Finite Group Databasebeta

← #28346403

Subgroups — #28346403

Order 13696.

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

normal subgroups only

This group has 462 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×#16812084C211yesmaximal, characteristic
2#117×D107C211yesmaximal, characteristic, complemented
3#20292954C212yesmaximal, complemented
4#20292953C212yesmaximal, complemented
5#20292947C212yesmaximal, complemented
6#20292942C212yesmaximal, complemented
7#20292667C211yesmaximal, characteristic, complemented
8#20292633C211yesmaximal, characteristic, complemented
9#20292631C211yesmaximal, characteristic, complemented
10#20292590C211yesmaximal, characteristic, complemented
11#428×C107C211yesmaximal, characteristic, complemented
12C22×D428—21no—
13C22×D428C2211yescharacteristic
14C22×C4×D107C2211yescharacteristic
15C2×#13133802C2211yescharacteristic
16#16812084—21no—
17#16812084—21no—
18#16812084C2212yes—
19#16812084C2212yes—
20#16812068C2211yescharacteristic
21#16812067C2211yescharacteristic, complemented
22#16812067C2211yescharacteristic, complemented
23#16812066—21no—
24#16812065—21no—
25C4⋊C4×D107C2211yescharacteristic, complemented
26C4⋊C4×D107C2211yescharacteristic, complemented
27C2×(C2×D107)⋊C4—21no—
28C22⋊C4×D107—21no—
29C42×D107—21no—
30#16812042C2212yescomplemented
31#16812040C2212yescomplemented
32#16812031C2212yescomplemented
33#16812031C2212yescomplemented
34#16812029C2212yescomplemented
35#16812029C2212yescomplemented
36(C2×D107)⋊C8C2212yescomplemented
37(C2×D107)⋊C8C2212yescomplemented
38C2×M16×C107C2211yescharacteristic
39#117×C107C2211yescharacteristic
40C2×C4⋊C4×C107C2211yescharacteristic
41#107×C107C2212yescomplemented
42#106×C107C2212yescomplemented
43C23×D107—21no—
44C23×D107D411yescharacteristic
45C2×D428—22no—
46C2×D428—22no—
47C2×D428—22no—
48C2×D428—22no—
49C2×D428—21no—
50C2×D428—44no—
51C2×D428C2312yes—
52C22×Dic107D411yescharacteristic
53C2×C4×D107—22no—
54C2×C4×D107—22no—
55C2×C4×D107—21no—
56C2×C4×D107—21no—
57C2×C4×D107D411yescharacteristic
58C2×C4×D107D411yescharacteristic
59C2×C4×D107C2311yescharacteristic
60C2×C4×D107C2311yescharacteristic
61#13133802—22no—
62C2×C107⋊C8C2312yes—
63C4⋊Dic107C2311yescharacteristic
64C4⋊Dic107C2311yescharacteristic
65Dic107⋊C4—21no—
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×Dic107—21no—
73C4×Dic107—21no—
74C4×Dic107—21no—
75C22×C4×C107—21no—
76C22×C4×C107C2311yescharacteristic
77M16×C107—22no—
78C2×C8×C107C2312yes—
79C4⋊C4×C107—21no—
80C4⋊C4×C107C2312yes—
81C4⋊C4×C107C2311yescharacteristic
82C4⋊C4×C107C2311yescharacteristic
83C22⋊C4×C107—21no—
84C42×C107—21no—
85C23×C107C2×D411yescharacteristic
86C22×D107—22no—
87C22×D107—22no—
88C22×D107—22no—
89C22×D107—41no—
90C22×D107—42no—
91C22×D107—42no—
92C22×D107—42no—
93C22×D107C2×D411yescharacteristic
94C22×D107C2×D411yescharacteristic
95C2×C4×C107—21no—
96C2×C4×C107—21no—
97C2×C4×C107—22no—
98C2×C4×C107—42no—
99C2×C4×C107C2×D411yescharacteristic
100C2×C4×C107#4112yes—
101C2×C4×C107#4112yes—
102C2×C4×C107C2411yescharacteristic
103C2×Dic107—22no—
104C2×Dic107—21no—
105C2×Dic107—21no—
106C2×Dic107C2×D411yescharacteristic
107C2×Dic107C2×D411yescharacteristic
108D428—22no—
109D428—22no—
110D428—44no—
111D428—44no—
112D428—42no—
113C4×D107—22no—
114C4×D107—22no—
115C4×D107—21no—
116C4×D107—21no—
117C4×D107—21no—
118C4×D107—21no—
119C4×D107—42no—
120C4×D107—42no—
121C8×C107—22no—
122C107⋊C8—22no—
123C22×C107—21no—
124C22×C107—21no—
125C22×C107#12312yes—
126C22×C107C22×D411yescharacteristic
127C2×D107—21no—
128C2×D107—21no—
129C2×D107—21no—
130C2×D107—21no—
131C2×D107—21no—
132C2×D107—44no—
133C2×D107—42no—
134C2×D107—42no—
135C2×D107—42no—
136C2×D107—84no—
137C2×D107AGL(1,ℤ/8)12yes—
138C4×C107—41no—
139C4×C107—42no—
140C4×C107—41no—
141C4×C107#12312yes—
142C4×C107C2×#4111yescharacteristic
143C4×C107C2×#4111yescharacteristic
144Dic107—21no—
145Dic107—21no—
146Dic107—41no—
147Dic107—41no—
148C2×C107—22no—
149C2×C107C2×#12311yescharacteristic
150C2×C107C2×AGL(1,ℤ/8)11yescharacteristic
151C2×C107#57511yescharacteristic
152D107—22no—
153D107—41no—
154D107—82no—
