Finite Group Databasebeta

← #28335753

Subgroups — #28335753

Order 13696.

230 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 45 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#20293375C211yesmaximal, characteristic, complemented
2#20293014C212yesmaximal, complemented
3#20292369C211yesmaximal, characteristic, complemented
4#20292366C212yesmaximal, complemented
5#490×C107C211yesmaximal, characteristic, complemented
6C2×#13133806—22no—
7C2×#13133803—22no—
8#16812097C2212yes—
9#16812094—22no—
10#16812087—22no—
11#16812084—22no—
12#16812020C2212yes—
13#127×C107C2211yescharacteristic
14#125×C107C2211yescharacteristic
15#102×C107—22no—
16#97×C107C2211yescharacteristic
17C2×Dic214D412yes—
18C2×D428—22no—
19C2×D428D412yes—
20C2×C4×D107—22no—
21#13133806—44no—
22#13133803—44no—
23C2×C107⋊C8—22no—
24Dic107⋊C4—22no—
25(C2×D107)⋊C4—22no—
26(C2×D107)⋊C4—22no—
27(C2×D107)⋊C4—22no—
28C2×D4×C107—22no—
29C2×D4×C107—21no—
30C2×C8×C107—21no—
31C4⋊C4×C107—22no—
32C4⋊C4×C107D412yes—
33C42×C107C2311yescharacteristic
34C23×C107—41no—
35C22×D107—22no—
36D4×C107—42no—
37D4×C107—42no—
38D4×C107—42no—
39C2×C4×C107—22no—
40C2×C4×C107C2×D412yes—
41C2×C4×C107C2×D411yescharacteristic
42C2×Dic107—22no—
43D428—42no—
44D428—82no—
45C4×D107—82no—
46Dic214—42no—
47C8×C107—41no—
48C107⋊C8—24no—
49C22×C107—82no—
50C22×C107—81no—
51C22×C107#12011yescharacteristic
52C2×D107—42no—
53C2×D107—42no—
54C2×D107—42no—
55C4×C107—22no—
56C4×C107—21no—
57C4×C107—42no—
58Dic107—42no—
59C2×C107—161no—
60C2×C107#45312yes—
61C2×C107#45411yescharacteristic
62D107—82no—
63#1656—1071nomaximal, Sylow
64C107#165611yescharacteristic, complemented, Sylow
65#562—1071no—
66#496—1072no—
67#490D10711yescharacteristic, complemented
68#330—1071no—
69#329—1072no—
70C2×SD16—2142no—
71C2×D8—2142no—
72#127C2×D10711yescharacteristic
73#125C2×D10711yescharacteristic
74#124—1072no—
75#123—2142no—
76C22⋊Q8—2142no—
77#121—2142no—
78#102—22no—
79#97—1072no—
80#97C2×D10711yescharacteristic
81C2×Q8—1072no—
82C2×D4—22no—
83C2×D4—21no—
84C2×D4—1072no—
85C2×D4—2142no—
86C22×C4—2142no—
87SD16—4284no—
88D8—4284no—
89C2×C8—21no—
90C2×C8—2142no—
91C4⋊C4—22no—
92C4⋊C4—2142no—
93C4⋊C4C22⋊D10712yes—
94C22⋊C4—2142no—
95C22⋊C4—2142no—
96C22⋊C4—2142no—
97C42C22×D10711yescharacteristic
98C23—41no—
99C23—2142no—
100Q8—4282no—
101D4—42no—
102D4—42no—
103D4—42no—
104D4—8562no—
105D4—4282no—
106C2×C4—22no—
107C2×C4—8562no—
108C2×C4—2142no—
109C2×C4D4×D10712yes—
110C2×C4C2×C22⋊D10711yescharacteristic
111C8—41no—
112C8—2144no—
113C22—82no—
114C22—81no—
115C22—4282no—
116C22—4282no—
117C22—4282no—
118C22#1681207711yescharacteristic
119C4—22no—
120C4—21no—
121C4—42no—
122C4—4282no—
123C2—161no—
124C2—8562no—
125C2#2029279312yes—
126C2#2029279911yescharacteristic
subgroupG/H
centerC22#16812077
commutator, cosocleC42×C107C23
FrattiniC42C22×D107
Fitting#490×C107C2
socleC22×C107#120
hypercenter#490D107
hypocenterC107#1656
LCSG ▷ C42×C107 ▷ C22×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #490

Complements

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

NH ≅ G/Ntype
1. #20293375C2semidirect
2. #20293014C2semidirect
3. #20292369C2semidirect
4. #20292366C2semidirect
5. #490×C107C2semidirect
64. C107#1656semidirect
67. #490D107semidirect

Maximal subgroups

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

subgroupindex
#202933752
#202930142
#202923692
#202923662
#490×C1072
#1656107