Finite Group Databasebeta

← #28346069

Subgroups — #28346069

Order 13696.

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

normal subgroups only

This group has 500 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×#16812129C211yesmaximal, characteristic, complemented
2C22×#13133801C211yesmaximal, characteristic, complemented
3#20293139C211yesmaximal, characteristic, complemented
4#20293134C211yesmaximal, characteristic, complemented
5#20292884C212yesmaximal, complemented
6#20292879C212yesmaximal, complemented
7#20292876C212yesmaximal, complemented
8#20292872C212yesmaximal, complemented
9#20292663C211yesmaximal, characteristic, complemented
10#20292624C211yesmaximal, characteristic, complemented
11#416×C107C211yesmaximal, characteristic, complemented
12C2×#13133818C2211yescharacteristic
13C22×C4×D107C2211yescharacteristic
14#16812129—24no—
15C2×#13133808C2211yescharacteristic, complemented
16C2×#13133806C2211yescharacteristic, complemented
17C2×#13133805C2211yescharacteristic, complemented
18C2×#13133803C2211yescharacteristic, complemented
19C2×#13133801—24no—
20C2×#13133801C2212yes—
21C2×#13133801C2212yes—
22C22×C107⋊C8C2211yescharacteristic
23#16812093—21no—
24#16812090—21no—
25#16812089C2212yes—
26#16812085C2212yes—
27#16812084C2212yes—
28#16812080C2212yes—
29#16812079—21no—
30#16812077—21no—
31C4×D428—22no—
32C4×C22⋊D107—22no—
33#16812068C2211yescharacteristic
34#16812042C2212yescomplemented
35#16812040C2212yescomplemented
36#16812034C2212yescomplemented
37#16812031C2212yescomplemented
38C2×#41×C107C2211yescharacteristic
39C22×C8×C107C2211yescharacteristic
40#117×C107C2211yescharacteristic
41#103×C107C2212yescomplemented
42#102×C107C2212yescomplemented
43C23×D107D411yescharacteristic
44#13133818—22no—
45#13133818—22no—
46C2×Dic214C2311yescharacteristic
47C2×D428—22no—
48C2×D428C2311yescharacteristic
49C2×C22⋊D107—21no—
50C2×C22⋊D107—22no—
51C22×Dic107D411yescharacteristic
52C2×C4×D107—22no—
53C2×C4×D107—22no—
54C2×C4×D107—21no—
55C2×C4×D107D412yes—
56C2×C4×D107C2312yes—
57#13133808—22no—
58#13133806—22no—
59#13133805—22no—
60#13133803—22no—
61#13133801—24no—
62#13133801—24no—
63C2×C107⋊C8D414yescomplemented
64C2×C107⋊C8C2311yescharacteristic
65C2×C107⋊C8C2311yescharacteristic
66C4⋊Dic107—22no—
67C4⋊Dic107C2312yes—
68Dic107⋊C4—22no—
69C22⋊Dic107—21no—
70C22⋊Dic107—22no—
71(C2×D107)⋊C4—22no—
72(C2×D107)⋊C4—22no—
73(C2×D107)⋊C4—22no—
74C4×Dic107—21no—
75#41×C107—22no—
76#41×C107—22no—
77C2×Q8×C107C2311yescharacteristic
78C2×D4×C107—21no—
79C2×D4×C107C2311yescharacteristic
80C22×C4×C107—21no—
81C22×C4×C107C2311yescharacteristic
82C2×C8×C107—22no—
83C2×C8×C107C2312yes—
84C4⋊C4×C107C2312yes—
85C22⋊C4×C107—21no—
86C42×C107—21no—
87C23×C107—21no—
88C23×C107C2×D411yescharacteristic
89C22×D107—22no—
90C22×D107—21no—
91C22×D107—42no—
92C22×D107C2×D412yes—
93Q8×C107—21no—
94Q8×C107—21no—
95D4×C107—21no—
96D4×C107—21no—
97D4×C107—42no—
98C2×C4×C107—22no—
99C2×C4×C107—21no—
100C2×C4×C107—42no—
101C2×C4×C107C2×D412yes—
102C2×C4×C107C2×D411yescharacteristic
103C2×C4×C107#4112yes—
104C2×C4×C107C2411yescharacteristic
105C22⋊D107—44no—
106C22⋊D107—42no—
107C2×Dic107—22no—
108C2×Dic107—22no—
109C2×Dic107—21no—
110C2×Dic107C2×D412yes—
111D428—21no—
112D428—21no—
113D428—44no—
114C4×D107—24no—
115C4×D107—22no—
116C4×D107—22no—
117C4×D107—42no—
118Dic214—21no—
119Dic214—21no—
120C8×C107—22no—
121C107⋊C8C2×D412yes—
122C107⋊C8C2×D412yes—
123C22×C107—22no—
124C22×C107—41no—
125C22×C107—42no—
126C22×C107#12112yes—
127C22×C107C22×D411yescharacteristic
128C2×D107—22no—
129C2×D107—22no—
130C2×D107—22no—
131C2×D107—44no—
132C2×D107—41no—
133C2×D107—42no—
134C4×C107—41no—
135C4×C107—42no—
136C4×C107#12112yes—
137C4×C107C22×D411yescharacteristic
138C4×C107C2×#4111yescharacteristic
139Dic107—22no—
140Dic107—41no—
141Dic107—42no—
142C2×C107—22no—
143C2×C107—81no—
144C2×C107C2×#12111yescharacteristic
145C2×C107#57512yes—
146D107—42no—
147D107—81no—
