Finite Group Databasebeta

← #28348108

Subgroups — #28348108

Order 13696.

472 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 133 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
1C2×#16812138C212yesmaximal, complemented
2Q8×D428C211yesmaximal, characteristic, complemented
3#20293281C211yesmaximal, characteristic
4#20293007C211yesmaximal, characteristic, complemented
5#20292886C212yesmaximal, complemented
6#20292837C211yesmaximal, characteristic, complemented
7#20292832C211yesmaximal, characteristic, complemented
8#20292790C212yesmaximal, complemented
9#20292784C212yesmaximal, complemented
10#20292706C211yesmaximal, characteristic, complemented
11C4×SD16×C107C211yesmaximal, characteristic, complemented
12C2×#13133817C2212yes—
13C2×Q8×D107C2212yes—
14#16812138—28no—
15C2×Dic428C2212yes—
16C2×Dic428C2212yes—
17C2×#13133808C2212yes—
18C2×#13133806C2212yes—
19C2×#13133801C2212yes—
20#16812109—21no—
21#16812109C2211yescharacteristic
22#16812109C2211yescharacteristic
23#16812093—22no—
24#16812089—22no—
25#16812089C2212yes—
26#16812089C2212yes—
27#16812083—21no—
28C4×D428—21no—
29C4×D428C2211yescharacteristic
30C2×C4⋊Dic107—21no—
31C4⋊(C107⋊C8)C2211yescharacteristic
32#16812032C2212yes—
33#16812032C2212yes—
34(C2×D107)⋊C8C2212yes—
35C2×SD16×C107C2211yescharacteristic
36C4×Q8×C107C2211yescharacteristic
37C4×D4×C107C2211yescharacteristic
38#106×C107C2211yescharacteristic
39#103×C107C2211yescharacteristic
40#102×C107C2211yescharacteristic
41C4×C8×C107C2211yescharacteristic
42#13133817—24no—
43#13133817—42no—
44C2×Dic214—22no—
45C2×Dic214C2312yes—
46C2×Dic214C2312yes—
47Q8×D107—24no—
48Q8×D107—42no—
49C2×D428D411yescharacteristic
50C2×C22⋊D107—22no—
51C22×Dic107—22no—
52C2×C4×D107—22no—
53C2×C4×D107C2312yes—
54Dic428—24no—
55Dic428—42no—
56#13133808—24no—
57#13133806—24no—
58#13133801—24no—
59C2×C107⋊C8C2312yes—
60C4⋊Dic107—21no—
61C4⋊Dic107—21no—
62C4⋊Dic107—21no—
63C4⋊Dic107—21no—
64C4⋊Dic107—22no—
65C4⋊Dic107—22no—
66C4⋊Dic107D411yescharacteristic
67(C2×D107)⋊C4—22no—
68(C2×D107)⋊C4—22no—
69(C2×D107)⋊C4D412yes—
70C2×Q8×C107C2311yescharacteristic
71C2×D4×C107C2311yescharacteristic
72C22×C4×C107—21no—
73SD16×C107D414yescomplemented
74C2×C8×C107C2311yescharacteristic
75C2×C8×C107C2311yescharacteristic
76C4⋊C4×C107—21no—
77C4⋊C4×C107C2311yescharacteristic
78C4⋊C4×C107C2311yescharacteristic
79C22⋊C4×C107—21no—
80C42×C107—21no—
81C42×C107C2311yescharacteristic
82C23×C107—21no—
83C22×D107C2×D412yes—
84Q8×C107—21no—
85Q8×C107C2×D412yes—
86D4×C107—21no—
87D4×C107C2×D412yes—
88C2×C4×C107—21no—
89C2×C4×C107—21no—
90C2×C4×C107—21no—
91C2×C4×C107—42no—
92C2×C4×C107C2×D411yescharacteristic
93C2×C4×C107C2×D411yescharacteristic
94C2×C4×C107C2411yescharacteristic
95C22⋊D107—44no—
96C2×Dic107—22no—
97C2×Dic107—22no—
98C2×Dic107—44no—
99C2×Dic107C2×D412yes—
100D428—22no—
101C4×D107—22no—
102C4×D107—22no—
103C4×D107—44no—
104Dic214—22no—
105Dic214—22no—
106Dic214—22no—
107Dic214—22no—
108Dic214—44no—
109C8×C107—21no—
110C8×C107C2×D412yes—
111C107⋊C8—22no—
112C22×C107—22no—
113C22×C107—41no—
114C22×C107—41no—
115C22×C107C22×D411yescharacteristic
116C2×D107—22no—
117C2×D107—22no—
118C2×D107—22no—
119C4×C107—22no—
120C4×C107—21no—
121C4×C107—41no—
122C4×C107—41no—
123C4×C107#13712yescomplemented
124C4×C107C22×D411yescharacteristic
125C4×C107H(5,𝔽2)11yescharacteristic
126Dic107—22no—
127Dic107—42no—
128Dic107—42no—
129C2×C107—42no—
130C2×C107D4211yescharacteristic
131C2×C107C2×#13711yescharacteristic
132C2×C107#57811yescharacteristic
133D107—42no—
134#3276—1071nomaximal, Sylow
135C107#327611yescharacteristic, complemented, Sylow
