×D10713yes—
96D4—21496no—
97C2×C4—10712no—
98C2×C4—21436no—
99C22—2128no—
100C22—224no—
101C22—2144no—
102C22—42824no—
103C22—21412no—
104C22#16812078116yesdirect product factor
105C22C22×C22⋊D107118yes—
106C22C22×C22⋊D107124yescomplemented
107C22C24×D10711yescharacteristic
108C4—21412no—
109C2—224no—
110C2—4284no—
111C2C2×#16812078112yesdirect product factor
112C2C23×C22⋊D10713yes—
subgroupG/H
centerC24C22×D107
commutator, cosocleC22×C107C25
FrattiniC22C24×D107
FittingC26×C107C2
socleC24×C107C23
hypercenterC26D107
hypocenterC107C22×#120
LCSG ▷ C22×C107 ▷ C107
UCSC1 ◁ C24 ◁ C26

Complements

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

NH ≅ G/Ntype
1. C23×C22⋊D107C2semidirect
2. C2×#16812078C2direct (N × H)
3. C22×C22⋊Dic107C2semidirect
4. C26×C107C2semidirect
5. C24×D107C22semidirect
7. C22×C22⋊D107C22semidirect
8. C23×Dic107C22semidirect
9. #16812078C22direct (N × H)
10. C2×C22⋊Dic107C22semidirect
13. C25×C107C22semidirect
15. C23×D107C23semidirect
17. C2×C22⋊D107C23semidirect
19. C22×Dic107C23semidirect
20. C22⋊Dic107C23semidirect
24. C24×C107D4semidirect
27. C24×C107C23semidirect
33. C23×C107C2×D4semidirect
38. C22×D107C24semidirect
41. C2×Dic107C24semidirect
45. C22×C107#120semidirect
47. C22×C107C22×D4semidirect
53. C2×C107C2×#120semidirect
57. C107C22×#120semidirect
58. C26D107semidirect
65. C25C2×D107semidirect
76. C24C22⋊D107semidirect
79. C24C22×D107semidirect
93. C23C2×C22⋊D107semidirect
104. C22#16812078direct (N × H)
106. C22C22×C22⋊D107semidirect
111. C2C2×#16812078direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C22⋊D1072
C2×#168120782
C22×C22⋊Dic1072
C26×C1072
C22×#120107