Finite Group Databasebeta

← #28348242

Subgroups — #28348242

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#20293444C211yesmaximal, characteristic, complemented
2#20293437C212yesmaximal, complemented
3#20293428C211yesmaximal, characteristic, complemented
4#20293010C211yesmaximal, characteristic, complemented
5#20292903C211yesmaximal, characteristic, complemented
6#20292889C212yesmaximal, complemented
7#20292807C212yesmaximal, complemented
8#20292798C212yesmaximal, complemented
9#20292743C211yesmaximal, characteristic, complemented
10#20292742C211yesmaximal, characteristic, complemented
11#472×C107C211yesmaximal, characteristic, complemented
12#16812156C2212yescomplemented
13#16812154C2212yes—
14#16812142—22no—
15#16812142C2212yes—
16#16812142C2212yescomplemented
17#16812138—22no—
18#16812135C2212yes—
19#16812133—22no—
20#16812133—22no—
21#16812131—21no—
22#16812131C2211yescharacteristic
23#16812129C2211yescharacteristic
24#16812128—21no—
25#16812128—22no—
26C2×Dic428—21no—
27C2×Dic428C2211yescharacteristic, complemented
28C2×#13133806C2212yescomplemented
29SD16×D107—22no—
30D8×D107—21no—
31#16812117C2212yescomplemented
32#16812115C2211yescharacteristic
33#16812096C2212yescomplemented
34#16812044C2211yescharacteristic, complemented
35#16812036C2211yescharacteristic, complemented
36#16812035C2211yescharacteristic
37#16812035C2212yescomplemented
38#16812027C2212yescomplemented
39C8×Dic107C2211yescharacteristic, complemented
40AGL(1,ℤ/8)×C107C2212yescomplemented
41C2×D8×C107C2211yescharacteristic
42#131×C107C2211yescharacteristic, complemented
43#108×C107C2211yescharacteristic, complemented
44#100×C107C2212yescomplemented
45#13133818—22no—
46#13133818C2311yescharacteristic
47#13133818C2312yes—
48#13133817—22no—
49#13133817—22no—
50#13133817—22no—
51#13133817—42no—
52C2×Dic214—22no—
53C2×Dic214C2312yes—
54Q8×D107—22no—
55C2×C22⋊D107—42no—
56D4×D107—22no—
57D4×D107—22no—
58Dic428—22no—
59Dic428—22no—
60Dic428—21no—
61Dic428—21no—
62Dic428D411yescharacteristic
63#13133808D411yescharacteristic
64#13133807—22no—
65#13133807—22no—
66#13133807D412yes—
67#13133806—22no—
68#13133806—22no—
69#13133806—22no—
70#13133806D411yescharacteristic
71#13133805D411yescharacteristic
72D856D411yescharacteristic
73#13133803—22no—
74#13133803D411yescharacteristic
75#13133802C2311yescharacteristic
76#13133801—21no—
77#13133801—22no—
78C2×C107⋊C8C2311yescharacteristic
79C8×D107—21no—
80C8×D107—22no—
81C22⋊Dic107—42no—
82C4×Dic107C2311yescharacteristic
83#41×C107C2311yescharacteristic
84C2×D4×C107C2312yes—
85SD16×C107—22no—
86D8×C107—22no—
87D8×C107—21no—
88D8×C107—21no—
89M16×C107—21no—
90M16×C107C2311yescharacteristic
91M16×C107C2312yes—
92C2×C8×C107—21no—
93C2×C8×C107C2311yescharacteristic
94C23×C107—22no—
95C22×D107—42no—
96Q8×C107C2×D411yescharacteristic
97D4×C107—21no—
98D4×C107—22no—
99D4×C107—22no—
100D4×C107C2×D411yescharacteristic
101C2×C4×C107—21no—
102C2×C4×C107C2411yescharacteristic
103C22⋊D107—21no—
104C22⋊D107—22no—
105C22⋊D107—42no—
106C22⋊D107—42no—
107C22⋊D107—42no—
108C2×Dic107—21no—
109C2×Dic107—22no—
110C2×Dic107—42no—
111C2×Dic107—42no—
112D428C2×D412yes—
113D428C2×D411yescharacteristic
114C4×D107—21no—
115C4×D107—22no—
116C4×D107—42no—
117Dic214—22no—
118Dic214—22no—
119Dic214—42no—
120Dic214C2×D412yes—
121Dic214C2×D411yescharacteristic
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×C107—42no—
131C22×C107—82no—
132C22×C107H(5,𝔽2)11yescharacteristic
133C2×D107—22no—
134C2×D107—21no—
135C2×D107—162no—
136C4×C107—21no—
137C4×C107C22×D411yescharacteristic
138C4×C107C22×D411yescharacteristic
139Dic107—22no—
140Dic107—21no—
141Dic107—41no—
142Dic107—42no—
143C2×C107—21no—
144C2×C107—41no—
145C2×C107—82no—
146C2×C107D4211yescharacteristic
147D107—42no—
148D107—41no—
149#3290—1071nomaximal, Sylow
150C107#329011yescharacteristic, complemented, Sylow
151#578—1071no—
152#577—1072no—
153#576—1071no—
154#495—1071no—
155#473—1071no—
156#472D10711yescharacteristic, complemented
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)—2142no—
170AGL(1,ℤ/8)—2142no—
171AGL(1,ℤ/8)C2×D10712yescomplemented
172#135—1071no—
173#135—1071no—
174#135—2141no—
175#135—2142no—
176#135—2141no—
177C2×Q16—1071no—
178C2×Q16—2141no—
179C2×SD16—1072no—
180C2×SD16—2142no—
181C2×D8—2141no—
182C2×D8C2×D10711yescharacteristic
183#131—1072no—
184#131—1071no—
185#131C2×D10711yescharacteristic, complemented
186#124—1072no—
187#108—1071no—
188#108C2×D10711yescharacteristic, complemented
189#104—1071no—
190#104—1071no—
191#104—1072no—
192#101—1072no—
193#100C2×D10712yescomplemented
194C4×C8—1071no—
195#41—1072no—
196#41—1071no—
197#41—4282no—
198#41—2142no—
199#41—2142no—
200#41—2142no—
201#41—2142no—
202#41C22×D10711yescharacteristic
203C2×Q8—1072no—
204C2×Q8—2142no—
205C2×Q8—2142no—
206C2×D4—4282no—
207C2×D4—2142no—
208C2×D4—2142no—
209C2×D4C22×D10712yes—
210Q16—1071no—
211Q16—1071no—
212Q16—2142no—
213Q16—2142no—
214Q16—2141no—
215Q16—2141no—
216SD16—22no—
217SD16—1071no—
218SD16—1071no—
219SD16—1072no—
220SD16—2142no—
221SD16—2142no—
222SD16—2142no—
223SD16—2142no—
224SD16—2142no—
225D8—22no—
226D8—21no—
227D8—21no—
228D8—1071no—
229D8—1071no—
230D8—2142no—
231M16—21no—
232M16—1071no—
233M16—2142no—
234M16—2141no—
235M16C22×D10711yescharacteristic
236M16C22×D10712yes—
237C2×C8—21no—
238C2×C8—1071no—
239C2×C8—2142no—
240C2×C8—2141no—
241C2×C8C22×D10711yescharacteristic
242C22⋊C4—4282no—
243C42—1071no—
244C23—22no—
245C23—4282no—
246Q8—1072no—
247Q8—1071no—
248Q8—4282no—
249Q8—2142no—
250Q8—2142no—
251Q8D4×D10711yescharacteristic
252D4—21no—
253D4—22no—
254D4—22no—
255D4—1072no—
256D4—1071no—
257D4—4282no—
258D4—4282no—
259D4—4282no—
260D4—2141no—
261D4—2142no—
262D4D4×D10711yescharacteristic
263C2×C4—21no—
264C2×C4—4282no—
265C2×C4—4282no—
266C2×C4—4282no—
267C2×C4—2141no—
268C2×C4—2142no—
269C2×C4—2141no—
270C2×C4—2142no—
271C2×C4C23×D10711yescharacteristic
272C8—21no—
273C8—22no—
274C8—1071no—
275C8—1071no—
276C8—2141no—
277C8D4×D10711yescharacteristic
278C8D4×D10711yescharacteristic
279C22—21no—
280C22—42no—
281C22—82no—
282C22—17122no—
283C22—2142no—
284C22—2141no—
285C22#1681215511yescharacteristic
286C4—21no—
287C4—4281no—
288C4—4282no—
289C4—2141no—
290C4—2142no—
291C4C2×D4×D10711yescharacteristic
292C4C2×D4×D10711yescharacteristic
293C2—21no—
294C2—41no—
295C2—82no—
296C2—4281no—
297C2—4282no—
298C2#2029326111yescharacteristic
subgroupG/H
centerC2#20293261
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#472×C107C2
socleC2×C107D42
hypercenter#472D107
hypocenterC107#3290
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C2 ◁ C2×C4 ◁ #472

