Finite Group Databasebeta

← #28334897

Subgroups — #28334897

Order 13696.

266 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 266 conjugacy classes of subgroups — too many to draw. Only its 73 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
1C4×#13133817C211yesmaximal, characteristic
2#20292649C211yesmaximal, characteristic, complemented
3C4×#13133801C211yesmaximal, characteristic, complemented
4#20292359C211yesmaximal, characteristic, complemented
5#20292357C211yesmaximal, characteristic, complemented
6D4⋊(C107⋊C8)C211yesmaximal, characteristic, complemented
7D4⋊C8×C107C211yesmaximal, characteristic, complemented
8C2×#13133817C411yescharacteristic
9C2×#13133801—21no—
10C2×#13133801—21no—
11C4×Dic214C2211yescharacteristic
12#16812074C411yescharacteristic
13D4×Dic107C411yescharacteristic
14C4×C22⋊D107—21no—
15(C22⋊D107)⋊C4—21no—
16#16812067C411yescharacteristic
17#16812065—21no—
18C2×C4×Dic107—21no—
19C42×D107C2211yescharacteristic
20C4⋊(C107⋊C8)C2211yescharacteristic
21Dic107⋊C8—21no—
22#16812019—21no—
23C4×C107⋊C8C2211yescharacteristic
24C4×D4×C107C2211yescharacteristic
25C4⋊C8×C107C2211yescharacteristic
26C4×C8×C107C2211yescharacteristic
27#13133817—22no—
28#13133817—22no—
29C2×Dic214C2×C411yescharacteristic
30C2×C22⋊D107—21no—
31C22×Dic107—21no—
32C22×Dic107—21no—
33C2×C4×D107C2×C411yescharacteristic
34C2×C4×D107D411yescharacteristic
35C2×C4×D107D411yescharacteristic
36#13133801—24no—
37#13133801—42no—
38C2×C107⋊C8—21no—
39C2×C107⋊C8—21no—
40C4⋊Dic107C2×C411yescharacteristic
41Dic107⋊C4—21no—
42Dic107⋊C4—21no—
43C22⋊Dic107—21no—
44(C2×D107)⋊C4—21no—
45C4×Dic107—21no—
46C4×Dic107—21no—
47C4×Dic107—21no—
48C4×Dic107C2×C411yescharacteristic
49C4×Dic107D411yescharacteristic
50C4×Dic107D411yescharacteristic
51C2×D4×C107C2×C411yescharacteristic
52C22×C4×C107—21no—
53C2×C8×C107—21no—
54C2×C8×C107—21no—
55C4⋊C4×C107C2×C411yescharacteristic
56C22⋊C4×C107—21no—
57C42×C107C2311yescharacteristic
58C23×C107—21no—
59C22×D107C22⋊C411yescharacteristic
60D4×C107—21no—
61D4×C107M1612yescomplemented
62C2×C4×C107—21no—
63C2×C4×C107—41no—
64C2×C4×C107—41no—
65C2×C4×C107C22×C411yescharacteristic
66C2×C4×C107C2×D411yescharacteristic
67C2×C4×C107C2×D411yescharacteristic
68C22⋊D107—24no—
69C2×Dic107—24no—
70C2×Dic107—21no—
71C2×Dic107—21no—
72C2×Dic107—42no—
73C2×Dic107C22⋊C411yescharacteristic
74C2×Dic107C22⋊C411yescharacteristic
75C2×Dic107C22⋊C411yescharacteristic
76C4×D107—21no—
77C4×D107—21no—
78C4×D107—21no—
79C4×D107—21no—
80C4×D107—22no—
81Dic214—21no—
82Dic214M1612yescomplemented
83C8×C107—22no—
84C8×C107—41no—
85C107⋊C8—22no—
86C107⋊C8—41no—
87C22×C107—22no—
88C22×C107—41no—
89C22×C107—41no—
90C22×C107C2×C22⋊C411yescharacteristic
91C2×D107—21no—
92C2×D107—21no—
93C2×D107—21no—
94C4×C107—21no—
95C4×C107—41no—
96C4×C107C2×M1611yescharacteristic
97C4×C107C2×M1611yescharacteristic
98C4×C107AGL(1,ℤ/8)11yescharacteristic
99C4×C107#13711yescharacteristic
100Dic107—22no—
101Dic107—21no—
102Dic107—21no—
103Dic107—41no—
104Dic107#10412yes—
105C2×C107—42no—
106C2×C107#40711yescharacteristic
107C2×C107#41711yescharacteristic
