Finite Group Databasebeta

← #28346411

Subgroups — #28346411

Order 13696.

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

normal subgroups only

This group has 410 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
1C2×#16812089C211yesmaximal, characteristic
2#117×D107C211yesmaximal, characteristic
3#20292971C212yesmaximal, complemented
4#20292970C212yesmaximal, complemented
5#20292964C212yesmaximal, complemented
6#20292959C212yesmaximal, complemented
7#20292667C211yesmaximal, characteristic, complemented
8#20292637C211yesmaximal, characteristic, complemented
9#20292635C211yesmaximal, characteristic, complemented
10#20292590C211yesmaximal, characteristic
11#428×C107C211yesmaximal, characteristic, complemented
12C22×Dic214C2211yescharacteristic
13C22×C4×D107C2211yescharacteristic
14C2×#13133802C2211yescharacteristic
15#16812089—21no—
16#16812089—21no—
17#16812089C2212yes—
18#16812089C2212yes—
19#16812068C2211yescharacteristic
20#16812067C2211yescharacteristic
21#16812067C2211yescharacteristic
22#16812066—21no—
23#16812065—21no—
24C2×C4⋊Dic107—21no—
25C4⋊C4×D107C2211yescharacteristic
26C4⋊C4×D107C2211yescharacteristic
27C2×(C2×D107)⋊C4—21no—
28C22⋊C4×D107—21no—
29C42×D107—21no—
30#16812042C2212yescomplemented
31#16812040C2212yescomplemented
32#16812034C2212yescomplemented
33#16812034C2212yescomplemented
34#16812032C2212yescomplemented
35#16812032C2212yescomplemented
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
44C2×Dic214—22no—
45C2×Dic214—22no—
46C2×Dic214—21no—
47C2×Dic214C2312yes—
48C22×Dic107—21no—
49C22×Dic107D411yescharacteristic
50C2×C4×D107—22no—
51C2×C4×D107—22no—
52C2×C4×D107—21no—
53C2×C4×D107—21no—
54C2×C4×D107D411yescharacteristic
55C2×C4×D107D411yescharacteristic
56C2×C4×D107C2311yescharacteristic
57C2×C4×D107C2311yescharacteristic
58#13133802—22no—
59C2×C107⋊C8C2312yes—
60C4⋊Dic107—22no—
61C4⋊Dic107—22no—
62C4⋊Dic107C2311yescharacteristic
63C4⋊Dic107C2311yescharacteristic
64Dic107⋊C4—21no—
65Dic107⋊C4—21no—
66C22⋊Dic107—21no—
67(C2×D107)⋊C4—21no—
68(C2×D107)⋊C4—21no—
69(C2×D107)⋊C4—22no—
70(C2×D107)⋊C4—22no—
71C4×Dic107—21no—
72C4×Dic107—21no—
73C4×Dic107—21no—
74C22×C4×C107—21no—
75C22×C4×C107C2311yescharacteristic
76M16×C107—22no—
77C2×C8×C107C2312yes—
78C4⋊C4×C107—21no—
79C4⋊C4×C107C2311yescharacteristic
80C4⋊C4×C107C2311yescharacteristic
81C4⋊C4×C107C2312yes—
82C22⋊C4×C107—21no—
83C42×C107—21no—
84C23×C107C2×D411yescharacteristic
85C22×D107—22no—
86C22×D107—22no—
87C22×D107—41no—
88C22×D107C2×D411yescharacteristic
89C22×D107C2×D411yescharacteristic
90C2×C4×C107—21no—
91C2×C4×C107—21no—
92C2×C4×C107—22no—
93C2×C4×C107—42no—
94C2×C4×C107C2×D411yescharacteristic
95C2×C4×C107#4112yes—
96C2×C4×C107#4112yes—
97C2×C4×C107C2411yescharacteristic
98C2×Dic107—22no—
99C2×Dic107—21no—
100C2×Dic107—21no—
101C2×Dic107—22no—
102C2×Dic107—42no—
103C2×Dic107C2×D411yescharacteristic
104C2×Dic107C2×D411yescharacteristic
105C4×D107—22no—
106C4×D107—22no—
107C4×D107—21no—
108C4×D107—21no—
109C4×D107—21no—
110C4×D107—21no—
111C4×D107—42no—
112C4×D107—42no—
113Dic214—22no—
114Dic214—22no—
115Dic214—42no—
116C8×C107—22no—
117C107⋊C8—22no—
118C22×C107—21no—
119C22×C107—21no—
120C22×C107#12312yes—
121C22×C107C22×D411yescharacteristic
122C2×D107—21no—
123C2×D107—21no—
124C2×D107—21no—
125C2×D107—21no—
126C2×D107—21no—
127C2×D107—44no—
128C2×D107#13712yes—
129C4×C107—41no—
130C4×C107—41no—
131C4×C107—42no—
132C4×C107#12312yes—
133C4×C107C2×#4111yescharacteristic
134C4×C107C2×#4111yescharacteristic
135Dic107—21no—
136Dic107—21no—
137Dic107—42no—
138Dic107—41no—
139Dic107—41no—
140C2×C107—22no—
141C2×C107C2×#12311yescharacteristic
142C2×C107C2×#13711yescharacteristic
143C2×C107#57511yescharacteristic
144D107—22no—
145D107—41no—
