Finite Group Databasebeta

← #28346458

Subgroups — #28346458

Order 13696.

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

normal subgroups only

This group has 492 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#20293172C211yesmaximal, characteristic, complemented
2C2×#16812071C211yesmaximal, characteristic, complemented
3#20292940C211yesmaximal, characteristic, complemented
4#20292906C212yesmaximal, complemented
5#20292821C212yesmaximal, complemented
6#20292763C211yesmaximal, characteristic, complemented
7#20292760C211yesmaximal, characteristic, complemented
8#20292709C212yesmaximal, complemented
9#20292707C212yesmaximal, complemented
10#20292677C211yesmaximal, characteristic, complemented
11#480×C107C211yesmaximal, characteristic, complemented
12C2×#13133817—21no—
13C22×D428C2211yescharacteristic
14C22×C4×D107—21no—
15C2×D856C2212yes—
16C2×#13133803C2212yes—
17#16812108C2211yescharacteristic, complemented
18#16812105C2211yescharacteristic, complemented
19#16812095—21no—
20#16812083C2211yescharacteristic
21#16812081—21no—
22D4×Dic107C2211yescharacteristic, complemented
23#16812071—21no—
24#16812071—21no—
25#16812071C2212yes—
26#16812071C2212yes—
27#16812071C2211yescharacteristic, complemented
28(C22⋊D107)⋊C4—21no—
29C2×(C2×D107)⋊C4—21no—
30C2×C4×Dic107C2211yescharacteristic
31#16812042C2212yescomplemented
32Dic107⋊C8C2212yescomplemented
33#16812031C2212yes—
34#16812029C2212yes—
35#16812029C2212yescomplemented
36C22⋊(C107⋊C8)C2211yescharacteristic
37C8×Dic107C2212yescomplemented
38#121×C107C2211yescharacteristic
39C2×C4⋊C4×C107C2211yescharacteristic
40#107×C107C2212yescomplemented
41#102×C107C2212yescomplemented
42C22⋊C8×C107C2211yescharacteristic
43C23×D107—21no—
44#13133817—44no—
45C2×Dic214—21no—
46C2×D428—21no—
47C2×D428—24no—
48C2×D428C2311yescharacteristic
49C2×D428C2312yes—
50C2×C22⋊D107—21no—
51C2×C22⋊D107—21no—
52C2×C22⋊D107—21no—
53C22×Dic107—21no—
54C22×Dic107D411yescharacteristic
55C22×Dic107D411yescharacteristic
56C2×C4×D107—21no—
57C2×C4×D107—22no—
58C2×C4×D107—22no—
59C2×C4×D107—44no—
60D856—24no—
61#13133803—24no—
62C2×C107⋊C8C2312yes—
63C4⋊Dic107C2311yescharacteristic
64Dic107⋊C4—21no—
65C22⋊Dic107—21no—
66(C2×D107)⋊C4—21no—
67(C2×D107)⋊C4—22no—
68(C2×D107)⋊C4—22no—
69(C2×D107)⋊C4—21no—
70C4×Dic107D411yescharacteristic
71C4×Dic107D411yescharacteristic
72C4×Dic107C2311yescharacteristic
73C4×Dic107C2311yescharacteristic
74C2×D4×C107—21no—
75C2×D4×C107C2311yescharacteristic
76C22×C4×C107—21no—
77C22×C4×C107C2311yescharacteristic
78C2×C8×C107C2312yes—
79C4⋊C4×C107—21no—
80C4⋊C4×C107C2312yes—
81C4⋊C4×C107C2311yescharacteristic
82C22⋊C4×C107—21no—
83C23×C107—21no—
84C23×C107C2×D411yescharacteristic
85C22×D107—24no—
86C22×D107—22no—
87C22×D107—21no—
88C22×D107—42no—
89C22×D107—42no—
90D4×C107—22no—
91D4×C107—42no—
92C2×C4×C107—22no—
93C2×C4×C107—21no—
94C2×C4×C107—22no—
95C2×C4×C107—42no—
96C2×C4×C107C2×D411yescharacteristic
97C2×C4×C107C2411yescharacteristic
98C22⋊D107—42no—
99C22⋊D107—42no—
100C22⋊D107—42no—
101C2×Dic107—21no—
102C2×Dic107—41no—
103C2×Dic107—42no—
104C2×Dic107D814yescomplemented
105C2×Dic107C2×D411yescharacteristic
106C2×Dic107C2×D411yescharacteristic
107C2×Dic107C2×D411yescharacteristic
108C2×Dic107C2×D411yescharacteristic
109D428—24no—
110D428—22no—
111D428—22no—
112D428#4114yescomplemented
113C4×D107—44no—
114C4×D107—44no—
115C4×D107—42no—
116Dic214—42no—
117C8×C107—22no—
118C107⋊C8—22no—
119C22×C107—21no—
120C22×C107—21no—
121C22×C107—41no—
122C22×C107—41no—
123C22×C107—41no—
124C22×C107C2×D812yes—
125C22×C107C22×D411yescharacteristic
126C2×D107—24no—
127C2×D107—48no—
128C2×D107—42no—
129C2×D107—42no—
130C2×D107—41no—
131C2×D107—41no—
132C2×D107—41no—
133C4×C107—21no—
134C4×C107—41no—
135C4×C107—42no—
136C4×C107C2×#4112yes—
137Dic107—21no—
138Dic107—21no—
139Dic107—41no—
140Dic107C2×D812yes—
141Dic107C2×D812yes—
142C2×C107—22no—
143C2×C107—81no—
144C2×C107#52511yescharacteristic
145C2×C107C22×D811yescharacteristic
146C2×C107#57511yescharacteristic
