Finite Group Databasebeta

← #29505513

Subgroups — #29505513

Order 15296.

252 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 91 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#21439957C211yesmaximal, characteristic, complemented
2#21439954C211yesmaximal, characteristic, complemented
3#21439945C213yesmaximal, complemented
4#21439935C213yesmaximal, complemented
5#21439933C213yesmaximal, complemented
6#21439930C213yesmaximal, complemented
7#126×C239C211yesmaximal, characteristic, complemented
8C23×D239C2211yescharacteristic
9C2×D956C2213yes—
10C2×D956C2213yes—
11C2×D956C2213yescomplemented
12C2×C22⋊D239C2213yes—
13C2×C4×D239C2213yescomplemented
14Dic239⋊C4C2213yescomplemented
15(C2×D239)⋊C4C2213yes—
16(C2×D239)⋊C4C2213yes—
17(C2×D239)⋊C4C2213yescomplemented
18C4⋊C4×C239C2213yescomplemented
19C22⋊C4×C239C2213yes—
20C42×C239C2211yescharacteristic, complemented
21C22×D239—43no—
22C22×D239C2311yescharacteristic
23C22×D239C2311yescharacteristic
24C22×D239C2313yes—
25D956—43no—
26D956—43no—
27D956—43no—
28C22⋊D239—43no—
29C2×Dic239C2313yes—
30C4×D239—43no—
31C23×C239C2311yescharacteristic
32C2×C4×C239C2313yes—
33C2×C4×C239C2313yes—
34C22×C239—23no—
35C22×C239C2411yescharacteristic
36C2×D239—23no—
37C2×D239—23no—
38C2×D239—23no—
39C2×D239—23no—
40C2×D239—23no—
41C2×D239—44no—
42C4×C239—23no—
43C4×C239—23no—
44Dic239—23no—
45C2×C239—41no—
46C2×C239H(5,𝔽2)13yes—
47D239—43no—
48D239—41no—
49D239—41no—
50C239#56011yescharacteristic, complemented, Sylow
51#560—2391nomaximal, Sylow
52#127—2391no—
53#126—2391no—
54#126D23911yescharacteristic, complemented
55#123—2393no—
56#121—2393no—
57#121—2393no—
58#120—2393no—
59C24—2391no—
60C2×D4—2393no—
61C2×D4—2393no—
62C2×D4—2393no—
63C2×D4—2393no—
64C22×C4—2393no—
65C4⋊C4—2393no—
66C4⋊C4C2×D23913yescomplemented
67C22⋊C4—2393no—
68C22⋊C4—2393no—
69C22⋊C4—2393no—
70C22⋊C4C2×D23913yes—
71C42C2×D23911yescharacteristic, complemented
72C23—9563no—
73C23—2391no—
74C23—2391no—
75C23—2393no—
76C23C22×D23911yescharacteristic
77D4—9563no—
78D4—9563no—
79D4—9563no—
80D4—9563no—
81C2×C4—9563no—
82C2×C4—2393no—
83C2×C4C22×D23913yes—
84C2×C4C22×D23913yes—
85C22—23no—
86C22—9564no—
87C22—4783no—
88C22—4783no—
89C22—4783no—
90C22—4783no—
91C22—4783no—
92C22C23×D23911yescharacteristic
93C4—23no—
94C4—23no—
95C4—4783no—
96C2—41no—
97C2—9561no—
98C2—9561no—
99C2—9563no—
100C2#2144000613yes—
subgroupG/H
center, FrattiniC22C23×D239
commutator, socle, cosocleC22×C239C24
Fitting#126×C239C2
hypercenter#126D239
hypocenterC239#560
LCSG ▷ C22×C239 ▷ C239
UCSC1 ◁ C22 ◁ #126

Complements

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

NH ≅ G/Ntype
1. #21439957C2semidirect
2. #21439954C2semidirect
3. #21439945C2semidirect
4. #21439935C2semidirect
5. #21439933C2semidirect
6. #21439930C2semidirect
7. #126×C239C2semidirect
11. C2×D956C22semidirect
13. C2×C4×D239C22semidirect
14. Dic239⋊C4C22semidirect
17. (C2×D239)⋊C4C22semidirect
18. C4⋊C4×C239C22semidirect
20. C42×C239C22semidirect
50. C239#560semidirect
54. #126D239semidirect
66. C4⋊C4C2×D239semidirect
71. C42C2×D239semidirect

Maximal subgroups

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

subgroupindex
#214399572
#214399542
#214399452
#214399352
#214399332
#214399302
#126×C2392
#560239