Finite Group Databasebeta

← #28348243

Subgroups — #28348243

Order 13696.

454 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 125 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#20293441C211yesmaximal, characteristic, complemented
2#20293438C212yesmaximal, complemented
3#20293010C211yesmaximal, characteristic, complemented
4#20292902C211yesmaximal, characteristic, complemented
5#20292894C211yesmaximal, characteristic, complemented
6#20292888C212yesmaximal, complemented
7#20292805C212yesmaximal, complemented
8#20292798C212yesmaximal, complemented
9#20292744C211yesmaximal, characteristic, complemented
10#20292742C211yesmaximal, characteristic, complemented
11#576×C107C211yesmaximal, characteristic, complemented
12#16812156C2212yescomplemented
13#16812139—22no—
14#16812139C2212yes—
15#16812139C2212yescomplemented
16#16812138—22no—
17#16812135—22no—
18#16812135C2212yescomplemented
19#16812129—22no—
20#16812129C2211yescharacteristic
21#16812128—21no—
22C2×Dic428C2211yescharacteristic, complemented
23Q16×D107—21no—
24C2×#13133806—22no—
25C2×#13133806C2212yescomplemented
26#16812116C2212yescomplemented
27#16812115C2211yescharacteristic, complemented
28#16812096C2212yescomplemented
29#16812045C2211yescharacteristic, complemented
30#16812044C2211yescharacteristic, complemented
31#16812036C2211yescharacteristic
32#16812036C2212yescomplemented
33#16812035C2211yescharacteristic, complemented
34#16812027C2212yescomplemented
35#16812026C2212yescomplemented
36C8×Dic107C2211yescharacteristic, complemented
37H(5,𝔽2)×C107C2212yes—
38AGL(1,ℤ/8)×C107—22no—
39AGL(1,ℤ/8)×C107C2212yes—
40#135×C107—21no—
41#135×C107C2211yescharacteristic
42C2×D8×C107—21no—
43C2×D8×C107C2211yescharacteristic
44#131×C107C2211yescharacteristic
45#13133818—22no—
46#13133818C2311yescharacteristic
47#13133817—22no—
48#13133817—42no—
49C2×Dic214—22no—
50C2×Dic214C2312yes—
51Q8×D107—22no—
52Dic428—21no—
53Dic428—21no—
54#13133808—22no—
55#13133808—22no—
56#13133808D411yescharacteristic
57#13133806—22no—
58#13133806—22no—
59#13133806—22no—
60#13133806—22no—
61#13133806—22no—
62#13133806D411yescharacteristic
63#13133805D411yescharacteristic
64#13133803—22no—
65#13133803D411yescharacteristic
66#13133802—22no—
67#13133802C2311yescharacteristic
68#13133802C2312yes—
69#13133801—21no—
70C2×C107⋊C8—22no—
71C2×C107⋊C8C2311yescharacteristic
72C8×D107—21no—
73C22⋊Dic107—42no—
74C4×Dic107C2311yescharacteristic
75#41×C107—22no—
76#41×C107—22no—
77#41×C107C2311yescharacteristic
78#41×C107C2312yes—
79C2×D4×C107—22no—
80C2×D4×C107—22no—
81C2×D4×C107—42no—
82C2×D4×C107C2312yes—
83Q16×C107D411yescharacteristic
84SD16×C107—22no—
85SD16×C107D412yes—
86D8×C107—22no—
87D8×C107—21no—
88D8×C107—21no—
89D8×C107D411yescharacteristic
90M16×C107—21no—
91M16×C107C2311yescharacteristic
92C2×C8×C107—21no—
93C2×C8×C107C2311yescharacteristic
94C23×C107—22no—
95C23×C107—42no—
96Q8×C107C2×D411yescharacteristic
97Q8×C107C2×D412yes—
98D4×C107—21no—
99D4×C107—22no—
100D4×C107—22no—
101D4×C107—22no—
102D4×C107—42no—
103D4×C107—42no—
104D4×C107C2×D411yescharacteristic
105D4×C107C2×D412yes—
106C2×C4×C107—21no—
107C2×C4×C107—22no—
108C2×C4×C107—42no—
109C2×C4×C107C2411yescharacteristic
110C22⋊D107—21no—
111C22⋊D107—42no—
112C2×Dic107—21no—
113C2×Dic107—22no—
114C2×Dic107—42no—
115D428C2×D411yescharacteristic
116C4×D107—21no—
117C4×D107—42no—
118Dic214—22no—
119Dic214—22no—
120Dic214—42no—
121Dic214C2×D411yescharacteristic
122C8×C107—21no—
123C8×C107C2×D411yescharacteristic
124C8×C107C2×D411yescharacteristic
125C107⋊C8—21no—
126C107⋊C8—22no—
127C107⋊C8C2×D411yescharacteristic
128C107⋊C8C2×D411yescharacteristic
129C22×C107—22no—
130C22×C107—21no—
131C22×C107—42no—
132C22×C107—82no—
133C22×C107—162no—
134C22×C107H(5,𝔽2)11yescharacteristic
135C2×D107—21no—
136C4×C107—22no—
137C4×C107—21no—
138C4×C107C22×D411yescharacteristic
139C4×C107C22×D411yescharacteristic
140Dic107—21no—
141Dic107—41no—
142Dic107—42no—
143C2×C107—21no—
144C2×C107—42no—
145C2×C107—41no—
146C2×C107—82no—
147C2×C107D4211yescharacteristic
