Finite Group Databasebeta

← #28346909

Subgroups — #28346909

Order 13696.

656 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 656 conjugacy classes of subgroups — too many to draw. Only its 153 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×#16812137C211yesmaximal, characteristic, complemented
2C2×#16812135C211yesmaximal, characteristic, complemented
3C2×#16812134C211yesmaximal, characteristic, complemented
4#20293173C211yesmaximal, characteristic, complemented
5C2×#16812105C211yesmaximal, characteristic, complemented
6#20293015C212yesmaximal, complemented
7#20293013C212yesmaximal, complemented
8#20293002C212yesmaximal, complemented
9#20292998C212yesmaximal, complemented
10M16×Dic107C211yesmaximal, characteristic, complemented
11C2×AGL(1,ℤ/8)×C107C211yesmaximal, characteristic, complemented
12C2×#13133818C2211yescharacteristic
13C22×D428C2211yescharacteristic
14C22×C22⋊D107—22no—
15#16812137—24no—
16#16812135—24no—
17#16812134—24no—
18C2×#13133807C2212yescomplemented
19C2×#13133806C2212yescomplemented
20C2×#13133805C2212yescomplemented
21C2×D856C2212yescomplemented
22C2×#13133803C2212yescomplemented
23C2×#13133803C2212yescomplemented
24C2×#13133802C2211yescharacteristic
25#16812107C2211yescharacteristic, complemented
26#16812105—22no—
27#16812105C2212yes—
28#16812105C2212yes—
29#16812105C2211yescharacteristic, complemented
30#16812098C2211yescharacteristic, complemented
31#16812096C2211yescharacteristic, complemented
32#16812081—22no—
33C4×C22⋊D107—22no—
34C2×C4×Dic107C2211yescharacteristic
35#16812019C2212yescomplemented
36C8×Dic107C2212yescomplemented
37C2×#41×C107C2211yescharacteristic
38C22×D4×C107C2211yescharacteristic
39AGL(1,ℤ/8)×C107—24no—
40C2×SD16×C107C2212yescomplemented
41C2×D8×C107C2212yescomplemented
42C2×M16×C107C2211yescharacteristic
43C23×D107—21no—
44#13133818—22no—
45#13133818—22no—
46C2×Dic214C2311yescharacteristic
47C2×D428—22no—
48C2×D428—22no—
49C2×D428—21no—
50C2×D428C2311yescharacteristic
51C2×D428C2312yes—
52C2×C22⋊D107—21no—
53C2×C22⋊D107—22no—
54C2×C22⋊D107—24no—
55C2×C22⋊D107—24no—
56C2×C22⋊D107—48no—
57C22×Dic107D412yes—
58C2×C4×D107—21no—
59#13133807—24no—
60#13133806—24no—
61#13133805—24no—
62D856—24no—
63#13133803—24no—
64#13133803—24no—
65#13133802D414yescomplemented
66C2×C107⋊C8C2312yes—
67Dic107⋊C4—22no—
68C22⋊Dic107—22no—
69(C2×D107)⋊C4—22no—
70C4×Dic107D412yes—
71C4×Dic107C2311yescharacteristic
72C4×Dic107C2311yescharacteristic
73C24×C107—21no—
74#41×C107—22no—
75#41×C107—22no—
76C2×Q8×C107C2311yescharacteristic
77C2×D4×C107—21no—
78C2×D4×C107—22no—
79C2×D4×C107—22no—
80C2×D4×C107—21no—
81C2×D4×C107C2311yescharacteristic
82C2×D4×C107C2312yes—
83C22×C4×C107—21no—
84C22×C4×C107C2311yescharacteristic
85SD16×C107—24no—
86D8×C107—24no—
87M16×C107D414yescomplemented
88C2×C8×C107C2312yes—
89C23×C107—21no—
90C23×C107—22no—
91C23×C107—44no—