155#3094—1071nomaximal, Sylow
156C107#309411yescharacteristic, complemented, Sylow
157C2×#121—1071no—
158C2×#117—1071no—
159#482—1072no—
160#482—1072no—
161#481—1072no—
162#480—1072no—
163#428—1071no—
164#428D10711yescharacteristic, complemented
165#418—1071no—
166#418—1071no—
167#407—1071no—
168C22×D4—1071no—
169C22×D4—2141no—
170C23×C4—1071no—
171C2×M16—1071no—
172C2×M16C2×D10711yescharacteristic
173#121—1072no—
174#121—1072no—
175#121—2141no—
176#121—2141no—
177#117—1071no—
178#117—1071no—
179#117—1071no—
180#117—2141no—
181#117—2141no—
182#117C2×D10711yescharacteristic
183C2×C4⋊C4—1071no—
184C2×C4⋊C4—1071no—
185C2×C4⋊C4C2×D10711yescharacteristic
186C2×C22⋊C4—2141no—
187C2×C22⋊C4—2141no—
188C2×C42—2141no—
189#107—1072no—
190#107C2×D10712yescomplemented
191#106—1072no—
192#106C2×D10712yescomplemented
193#102—1072no—
194#102—1072no—
195#102—1072no—
196#102—1072no—
197C22⋊C8—1072no—
198C22⋊C8—1072no—
199C24—1071no—
200C24—2141no—
201C2×D4—1072no—
202C2×D4—4284no—
203C2×D4—2142no—
204C2×D4—2142no—
205C2×D4—2141no—
206C2×D4—2142no—
207C2×D4—2142no—
208C22×C4—21no—
209C22×C4—1071no—
210C22×C4—1071no—
211C22×C4—1071no—
212C22×C4—1071no—
213C22×C4—1071no—
214C22×C4—2141no—
215C22×C4—2141no—
216C22×C4—2142no—
217C22×C4—2142no—
218C22×C4C22×D10711yescharacteristic
219M16—22no—
220M16—2142no—
221C2×C8—1072no—
222C2×C8C22×D10712yes—
223C4⋊C4—21no—
224C4⋊C4—1071no—
225C4⋊C4—1071no—
226C4⋊C4—2141no—
227C4⋊C4—2141no—
228C4⋊C4C22×D10711yescharacteristic
229C4⋊C4C22×D10711yescharacteristic
230C4⋊C4C22×D10712yes—
231C22⋊C4—21no—
232C22⋊C4—2142no—
233C22⋊C4—2142no—
234C22⋊C4—2141no—
235C22⋊C4—2141no—
236C22⋊C4—2141no—
237C42—21no—
238C42—2141no—
239C42—2141no—
240C42—2141no—
241C23—1071no—
242C23—1071no—
243C23—4282no—
244C23—4282no—
245C23—4281no—
246C23—4282no—
247C23—2142no—
248C23—2142no—
249C23—2142no—
250C23D4×D10711yescharacteristic
251D4—4284no—
252D4—4284no—
253D4—4282no—
254D4—2142no—
255D4—2142no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—22no—
259C2×C4—42no—
260C2×C4—1071no—
261C2×C4—1071no—
262C2×C4—4282no—
263C2×C4—4282no—
264C2×C4—2141no—
265C2×C4—2141no—
266C2×C4—2142no—
267C2×C4—2142no—
268C2×C4—2142no—
269C2×C4—2141no—
270C2×C4—2141no—
271C2×C4—2141no—
272C2×C4—2141no—
273C2×C4D4×D10711yescharacteristic
274C2×C4#1313381812yes—
275C2×C4#1313381912yes—
276C2×C4C23×D10711yescharacteristic
277C8—22no—
278C8—2142no—
279C22—21no—
280C22—21no—
281C22—8564no—
282C22—1072no—
283C22—4284no—
284C22—4282no—
285C22—4282no—
286C22—4282no—
287C22—2141no—
288C22—2141no—
289C22—2141no—
290C22—2141no—
291C22—2141no—
292C22#1681209412yes—
293C22C2×D4×D10711yescharacteristic
294C4—42no—
295C4—41no—
296C4—41no—
297C4—4281no—
298C4—4281no—
299C4—2141no—
300C4—2141no—
301C4#1681209412yes—
302C4C2×#1313381811yescharacteristic
303C4C2×#1313381911yescharacteristic
304C2—22no—
305C2—8562no—
306C2—4281no—
307C2—2142no—
308C2C2×#1681209411yescharacteristic
309C2AGL(1,ℤ/8)×D10711yescharacteristic
310C2#2029342611yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#428×C107C2
socleC22×C107C22×D4
prime radicalC2×#16812084C2
hypercenter#428D107
hypocenterC107#3094
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. #117×D107C2semidirect
3. #20292954C2semidirect
4. #20292953C2semidirect
5. #20292947C2semidirect
6. #20292942C2semidirect
7. #20292667C2semidirect
8. #20292633C2semidirect
9. #20292631C2semidirect
10. #20292590C2semidirect
11. #428×C107C2semidirect
21. #16812067C22semidirect
22. #16812067C22semidirect
25. C4⋊C4×D107C22semidirect
26. C4⋊C4×D107C22semidirect
30. #16812042C22semidirect
31. #16812040C22semidirect
32. #16812031C22semidirect
33. #16812031C22semidirect
34. #16812029C22semidirect
35. #16812029C22semidirect
36. (C2×D107)⋊C8C22semidirect
37. (C2×D107)⋊C8C22semidirect
41. #107×C107C22semidirect
42. #106×C107C22semidirect
156. C107#3094semidirect
164. #428D107semidirect
190. #107C2×D107semidirect
192. #106C2×D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168120842
#117×D1072
#202929542
#202929532
#202929472
#202929422
#202926672
#202926332
#202926312
#202925902
#428×C1072
#3094107