Finite Group Databasebeta

← #28345719

Subgroups — #28345719

Order 13696.

754 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 754 conjugacy classes of subgroups — too many to draw. Only its 153 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
1C22×#13133818C211yesmaximal, characteristic
2C2×#16812140C212yesmaximal, complemented
3#20292787C214yesmaximal, complemented
4#20292782C212yesmaximal, complemented
5#20292777C212yesmaximal, complemented
6#20292626C212yesmaximal, complemented
7#20292591C211yesmaximal, characteristic, complemented
8C2×C22⋊Q8×C107C211yesmaximal, characteristic, complemented
9C2×#13133818—28no—
10C2×#13133818—21no—
11C2×#13133818—21no—
12C2×#13133818C2212yes—
13C2×#13133818C2212yes—
14C22×Dic214C2211yescharacteristic
15C22×D428C2211yescharacteristic
16C22×C22⋊D107—21no—
17C22×C4×D107—21no—
18#16812140—28no—
19C2×#13133808C2214yescomplemented
20C2×#13133805C2214yescomplemented
21C2×#13133802C2212yescomplemented
22#16812032C2214yescomplemented
23#16812029C2214yescomplemented
24C22⋊(C107⋊C8)C2212yes—
25C22⋊(C107⋊C8)C2212yescomplemented
26C22×Q8×C107C2211yescharacteristic, complemented
27C23×C4×C107C2211yescharacteristic
28C22⋊Q8×C107—21no—
29C22⋊Q8×C107—21no—
30C22⋊Q8×C107C2212yes—
31C22⋊Q8×C107C2212yescomplemented
32C2×C4⋊C4×C107—21no—
33C2×C4⋊C4×C107C2211yescharacteristic, complemented
34C2×C22⋊C4×C107—21no—
35C23×D107—21no—
36#13133818—28no—
37#13133818—28no—
38#13133818—24no—
39#13133818—24no—
40#13133818—24no—
41#13133818D418yes—
42C2×Dic214—21no—
43C2×Dic214—24no—
44C2×Dic214C2312yes—
45C2×D428—21no—
46C2×D428—24no—
47C2×D428C2312yes—
48C2×C22⋊D107—28no—
49C2×C22⋊D107—22no—
50C2×C22⋊D107—22no—
51C22×Dic107—21no—
52C2×C4×D107—28no—
53C2×C4×D107—22no—
54C2×C4×D107—22no—
55#13133808—28no—
56#13133805—28no—
57#13133802—24no—
58C2×C107⋊C8C2314yes—
59C24×C107D411yescharacteristic
60C2×Q8×C107—21no—
61C2×Q8×C107—24no—
62C2×Q8×C107C2312yes—
63C22×C4×C107—24no—
64C22×C4×C107—21no—
65C22×C4×C107—21no—
66C22×C4×C107D411yescharacteristic
67C22×C4×C107D411yescharacteristic
68C22×C4×C107D411yescharacteristic
69C22×C4×C107C2311yescharacteristic
70C22×C4×C107C2311yescharacteristic
71C22×C4×C107C2311yescharacteristic
72C4⋊C4×C107—22no—
73C4⋊C4×C107—22no—
74C4⋊C4×C107—21no—
75C4⋊C4×C107C2312yes—
76C22⋊C4×C107—22no—
77C22⋊C4×C107—22no—
78C23×C107—22no—
79C23×C107—22no—
80C23×C107—41no—
81C23×C107C2×D411yescharacteristic
82C23×C107C2×D411yescharacteristic
83C23×C107C2×D411yescharacteristic
84C22×D107—24no—
85C22×D107—22no—
86C22×D107—42no—
87C22×D107—42no—
88Q8×C107—24no—
89Q8×C107—42no—
90C2×C4×C107—22no—
91C2×C4×C107—22no—
92C2×C4×C107—22no—
93C2×C4×C107—22no—
94C2×C4×C107—22no—
95C2×C4×C107—22no—
96C2×C4×C107—22no—
97C2×C4×C107—42no—
98C2×C4×C107—42no—
99C2×C4×C107C2×D414yes—
100C2×C4×C107C2×D411yescharacteristic
101C2×C4×C107C2×D411yescharacteristic
102C2×C4×C107C2×D411yescharacteristic
103C2×C4×C107C2411yescharacteristic
104C22⋊D107—28no—
105C22⋊D107—28no—
106C2×Dic107—24no—
107C2×Dic107—22no—
108D428—24no—
109D428—22no—
110D428C2×D414yes—
111C4×D107—28no—
112C4×D107—28no—
113Dic214—24no—
114Dic214—22no—
115Dic214C2×D414yes—
116C107⋊C8—24no—
117C22×C107—21no—
118C22×C107—21no—
119C22×C107—21no—
120C22×C107—21no—
121C22×C107—21no—
122C22×C107—21no—
123C22×C107—21no—
124C22×C107—44no—
125C22×C107#12012yes—
126C22×C107#13712yescomplemented
127C22×C107C22×D411yescharacteristic
128C2×D107—24no—
129C2×D107—48no—
130C2×D107—42no—
131C2×D107—42no—
132C4×C107—21no—
133C4×C107—21no—
134C4×C107—42no—
135C4×C107—42no—
136C4×C107#12012yes—
137C4×C107C22×D412yes—
138Dic107—24no—
139C2×C107—22no—
140C2×C107—22no—
141C2×C107—41no—
142C2×C107C2×#12011yescharacteristic
143C2×C107C2×#13711yescharacteristic
144C2×C107#57511yescharacteristic
145D107—44no—
146#3007—1071nomaximal, Sylow
147C107#300711yescharacteristic, complemented, Sylow
148C22×#41—1071no—