147D107—44no—
148D107—81no—
149#3100—1071nomaximal, Sylow
150C107#310011yescharacteristic, complemented, Sylow
151#532—1071no—
152C2×C4×D4—1071no—
153#480—1071no—
154#480D10711yescharacteristic, complemented
155#474—1072no—
156#459—1072no—
157#447—1071no—
158#447—1071no—
159C4×D8—1072no—
160C4×D8—1072no—
161#431—1071no—
162C2×#41—2141no—
163C22×D4—1071no—
164C23×C4—2141no—
165C2×D8—1072no—
166C2×D8—1072no—
167C4⋊Q8—1071no—
168#127—1071no—
169#124—2141no—
170#121—1071no—
171#121—2141no—
172#121C2×D10711yescharacteristic
173C4×D4—1071no—
174C4×D4—1072no—
175C4×D4—1072no—
176C4×D4—1071no—
177C4×D4—2141no—
178C4×D4—2141no—
179C4×D4—2141no—
180C2×C4⋊C4C2×D10711yescharacteristic
181C2×C22⋊C4—2141no—
182C2×C42—1071no—
183#107—1072no—
184#107C2×D10712yescomplemented
185C4⋊C8—1072no—
186#102—1072no—
187#102—1072no—
188#102—1072no—
189#102C2×D10712yescomplemented
190C22⋊C8—1071no—
191C22⋊C8C2×D10711yescharacteristic
192C4×C8—1072no—
193C24—2141no—
194#41—4284no—
195C2×Q8—2141no—
196C2×D4—21no—
197C2×D4—1072no—
198C2×D4—1071no—
199C2×D4—2144no—
200C2×D4—2141no—
201C2×D4—2141no—
202C2×D4—2141no—
203C2×D4—2141no—
204C2×D4C22×D10711yescharacteristic
205C22×C4—21no—
206C22×C4—1071no—
207C22×C4—1071no—
208C22×C4—4284no—
209C22×C4—2142no—
210C22×C4—2142no—
211C22×C4—2141no—
212C22×C4—2141no—
213C22×C4C22×D10711yescharacteristic
214D8—2144no—
215D8—2144no—
216C2×C8—1072no—
217C2×C8C22×D10712yes—
218C4⋊C4—21no—
219C4⋊C4—1071no—
220C4⋊C4—2141no—
221C4⋊C4C22×D10711yescharacteristic
222C4⋊C4C22×D10712yes—
223C22⋊C4—21no—
224C22⋊C4—2142no—
225C22⋊C4—2142no—
226C22⋊C4—2141no—
227C22⋊C4—2141no—
228C22⋊C4—2141no—
229C42—1071no—
230C42—1071no—
231C42—1071no—
232C42—1071no—
233C23—21no—
234C23—4282no—
235C23—4282no—
236C23—2144no—
237C23—2141no—
238C23—2142no—
239C23D4×D10711yescharacteristic
240Q8—4282no—
241D4—22no—
242D4—42no—
243D4—1074no—
244D4—4282no—
245D4—4282no—
246D4—4282no—
247D4—2144no—
248D4—2142no—
249D4—2142no—
250C2×C4—22no—
251C2×C4—21no—
252C2×C4—22no—
253C2×C4—42no—
254C2×C4—1074no—
255C2×C4—1071no—
256C2×C4—1071no—
257C2×C4—1071no—
258C2×C4—1071no—
259C2×C4—4284no—
260C2×C4—4284no—
261C2×C4—4282no—
262C2×C4—4282no—
263C2×C4—4281no—
264C2×C4—2141no—
265C2×C4D4×D10711yescharacteristic
266C2×C4C23×D10711yescharacteristic
267C8—22no—
268C8—2142no—
269C22—21no—
270C22—21no—
271C22—41no—
272C22—41no—
273C22—41no—
274C22—4288no—
275C22—4281no—
276C22—4281no—
277C22—4281no—
278C22—4282no—
279C22—4282no—
280C22—2144no—
281C22D8×D10712yes—
282C22C2×D4×D10711yescharacteristic
283C4—21no—
284C4—42no—
285C4—41no—
286C4—1072no—
287C4—1072no—
288C4—4281no—
289C4—2141no—
290C4—2141no—
291C4#41×D10712yes—
292C2—22no—
293C2—81no—
294C2—8561no—
295C2—4284no—
296C2#2029313511yescharacteristic
297C2C2×D8×D10711yescharacteristic
298C2#2029342611yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#480×C107C2
socleC22×C107C22×D4
hypercenter#480D107
hypocenterC107#3100
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #480

Complements

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

NH ≅ G/Ntype
1. #20293172C2semidirect
2. C2×#16812071C2semidirect
3. #20292940C2semidirect
4. #20292906C2semidirect
5. #20292821C2semidirect
6. #20292763C2semidirect
7. #20292760C2semidirect
8. #20292709C2semidirect
9. #20292707C2semidirect
10. #20292677C2semidirect
11. #480×C107C2semidirect
17. #16812108C22semidirect
18. #16812105C22semidirect
22. D4×Dic107C22semidirect
27. #16812071C22semidirect
31. #16812042C22semidirect
32. Dic107⋊C8C22semidirect
35. #16812029C22semidirect
37. C8×Dic107C22semidirect
40. #107×C107C22semidirect
41. #102×C107C22semidirect
104. C2×Dic107D8semidirect
112. D428#41semidirect
150. C107#3100semidirect
154. #480D107semidirect
184. #107C2×D107semidirect
189. #102C2×D107semidirect

Maximal subgroups

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

subgroupindex
#202931722
C2×#168120712
#202929402
#202929062
#202928212
#202927632
#202927602
#202927092
#202927072
#202926772
#480×C1072
#3100107