146#3095—1071nomaximal, Sylow
147C107#309511yescharacteristic, complemented, Sylow
148C2×C22⋊Q8—1071no—
149C2×#117—1071no—
150#485—1072no—
151#485—1072no—
152#484—1072no—
153#483—1072no—
154#428—1071no—
155#428D10711yescharacteristic, complemented
156#419—1071no—
157#419—1071no—
158#407—1071no—
159C22×Q8—1071no—
160C23×C4—1071no—
161C2×M16—1071no—
162C2×M16C2×D10711yescharacteristic
163C22⋊Q8—1072no—
164C22⋊Q8—1072no—
165C22⋊Q8—2141no—
166C22⋊Q8—2141no—
167#117—1071no—
168#117—1071no—
169#117—1071no—
170#117—2141no—
171#117—2141no—
172#117C2×D10711yescharacteristic
173C2×C4⋊C4—1071no—
174C2×C4⋊C4—1071no—
175C2×C4⋊C4—2141no—
176C2×C4⋊C4C2×D10711yescharacteristic
177C2×C22⋊C4—2141no—
178C2×C22⋊C4—2141no—
179C2×C42—2141no—
180#107—1072no—
181#107C2×D10712yescomplemented
182#106—1072no—
183#106C2×D10712yescomplemented
184#103—1072no—
185#103—1072no—
186#103—1072no—
187#103—1072no—
188C22⋊C8—1072no—
189C22⋊C8—1072no—
190C24—1071no—
191C2×Q8—1072no—
192C2×Q8—2142no—
193C2×Q8—2142no—
194C2×Q8—2141no—
195C22×C4—21no—
196C22×C4—1071no—
197C22×C4—1071no—
198C22×C4—1071no—
199C22×C4—1071no—
200C22×C4—1071no—
201C22×C4—2141no—
202C22×C4—2141no—
203C22×C4—2141no—
204C22×C4—2142no—
205C22×C4—2142no—
206C22×C4C22×D10711yescharacteristic
207M16—22no—
208M16—2142no—
209C2×C8—1072no—
210C2×C8C22×D10712yes—
211C4⋊C4—21no—
212C4⋊C4—1071no—
213C4⋊C4—1071no—
214C4⋊C4—2141no—
215C4⋊C4—2141no—
216C4⋊C4—2142no—
217C4⋊C4—2142no—
218C4⋊C4C22×D10711yescharacteristic
219C4⋊C4C22×D10711yescharacteristic
220C4⋊C4C22×D10712yes—
221C22⋊C4—21no—
222C22⋊C4—2141no—
223C22⋊C4—2141no—
224C22⋊C4—2142no—
225C22⋊C4—2142no—
226C22⋊C4—2141no—
227C42—21no—
228C42—2141no—
229C42—2141no—
230C42—2141no—
231C23—1071no—
232C23—1071no—
233C23—4281no—
234C23—2142no—
235C23—2142no—
236C23D4×D10711yescharacteristic
237Q8—4282no—
238Q8—2142no—
239Q8—2142no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—22no—
243C2×C4—42no—
244C2×C4—1071no—
245C2×C4—1071no—
246C2×C4—4282no—
247C2×C4—4282no—
248C2×C4—4282no—
249C2×C4—2141no—
250C2×C4—2141no—
251C2×C4—2142no—
252C2×C4—2142no—
253C2×C4—2142no—
254C2×C4—2142no—
255C2×C4—2141no—
256C2×C4—2141no—
257C2×C4—2141no—
258C2×C4—2141no—
259C2×C4D4×D10711yescharacteristic
260C2×C4#1313381812yes—
261C2×C4#1313381912yes—
262C2×C4C23×D10711yescharacteristic
263C8—22no—
264C8—2142no—
265C22—21no—
266C22—21no—
267C22—1072no—
268C22—4284no—
269C22—2141no—
270C22—2141no—
271C22—2141no—
272C22—2141no—
273C22—2141no—
274C22#1681209412yes—
275C22C2×D4×D10711yescharacteristic
276C4—41no—
277C4—42no—
278C4—41no—
279C4—4282no—
280C4—4281no—
281C4—4281no—
282C4—2141no—
283C4—2141no—
284C4#1681209412yes—
285C4C2×#1313381811yescharacteristic
286C4C2×#1313381911yescharacteristic
287C2—22no—
288C2—4281no—
289C2—2142no—
290C2C2×#1681209411yescharacteristic
291C2#137×D10711yescharacteristic
292C2#2029342311yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#428×C107C2
socleC22×C107C22×D4
prime radicalC23×D107D4
hypercenter#428D107
hypocenterC107#3095
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
3. #20292971C2semidirect
4. #20292970C2semidirect
5. #20292964C2semidirect
6. #20292959C2semidirect
7. #20292667C2semidirect
8. #20292637C2semidirect
9. #20292635C2semidirect
11. #428×C107C2semidirect
30. #16812042C22semidirect
31. #16812040C22semidirect
32. #16812034C22semidirect
33. #16812034C22semidirect
34. #16812032C22semidirect
35. #16812032C22semidirect
41. #107×C107C22semidirect
42. #106×C107C22semidirect
147. C107#3095semidirect
155. #428D107semidirect
181. #107C2×D107semidirect
183. #106C2×D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168120892
#117×D1072
#202929712
#202929702
#202929642
#202929592
#202926672
#202926372
#202926352
#202925902
#428×C1072
#3095107