Finite Group Databasebeta

← #28335649

Subgroups — #28335649

Order 13696.

212 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 53 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#20293362C211yesmaximal, characteristic
2#20292974C211yesmaximal, characteristic
3#20292913C211yesmaximal, characteristic, complemented
4#20292837C211yesmaximal, characteristic, complemented
5#20292370C211yesmaximal, characteristic
6#20292359C211yesmaximal, characteristic, complemented
7Q8⋊C8×C107C211yesmaximal, characteristic, complemented
8C2×#13133808—21no—
9#16812109C2211yescharacteristic
10#16812106C2211yescharacteristic
11#16812098—21no—
12#16812097—21no—
13#16812094—22no—
14C4×Dic214C2211yescharacteristic
15C4×D428—21no—
16#16812071—21no—
17#16812041—21no—
18C4⋊(C107⋊C8)C2211yescharacteristic
19#16812034—21no—
20#16812032—21no—
21#16812031—22no—
22C4×Q8×C107C2211yescharacteristic
23C4⋊C8×C107C2211yescharacteristic
24C4×C8×C107C2211yescharacteristic
25C2×Dic214—21no—
26C2×Dic214D411yescharacteristic
27C2×D428—21no—
28C2×D428—21no—
29C2×D428D412yes—
30C2×C4×D107—22no—
31#13133808—42no—
32C2×C107⋊C8—21no—
33C4⋊Dic107—21no—
34C4⋊Dic107—21no—
35C4⋊Dic107D411yescharacteristic
36Dic107⋊C4—21no—
37(C2×D107)⋊C4—22no—
38(C2×D107)⋊C4—22no—
39(C2×D107)⋊C4—22no—
40C4×Dic107—21no—
41C2×Q8×C107D411yescharacteristic
42C2×C8×C107—21no—
43C2×C8×C107—21no—
44C4⋊C4×C107—21no—
45C4⋊C4×C107D411yescharacteristic
46C42×C107—21no—
47C42×C107C2311yescharacteristic
48C22×D107—22no—
49Q8×C107—21no—
50Q8×C107—21no—
51C2×C4×C107—21no—
52C2×C4×C107—21no—
53C2×C4×C107C2×D411yescharacteristic
54C2×C4×C107C2×D411yescharacteristic
55C2×C4×C107C2×D411yescharacteristic
56C2×Dic107—21no—
57C2×Dic107—21no—
58C2×Dic107—41no—
59D428—42no—
60D428—41no—
61D428—41no—
62D428—41no—
63D428—41no—
64C4×D107—82no—
65Dic214—21no—
66Dic214—21no—
67Dic214—41no—
68Dic214—41no—
69C8×C107—41no—
70C8×C107—41no—
71C107⋊C8—41no—
72C22×C107#12011yescharacteristic
73C2×D107—42no—
74C2×D107—42no—
75C2×D107—42no—
76C4×C107—21no—
77C4×C107—41no—
78C4×C107—41no—
79C4×C107#13511yescharacteristic
80C4×C107#13511yescharacteristic
81C4×C107AGL(1,ℤ/8)11yescharacteristic
82C4×C107#13711yescharacteristic
83Dic107—41no—
84Dic107—41no—
85Dic107—81no—
86C2×C107#44911yescharacteristic
87C2×C107#45211yescharacteristic
88C2×C107#45511yescharacteristic
89D107—82no—
90#1640—1071nomaximal, Sylow
91C107#164011yescharacteristic, complemented, Sylow
92#559—1071no—
93#486—1071no—
94#475—1071no—
95#462—1071no—
96#331—1071no—
97Q8⋊C8—1071no—
98Q8⋊C8D10711yescharacteristic, complemented
99C2×Q16—2141no—
100C4⋊Q8—1071no—
101#127—1071no—
102#124—2141no—
103#124—2141no—
104#123—2142no—
105C4×Q8—1071no—
106C4×Q8C2×D10711yescharacteristic
107C4×D4—2141no—
108C4×D4—2141no—
109#106—2141no—
110C4⋊C8—1071no—
111C4⋊C8C2×D10711yescharacteristic
112#103—2141no—
113#103—2141no—
114#102—2142no—
115C4×C8C2×D10711yescharacteristic
116C2×Q8—1071no—
117C2×Q8—2141no—
118C2×Q8D42811yescharacteristic
119C2×D4—1072no—
120C2×D4—2141no—
121C2×D4—2141no—
122C22×C4—2142no—
123Q16—4282no—
124C2×C8—21no—
125C2×C8—21no—
126C2×C8—2141no—
127C4⋊C4—21no—
128C4⋊C4—1071no—
129C4⋊C4—2141no—
130C4⋊C4—2141no—
131C4⋊C4—2141no—
132C4⋊C4D42811yescharacteristic
133C22⋊C4—2142no—
134C22⋊C4—2142no—
135C22⋊C4—2142no—
136C42—21no—
137C42—2141no—
138C42C22×D10711yescharacteristic
139C23—2142no—
140Q8—21no—
141Q8—21no—
142Q8—4281no—
143Q8—4281no—
144Q8—2141no—
145Q8—2141no—
146D4—4282no—
147D4—4281no—
148D4—4281no—
149D4—4281no—
150D4—4281no—
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—8562no—
154C2×C4—4281no—
155C2×C4—2141no—
156C2×C4—2141no—
157C2×C4D4×D10711yescharacteristic
158C2×C4D4×D10711yescharacteristic
159C2×C4C2×D42811yescharacteristic
160C8—41no—
161C8—41no—
162C8—4281no—
163C22—4282no—
164C22—4282no—
165C22—4282no—
166C22#1681207911yescharacteristic
167C4—21no—
168C4—41no—
169C4—41no—
170C4—8561no—
171C4—4281no—
172C4—4281no—
173C4#1681213011yescharacteristic
174C4#1681213111yescharacteristic
175C4#1681213711yescharacteristic
176C4#1681214111yescharacteristic
177C2—8562no—
178C2#2029277411yescharacteristic
179C2#2029279211yescharacteristic
180C2#2029280611yescharacteristic
subgroupG/H
centerC22#16812079
commutator, cosocleC42×C107C23
FrattiniC42C22×D107
FittingQ8⋊C8×C107C2
socleC22×C107#120
prime radical#16812106C22
hypercenterQ8⋊C8D107
hypocenterC107#1640
LCSG ▷ C42×C107 ▷ C22×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ Q8⋊C8

Complements

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

NH ≅ G/Ntype
3. #20292913C2semidirect
4. #20292837C2semidirect
6. #20292359C2semidirect
7. Q8⋊C8×C107C2semidirect
91. C107#1640semidirect
98. Q8⋊C8D107semidirect

Maximal subgroups

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

subgroupindex
#202933622
#202929742
#202929132
#202928372
#202923702
#202923592
Q8⋊C8×C1072
#1640107