149C2×#137—1072no—
150C2×C22⋊Q8D10711yescharacteristic, complemented
151#452—1074no—
152#451—1072no—
153#450—1072no—
154#417—1072no—
155#407—1071no—
156C2×#41—1072no—
157C2×#41—1072no—
158C2×#41—2148no—
159C2×#41—2141no—
160C2×#41—2141no—
161C22×Q8—1071no—
162C22×Q8C2×D10711yescharacteristic, complemented
163C22×D4—1071no—
164C22×D4—2141no—
165C23×C4—2141no—
166C23×C4C2×D10711yescharacteristic
167#137—2148no—
168C2×Q16—1074no—
169C2×SD16—1074no—
170C2×M16—1072no—
171C22⋊Q8—21no—
172C22⋊Q8—21no—
173C22⋊Q8C2×D10712yes—
174C22⋊Q8C2×D10712yescomplemented
175C2×C4⋊C4—21no—
176C2×C4⋊C4C2×D10711yescharacteristic, complemented
177C2×C22⋊C4—21no—
178#103—1074no—
179#102—1074no—
180C22⋊C8—1072no—
181C22⋊C8—1072no—
182C24—2141no—
183C24C22⋊D10711yescharacteristic
184#41—1078no—
185#41—2144no—
186#41—2144no—
187#41—2148no—
188#41—2148no—
189#41—2144no—
190C2×Q8—21no—
191C2×Q8—24no—
192C2×Q8—1072no—
193C2×Q8—2144no—
194C2×Q8—2141no—
195C2×Q8C22×D10712yes—
196C2×D4—1072no—
197C2×D4—2144no—
198C2×D4—2148no—
199C2×D4—2142no—
200C2×D4—2142no—
201C2×D4—2141no—
202C22×C4—24no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—2148no—
206C22×C4—2141no—
207C22×C4—2142no—
208C22×C4—2142no—
209C22×C4C22⋊D10711yescharacteristic
210C22×C4C22⋊D10711yescharacteristic
211C22×C4C22⋊D10711yescharacteristic
212C22×C4C22×D10711yescharacteristic
213C22×C4C22×D10711yescharacteristic
214C22×C4C22×D10711yescharacteristic
215Q16—2148no—
216SD16—2148no—
217M16—2144no—
218C2×C8—1074no—
219C4⋊C4—22no—
220C4⋊C4—22no—
221C4⋊C4—21no—
222C4⋊C4C22×D10712yes—
223C22⋊C4—22no—
224C22⋊C4—22no—
225C23—22no—
226C23—22no—
227C23—41no—
228C23—4282no—
229C23—4282no—
230C23—2144no—
231C23—2142no—
232C23C2×C22⋊D10711yescharacteristic
233C23C2×C22⋊D10711yescharacteristic
234C23C2×C22⋊D10711yescharacteristic
235Q8—24no—
236Q8—42no—
237Q8—1074no—
238Q8—2144no—
239Q8—2142no—
240D4—1074no—
241D4—2144no—
242D4—2148no—
243D4—2148no—
244D4—2142no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—2148no—
255C2×C4—2144no—
256C2×C4—2142no—
257C2×C4—2148no—
258C2×C4D4×D10714yes—
259C2×C4C2×C22⋊D10711yescharacteristic
260C2×C4C2×C22⋊D10711yescharacteristic
261C2×C4C2×C22⋊D10711yescharacteristic
262C2×C4C23×D10711yescharacteristic
263C8—2144no—
264C22—21no—
265C22—21no—
266C22—21no—
267C22—21no—
268C22—21no—
269C22—21no—
270C22—21no—
271C22—44no—
272C22—4288no—
273C22—4282no—
274C22—4282no—
275C22—2144no—
276C22#1681207712yes—
277C22#1681214012yescomplemented
278C22C22×C22⋊D10711yescharacteristic
279C4—21no—
280C4—21no—
281C4—42no—
282C4—42no—
283C4—2144no—
284C4#1681207712yes—
285C4C2×D4×D10712yes—
286C2—22no—
287C2—22no—
288C2—41no—
289C2—4284no—
290C2C2×#1681207711yescharacteristic
291C2C2×#1681214011yescharacteristic
292C2#2029342511yescharacteristic
subgroupG/H
centerC22C22×C22⋊D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
FittingC2×C22⋊Q8×C107C2
socleC22×C107C22×D4
prime radicalC22×#13133818C2
hypercenterC2×C22⋊Q8D107
hypocenterC107#3007
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. C2×#16812140C2semidirect
3. #20292787C2semidirect
4. #20292782C2semidirect
5. #20292777C2semidirect
6. #20292626C2semidirect
7. #20292591C2semidirect
8. C2×C22⋊Q8×C107C2semidirect
19. C2×#13133808C22semidirect
20. C2×#13133805C22semidirect
21. C2×#13133802C22semidirect
22. #16812032C22semidirect
23. #16812029C22semidirect
25. C22⋊(C107⋊C8)C22semidirect
26. C22×Q8×C107C22semidirect
31. C22⋊Q8×C107C22semidirect
33. C2×C4⋊C4×C107C22semidirect
126. C22×C107#137semidirect
147. C107#3007semidirect
150. C2×C22⋊Q8D107semidirect
162. C22×Q8C2×D107semidirect
174. C22⋊Q8C2×D107semidirect
176. C2×C4⋊C4C2×D107semidirect
277. C22#16812140semidirect

Maximal subgroups

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

subgroupindex
C22×#131338182
C2×#168121402
#202927872
#202927822
#202927772
#202926262
#202925912
C2×C22⋊Q8×C1072
#3007107