Finite Group Databasebeta

← #28346463

Subgroups — #28346463

Order 13696.

492 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 492 conjugacy classes of subgroups — too many to draw. Only its 127 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#20293178C211yesmaximal, characteristic, complemented
2C4×D4×D107C211yesmaximal, characteristic, complemented
3#20292942C212yesmaximal, complemented
4#20292904C211yesmaximal, characteristic, complemented
5#20292823C211yesmaximal, characteristic, complemented
6#20292822C211yesmaximal, characteristic, complemented
7#20292762C212yesmaximal, complemented
8#20292709C212yesmaximal, complemented
9#20292707C212yesmaximal, complemented
10#20292679C211yesmaximal, characteristic, complemented
11#474×C107C211yesmaximal, characteristic, complemented
12C2×#13133819—21no—
13C2×D4×D107C2211yescharacteristic
14C22×C4×D107—21no—
15C2×D856C2212yes—
16C2×#13133803C2212yes—
17#16812106C2211yescharacteristic, complemented
18#16812098—21no—
19#16812084—21no—
20#16812084C2212yes—
21C4×D428C2211yescharacteristic
22D4×Dic107C2211yescharacteristic
23#16812071—21no—
24#16812071C2211yescharacteristic
25#16812071C2212yescomplemented
26C4×C22⋊D107—21no—
27(C22⋊D107)⋊C4—21no—
28C4⋊C4×D107C2211yescharacteristic
29C22⋊C4×D107—21no—
30C42×D107C2211yescharacteristic
31#16812042C2212yescomplemented
32C4⋊(C107⋊C8)C2211yescharacteristic, complemented
33#16812031C2212yescomplemented
34#16812029C2212yes—
35#16812029C2212yescomplemented
36(C2×D107)⋊C8C2212yes—
37C8×Dic107C2212yescomplemented
38C4⋊Q8×C107C2211yescharacteristic, complemented
39C4×D4×C107C2211yescharacteristic
40#107×C107C2212yescomplemented
41C4⋊C8×C107C2211yescharacteristic, complemented
42#102×C107C2212yes—
43C23×D107—21no—
44#13133819—44no—
45C2×D428—22no—
46C2×D428—21no—
47C2×D428—21no—
48C2×D428C2312yes—
49C2×D428C2311yescharacteristic
50C2×C22⋊D107—21no—
51D4×D107—22no—
52D4×D107—22no—
53C22×Dic107—21no—
54C2×C4×D107—21no—
55C2×C4×D107—21no—
56C2×C4×D107—21no—
57C2×C4×D107—22no—
58C2×C4×D107—44no—
59C2×C4×D107D411yescharacteristic
60C2×C4×D107D411yescharacteristic
61C2×C4×D107C2311yescharacteristic
62D856—24no—
63#13133803—24no—
64C2×C107⋊C8C2312yes—
65C4⋊Dic107C2311yescharacteristic
66Dic107⋊C4—21no—
67C22⋊Dic107—21no—
68(C2×D107)⋊C4—21no—
69(C2×D107)⋊C4—21no—
70(C2×D107)⋊C4—22no—
71(C2×D107)⋊C4—22no—
72C4×Dic107D411yescharacteristic
73C4×Dic107D411yescharacteristic
74C4×Dic107C2311yescharacteristic
75C2×Q8×C107—21no—
76C2×D4×C107C2311yescharacteristic
77C22×C4×C107—21no—
78C2×C8×C107C2312yes—
79C4⋊C4×C107—21no—
80C4⋊C4×C107C2312yes—
81C4⋊C4×C107C2311yescharacteristic
82C22⋊C4×C107—21no—
83C42×C107C2311yescharacteristic
84C23×C107—21no—
85C22×D107—24no—
86C22×D107—21no—
87C22×D107—22no—
88C22×D107—42no—
89C22×D107—42no—
90C22×D107C2×D411yescharacteristic
91Q8×C107—42no—
92D4×C107—21no—
93D4×C107#4112yescomplemented
94C2×C4×C107—21no—
95C2×C4×C107—22no—
96C2×C4×C107—42no—
97C2×C4×C107C2×D411yescharacteristic
98C2×C4×C107C2×D411yescharacteristic
99C2×C4×C107C2411yescharacteristic
100C22⋊D107—24no—
101C2×Dic107—21no—
102C2×Dic107—42no—
103C2×Dic107C2×D411yescharacteristic
104C2×Dic107C2×D411yescharacteristic
105C2×Dic107C2×D411yescharacteristic
106D428—21no—
107D428—22no—
108D428—22no—
109D428—42no—
110D428—44no—
111D428—42no—
112D428#4112yescomplemented
113C4×D107—21no—
114C4×D107—21no—
115C4×D107—41no—
116C4×D107—42no—
117C4×D107—42no—
118C4×D107—42no—
119C4×D107—44no—
120C4×D107D814yescomplemented
121C8×C107—22no—
122C107⋊C8—22no—
123C22×C107—22no—
124C22×C107—41no—
125C22×C107—41no—
126C22×C107C22×D411yescharacteristic
127C2×D107—21no—
128C2×D107—21no—