Complements

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

NH ≅ G/Ntype
1. #20293444C2semidirect
2. #20293437C2semidirect
3. #20293428C2semidirect
4. #20293010C2semidirect
5. #20292903C2semidirect
6. #20292889C2semidirect
7. #20292807C2semidirect
8. #20292798C2semidirect
9. #20292743C2semidirect
10. #20292742C2semidirect
11. #472×C107C2semidirect
12. #16812156C22semidirect
16. #16812142C22semidirect
27. C2×Dic428C22semidirect
28. C2×#13133806C22semidirect
31. #16812117C22semidirect
33. #16812096C22semidirect
34. #16812044C22semidirect
35. #16812036C22semidirect
37. #16812035C22semidirect
38. #16812027C22semidirect
39. C8×Dic107C22semidirect
40. AGL(1,ℤ/8)×C107C22semidirect
42. #131×C107C22semidirect
43. #108×C107C22semidirect
44. #100×C107C22semidirect
150. C107#3290semidirect
156. #472D107semidirect
171. AGL(1,ℤ/8)C2×D107semidirect
185. #131C2×D107semidirect
188. #108C2×D107semidirect
193. #100C2×D107semidirect

Maximal subgroups

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

subgroupindex
#202934442
#202934372
#202934282
#202930102
#202929032
#202928892
#202928072
#202927982
#202927432
#202927422
#472×C1072
#3290107