Finite Group Databasebeta

← #28346395

Subgroups — #28346395

Order 13696.

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

normal subgroups only

This group has 414 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#20293139C211yesmaximal, characteristic
2C2×#16812067C211yesmaximal, characteristic, complemented
3#20292971C212yesmaximal, complemented
4#20292970C212yesmaximal, complemented
5#20292954C212yesmaximal, complemented
6#20292953C212yesmaximal, complemented
7#20292670C211yesmaximal, characteristic, complemented
8#20292629C211yesmaximal, characteristic, complemented
9#20292626C211yesmaximal, characteristic, complemented
10#20292590C211yesmaximal, characteristic, complemented
11#428×C107C211yesmaximal, characteristic, complemented
12C2×#13133818C2211yescharacteristic
13C22×C4×D107C2211yescharacteristic
14C2×#13133802C2211yescharacteristic
15#16812093—21no—
16#16812089C2211yescharacteristic
17#16812089C2211yescharacteristic
18#16812084C2211yescharacteristic
19#16812084C2211yescharacteristic
20#16812079—21no—
21C4×D428—21no—
22C4×D428—21no—
23#16812067—21no—
24#16812067—21no—
25#16812067C2212yes—
26#16812067C2212yes—
27C2×C4⋊Dic107C2211yescharacteristic
28C2×(C2×D107)⋊C4—21no—
29C2×C4×Dic107—21no—
30#16812042C2212yescomplemented
31#16812040C2212yescomplemented
32#16812034C2212yescomplemented
33#16812032C2212yescomplemented
34#16812031C2212yescomplemented
35#16812029C2212yescomplemented
36(C2×D107)⋊C8C2212yes—
37(C2×D107)⋊C8C2212yes—
38C2×M16×C107C2211yescharacteristic
39#117×C107C2211yescharacteristic
40C2×C4⋊C4×C107C2211yescharacteristic
41#107×C107C2212yescomplemented
42#106×C107C2212yescomplemented
43C23×D107D411yescharacteristic
44#13133818—22no—
45#13133818—22no—
46C2×Dic214C2311yescharacteristic
47C2×D428—21no—
48C2×D428—21no—
49C2×D428C2311yescharacteristic
50C2×C22⋊D107—21no—
51C22×Dic107—21no—
52C22×Dic107D411yescharacteristic
53C2×C4×D107—22no—
54C2×C4×D107—22no—
55C2×C4×D107—21no—
56C2×C4×D107D411yescharacteristic
57C2×C4×D107D411yescharacteristic
58C2×C4×D107C2311yescharacteristic
59C2×C4×D107C2311yescharacteristic
60#13133802—22no—
61C2×C107⋊C8C2312yes—
62C4⋊Dic107—21no—
63C4⋊Dic107—21no—
64C4⋊Dic107—21no—
65C4⋊Dic107C2312yes—
66(C2×D107)⋊C4—22no—
67(C2×D107)⋊C4—22no—
68(C2×D107)⋊C4—21no—
69(C2×D107)⋊C4—21no—
70(C2×D107)⋊C4—21no—
71(C2×D107)⋊C4—21no—
72C4×Dic107—22no—
73C4×Dic107—22no—
74C22×C4×C107—21no—
75C22×C4×C107C2311yescharacteristic
76M16×C107—22no—
77C2×C8×C107C2312yes—
78C4⋊C4×C107—21no—
79C4⋊C4×C107C2312yes—
80C4⋊C4×C107C2311yescharacteristic
81C4⋊C4×C107C2311yescharacteristic
82C22⋊C4×C107—21no—
83C42×C107—21no—
84C23×C107C2×D411yescharacteristic
85C22×D107—22no—
86C22×D107—21no—
87C22×D107—41no—
88C22×D107—41no—
89C22×D107C2×D411yescharacteristic
90C22×D107C2×D411yescharacteristic
91C2×C4×C107—21no—
92C2×C4×C107—21no—
93C2×C4×C107—22no—
94C2×C4×C107—42no—
95C2×C4×C107C2×D411yescharacteristic
96C2×C4×C107#4112yes—
97C2×C4×C107#4112yes—
98C2×C4×C107C2411yescharacteristic
99C22⋊D107—42no—
100C2×Dic107—22no—
101C2×Dic107—22no—
102C2×Dic107—21no—
103C2×Dic107—42no—
104C2×Dic107C2×D411yescharacteristic
105C2×Dic107C2×D411yescharacteristic
106D428—21no—
107D428—21no—
108D428—42no—
109D428—42no—
110C4×D107—22no—
111C4×D107—22no—
112C4×D107—21no—
113C4×D107—21no—
114C4×D107—21no—
115C4×D107—21no—
116C4×D107—42no—
117Dic214—21no—
118Dic214—21no—
119C8×C107—22no—
120C107⋊C8—22no—
121C22×C107—21no—
122C22×C107—21no—
123C22×C107#12312yes—
124C22×C107C22×D411yescharacteristic
125C2×D107—21no—
126C2×D107—21no—
127C2×D107—21no—
128C2×D107—21no—
129C2×D107—21no—
130C2×D107—21no—
131C2×D107—44no—
132C2×D107—41no—
133C2×D107—41no—
134C2×D107—41no—
135C4×C107—41no—
136C4×C107—41no—
137C4×C107—42no—
138C4×C107#12312yes—
139C4×C107C2×#4111yescharacteristic
140C4×C107C2×#4111yescharacteristic
141Dic107—21no—
142Dic107—21no—
143Dic107—41no—
144Dic107—42no—
145C2×C107—22no—
146C2×C107C2×#12311yescharacteristic
147C2×C107#57511yescharacteristic