129C2×D107—22no—
130C2×D107—48no—
131C2×D107—41no—
132C2×D107—41no—
133C2×D107—42no—
134C2×D107—42no—
135C2×D107—42no—
136C2×D107C2×D812yes—
137C4×C107—21no—
138C4×C107—41no—
139C4×C107—42no—
140C4×C107C2×D812yes—
141C4×C107C2×#4111yescharacteristic
142C4×C107C2×#4111yescharacteristic
143Dic107—21no—
144Dic107—21no—
145Dic107—41no—
146Dic107C2×D812yes—
147C2×C107—42no—
148C2×C107#52511yescharacteristic
149C2×C107C22×D811yescharacteristic
150C2×C107#57511yescharacteristic
151D107—22no—
152D107—42no—
153D107—82no—
154#3100—1071nomaximal, Sylow
155C107#310011yescharacteristic, complemented, Sylow
156#532—1071no—
157C2×C4×D4—1071no—
158#480—1072no—
159#474—1071no—
160#474D10711yescharacteristic, complemented
161#459—1071no—
162#459—1071no—
163#447—1072no—
164C4×D8—1072no—
165C4×D8—1072no—
166#431—1071no—
167C2×#41—2141no—
168C22×D4—1071no—
169C23×C4—2141no—
170C2×D8—1072no—
171C2×D8—1072no—
172C4⋊Q8C2×D10711yescharacteristic, complemented
173#127—1071no—
174#124—2141no—
175#121—1072no—
176#121—2141no—
177C4×D4—1071no—
178C4×D4—1071no—
179C4×D4—1072no—
180C4×D4—1071no—
181C4×D4—2141no—
182C4×D4—2141no—
183C4×D4—2141no—
184C4×D4C2×D10711yescharacteristic
185C2×C4⋊C4—1071no—
186C2×C22⋊C4—2141no—
187C2×C42—1071no—
188#107—1072no—
189#107C2×D10712yescomplemented
190C4⋊C8—1071no—
191C4⋊C8C2×D10711yescharacteristic, complemented
192#102—1072no—
193#102—1072no—
194#102—1072no—
195#102C2×D10712yes—
196C22⋊C8—1072no—
197C4×C8—1072no—
198C24—2141no—
199#41—4284no—
200C2×Q8—21no—
201C2×D4—1071no—
202C2×D4—1072no—
203C2×D4—2142no—
204C2×D4—2141no—
205C2×D4—2142no—
206C2×D4—2142no—
207C2×D4—2141no—
208C2×D4—2141no—
209C2×D4C22×D10711yescharacteristic
210C22×C4—21no—
211C22×C4—1071no—
212C22×C4—1071no—
213C22×C4—1071no—
214C22×C4—4284no—
215C22×C4—2141no—
216C22×C4—2142no—
217C22×C4—2141no—
218C22×C4—2141no—
219C22×C4—2141no—
220D8—2144no—
221D8—2144no—
222C2×C8—1072no—
223C2×C8C22×D10712yes—
224C4⋊C4—21no—
225C4⋊C4—1071no—
226C4⋊C4—2141no—
227C4⋊C4C22×D10711yescharacteristic
228C4⋊C4C22×D10712yes—
229C22⋊C4—21no—
230C22⋊C4—2141no—
231C22⋊C4—2142no—
232C22⋊C4—2142no—
233C22⋊C4—2141no—
234C22⋊C4—2141no—
235C42—1071no—
236C42—1071no—
237C42—1071no—
238C42C22×D10711yescharacteristic
239C23—21no—
240C23—1071no—
241C23—4282no—
242C23—4282no—
243C23—2144no—
244C23—2142no—
245C23—2141no—
246Q8—42no—
247D4—21no—
248D4—1072no—
249D4—4282no—
250D4—4284no—
251D4—4282no—
252D4—2144no—
253D4—2142no—
254D4—2142no—
255D4—2141no—
256D4#1313381912yescomplemented
257C2×C4—21no—
258C2×C4—22no—
259C2×C4—42no—
260C2×C4—1071no—
261C2×C4—1074no—
262C2×C4—1071no—
263C2×C4—1071no—
264C2×C4—4281no—
265C2×C4—4282no—
266C2×C4—4284no—
267C2×C4—4282no—
268C2×C4—4282no—
269C2×C4—4282no—
270C2×C4—2141no—
271C2×C4—2141no—
272C2×C4—2141no—
273C2×C4D4×D10711yescharacteristic
274C2×C4D4×D10711yescharacteristic
275C2×C4C23×D10711yescharacteristic
276C8—22no—
277C8—2142no—
278C22—22no—
279C22—41no—
280C22—41no—
281C22—1072no—
282C22—4288no—
283C22—4282no—
284C22—4282no—
285C22—4282no—
286C22—4281no—
287C22—4281no—
288C22—2142no—
289C22—2141no—
290C22—2141no—
291C22C2×D4×D10711yescharacteristic
292C4—21no—
293C4—41no—
294C4—42no—
295C4—1072no—
296C4—4281no—
297C4—2141no—
298C4—2141no—
299C4D8×D10712yes—
300C4#41×D10711yescharacteristic
301C4C2×#1313381911yescharacteristic
302C2—42no—
303C2—8562no—
304C2—4282no—
305C2—2142no—
306C2#2029314011yescharacteristic
307C2C2×D8×D10711yescharacteristic
308C2#2029342611yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#474×C107C2
socleC22×C107C22×D4
hypercenter#474D107
hypocenterC107#3100
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #474

