Finite Group Databasebeta

← #28346937

Subgroups — #28346937

Order 13696.

564 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 564 conjugacy classes of subgroups — too many to draw. Only its 165 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×#16812142C211yesmaximal, characteristic, complemented
2C2×#16812137C211yesmaximal, characteristic, complemented
3C2×#16812131C211yesmaximal, characteristic, complemented
4C22×#13133807C211yesmaximal, characteristic, complemented
5#20293203C211yesmaximal, characteristic, complemented
6#20293196C211yesmaximal, characteristic, complemented
7#20293021C212yesmaximal, complemented
8#20293016C212yesmaximal, complemented
9#20293011C212yesmaximal, complemented
10#20293000C212yesmaximal, complemented
11#405×C107C211yesmaximal, characteristic, complemented
12C2×#13133818C2211yescharacteristic
13C2×#13133818C2211yescharacteristic
14C22×Dic214C2211yescharacteristic
15C22×D428C2211yescharacteristic
16#16812142—24no—
17#16812137—24no—
18#16812131—24no—
19C2×Dic428C2211yescharacteristic, complemented
20C2×Dic428C2212yescomplemented
21C2×#13133807—24no—
22C2×#13133807C2212yes—
23C2×#13133807C2212yes—
24C2×#13133807C2212yes—
25C2×#13133807C2211yescharacteristic, complemented
26C2×#13133807C2211yescharacteristic, complemented
27C2×#13133807C2212yescomplemented
28C2×D856C2212yes—
29C2×D856C2211yescharacteristic, complemented
30#16812109C2212yescomplemented
31#16812106C2212yes—
32#16812097C2212yes—
33#16812097C2212yescomplemented
34#16812093—22no—
35#16812089—22no—
36#16812084—22no—
37#16812079—22no—
38C2×M16×C107C2211yescharacteristic
39C22×C8×C107C2211yescharacteristic
40#117×C107C2211yescharacteristic
41#97×C107C2212yescomplemented
42C4×C8×C107C2212yescomplemented
43C23×D107—21no—
44#13133818—22no—
45#13133818—22no—
46#13133818—22no—
47#13133818—22no—
48C2×Dic214—22no—
49C2×Dic214—22no—
50C2×Dic214—21no—
51C2×Dic214C2311yescharacteristic
52C2×Dic214C2312yes—
53C2×Dic214C2311yescharacteristic
54C2×D428—22no—
55C2×D428—22no—
56C2×D428—21no—
57C2×D428—24no—
58C2×D428C2312yes—
59C2×D428C2311yescharacteristic
60C2×D428C2311yescharacteristic
61C2×C22⋊D107—21no—
62C2×C22⋊D107—21no—
63C22×Dic107—21no—
64C2×C4×D107—21no—
65C2×C4×D107—21no—
66Dic428—22no—
67Dic428—24no—
68#13133807—24no—
69#13133807—24no—
70#13133807—24no—
71#13133807—22no—
72#13133807—22no—
73#13133807—24no—
74D856—24no—
75D856—22no—
76C4⋊Dic107—24no—
77(C2×D107)⋊C4—24no—
78(C2×D107)⋊C4—24no—
79C22×C4×C107C2311yescharacteristic
80M16×C107D414yescomplemented
81C2×C8×C107D414yescomplemented
82C2×C8×C107C2312yes—
83C2×C8×C107C2311yescharacteristic
84C2×C8×C107C2311yescharacteristic
85C4⋊C4×C107D412yes—
86C22⋊C4×C107D412yes—
87C42×C107C2312yes—
88C23×C107C2×D411yescharacteristic
89C22×D107—21no—
90C22×D107—22no—
91C22×D107—21no—
92C22×D107—42no—
93C22×D107—82no—
94C2×C4×C107C2×D414yes—
95C2×C4×C107C2×D411yescharacteristic
96C2×C4×C107C2×D412yes—
97C2×C4×C107C2×D412yes—
98C2×C4×C107C2411yescharacteristic
99C22⋊D107—42no—
100C22⋊D107—42no—
101C2×Dic107—21no—
102C2×Dic107—22no—
103C2×Dic107—21no—
104C2×Dic107—42no—
105D428—21no—
106D428—22no—
107D428—21no—
108D428—21no—
109D428—22no—
110D428—21no—
111D428—42no—
112D428—84no—
113C4×D107—42no—
114C4×D107—42no—
115Dic214—22no—
116Dic214—21no—
117Dic214—22no—
118Dic214—21no—
119Dic214—21no—
120Dic214—21no—
121Dic214—42no—
122C8×C107C2×D412yes—
123C8×C107C2×D412yes—
124C8×C107C2×D414yes—
125C22×C107—22no—
126C22×C107#12712yes—
127C22×C107C22×D411yescharacteristic
128C2×D107—41no—
129C2×D107—41no—
130C2×D107—42no—
131C2×D107—42no—
132C2×D107—44no—
133C2×D107—42no—
134C2×D107—84no—
135C4×C107—24no—
136C4×C107#12712yes—
137C4×C107C22×D411yescharacteristic
138C4×C107C22×D411yescharacteristic
139Dic107—42no—
140Dic107—41no—
141Dic107—41no—
142C2×C107—22no—
143C2×C107C2×#12711yescharacteristic
144C2×C107#57712yes—
145D107—81no—
146D107—81no—
147D107—82no—
148#3154—1071nomaximal, Sylow
149C107#315411yescharacteristic, complemented, Sylow
150C2×#137—1071no—
151C2×AGL(1,ℤ/8)—1071no—
152C2×#135—1071no—
153C22×SD16—1071no—
154#536—1071no—
155#535—1071no—
156#497—1072no—
157#496—1072no—
158#495—1072no—
159#492—1072no—
160#405D10711yescharacteristic, complemented
161C2×#41—1071no—
162C2×#41—1071no—
163C22×Q8—1071no—
164C22×D4—1071no—
165#137—2144no—
166AGL(1,ℤ/8)—2144no—
167#135—2144no—
168C2×Q16—1071no—
169C2×Q16—1072no—
170C2×SD16—1072no—
171C2×SD16—1072no—
172C2×SD16—1071no—
173C2×SD16—1071no—
174C2×SD16—1072no—
175C2×SD16—1072no—
176C2×SD16—2144no—
177C2×D8—1071no—
178C2×D8—1072no—
179C2×M16C2×D10711yescharacteristic
180C22×C8C2×D10711yescharacteristic
181C4⋊Q8—1072no—
182#127—1072no—
183#124—1072no—
184#124—1072no—
185#123—2142no—
186C22⋊Q8—2142no—
187#121—2142no—
188#120—2142no—
189#117C2×D10711yescharacteristic
190#97C2×D10712yescomplemented
191C4×C8C2×D10712yescomplemented
192C24—2141no—
193#41—2142no—
194#41—2142no—
195#41—2142no—
196#41—2142no—
197C2×Q8—1071no—
198C2×Q8—1072no—
199C2×Q8—1071no—
200C2×Q8—2142no—
201C2×Q8—2142no—
202C2×Q8—2141no—
203C2×D4—1071no—
204C2×D4—1072no—
205C2×D4—1071no—
206C2×D4—2141no—
207C2×D4—2141no—
208C2×D4—2144no—
209C2×D4—2141no—
210C2×D4—2142no—
211C2×D4—2142no—
212C22×C4—2141no—
213C22×C4—2141no—
214C22×C4—2141no—
215C22×C4C22×D10711yescharacteristic
216Q16—2142no—
217Q16—2144no—
218SD16—2144no—
219SD16—2144no—
220SD16—2142no—
221SD16—2142no—
222SD16—2144no—
223SD16—2144no—
224D8—2142no—
225D8—2144no—
226M16D42814yescomplemented
227C2×C8D42814yescomplemented
228C2×C8C22×D10711yescharacteristic
229C2×C8C22×D10712yes—
230C2×C8C22×D10711yescharacteristic
231C4⋊C4—2144no—
232C4⋊C4D42812yes—
233C22⋊C4—2144no—
234C22⋊C4—2144no—
235C22⋊C4D42812yes—
236C42C22×D10712yes—
237C23—8562no—
238C23—4282no—
239C23—2141no—
240C23—2142no—
241C23—2141no—
242C23C2×D42811yescharacteristic
243Q8—4282no—
244Q8—2142no—
245Q8—2141no—
246Q8—2142no—
247Q8—2141no—
248Q8—2141no—
249Q8—2141no—
250D4—8564no—
251D4—4282no—
252D4—4282no—
253D4—4282no—
254D4—2141no—
255D4—2142no—
256D4—2141no—
257D4—2141no—
258D4—2142no—
259D4—2141no—
260C2×C4—4282no—
261C2×C4—4282no—
262C2×C4—4282no—
263C2×C4—2141no—
264C2×C4—2142no—
265C2×C4—2141no—
266C2×C4C2×D42814yes—
267C2×C4C2×D42811yescharacteristic
268C2×C4C2×D42812yes—
269C2×C4C2×D42812yes—
270C2×C4C23×D10711yescharacteristic
271C8C2×D42812yes—
272C8C2×D42812yes—
273C8C2×D42814yes—
274C22—22no—
275C22—8564no—
276C22—4281no—
277C22—4281no—
278C22—4282no—
279C22—4282no—
280C22—4284no—
281C22—4282no—
282C22#1681210612yes—
283C22C22×D42811yescharacteristic
284C4—24no—
285C4—4281no—
286C4—4281no—
287C4—4282no—
288C4#1681210612yes—
289C4C22×D42811yescharacteristic
290C4C22×D42811yescharacteristic
291C2—22no—
292C2—8562no—
293C2—8561no—
294C2—8561no—
295C2C2×#1681210611yescharacteristic
296C2#2029343912yes—
subgroupG/H
centerC22C22×D428
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#405×C107C2
socleC22×C107C22×D4
hypercenter#405D107
hypocenterC107#3154
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ #117 ◁ #405