92C23×C107—42no—
93C23×C107C2×D411yescharacteristic
94C22×D107—21no—
95C22×D107—22no—
96C22×D107—44no—
97C22×D107—42no—
98Q8×C107—21no—
99Q8×C107—21no—
100D4×C107—21no—
101D4×C107—22no—
102D4×C107—22no—
103D4×C107—21no—
104D4×C107—42no—
105D4×C107—42no—
106C2×C4×C107—21no—
107C2×C4×C107—42no—
108C2×C4×C107C2×D411yescharacteristic
109C2×C4×C107C2×D412yes—
110C2×C4×C107C2×D412yes—
111C2×C4×C107C2411yescharacteristic
112C22⋊D107—48no—
113C22⋊D107—42no—
114C22⋊D107—44no—
115C22⋊D107—48no—
116C2×Dic107—24no—
117C2×Dic107—21no—
118C2×Dic107C2×D412yes—
119C2×Dic107C2×D412yes—
120D428—21no—
121D428—22no—
122D428—21no—
123D428—22no—
124D428—42no—
125C4×D107—42no—
126Dic214—21no—
127Dic214—21no—
128C8×C107C2×D414yescomplemented
129C107⋊C8C2×D414yescomplemented
130C22×C107—22no—
131C22×C107—41no—
132C22×C107—42no—
133C22×C107—44no—
134C22×C107—42no—
135C22×C107—84no—
136C22×C107#12712yes—
137C22×C107C22×D411yescharacteristic
138C2×D107—41no—
139C2×D107—42no—
140C2×D107—44no—
141C2×D107—42no—
142C2×D107—84no—
143C4×C107—41no—
144C4×C107#12712yes—
145C4×C107C22×D411yescharacteristic
146C4×C107C22×D411yescharacteristic
147Dic107—22no—
148Dic107—41no—
149Dic107AGL(1,ℤ/8)14yescomplemented
150C2×C107—22no—
151C2×C107—81no—
152C2×C107—82no—
153C2×C107C2×#12711yescharacteristic
154C2×C107C2×AGL(1,ℤ/8)12yes—
155D107—81no—
156D107—82no—
157#3151—1071nomaximal, Sylow
158C107#315111yescharacteristic, complemented, Sylow
159C2×AGL(1,ℤ/8)—1071no—
160C2×AGL(1,ℤ/8)—1071no—
161C2×AGL(1,ℤ/8)—1071no—
162C2×AGL(1,ℤ/8)D10711yescharacteristic, complemented
163#532—1071no—
164C2×#127—1071no—
165#496—1072no—
166#496—1072no—
167#493—1072no—
168#492—1072no—
169C4×M16—1071no—
170C2×#41—1071no—
171C2×#41C2×D10711yescharacteristic
172C22×D4—1071no—
173C22×D4—2142no—
174C22×D4C2×D10711yescharacteristic
175AGL(1,ℤ/8)—24no—
176AGL(1,ℤ/8)—2144no—
177AGL(1,ℤ/8)—2144no—
178AGL(1,ℤ/8)—2144no—
179C2×SD16—1072no—
180C2×SD16—1072no—
181C2×SD16—1072no—
182C2×SD16C2×D10712yescomplemented
183C2×D8—1072no—
184C2×D8—1072no—
185C2×D8—1072no—
186C2×D8C2×D10712yescomplemented
187C2×M16—1071no—
188C2×M16C2×D10711yescharacteristic
189C4⋊Q8—1071no—
190#127—1072no—
191#127—1072no—
192#127—1071no—
193#127—2142no—
194#124—1071no—
195#124—1071no—
196#121—2142no—
197C4×D4—2142no—
198C2×C42—1071no—
199#97—1072no—
200C4×C8—1072no—
201C24—21no—
202C24—2141no—
203#41—22no—
204#41—22no—
205#41—2142no—
206#41—2142no—
207C2×Q8—1071no—
208C2×Q8C22×D10711yescharacteristic
209C2×D4—21no—
210C2×D4—22no—
211C2×D4—21no—
212C2×D4—22no—
213C2×D4—1071no—
214C2×D4—1072no—
215C2×D4—4288no—
216C2×D4—2142no—
217C2×D4—2144no—
218C2×D4—2144no—
219C2×D4—2141no—
220C2×D4—2141no—
221C2×D4—2142no—
222C2×D4—2142no—
223C2×D4C22×D10711yescharacteristic
224C2×D4C22×D10712yes—
225C22×C4—21no—
226C22×C4—1072no—
227C22×C4—2141no—
228C22×C4C22×D10711yescharacteristic
229SD16—24no—
