Finite Group Databasebeta

← #28348250

Subgroups — #28348250

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#20293436C212yesmaximal, complemented
3#20293432C211yesmaximal, characteristic, complemented
4#20293012C211yesmaximal, characteristic, complemented
5#20292898C211yesmaximal, characteristic, complemented
6#20292893C212yesmaximal, complemented
7#20292803C212yesmaximal, complemented
8#20292802C212yesmaximal, complemented
9#20292743C211yesmaximal, characteristic, complemented
10#20292742C211yesmaximal, characteristic, complemented
11#473×C107C211yesmaximal, characteristic, complemented
12#16812157C2212yescomplemented
13#16812155C2212yes—
14#16812141—22no—
15#16812141—22no—
16#16812140C2212yescomplemented
17#16812137—22no—
18#16812137C2212yes—
19#16812137C2212yescomplemented
20#16812136—22no—
21#16812132—22no—
22#16812132—21no—
23#16812131—21no—
24#16812131C2211yescharacteristic
25#16812129C2211yescharacteristic
26Q16×D107—21no—
27C2×#13133805C2212yescomplemented
28SD16×D107—22no—
29C2×D856—21no—
30C2×D856C2211yescharacteristic
31#16812117C2212yescomplemented
32#16812115C2211yescharacteristic, complemented
33#16812098C2212yescomplemented
34#16812044C2211yescharacteristic, complemented
35#16812036C2211yescharacteristic
36#16812035C2211yescharacteristic, complemented
37#16812035C2212yescomplemented
38#16812025C2212yescomplemented
39C8×Dic107C2211yescharacteristic, complemented
40#137×C107C2212yes—
41C2×Q16×C107C2211yescharacteristic, complemented
42#131×C107C2211yescharacteristic
43#108×C107C2211yescharacteristic, complemented
44#101×C107C2212yescomplemented
45#13133819—22no—
46#13133819—22no—
47#13133819—22no—
48#13133818—22no—
49#13133818—42no—
50#13133818C2311yescharacteristic
51#13133818C2312yes—
52Q8×D107—22no—
53Q8×D107—22no—
54C2×D428—22no—
55C2×D428C2312yes—
56D4×D107—22no—
57D4×D107—42no—
58Dic428D411yescharacteristic
59#13133808—22no—
60#13133808D411yescharacteristic
61#13133807—22no—
62#13133807—22no—
63#13133807D412yes—
64#13133806D411yescharacteristic
65#13133805—22no—
66#13133805—22no—
67#13133805—22no—
68#13133805D411yescharacteristic
69D856—22no—
70D856—22no—
71D856—21no—
72D856—21no—
73D856D411yescharacteristic
74#13133803D411yescharacteristic
75#13133802C2311yescharacteristic
76#13133801—21no—
77#13133801—22no—
78C2×C107⋊C8C2311yescharacteristic
79C8×D107—21no—
80C8×D107—22no—
81(C2×D107)⋊C4—42no—
82C4×Dic107C2311yescharacteristic
83#41×C107C2311yescharacteristic
84C2×Q8×C107C2312yes—
85Q16×C107—22no—
86Q16×C107—21no—
87Q16×C107—21no—
88SD16×C107—22no—
89M16×C107—21no—
90M16×C107C2311yescharacteristic
91M16×C107C2312yes—
92C2×C8×C107—21no—
93C2×C8×C107C2311yescharacteristic
94C22×D107—22no—
95C22×D107—42no—
96Q8×C107—22no—
97Q8×C107—22no—
98Q8×C107C2×D411yescharacteristic
99D4×C107—21no—
100D4×C107C2×D411yescharacteristic
101C2×C4×C107—21no—
102C2×C4×C107—22no—
103C2×C4×C107C2411yescharacteristic
104C22⋊D107—22no—
105C22⋊D107—21no—
106C22⋊D107—42no—
107C2×Dic107—21no—
108D428—22no—
109D428—22no—
110D428—42no—
111D428—42no—
112D428C2×D411yescharacteristic
113D428C2×D412yes—
114C4×D107—22no—
115C4×D107—21no—
116C4×D107—42no—
117C4×D107—42no—
118C4×D107—42no—
119Dic214—42no—
120Dic214C2×D411yescharacteristic
121Dic214C2×D412yes—
122C8×C107—21no—
123C8×C107—22no—
124C8×C107C2×D411yescharacteristic
125C8×C107C2×D411yescharacteristic
126C107⋊C8—21no—
127C107⋊C8C2×D411yescharacteristic
128C107⋊C8C2×D411yescharacteristic
129C22×C107—21no—
130C22×C107H(5,𝔽2)11yescharacteristic
131C2×D107—21no—
132C2×D107—22no—
133C2×D107—42no—
134C2×D107—82no—
135C2×D107—162no—
136C4×C107—21no—
137C4×C107—42no—
138C4×C107C22×D411yescharacteristic
139C4×C107C22×D411yescharacteristic
140Dic107—21no—
141Dic107—22no—
142Dic107—41no—
143C2×C107—21no—
144C2×C107—41no—
145C2×C107D4211yescharacteristic
146D107—41no—
147D107—42no—
148D107—82no—
149#3290—1071nomaximal, Sylow
150C107#329011yescharacteristic, complemented, Sylow
151#578—1071no—
152#577—1072no—
153#576—1071no—
154#495—1071no—
155#473D10711yescharacteristic, complemented
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—2142no—
166#137—2142no—
167#137C2×D10712yes—
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—1072no—
180C2×SD16—2142no—
181C2×D8—1071no—
182C2×D8—2141no—
183#131—1072no—
184#131—1071no—
185#131C2×D10711yescharacteristic
186#124—1072no—
187#108—1071no—
188#108C2×D10711yescharacteristic, complemented
189#104—1071no—
190#104—1071no—
191#104—1072no—
192#101C2×D10712yescomplemented
193#100—1072no—
194C4×C8—1071no—
195#41—1071no—
196#41—1072no—
197#41—4282no—
198#41—2142no—
199#41—2142no—
200#41—2142no—
201#41—2142no—
202#41C22×D10711yescharacteristic
203C2×Q8—2142no—
204C2×Q8—2142no—
205C2×Q8C22×D10712yes—
206C2×D4—1072no—
207C2×D4—4282no—
208C2×D4—2142no—
209C2×D4—2142no—
210Q16—22no—
211Q16—21no—
212Q16—21no—
213Q16—1071no—
214Q16—1071no—
215Q16—2142no—
216SD16—22no—
217SD16—1071no—
218SD16—1071no—
219SD16—1072no—
220SD16—2142no—
221SD16—2142no—
222SD16—2142no—
223SD16—2142no—
224SD16—2142no—
225D8—1071no—
226D8—1071no—
227D8—2142no—
228D8—2142no—
229D8—2141no—
230D8—2141no—
231M16—21no—
232M16—1071no—
233M16—2142no—
234M16—2141no—
235M16C22×D10712yes—
236M16C22×D10711yescharacteristic
237C2×C8—21no—
238C2×C8—1071no—
239C2×C8—2142no—
240C2×C8—2141no—
241C2×C8C22×D10711yescharacteristic
242C22⋊C4—4282no—
243C42—1071no—
244C23—4282no—
245C23—2142no—
246Q8—22no—
247Q8—22no—
248Q8—1071no—
249Q8—1072no—
250Q8—4282no—
251Q8D4×D10711yescharacteristic
252D4—21no—
253D4—1071no—
254D4—1072no—
255D4—4282no—
256D4—4282no—
257D4—4282no—
258D4—2142no—
259D4—2141no—
260D4—2142no—
261D4—2142no—
262D4D4×D10711yescharacteristic
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—21no—
273C8—22no—
274C8—1071no—
275C8—1071no—
276C8—2141no—
277C8D4×D10711yescharacteristic
278C8D4×D10711yescharacteristic
279C22—21no—
280C22—8562no—
281C22—4282no—
282C22—17122no—
283C22—2141no—
284C22—2142no—
285C22#1681215511yescharacteristic
286C4—21no—
287C4—42no—
288C4—4281no—
289C4—2142no—
290C4—2141no—
291C4C2×D4×D10711yescharacteristic
292C4C2×D4×D10711yescharacteristic
293C2—21no—
294C2—41no—
295C2—8562no—
296C2—4282no—
297C2—4281no—
298C2#2029326111yescharacteristic
subgroupG/H
centerC2#20293261
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#473×C107C2
socleC2×C107D42
hypercenter#473D107
hypocenterC107#3290
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C2 ◁ C2×C4 ◁ #473

