Finite Group Databasebeta

← #28346946

Subgroups — #28346946

Order 13696.

596 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 596 conjugacy classes of subgroups — too many to draw. Only its 149 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×#16812129C211yesmaximal, characteristic, complemented
3C22×#13133803C211yesmaximal, characteristic, complemented
4#20293194C211yesmaximal, characteristic, complemented
5#20293191C211yesmaximal, characteristic, complemented
6#20293015C212yesmaximal, complemented
7#20293013C212yesmaximal, complemented
8#20293008C212yesmaximal, complemented
9#20293002C212yesmaximal, complemented
10#20292585C211yesmaximal, characteristic, complemented
11C2×AGL(1,ℤ/8)×C107C211yesmaximal, characteristic, complemented
12C2×#13133818C2211yescharacteristic
13C22×D428C2211yescharacteristic
14#16812137—24no—
15#16812129—24no—
16C2×#13133808C2211yescharacteristic, complemented
17C2×#13133807C2212yescomplemented
18C2×#13133806C2211yescharacteristic, complemented
19C2×#13133805C2211yescharacteristic, complemented
20C2×D856C2212yescomplemented
21C2×#13133803—24no—
22C2×#13133803C2212yes—
23C2×#13133803C2212yes—
24C2×#13133803C2211yescharacteristic, complemented
25C22×C107⋊C8C2211yescharacteristic
26#16812105C2212yescomplemented
27#16812105C2212yescomplemented
28#16812098C2212yescomplemented
29#16812096C2212yescomplemented
30#16812083—22no—
31#16812080—22no—
32#16812078—22no—
33#16812077—22no—
34#16812068C2211yescharacteristic
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×D428C2312yes—
51C2×D428C2311yescharacteristic
52C2×C22⋊D107—21no—
53C2×C22⋊D107—24no—
54C2×C22⋊D107—24no—
55C2×C4×D107—21no—
56#13133808—22no—
57#13133807—24no—
58#13133806—22no—
59#13133805—22no—
60D856—24no—
61#13133803—24no—
62#13133803—24no—
63#13133803—22no—
64C2×C107⋊C8D414yescomplemented
65C2×C107⋊C8C2311yescharacteristic
66C2×C107⋊C8C2311yescharacteristic
67C4⋊Dic107D412yes—
68C22⋊Dic107—24no—
69C22⋊Dic107D412yes—
70(C2×D107)⋊C4—24no—
71C4×Dic107C2312yes—
72C24×C107—21no—
73#41×C107—22no—
74#41×C107—22no—
75C2×Q8×C107C2311yescharacteristic
76C2×D4×C107—21no—
77C2×D4×C107—22no—
78C2×D4×C107—22no—
79C2×D4×C107—21no—
80C2×D4×C107C2312yes—
81C2×D4×C107C2311yescharacteristic
82C22×C4×C107—21no—
83C22×C4×C107C2311yescharacteristic
84SD16×C107—24no—
85D8×C107—24no—
86M16×C107D414yescomplemented
87C2×C8×C107C2312yes—
88C23×C107—21no—
89C23×C107—22no—
90C23×C107—42no—
91C23×C107—82no—
92C23×C107C2×D411yescharacteristic
93C22×D107—22no—
94C22×D107—21no—
95C22×D107—42no—
96C22×D107—82no—
97Q8×C107—21no—
98Q8×C107—21no—
99D4×C107—21no—
100D4×C107—22no—
101D4×C107—22no—
102D4×C107—21no—
103D4×C107—42no—
104D4×C107—42no—
105C2×C4×C107—21no—
106C2×C4×C107—42no—
107C2×C4×C107C2×D411yescharacteristic
108C2×C4×C107C2×D412yes—
109C2×C4×C107C2×D412yes—
110C2×C4×C107C2411yescharacteristic
111C22⋊D107—42no—
112C22⋊D107—84no—
113C22⋊D107—84no—
114C2×Dic107—21no—
115C2×Dic107C2×D414yes—
116D428—22no—
117D428—21no—
118D428—21no—
119D428—22no—
120D428—42no—
121C4×D107—42no—
122Dic214—21no—
123Dic214—21no—
124C8×C107C2×D414yes—
125C107⋊C8C2×D412yes—
126C107⋊C8C2×D412yes—
127C22×C107—22no—
128C22×C107—41no—
129C22×C107—42no—
130C22×C107—42no—
131C22×C107—44no—
132C22×C107—84no—
133C22×C107#12712yes—
134C22×C107C22×D411yescharacteristic
135C2×D107—41no—
136C2×D107—42no—
137C2×D107—42no—
138C2×D107—44no—
139C2×D107—84no—
140C4×C107—41no—
141C4×C107#12712yes—
142C4×C107C22×D411yescharacteristic
143C4×C107C22×D411yescharacteristic
144Dic107—24no—
145Dic107—41no—
146C2×C107—22no—
147C2×C107—81no—
148C2×C107—82no—
149C2×C107C2×#12711yescharacteristic
150C2×C107#57612yes—
151D107—81no—
152D107—82no—
153#3155—1071nomaximal, Sylow
154C107#315511yescharacteristic, complemented, Sylow
155C2×AGL(1,ℤ/8)—1071no—
156C2×AGL(1,ℤ/8)D10711yescharacteristic, complemented
157C2×#135—1071no—