148C2×C107#57511yescharacteristic
149D107—41no—
150D107—41no—
151D107—81no—
152#3093—1071nomaximal, Sylow
153C107#309311yescharacteristic, complemented, Sylow
154#525—1071no—
155C2×#117—1071no—
156#485—1072no—
157#485—1072no—
158#482—1072no—
159#482—1072no—
160#428—1071no—
161#428D10711yescharacteristic, complemented
162#417—1071no—
163#417—1071no—
164#407—1071no—
165C2×#41—1071no—
166C23×C4—1071no—
167C2×M16—1071no—
168C2×M16C2×D10711yescharacteristic
169#123—2141no—
170C22⋊Q8—1071no—
171C22⋊Q8—1071no—
172#121—1071no—
173#121—1071no—
174#120—2141no—
175C4×D4—2141no—
176C4×D4—2141no—
177#117—1072no—
178#117—1072no—
179#117—2141no—
180#117—2141no—
181#117C2×D10711yescharacteristic
182C2×C4⋊C4—1071no—
183C2×C4⋊C4C2×D10711yescharacteristic
184C2×C22⋊C4—2141no—
185C2×C42—2141no—
186#107—1072no—
187#107C2×D10712yescomplemented
188#106—1072no—
189#106C2×D10712yescomplemented
190#103—1072no—
191#103—1072no—
192#102—1072no—
193#102—1072no—
194C22⋊C8—1072no—
195C22⋊C8—1072no—
196C24—1071no—
197#41—2142no—
198#41—2142no—
199C2×Q8—1071no—
200C2×D4—1071no—
201C2×D4—2141no—
202C2×D4—2141no—
203C2×D4—2141no—
204C22×C4—21no—
205C22×C4—1071no—
206C22×C4—1071no—
207C22×C4—1071no—
208C22×C4—1071no—
209C22×C4—1071no—
210C22×C4—2141no—
211C22×C4—2141no—
212C22×C4—2142no—
213C22×C4—2142no—
214C22×C4C22×D10711yescharacteristic
215M16—22no—
216M16—2142no—
217C2×C8—1072no—
218C2×C8C22×D10712yes—
219C4⋊C4—21no—
220C4⋊C4—1072no—
221C4⋊C4—2141no—
222C4⋊C4—2141no—
223C4⋊C4—2141no—
224C4⋊C4C22×D10712yes—
225C4⋊C4C22×D10711yescharacteristic
226C4⋊C4C22×D10711yescharacteristic
227C22⋊C4—21no—
228C22⋊C4—2141no—
229C22⋊C4—2141no—
230C22⋊C4—2142no—
231C22⋊C4—2142no—
232C22⋊C4—2141no—
233C22⋊C4—2141no—
234C42—21no—
235C42—2142no—
236C42—2142no—
237C23—1071no—
238C23—1071no—
239C23—4281no—
240C23—4281no—
241C23—2141no—
242C23—2142no—
243C23D4×D10711yescharacteristic
244Q8—2141no—
245Q8—2141no—
246D4—4282no—
247D4—4282no—
248D4—4282no—
249D4—2141no—
250D4—2141no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—22no—
254C2×C4—42no—
255C2×C4—1071no—
256C2×C4—1071no—
257C2×C4—4282no—
258C2×C4—4282no—
259C2×C4—2142no—
260C2×C4—2141no—
261C2×C4—2142no—
262C2×C4—2142no—
263C2×C4—2142no—
264C2×C4—2141no—
265C2×C4—2141no—
266C2×C4—2141no—
267C2×C4—2141no—
268C2×C4D4×D10711yescharacteristic
269C2×C4#1313381812yes—
270C2×C4#1313381912yes—
271C2×C4C23×D10711yescharacteristic
272C8—22no—
273C8—2142no—
274C22—21no—
275C22—21no—
276C22—4284no—
277C22—4281no—
278C22—4281no—
279C22—4281no—
280C22—2141no—
281C22—2141no—
282C22—2141no—
283C22—2141no—
284C22—2141no—
285C22—2141no—
286C22#1681209412yes—
287C22C2×D4×D10711yescharacteristic
288C4—42no—
289C4—41no—
290C4—41no—
291C4—4281no—
292C4—4282no—
293C4—2141no—
294C4—2141no—
295C4#1681209412yes—
296C4C2×#1313381811yescharacteristic
297C4C2×#1313381911yescharacteristic
298C2—22no—
299C2—8561no—
300C2—4281no—
301C2—4281no—
302C2C2×#1681209411yescharacteristic
303C2#2029342311yescharacteristic
304C2#2029342611yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#428×C107C2
socleC22×C107C22×D4
prime radical#20293139C2
hypercenter#428D107
hypocenterC107#3093
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
2. C2×#16812067C2semidirect
3. #20292971C2semidirect
4. #20292970C2semidirect
5. #20292954C2semidirect
6. #20292953C2semidirect
7. #20292670C2semidirect
8. #20292629C2semidirect
9. #20292626C2semidirect
10. #20292590C2semidirect
11. #428×C107C2semidirect
30. #16812042C22semidirect
31. #16812040C22semidirect
32. #16812034C22semidirect
33. #16812032C22semidirect
34. #16812031C22semidirect
35. #16812029C22semidirect
41. #107×C107C22semidirect
42. #106×C107C22semidirect
153. C107#3093semidirect
161. #428D107semidirect
187. #107C2×D107semidirect
189. #106C2×D107semidirect

Maximal subgroups

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

subgroupindex
#202931392
C2×#168120672
#202929712
#202929702
#202929542
#202929532
#202926702
#202926292
#202926262
#202925902
#428×C1072
#3093107