Complements

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

NH ≅ G/Ntype
1. #20293440C2semidirect
2. #20293436C2semidirect
3. #20293432C2semidirect
4. #20293012C2semidirect
5. #20292898C2semidirect
6. #20292893C2semidirect
7. #20292803C2semidirect
8. #20292802C2semidirect
9. #20292743C2semidirect
10. #20292742C2semidirect
11. #473×C107C2semidirect
12. #16812157C22semidirect
16. #16812140C22semidirect
19. #16812137C22semidirect
27. C2×#13133805C22semidirect
31. #16812117C22semidirect
32. #16812115C22semidirect
33. #16812098C22semidirect
34. #16812044C22semidirect
36. #16812035C22semidirect
37. #16812035C22semidirect
38. #16812025C22semidirect
39. C8×Dic107C22semidirect
41. C2×Q16×C107C22semidirect
43. #108×C107C22semidirect
44. #101×C107C22semidirect
150. C107#3290semidirect
155. #473D107semidirect
178. C2×Q16C2×D107semidirect
188. #108C2×D107semidirect
192. #101C2×D107semidirect

Maximal subgroups

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

subgroupindex
#202934402
#202934362
#202934322
#202930122
#202928982
#202928932
#202928032
#202928022
#202927432
#202927422
#473×C1072
#3290107