230SD16—2144no—
231SD16—2144no—
232SD16—2144no—
233D8—24no—
234D8—2144no—
235D8—2144no—
236D8—2144no—
237M16—1074no—
238M16C22⋊D10714yescomplemented
239C2×C8—1072no—
240C2×C8C22×D10712yes—
241C4⋊C4—2142no—
242C22⋊C4—2142no—
243C22⋊C4—2142no—
244C42—1072no—
245C42—1071no—
246C42—1071no—
247C23—21no—
248C23—22no—
249C23—44no—
250C23—42no—
251C23—4284no—
252C23—4282no—
253C23—2141no—
254C23—2142no—
255C23D4×D10711yescharacteristic
256Q8—21no—
257Q8—21no—
258Q8—2141no—
259Q8—2141no—
260D4—21no—
261D4—22no—
262D4—21no—
263D4—22no—
264D4—42no—
265D4—42no—
266D4—4288no—
267D4—4284no—
268D4—4282no—
269D4—4288no—
270D4—4282no—
271D4—2141no—
272D4—2142no—
273D4—2141no—
274D4—2142no—
275C2×C4—21no—
276C2×C4—42no—
277C2×C4—1072no—
278C2×C4—1072no—
279C2×C4—4282no—
280C2×C4—2144no—
281C2×C4—2141no—
282C2×C4D4×D10711yescharacteristic
283C2×C4D4×D10712yes—
284C2×C4C2×C22⋊D10712yes—
285C2×C4C23×D10711yescharacteristic
286C8—1074no—
287C8C2×C22⋊D10714yescomplemented
288C22—22no—
289C22—44no—
290C22—42no—
291C22—41no—
292C22—42no—
293C22—84no—
294C22—8564no—
295C22—4284no—
296C22—4282no—
297C22—4281no—
298C22—4282no—
299C22#1681210512yes—
300C22C2×D4×D10711yescharacteristic
301C4—41no—
302C4—1074no—
303C4—4281no—
304C4—2142no—
305C4#1681210512yes—
306C4C2×D4×D10711yescharacteristic
307C4C22×C22⋊D10711yescharacteristic
308C2—22no—
309C2—82no—
310C2—81no—
311C2—8562no—
312C2—8561no—
313C2C2×#1681210511yescharacteristic
314C2AGL(1,ℤ/8)×D10712yes—
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
FittingC2×AGL(1,ℤ/8)×C107C2
socleC22×C107C22×D4
hypercenterC2×AGL(1,ℤ/8)D107
hypocenterC107#3151
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×#16812137C2semidirect
2. C2×#16812135C2semidirect
3. C2×#16812134C2semidirect
4. #20293173C2semidirect
5. C2×#16812105C2semidirect
6. #20293015C2semidirect
7. #20293013C2semidirect
8. #20293002C2semidirect
9. #20292998C2semidirect
10. M16×Dic107C2semidirect
11. C2×AGL(1,ℤ/8)×C107C2semidirect
18. C2×#13133807C22semidirect
19. C2×#13133806C22semidirect
20. C2×#13133805C22semidirect
21. C2×D856C22semidirect
22. C2×#13133803C22semidirect
23. C2×#13133803C22semidirect
25. #16812107C22semidirect
29. #16812105C22semidirect
30. #16812098C22semidirect
31. #16812096C22semidirect
35. #16812019C22semidirect
36. C8×Dic107C22semidirect
40. C2×SD16×C107C22semidirect
41. C2×D8×C107C22semidirect
65. #13133802D4semidirect
87. M16×C107D4semidirect
128. C8×C107C2×D4semidirect
129. C107⋊C8C2×D4semidirect
149. Dic107AGL(1,ℤ/8)semidirect
158. C107#3151semidirect
162. C2×AGL(1,ℤ/8)D107semidirect
182. C2×SD16C2×D107semidirect
186. C2×D8C2×D107semidirect
238. M16C22⋊D107semidirect
287. C8C2×C22⋊D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168121372
C2×#168121352
C2×#168121342
#202931732
C2×#168121052
#202930152
#202930132
#202930022
#202929982
M16×Dic1072
C2×AGL(1,ℤ/8)×C1072
#3151107