Complements

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

NH ≅ G/Ntype
1. #20293178C2semidirect
2. C4×D4×D107C2semidirect
3. #20292942C2semidirect
4. #20292904C2semidirect
5. #20292823C2semidirect
6. #20292822C2semidirect
7. #20292762C2semidirect
8. #20292709C2semidirect
9. #20292707C2semidirect
10. #20292679C2semidirect
11. #474×C107C2semidirect
17. #16812106C22semidirect
25. #16812071C22semidirect
31. #16812042C22semidirect
32. C4⋊(C107⋊C8)C22semidirect
33. #16812031C22semidirect
35. #16812029C22semidirect
37. C8×Dic107C22semidirect
38. C4⋊Q8×C107C22semidirect
40. #107×C107C22semidirect
41. C4⋊C8×C107C22semidirect
93. D4×C107#41semidirect
112. D428#41semidirect
120. C4×D107D8semidirect
155. C107#3100semidirect
160. #474D107semidirect
172. C4⋊Q8C2×D107semidirect
189. #107C2×D107semidirect
191. C4⋊C8C2×D107semidirect
256. D4#13133819semidirect

Maximal subgroups

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

subgroupindex
#202931782
C4×D4×D1072
#202929422
#202929042
#202928232
#202928222
#202927622
#202927092
#202927072
#202926792
#474×C1072
#3100107