158C22×D8—1071no—
159#535—1071no—
160#535—1071no—
161#496—1072no—
162#496—1072no—
163#495—1072no—
164#493—1072no—
165#405—1071no—
166C2×#41—1071no—
167C2×#41C2×D10711yescharacteristic
168C22×D4—1071no—
169C22×D4C2×D10711yescharacteristic
170AGL(1,ℤ/8)—24no—
171AGL(1,ℤ/8)—2144no—
172#135—2144no—
173C2×Q16—1071no—
174C2×SD16—1071no—
175C2×SD16—1071no—
176C2×SD16—1072no—
177C2×SD16C2×D10712yescomplemented
178C2×D8—1072no—
179C2×D8—1072no—
180C2×D8—1071no—
181C2×D8—1072no—
182C2×D8—2144no—
183C2×D8C2×D10712yescomplemented
184C2×M16C2×D10711yescharacteristic
185C22×C8—1071no—
186#127—1072no—
187#127—1072no—
188#124—1072no—
189#124—1072no—
190#121—2142no—
191#121—2142no—
192#120—2142no—
193#120—2142no—
194#117—1071no—
195#97—1072no—
196C4×C8—1072no—
197C24—21no—
198C24—2141no—
199#41—22no—
200#41—22no—
201#41—2142no—
202#41—2142no—
203C2×Q8—1071no—
204C2×Q8C22×D10711yescharacteristic
205C2×D4—21no—
206C2×D4—22no—
207C2×D4—21no—
208C2×D4—22no—
209C2×D4—1072no—
210C2×D4—1071no—
211C2×D4—2144no—
212C2×D4—2144no—
213C2×D4—2141no—
214C2×D4—2141no—
215C2×D4—2142no—
216C2×D4—2142no—
217C2×D4C22×D10712yes—
218C2×D4C22×D10711yescharacteristic
219C22×C4—21no—
220C22×C4—2141no—
221C22×C4C22×D10711yescharacteristic
222Q16—2142no—
223SD16—24no—
224SD16—2142no—
225SD16—2142no—
226SD16—2144no—
227D8—24no—
228D8—2144no—
229D8—2144no—
230D8—2142no—
231D8—2144no—
232M16C22⋊D10714yescomplemented
233C2×C8—1074no—
234C2×C8—1071no—
235C2×C8—1071no—
236C2×C8C22×D10712yes—
237C4⋊C4—1072no—
238C22⋊C4—1072no—
239C22⋊C4—2144no—
240C22⋊C4—2144no—
241C42—1072no—
242C23—22no—
243C23—21no—
244C23—42no—
245C23—82no—
246C23—8562no—
247C23—4282no—
248C23—2142no—
249C23—2141no—
250C23D4×D10711yescharacteristic
251Q8—21no—
252Q8—21no—
253Q8—2141no—
254Q8—2141no—
255D4—22no—
256D4—21no—
257D4—21no—
258D4—22no—
259D4—42no—
260D4—42no—
261D4—8564no—
262D4—8564no—
263D4—4282no—
264D4—4282no—
265D4—2142no—
266D4—2141no—
267D4—2141no—
268D4—2142no—
269C2×C4—21no—
270C2×C4—42no—
271C2×C4—1074no—
272C2×C4—4282no—
273C2×C4—2141no—
274C2×C4D4×D10711yescharacteristic
275C2×C4D4×D10712yes—
276C2×C4C2×C22⋊D10712yes—
277C2×C4C23×D10711yescharacteristic
278C8—1072no—
279C8—1072no—
280C8C2×C22⋊D10714yes—
281C22—22no—
282C22—41no—
283C22—42no—
284C22—42no—
285C22—44no—
286C22—84no—
287C22—8564no—
288C22—4281no—
289C22—4282no—
290C22—4282no—
291C22—4284no—
292C22#1681210512yes—
293C22C2×D4×D10711yescharacteristic
294C4—41no—
295C4—4281no—
296C4—2144no—
297C4#1681210512yes—
298C4C2×D4×D10711yescharacteristic
299C4C22×C22⋊D10711yescharacteristic
300C2—22no—
301C2—82no—
302C2—81no—
303C2—8562no—
304C2—8561no—
305C2C2×#1681210511yescharacteristic
306C2#2029343112yes—
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#3155
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×#16812129C2semidirect
3. C22×#13133803C2semidirect
4. #20293194C2semidirect
5. #20293191C2semidirect
6. #20293015C2semidirect
7. #20293013C2semidirect
8. #20293008C2semidirect
9. #20293002C2semidirect
10. #20292585C2semidirect
11. C2×AGL(1,ℤ/8)×C107C2semidirect
16. C2×#13133808C22semidirect
17. C2×#13133807C22semidirect
18. C2×#13133806C22semidirect
19. C2×#13133805C22semidirect
20. C2×D856C22semidirect
24. C2×#13133803C22semidirect
26. #16812105C22semidirect
27. #16812105C22semidirect
28. #16812098C22semidirect
29. #16812096C22semidirect
35. #16812019C22semidirect
36. C8×Dic107C22semidirect
40. C2×SD16×C107C22semidirect
41. C2×D8×C107C22semidirect
64. C2×C107⋊C8D4semidirect
86. M16×C107D4semidirect
154. C107#3155semidirect
156. C2×AGL(1,ℤ/8)D107semidirect
177. C2×SD16C2×D107semidirect
183. C2×D8C2×D107semidirect
232. M16C22⋊D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168121372
C2×#168121292
C22×#131338032
#202931942
#202931912
#202930152
#202930132
#202930082
#202930022
#202925852
C2×AGL(1,ℤ/8)×C1072
#3155107