148#3054—1071nomaximal, Sylow
149C107#305411yescharacteristic, complemented, Sylow
150C2×#135—1071no—
151C22×M16—1071no—
152#525—1071no—
153#525—1071no—
154#470—1072no—
155#469—1072no—
156#468—1072no—
157#468—1072no—
158#427—1071no—
159#416—1071no—
160#416D10711yescharacteristic, complemented
161C2×#41—1071no—
162C2×#41C2×D10711yescharacteristic
163C23×C4—1071no—
164#135—2144no—
165C2×Q16—1071no—
166C2×SD16—1071no—
167C2×SD16—1071no—
168C2×D8—1071no—
169C2×M16—1072no—
170C2×M16—1072no—
171C2×M16—2144no—
172C22×C8—1071no—
173C22×C8C2×D10711yescharacteristic
174#123—2141no—
175#123—2141no—
176C22⋊Q8—1072no—
177C22⋊Q8—1072no—
178#121—1072no—
179#121—1072no—
180#120—2141no—
181#120—2141no—
182C4×D4—2142no—
183C4×D4—2142no—
184#117—1071no—
185#117C2×D10711yescharacteristic
186#107—1072no—
187#106—1072no—
188#103—1072no—
189#103C2×D10712yescomplemented
190#102—1072no—
191#102C2×D10712yescomplemented
192C24—1071no—
193#41—22no—
194#41—22no—
195#41—2142no—
196#41—2142no—
197C2×Q8—1071no—
198C2×Q8C22×D10711yescharacteristic
199C2×D4—21no—
200C2×D4—1071no—
201C2×D4—2142no—
202C2×D4—2142no—
203C2×D4—2141no—
204C2×D4C22×D10711yescharacteristic
205C22×C4—21no—
206C22×C4—1071no—
207C22×C4—1072no—
208C22×C4—1072no—
209C22×C4—2141no—
210C22×C4—2142no—
211C22×C4—2142no—
212C22×C4C22×D10711yescharacteristic
213Q16—2142no—
214SD16—2142no—
215SD16—2142no—
216D8—2142no—
217M16—2144no—
218M16—2144no—
219C2×C8—22no—
220C2×C8—1074no—
221C2×C8—1071no—
222C2×C8—1071no—
223C2×C8C22×D10712yes—
224C4⋊C4—1072no—
225C4⋊C4—2142no—
226C4⋊C4—2142no—
227C4⋊C4C22×D10712yes—
228C22⋊C4—21no—
229C22⋊C4—2142no—
230C22⋊C4—2142no—
231C22⋊C4—2142no—
232C22⋊C4—2142no—
233C22⋊C4—2141no—
234C42—21no—
235C42—2141no—
236C23—21no—
237C23—1072no—
238C23—4282no—
239C23—2141no—
240C23—2142no—
241C23D4×D10711yescharacteristic
242Q8—21no—
243Q8—21no—
244Q8—2141no—
245Q8—2141no—
246D4—21no—
247D4—21no—
248D4—42no—
249D4—4284no—
250D4—4284no—
251D4—4282no—
252D4—2141no—
253D4—2141no—
254C2×C4—22no—
255C2×C4—21no—
256C2×C4—42no—
257C2×C4—1072no—
258C2×C4—4282no—
259C2×C4—2142no—
260C2×C4—2141no—
261C2×C4—2142no—
262C2×C4—2144no—
263C2×C4—2142no—
264C2×C4—2142no—
265C2×C4D4×D10712yes—
266C2×C4D4×D10711yescharacteristic
267C2×C4#1313381812yes—
268C2×C4C23×D10711yescharacteristic
269C8—22no—
270C8—1072no—
271C8—1072no—
272C22—22no—
273C22—42no—
274C22—41no—
275C22—4284no—
276C22—4282no—
277C22—4281no—
278C22—2142no—
279C22—2142no—
280C22—2142no—
281C22#1681208212yes—
282C22C2×D4×D10711yescharacteristic
283C4—42no—
284C4—41no—
285C4—4282no—
286C4—4281no—
287C4—2142no—
288C4#1681208212yes—
289C4C2×D4×D10711yescharacteristic
290C4C2×#1313381811yescharacteristic
291C2—22no—
292C2—81no—
293C2—8561no—
294C2—4282no—
295C2C2×#1681208211yescharacteristic
296C2#2029342412yes—
subgroupG/H
centerC2×C4D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#416×C107C2
socleC22×C107C22×D4
hypercenter#416D107
hypocenterC107#3054
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ #416

Complements

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

NH ≅ G/Ntype
1. C2×#16812129C2semidirect
2. C22×#13133801C2semidirect
3. #20293139C2semidirect
4. #20293134C2semidirect
5. #20292884C2semidirect
6. #20292879C2semidirect
7. #20292876C2semidirect
8. #20292872C2semidirect
9. #20292663C2semidirect
10. #20292624C2semidirect
11. #416×C107C2semidirect
15. C2×#13133808C22semidirect
16. C2×#13133806C22semidirect
17. C2×#13133805C22semidirect
18. C2×#13133803C22semidirect
34. #16812042C22semidirect
35. #16812040C22semidirect
36. #16812034C22semidirect
37. #16812031C22semidirect
41. #103×C107C22semidirect
42. #102×C107C22semidirect
63. C2×C107⋊C8D4semidirect
149. C107#3054semidirect
160. #416D107semidirect
189. #103C2×D107semidirect
191. #102C2×D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168121292
C22×#131338012
#202931392
#202931342
#202928842
#202928792
#202928762
#202928722
#202926632
#202926242
#416×C1072
#3054107