Finite Group Databasebeta

← #28335754

Subgroups — #28335754

Order 13696.

230 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 49 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#20293017C212yesmaximal, complemented
2#20292993C211yesmaximal, characteristic, complemented
3#20292369C211yesmaximal, characteristic, complemented
4#20292367C212yesmaximal, complemented
5#562×C107C211yesmaximal, characteristic, complemented
6C2×#13133805—22no—
7C2×#13133803—22no—
8#16812106C2211yescharacteristic
9#16812029—22no—
10#16812020C2211yescharacteristic
11#16812020C2212yes—
12#125×C107C2211yescharacteristic
13#124×C107C2212yes—
14#123×C107—22no—
15C22⋊Q8×C107—22no—
16#121×C107—22no—
17C2×D428—22no—
18C2×D428—21no—
19#13133805—44no—
20#13133803—44no—
21C2×C107⋊C8—22no—
22C2×C107⋊C8—21no—
23C2×Q8×C107D412yes—
24C2×D4×C107—22no—
25C2×D4×C107D412yes—
26C22×C4×C107—22no—
27C4⋊C4×C107—22no—
28C4⋊C4×C107—22no—
29C4⋊C4×C107D412yes—
30C22⋊C4×C107—22no—
31C22⋊C4×C107—22no—
32C22⋊C4×C107—22no—
33C42×C107C2311yescharacteristic
34C23×C107—22no—
35C22×D107—41no—
36Q8×C107—42no—
37D4×C107—42no—
38D4×C107—82no—
39C2×C4×C107—22no—
40C2×C4×C107—22no—
41C2×C4×C107—82no—
42C2×C4×C107C2×D411yescharacteristic
43C2×C4×C107C2×D412yes—
44D428—42no—
45D428—42no—
46D428—42no—
47C107⋊C8—24no—
48C107⋊C8—41no—
49C22×C107—42no—
50C22×C107—42no—
51C22×C107—42no—
52C22×C107#12011yescharacteristic
53C2×D107—82no—
54C2×D107—81no—
55C4×C107—22no—
56C4×C107—21no—
57C4×C107—42no—
58C4×C107—42no—
59C2×C107—82no—
60C2×C107#45312yes—
61C2×C107#45411yescharacteristic
62D107—161no—
63#1656—1071nomaximal, Sylow
64C107#165611yescharacteristic, complemented, Sylow
65#562D10711yescharacteristic, complemented
66#496—1072no—
67#490—1071no—
68#330—1071no—
69#329—1072no—
70C2×SD16—2142no—
71C2×D8—2142no—
72#127—1071no—
73#125C2×D10711yescharacteristic
74#124C2×D10712yes—
75#123—22no—
76C22⋊Q8—22no—
77#121—22no—
78#102—2142no—
79#97—1071no—
80#97—1072no—
81C2×Q8C22⋊D10712yes—
82C2×D4—22no—
83C2×D4—2141no—
84C2×D4—2142no—
85C2×D4C22⋊D10712yes—
86C22×C4—22no—
87SD16—4284no—
88D8—4284no—
89C2×C8—2141no—
90C2×C8—2142no—
91C4⋊C4—22no—
92C4⋊C4—22no—
93C4⋊C4C22⋊D10712yes—
94C22⋊C4—22no—
95C22⋊C4—22no—
96C22⋊C4—22no—
97C42C22×D10711yescharacteristic
98C23—22no—
99C23—4281no—
100Q8—42no—
101D4—42no—
102D4—82no—
103D4—4282no—
104D4—4282no—
105D4—4282no—
106C2×C4—22no—
107C2×C4—22no—
108C2×C4—82no—
109C2×C4C2×C22⋊D10711yescharacteristic
110C2×C4C2×C22⋊D10712yes—
111C8—4281no—
112C8—2144no—
113C22—42no—
114C22—42no—
115C22—42no—
116C22—8562no—
117C22—8561no—
118C22#1681207811yescharacteristic
119C4—22no—
120C4—21no—
121C4—42no—
122C4—42no—
123C2—82no—
124C2—17121no—
125C2#2029279512yes—
126C2#2029280011yescharacteristic
subgroupG/H
centerC22#16812078
commutator, cosocleC42×C107C23
FrattiniC42C22×D107
Fitting#562×C107C2
socleC22×C107#120
hypercenter#562D107
hypocenterC107#1656
LCSG ▷ C42×C107 ▷ C22×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #562

Complements

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

NH ≅ G/Ntype
1. #20293017C2semidirect
2. #20292993C2semidirect
3. #20292369C2semidirect
4. #20292367C2semidirect
5. #562×C107C2semidirect
64. C107#1656semidirect
65. #562D107semidirect

Maximal subgroups

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

subgroupindex
#202930172
#202929932
#202923692
#202923672
#562×C1072
#1656107