Complements

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

NH ≅ G/Ntype
1. C2×#16812142C2semidirect
2. C2×#16812137C2semidirect
3. C2×#16812131C2semidirect
4. C22×#13133807C2semidirect
5. #20293203C2semidirect
6. #20293196C2semidirect
7. #20293021C2semidirect
8. #20293016C2semidirect
9. #20293011C2semidirect
10. #20293000C2semidirect
11. #405×C107C2semidirect
19. C2×Dic428C22semidirect
20. C2×Dic428C22semidirect
25. C2×#13133807C22semidirect
26. C2×#13133807C22semidirect
27. C2×#13133807C22semidirect
29. C2×D856C22semidirect
30. #16812109C22semidirect
33. #16812097C22semidirect
41. #97×C107C22semidirect
42. C4×C8×C107C22semidirect
80. M16×C107D4semidirect
81. C2×C8×C107D4semidirect
149. C107#3154semidirect
160. #405D107semidirect
190. #97C2×D107semidirect
191. C4×C8C2×D107semidirect
226. M16D428semidirect
227. C2×C8D428semidirect

Maximal subgroups

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

subgroupindex
C2×#168121422
C2×#168121372
C2×#168121312
C22×#131338072
#202932032
#202931962
#202930212
#202930162
#202930112
#202930002
#405×C1072
#3154107