Finite Group Databasebeta

← #28348246

Subgroups — #28348246

Order 13696.

454 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 117 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#20293440C211yesmaximal, characteristic, complemented
2#20293433C212yesmaximal, complemented
3#20293428C211yesmaximal, characteristic, complemented
4#20292899C211yesmaximal, characteristic, complemented
5#20292894C211yesmaximal, characteristic, complemented
6#20292887C212yesmaximal, complemented
7#20292804C212yesmaximal, complemented
8#20292799C212yesmaximal, complemented
9#20292745C211yesmaximal, characteristic, complemented
10#20292745C211yesmaximal, characteristic, complemented
11#495×C107C211yesmaximal, characteristic, complemented
12#16812157C2212yescomplemented
13#16812154C2212yes—
14#16812141—22no—
15#16812140C2212yes—
16#16812140C2212yescomplemented
17#16812138—22no—
18#16812136—22no—
19#16812135C2212yes—
20#16812135C2212yescomplemented
21#16812133—22no—
22#16812132—21no—
23#16812131C2211yescharacteristic
24#16812131C2211yescharacteristic
25#16812130—22no—
26#16812128—21no—
27Q16×D107—21no—
28SD16×D107—22no—
29D8×D107—21no—
30#16812115C2211yescharacteristic
31#16812115C2212yescomplemented
32#16812115C2211yescharacteristic, complemented
33#16812045C2211yescharacteristic, complemented
34#16812045C2211yescharacteristic, complemented
35#16812037C2211yescharacteristic
36#16812037C2211yescharacteristic, complemented
37#16812037C2212yescomplemented
38#16812028C2212yescomplemented
39#16812026C2212yescomplemented
40C2×Q16×C107C2211yescharacteristic, complemented
41C2×SD16×C107C2212yescomplemented
42C2×D8×C107C2211yescharacteristic
43#124×C107C2212yescomplemented
44C4×C8×C107C2211yescharacteristic, complemented
45#13133819—22no—
46#13133819—22no—
47#13133818—42no—
48#13133818C2311yescharacteristic
49#13133818C2312yes—
50#13133818C2311yescharacteristic
51#13133817—22no—
52#13133817—22no—
53Q8×D107—22no—
54Q8×D107—22no—
55C2×C22⋊D107—42no—
56D4×D107—22no—
57D4×D107—22no—
58Dic428D411yescharacteristic
59Dic428D411yescharacteristic
60#13133808—22no—
61#13133808—22no—
62#13133807D411yescharacteristic
63#13133807D411yescharacteristic
64#13133807D412yes—
65#13133806—22no—
66#13133806—22no—
67#13133805—22no—
68#13133805—22no—
69D856D411yescharacteristic
70D856D411yescharacteristic
71#13133803—22no—
72#13133803—22no—
73#13133802C2311yescharacteristic
74#13133802C2311yescharacteristic
75#13133802C2312yes—
76#13133801—21no—
77#13133801—22no—
78#13133801—21no—
79C8×D107—21no—
80C8×D107—22no—
81C8×D107—21no—
82C2×Q8×C107C2312yes—
83C2×D4×C107C2312yes—
84Q16×C107—21no—
85Q16×C107—21no—
86SD16×C107—22no—
87SD16×C107—22no—
88D8×C107—21no—
89D8×C107—21no—
90C2×C8×C107C2311yescharacteristic
91C2×C8×C107C2311yescharacteristic
92C22⋊C4×C107—42no—
93C42×C107C2311yescharacteristic
94C23×C107—22no—
95C22×D107—42no—
96Q8×C107—22no—
97Q8×C107—22no—
98D4×C107—22no—
99D4×C107—22no—
100C2×C4×C107—21no—
101C2×C4×C107—22no—
102C2×C4×C107C2411yescharacteristic
103C22⋊D107—21no—
104C22⋊D107—22no—
105C22⋊D107—21no—
106C22⋊D107—42no—
107C22⋊D107—42no—
108C2×Dic107—42no—
109D428—42no—
110D428C2×D411yescharacteristic
111D428C2×D412yes—
112D428C2×D411yescharacteristic
113C4×D107—21no—
114C4×D107—22no—
115C4×D107—21no—
116C4×D107—42no—
117C4×D107—42no—
118Dic214—42no—
119Dic214C2×D411yescharacteristic
120Dic214C2×D412yes—
121Dic214C2×D411yescharacteristic
122C8×C107C2×D411yescharacteristic
123C8×C107C2×D411yescharacteristic
124C8×C107C2×D411yescharacteristic
125C8×C107C2×D411yescharacteristic
126C107⋊C8—21no—
127C107⋊C8—22no—
128C107⋊C8—21no—
129C22×C107—42no—
130C22×C107—82no—
131C22×C107H(5,𝔽2)11yescharacteristic
132C2×D107—21no—
133C2×D107—22no—
134C2×D107—21no—
135C2×D107—162no—
136C4×C107—41no—
137C4×C107—42no—
138C4×C107C22×D411yescharacteristic
139C4×C107C22×D411yescharacteristic
140Dic107—21no—
141Dic107—22no—
142Dic107—21no—
143C2×C107—21no—
144C2×C107—82no—
145C2×C107D4211yescharacteristic
146D107—41no—
147D107—42no—
148D107—41no—
149#3290—1071nomaximal, Sylow
150C107#329011yescharacteristic, complemented, Sylow
151#578—1071no—
152#577—1072no—
153#576—1071no—
154#495D10711yescharacteristic, complemented
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)—1072no—
164#137—1072no—
165#137—1072no—
166#137—2142no—
167#137—2142no—
168AGL(1,ℤ/8)—1072no—
169AGL(1,ℤ/8)—1072no—
170AGL(1,ℤ/8)—2142no—
171AGL(1,ℤ/8)—2142no—
172#135—1071no—
173#135—1071no—
174#135—2141no—
175#135—2141no—
176#135—2142no—
177C2×Q16—2141no—
178C2×Q16C2×D10711yescharacteristic, complemented
179C2×SD16—2142no—
180C2×SD16C2×D10712yescomplemented
181C2×D8—2141no—
182C2×D8C2×D10711yescharacteristic
183#131—1071no—
184#131—1072no—
185#131—1071no—
186#124C2×D10712yescomplemented
187#108—1071no—
188#108—1071no—
189#104—1071no—
190#104—1071no—
191#104—1072no—
192#101—1072no—
193#100—1072no—
194C4×C8C2×D10711yescharacteristic, complemented
195#41—1071no—
196#41—1072no—
197#41—1071no—
198#41—4282no—
199#41—2142no—
200#41—2142no—
201#41—2142no—
202#41—2142no—
203C2×Q8—2142no—
204C2×Q8—2142no—
205C2×Q8C22×D10712yes—
206C2×D4—4282no—
207C2×D4—2142no—
208C2×D4—2142no—
209C2×D4C22×D10712yes—
210Q16—21no—
211Q16—21no—
212Q16—1071no—
213Q16—1071no—
214Q16—2142no—
215Q16—2142no—
216SD16—22no—
217SD16—22no—
218SD16—1071no—
219SD16—1071no—
220SD16—1072no—
221SD16—2142no—
222SD16—2142no—
223SD16—2142no—
224SD16—2142no—
225D8—21no—
226D8—21no—
227D8—1071no—
228D8—1071no—
229D8—2142no—
230D8—2142no—
231M16—1071no—
232M16—1072no—
233M16—1071no—
234M16—2141no—
235M16—2142no—
236M16—2141no—
237C2×C8—2141no—
238C2×C8—2142no—
239C2×C8—2141no—
240C2×C8C22×D10711yescharacteristic
241C2×C8C22×D10711yescharacteristic
242C22⋊C4—42no—
243C42C22×D10711yescharacteristic
244C23—22no—
245C23—4282no—
246Q8—22no—
247Q8—22no—
248Q8—1071no—
249Q8—1072no—
250Q8—1071no—
251Q8—4282no—
252D4—22no—
253D4—22no—
254D4—1071no—
255D4—1072no—
256D4—1071no—
257D4—4282no—
258D4—4282no—
259D4—4282no—
260D4—2141no—
261D4—2142no—
262D4—2141no—
263C2×C4—21no—
264C2×C4—22no—
265C2×C4—4282no—
266C2×C4—4282no—
267C2×C4—4282no—
268C2×C4—2141no—
269C2×C4—2142no—
270C2×C4—2141no—
271C2×C4C23×D10711yescharacteristic
272C8—2141no—
273C8—2142no—
274C8—2141no—
275C8D4×D10711yescharacteristic
276C8D4×D10711yescharacteristic
277C8D4×D10711yescharacteristic
278C8D4×D10711yescharacteristic
279C22—42no—
280C22—82no—
281C22—17122no—
282C22—2141no—
283C22—2142no—
284C22—2141no—
285C22#1681215411yescharacteristic
286C4—41no—
287C4—42no—
288C4—2141no—
289C4—2142no—
290C4—2141no—
291C4C2×D4×D10711yescharacteristic
292C4C2×D4×D10711yescharacteristic
293C2—21no—
294C2—82no—
295C2—4281no—
296C2—4282no—
297C2—4281no—
298C2#2029326211yescharacteristic
subgroupG/H
centerC2#20293262
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#495×C107C2
socleC2×C107D42
hypercenter#495D107
hypocenterC107#3290
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C2 ◁ C2×C4 ◁ #495

