Finite Group Databasebeta

← #28405384

Subgroups — #28405384

Order 14464.

5992 conjugacy classes of subgroups; only the 3201 normal subgroups are shown.

normal subgroups only

This group has 5992 conjugacy classes of subgroups — too many to draw. Only its 3201 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
1#20299434C2112yesmaximal, complemented
2#20299433C2116yesmaximal, complemented
3C2×#16813667C2118yesmaximal, complemented
4H(5,𝔽2)×D113C2116yesmaximal, complemented
5C2×H(5,𝔽2)×C113C211yesmaximal, characteristic, complemented
6#16813670C2218yescomplemented
7#16813669C22148yescomplemented
8#16813668C22148yescomplemented
9#16813667C221144yescomplemented
10#16813667C22172yescomplemented
11C2×#14268647C22136yescomplemented
12C2×#14268646C22172yescomplemented
13#41×D113C22196yescomplemented
14C22×C22⋊D113C22124yescomplemented
15C2×D4×D113C22172yescomplemented
16H(5,𝔽2)×C113C22116yescomplemented
17C2×#41×C113C2216yescomplemented
18C22×D4×C113C2219yescomplemented
19C23×D113C23124yescomplemented
20#14268648C23132yescomplemented
21#14268647C231144yescomplemented
22#14268647C23148yescomplemented
23#14268646C231288yescomplemented
24C2×Dic226C23136yescomplemented
25Q8×D113C23132yescomplemented
26C2×D452C23136yescomplemented
27C2×C22⋊D113C231144yescomplemented
28C2×C22⋊D113C23172yescomplemented
29D4×D113C231288yescomplemented
30C22×Dic113C23124yescomplemented
31C2×C4×D113C23172yescomplemented
32C24×C113C2316yescomplemented
33#41×C113C23148yescomplemented
34C2×Q8×C113C2312yescomplemented
35C2×D4×C113C23172yescomplemented
36C2×D4×C113C23118yescomplemented
37C22×C4×C113C2319yescomplemented
38C23×C113—86no—
39C23×C113C2419yes—
40C23×C113C24124yes—
41C22×D113—824no—
42C22×D113C24172yes—
43Q8×C113C2418yes—
44D4×C113C24172yes—
45C2×C4×C113C2416yes—
46C2×C4×C113C24136yes—
47C22⋊D113C241144yes—
48C22⋊D113C24116yes—
49C2×Dic113C24172yes—
50D452C24148yes—
51C4×D113C24196yes—
52Dic226C24148yes—
53C22×C113—424no—
54C22×C113—49no—
55C22×C113C25118yes—
56C22×C113C2511yescharacteristic
57C2×D113—472no—
58C2×D113C25116yes—
59C4×C113C25112yes—
60Dic113C25116yes—
61C2×C113—218no—
62C2×C113—21no—
63C2×C113C2611yescharacteristic
64D113—216no—
65H(7,𝔽2)—1131nomaximal, Sylow
66C113H(7,𝔽2)11yescharacteristic, complemented, Sylow
67#585—11316no—
68#585—11312no—
69C2×H(5,𝔽2)—11316no—
70C2×H(5,𝔽2)—11318no—
71C2×H(5,𝔽2)D11311yescharacteristic, complemented
722−1+4—11348no—
732−1+4—1138no—
74H(5,𝔽2)—11348no—
75H(5,𝔽2)—113144no—
76H(5,𝔽2)—11372no—
77H(5,𝔽2)C2×D113116yescomplemented
78C2×#41—11396no—
79C2×#41—11372no—
80C2×#41—11336no—
81C2×#41C2×D11316yescomplemented
82C22×D4—11324no—
83C22×D4—11372no—
84C22×D4C2×D11319yescomplemented
85C24—11324no—
86C24C22×D11316yescomplemented
87#41—11332no—
88#41—113288no—
89#41—113144no—
90#41—11348no—
91#41C22×D113148yescomplemented
92C2×Q8—11332no—
93C2×Q8—11336no—
94C2×Q8C22×D11312yescomplemented
95C2×D4—113144no—
96C2×D4—11372no—
97C2×D4—113288no—
98C2×D4—11336no—
99C2×D4C22×D113172yescomplemented
100C2×D4C22×D113118yescomplemented
101C22×C4—11324no—
102C22×C4—11372no—
103C22×C4C22×D11319yescomplemented
104C23—86no—
105C23—11372no—
106C23—90424no—
107C23C23×D113124yes—
108C23C23×D11319yes—
109Q8—11348no—
110Q8C23×D11318yes—
111D4—113144no—
112D4—11316no—
113D4—11348no—
114D4C23×D113172yes—
115C2×C4—11372no—
116C2×C4—11396no—
117C2×C4C23×D113136yes—
118C2×C4C23×D11316yes—
119C22—49no—
120C22—424no—
121C22—11316no—
122C22—45272no—
123C22C24×D113118yes—
124C22C24×D11311yescharacteristic
125C4—11316no—
126C4C24×D113112yes—
127C2—21no—
128C2—218no—
129C2—22616no—
130C2C25×D11311yescharacteristic
subgroupG/H
center, FrattiniC2C25×D113
commutator, socle, cosocleC2×C113C26
FittingC2×H(5,𝔽2)×C113C2
hypercenterC2×H(5,𝔽2)D113
hypocenterC113H(7,𝔽2)
LCSG ▷ C2×C113 ▷ C113
UCSC1 ◁ C2 ◁ C2×H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. #20299434C2semidirect
2. #20299433C2semidirect
3. C2×#16813667C2semidirect
4. H(5,𝔽2)×D113C2semidirect
5. C2×H(5,𝔽2)×C113C2semidirect
6. #16813670C22semidirect
7. #16813669C22semidirect
8. #16813668C22semidirect
9. #16813667C22semidirect
10. #16813667C22semidirect
11. C2×#14268647C22semidirect
12. C2×#14268646C22semidirect
13. #41×D113C22semidirect
14. C22×C22⋊D113C22semidirect
15. C2×D4×D113C22semidirect
16. H(5,𝔽2)×C113C22semidirect
17. C2×#41×C113C22semidirect
18. C22×D4×C113C22semidirect
19. C23×D113C23semidirect
20. #14268648C23semidirect
21. #14268647C23semidirect
22. #14268647C23semidirect
23. #14268646C23semidirect
24. C2×Dic226C23semidirect
25. Q8×D113C23semidirect
26. C2×D452C23semidirect
27. C2×C22⋊D113C23semidirect
28. C2×C22⋊D113C23semidirect
29. D4×D113C23semidirect
30. C22×Dic113C23semidirect
31. C2×C4×D113C23semidirect
32. C24×C113C23semidirect
33. #41×C113C23semidirect
34. C2×Q8×C113C23semidirect
35. C2×D4×C113C23semidirect
36. C2×D4×C113C23semidirect
37. C22×C4×C113C23semidirect
66. C113H(7,𝔽2)semidirect
71. C2×H(5,𝔽2)D113semidirect
77. H(5,𝔽2)C2×D113semidirect
81. C2×#41C2×D113semidirect
84. C22×D4C2×D113semidirect
86. C24C22×D113semidirect
91. #41C22×D113semidirect
94. C2×Q8C22×D113semidirect
99. C2×D4C22×D113semidirect
100. C2×D4C22×D113semidirect
103. C22×C4C22×D113semidirect

Maximal subgroups

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

subgroupindex
#202994342
#202994332
C2×#168136672
H(5,𝔽2)×D1132
C2×H(5,𝔽2)×C1132
H(7,𝔽2)113