108C2×C107C2×#10411yescharacteristic
109D107—41no—
110#1489—1071nomaximal, Sylow
111C107#148911yescharacteristic, complemented, Sylow
112C4×#41—1071no—
113#423—1071no—
114C4×M16—1071no—
115Q8⋊C8—1071no—
116Q8⋊C8—1071no—
117D4⋊C8—1071no—
118D4⋊C8D10711yescharacteristic, complemented
119C2×#41—1071no—
120C2×M16—2141no—
121C2×M16—2141no—
122C4×Q8—1071no—
123C4×Q8—1071no—
124C4×D4—1071no—
125C4×D4—2141no—
126C4×D4—2141no—
127C4×D4C2×D10711yescharacteristic
128#117—1071no—
129#117—2141no—
130C2×C42—1071no—
131C2×C42—2141no—
132C4⋊C8—1071no—
133C4⋊C8—2141no—
134C4⋊C8C2×D10711yescharacteristic
135#97—2141no—
136C4×C8—1071no—
137C4×C8C2×D10711yescharacteristic
138#41—2142no—
139#41—2142no—
140C2×Q8—1071no—
141C2×D4—2141no—
142C2×D4C4×D10711yescharacteristic
143C22×C4—21no—
144C22×C4—1071no—
145C22×C4—1071no—
146C22×C4—1071no—
147C22×C4—2141no—
148C22×C4—2141no—
149M16—4282no—
150M16—2144no—
151C2×C8—21no—
152C2×C8—21no—
153C2×C8—2141no—
154C2×C8—2141no—
155C4⋊C4—1071no—
156C4⋊C4—2141no—
157C4⋊C4—2141no—
158C4⋊C4C4×D10711yescharacteristic
159C22⋊C4—21no—
160C22⋊C4—2141no—
161C22⋊C4—2141no—
162C42—1071no—
163C42—1071no—
164C42—1071no—
165C42—2141no—
166C42—2141no—
167C42—2141no—
168C42C22×D10711yescharacteristic
169C23—21no—
170C23—1071no—
171Q8—1072no—
172Q8—2141no—
173D4—21no—
174D4—2144no—
175D4#1313380112yescomplemented
176C2×C4—21no—
177C2×C4—41no—
178C2×C4—41no—
179C2×C4—1071no—
180C2×C4—1071no—
181C2×C4—1071no—
182C2×C4—4282no—
183C2×C4—2142no—
184C2×C4—2141no—
185C2×C4—2144no—
186C2×C4—2141no—
187C2×C4—2141no—
188C2×C4—2141no—
189C2×C4—2141no—
190C2×C4—2141no—
191C2×C4C2×C4×D10711yescharacteristic
192C2×C4D4×D10711yescharacteristic
193C2×C4D4×D10711yescharacteristic
194C8—22no—
195C8—41no—
196C8—4281no—
197C8—2142no—
198C22—22no—
199C22—41no—
200C22—41no—
201C22—2141no—
202C22—2141no—
203C22—2141no—
204C22C22⋊C4×D10711yescharacteristic
205C4—21no—
206C4—41no—
207C4—1072no—
208C4—4281no—
209C4—2141no—
210C4—2141no—
211C4—2142no—
212C4M16×D10711yescharacteristic
213C4C2×#1313380111yescharacteristic
214C4#1681213311yescharacteristic
215C4#1681213811yescharacteristic
216C2—42no—
217C2—4281no—
218C2#2029258911yescharacteristic
219C2#2029262511yescharacteristic
220C2#104×D10711yescharacteristic
subgroupG/H
centerC2×C4D4×D107
commutatorC2×C4×C107C22×C4
FrattiniC42C22×D107
FittingD4⋊C8×C107C2
socleC22×C107C2×C22⋊C4
prime radicalC2×#13133817C4
cosocleC42×C107C23
hypercenterD4⋊C8D107
hypocenterC107#1489
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C2×C4 ◁ C42 ◁ D4⋊C8

Complements

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

NH ≅ G/Ntype
2. #20292649C2semidirect
3. C4×#13133801C2semidirect
4. #20292359C2semidirect
5. #20292357C2semidirect
6. D4⋊(C107⋊C8)C2semidirect
7. D4⋊C8×C107C2semidirect
61. D4×C107M16semidirect
82. Dic214M16semidirect
111. C107#1489semidirect
118. D4⋊C8D107semidirect
175. D4#13133801semidirect

Maximal subgroups

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

subgroupindex
C4×#131338172
#202926492
C4×#131338012
#202923592
#202923572
D4⋊(C107⋊C8)2
D4⋊C8×C1072
#1489107