148D107—41no—
149#3290—1071nomaximal, Sylow
150C107#329011yescharacteristic, complemented, Sylow
151#578—1071no—
152#577—1072no—
153#576D10711yescharacteristic, complemented
154#495—1071no—
155#473—1071no—
156#472—1071no—
157#471—1072no—
158#455—1072no—
159#454—1072no—
160#443—1071no—
161#443—1071no—
1622−1+4—1072no—
163H(5,𝔽2)C2×D10712yes—
164#137—1072no—
165#137—1072no—
166#137—2142no—
167#137—2142no—
168AGL(1,ℤ/8)—22no—
169AGL(1,ℤ/8)—1072no—
170AGL(1,ℤ/8)—2142no—
171AGL(1,ℤ/8)C2×D10712yes—
172#135—21no—
173#135—1071no—
174#135—2142no—
175#135—2141no—
176#135C2×D10711yescharacteristic
177C2×Q16—1071no—
178C2×Q16—2141no—
179C2×SD16—1072no—
180C2×SD16—2142no—
181C2×D8—21no—
182C2×D8C2×D10711yescharacteristic
183#131—1072no—
184#131—1071no—
185#131C2×D10711yescharacteristic
186#124—1072no—
187#108—1071no—
188#108—1071no—
189#104—1071no—
190#104—1072no—
191#104—1071no—
192#101—1072no—
193#100—1072no—
194C4×C8—1071no—
195#41—22no—
196#41—22no—
197#41—1071no—
198#41—4282no—
199#41—2142no—
200#41—2142no—
201#41C22×D10711yescharacteristic
202#41C22×D10712yes—
203C2×Q8—1072no—
204C2×Q8—2142no—
205C2×Q8—2142no—
206C2×D4—22no—
207C2×D4—22no—
208C2×D4—42no—
209C2×D4C22×D10712yes—
210Q16—1071no—
211Q16—2142no—
212Q16—2142no—
213Q16—2141no—
214Q16—2141no—
215Q16C22⋊D10711yescharacteristic
216SD16—22no—
217SD16—1071no—
218SD16—1071no—
219SD16—2142no—
220SD16—2142no—
221SD16—2142no—
222SD16—2142no—
223SD16—2142no—
224SD16C22⋊D10712yes—
225D8—22no—
226D8—21no—
227D8—21no—
228D8—1071no—
229D8—2142no—
230D8C22⋊D10711yescharacteristic
231M16—21no—
232M16—1071no—
233M16—1072no—
234M16—2142no—
235M16—2141no—
236M16C22×D10711yescharacteristic
237C2×C8—21no—
238C2×C8—1071no—
239C2×C8—2142no—
240C2×C8—2141no—
241C2×C8C22×D10711yescharacteristic
242C22⋊C4—4282no—
243C42—1071no—
244C23—22no—
245C23—42no—
246Q8—1071no—
247Q8—4282no—
248Q8—2142no—
249Q8—2142no—
250Q8D4×D10711yescharacteristic
251Q8C2×C22⋊D10712yes—
252D4—21no—
253D4—22no—
254D4—22no—
255D4—22no—
256D4—42no—
257D4—42no—
258D4—1071no—
259D4—4282no—
260D4—2141no—
261D4D4×D10711yescharacteristic
262D4C2×C22⋊D10712yes—
263C2×C4—22no—
264C2×C4—21no—
265C2×C4—42no—
266C2×C4—4282no—
267C2×C4—4282no—
268C2×C4—2141no—
269C2×C4—2142no—
270C2×C4—2141no—
271C2×C4C23×D10711yescharacteristic
272C8—21no—
273C8—1071no—
274C8—1071no—
275C8—2142no—
276C8—2141no—
277C8C2×C22⋊D10711yescharacteristic
278C8C2×C22⋊D10711yescharacteristic
279C22—22no—
280C22—21no—
281C22—42no—
282C22—82no—
283C22—162no—
284C22—2141no—
285C22#1681215411yescharacteristic
286C4—21no—
287C4—22no—
288C4—4281no—
289C4—4282no—
290C4—2141no—
291C4C2×D4×D10711yescharacteristic
292C4C22×C22⋊D10711yescharacteristic
293C2—21no—
294C2—41no—
295C2—42no—
296C2—82no—
297C2—4281no—
298C2D4×C22⋊D10711yescharacteristic
subgroupG/H
centerC2D4×C22⋊D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#576×C107C2
socleC2×C107D42
hypercenter#576D107
hypocenterC107#3290
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C2 ◁ C2×C4 ◁ #576

Complements

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

NH ≅ G/Ntype
1. #20293441C2semidirect
2. #20293438C2semidirect
3. #20293010C2semidirect
4. #20292902C2semidirect
5. #20292894C2semidirect
6. #20292888C2semidirect
7. #20292805C2semidirect
8. #20292798C2semidirect
9. #20292744C2semidirect
10. #20292742C2semidirect
11. #576×C107C2semidirect
12. #16812156C22semidirect
15. #16812139C22semidirect
18. #16812135C22semidirect
22. C2×Dic428C22semidirect
25. C2×#13133806C22semidirect
26. #16812116C22semidirect
27. #16812115C22semidirect
28. #16812096C22semidirect
29. #16812045C22semidirect
30. #16812044C22semidirect
32. #16812036C22semidirect
33. #16812035C22semidirect
34. #16812027C22semidirect
35. #16812026C22semidirect
36. C8×Dic107C22semidirect
150. C107#3290semidirect
153. #576D107semidirect

Maximal subgroups

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

subgroupindex
#202934412
#202934382
#202930102
#202929022
#202928942
#202928882
#202928052
#202927982
#202927442
#202927422
#576×C1072
#3290107