Complements

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

NH ≅ G/Ntype
1. #20293440C2semidirect
2. #20293433C2semidirect
3. #20293428C2semidirect
4. #20292899C2semidirect
5. #20292894C2semidirect
6. #20292887C2semidirect
7. #20292804C2semidirect
8. #20292799C2semidirect
9. #20292745C2semidirect
10. #20292745C2semidirect
11. #495×C107C2semidirect
12. #16812157C22semidirect
16. #16812140C22semidirect
20. #16812135C22semidirect
31. #16812115C22semidirect
32. #16812115C22semidirect
33. #16812045C22semidirect
34. #16812045C22semidirect
36. #16812037C22semidirect
37. #16812037C22semidirect
38. #16812028C22semidirect
39. #16812026C22semidirect
40. C2×Q16×C107C22semidirect
41. C2×SD16×C107C22semidirect
43. #124×C107C22semidirect
44. C4×C8×C107C22semidirect
150. C107#3290semidirect
154. #495D107semidirect
178. C2×Q16C2×D107semidirect
180. C2×SD16C2×D107semidirect
186. #124C2×D107semidirect
194. C4×C8C2×D107semidirect

Maximal subgroups

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

subgroupindex
#202934402
#202934332
#202934282
#202928992
#202928942
#202928872
#202928042
#202927992
#202927452
#495×C1072
#3290107