136C2×#137—1072no—
137D4×Q8—1071no—
138#547—1071no—
139#494—1071no—
140#470—1072no—
141#462—1071no—
142#461—1071no—
143#452—1072no—
144#451—1072no—
145C4×SD16D10711yescharacteristic, complemented
146#436—1071no—
147C2×#41—1072no—
148C22×Q8—1072no—
149#137—2148no—
150C2×Q16—1072no—
151C2×Q16—1072no—
152C2×Q16—1072no—
153C2×SD16—1072no—
154C2×SD16C2×D10711yescharacteristic
155C2×M16—1072no—
156C4⋊Q8—1071no—
157C4⋊Q8—1071no—
158C4⋊Q8—2141no—
159#123—2142no—
160C22⋊Q8—1072no—
161C22⋊Q8—1072no—
162C22⋊Q8—2142no—
163#121—2141no—
164C4×Q8C2×D10711yescharacteristic
165C4×D4—1071no—
166C4×D4—2141no—
167C4×D4C2×D10711yescharacteristic
168C2×C4⋊C4—2141no—
169#106C2×D10711yescharacteristic
170C4⋊C8—1071no—
171#103—1072no—
172#103—1072no—
173#103C2×D10711yescharacteristic
174#102C2×D10711yescharacteristic
175C22⋊C8—1072no—
176C4×C8C2×D10711yescharacteristic
177#41—4282no—
178#41—2144no—
179C2×Q8—1072no—
180C2×Q8—1072no—
181C2×Q8—4282no—
182C2×Q8—2142no—
183C2×Q8—2144no—
184C2×Q8C22×D10711yescharacteristic
185C2×D4—1071no—
186C2×D4—2142no—
187C2×D4C22×D10711yescharacteristic
188C22×C4—21no—
189C22×C4—1072no—
190C22×C4—2142no—
191C22×C4—2142no—
192Q16—4282no—
193Q16—2144no—
194Q16—2144no—
195SD16—2144no—
196SD16D42814yescomplemented
197M16—2144no—
198C2×C8—1072no—
199C2×C8C22×D10711yescharacteristic
200C2×C8C22×D10711yescharacteristic
201C4⋊C4—21no—
202C4⋊C4—1071no—
203C4⋊C4—2142no—
204C4⋊C4—2142no—
205C4⋊C4—2141no—
206C4⋊C4—2141no—
207C4⋊C4—2141no—
208C4⋊C4—2141no—
209C4⋊C4C22×D10711yescharacteristic
210C4⋊C4C22×D10711yescharacteristic
211C22⋊C4—21no—
212C22⋊C4—1072no—
213C22⋊C4—2142no—
214C22⋊C4—2142no—
215C42—21no—
216C42C22×D10711yescharacteristic
217C23—21no—
218C23—1072no—
219Q8—21no—
220Q8—4284no—
221Q8—2142no—
222Q8—2142no—
223Q8—2142no—
224Q8—2142no—
225Q8C2×D42812yes—
226D4—21no—
227D4—4284no—
228D4—2142no—
229D4C2×D42812yes—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—42no—
234C2×C4—1072no—
235C2×C4—4284no—
236C2×C4—4284no—
237C2×C4—2142no—
238C2×C4—2142no—
239C2×C4—2142no—
240C2×C4—2142no—
241C2×C4D4×D10711yescharacteristic
242C2×C4D4×D10711yescharacteristic
243C2×C4C23×D10711yescharacteristic
244C8—21no—
245C8—2142no—
246C8C2×D42812yes—
247C22—22no—
248C22—41no—
249C22—41no—
250C22—2142no—
251C22—2142no—
252C22—2142no—
253C22C2×D4×D10711yescharacteristic
254C4—21no—
255C4—22no—
256C4—41no—
257C4—41no—
258C4—4282no—
259C4—4282no—
260C4—2142no—
261C4#1681213812yescomplemented
262C4C22×D42811yescharacteristic
263C4#1681215511yescharacteristic
264C2—42no—
265C2—4282no—
266C2D4×D42811yescharacteristic
267C2C2×#1681213811yescharacteristic
268C2#2029344111yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
FittingC4×SD16×C107C2
socleC22×C107C22×D4
prime radical#20293281C2
hypercenterC4×SD16D107
hypocenterC107#3276
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ C4×SD16

Complements

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

NH ≅ G/Ntype
1. C2×#16812138C2semidirect
2. Q8×D428C2semidirect
4. #20293007C2semidirect
5. #20292886C2semidirect
6. #20292837C2semidirect
7. #20292832C2semidirect
8. #20292790C2semidirect
9. #20292784C2semidirect
10. #20292706C2semidirect
11. C4×SD16×C107C2semidirect
73. SD16×C107D4semidirect
123. C4×C107#137semidirect
135. C107#3276semidirect
145. C4×SD16D107semidirect
196. SD16D428semidirect
261. C4#16812138semidirect

Maximal subgroups

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

subgroupindex
C2×#168121382
Q8×D4282
#202932812
#202930072
#202928862
#202928372
#202928322
#202927902
#202927842
#202927062
C4×SD16×C1072
#3276107