Finite Group Databasebeta

← #28346935

Subgroups — #28346935

Order 13696.

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

normal subgroups only

This group has 564 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×#16812142C211yesmaximal, characteristic, complemented
2C2×#16812129C211yesmaximal, characteristic, complemented
3C22×#13133806C211yesmaximal, characteristic, complemented
4#20293198C211yesmaximal, characteristic, complemented
5#20293194C211yesmaximal, characteristic, complemented
6#20293018C212yesmaximal, complemented
7#20293015C212yesmaximal, complemented
8#20293010C212yesmaximal, complemented
9#20292998C212yesmaximal, complemented
10#20292585C211yesmaximal, characteristic, complemented
11C2×AGL(1,ℤ/8)×C107C211yesmaximal, characteristic, complemented
12C2×#13133818C2211yescharacteristic
13C22×Dic214C2211yescharacteristic
14#16812142—24no—
15#16812129—24no—
16C2×Dic428C2212yescomplemented
17C2×#13133808C2211yescharacteristic, complemented
18C2×#13133807C2212yescomplemented
19C2×#13133806—24no—
20C2×#13133806C2212yes—
21C2×#13133806C2212yes—
22C2×#13133806C2211yescharacteristic, complemented
23C2×#13133805C2211yescharacteristic, complemented
24C2×#13133803C2211yescharacteristic, complemented
25C22×C107⋊C8C2211yescharacteristic
26#16812107C2212yescomplemented
27#16812105C2212yes—
28#16812096C2212yes—
29#16812096C2212yescomplemented
30#16812090—22no—
31#16812088—22no—
32#16812080—22no—
33#16812078—22no—
34#16812068C2211yescharacteristic
35#16812019C2212yescomplemented
36C8×Dic107C2212yescomplemented
37C2×#41×C107C2211yescharacteristic
38C22×D4×C107C2211yescharacteristic
39AGL(1,ℤ/8)×C107—24no—
40C2×SD16×C107C2212yes—
41C2×D8×C107C2212yes—
42C2×M16×C107C2211yescharacteristic
43#13133818—22no—
44#13133818—22no—
45C2×Dic214—22no—
46C2×Dic214—22no—
47C2×Dic214—21no—
48C2×Dic214C2311yescharacteristic
49C2×Dic214C2312yes—
50C2×D428C2311yescharacteristic
51C2×C22⋊D107—21no—
52C2×C22⋊D107—24no—
53C22×Dic107—21no—
54C2×C4×D107—21no—
55Dic428—24no—
56#13133808—22no—
57#13133807—24no—
58#13133806—24no—
59#13133806—24no—
60#13133806—22no—
61#13133805—22no—
62#13133803—22no—
63C2×C107⋊C8D414yescomplemented
64C2×C107⋊C8C2311yescharacteristic
65C2×C107⋊C8C2311yescharacteristic
66C4⋊Dic107D412yes—
67Dic107⋊C4—24no—
68C22⋊Dic107—24no—
69C22⋊Dic107—24no—
70C22⋊Dic107D412yes—
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—21no—
94Q8×C107—21no—
95Q8×C107—21no—
96D4×C107—21no—
97D4×C107—22no—
98D4×C107—22no—
99D4×C107—21no—
100D4×C107—42no—
101D4×C107—42no—
102C2×C4×C107—21no—
103C2×C4×C107—42no—
104C2×C4×C107C2×D411yescharacteristic
105C2×C4×C107C2×D412yes—
106C2×C4×C107C2×D412yes—
107C2×C4×C107C2411yescharacteristic
108C22⋊D107—42no—
109C22⋊D107—84no—
110C2×Dic107—21no—
111C2×Dic107—22no—
112C2×Dic107—42no—
113C2×Dic107C2×D414yes—
114D428—21no—
115D428—21no—
116C4×D107—42no—
117Dic214—22no—
118Dic214—21no—
119Dic214—22no—
120Dic214—21no—
121Dic214—42no—
122C8×C107C2×D414yes—
123C107⋊C8C2×D412yes—
124C107⋊C8C2×D412yes—
125C22×C107—22no—
126C22×C107—41no—
127C22×C107—42no—
128C22×C107—44no—
129C22×C107—42no—
130C22×C107—84no—
131C22×C107#12712yes—
132C22×C107C22×D411yescharacteristic
133C2×D107—41no—
134C2×D107—42no—
135C4×C107—41no—
136C4×C107#12712yes—
137C4×C107C22×D411yescharacteristic
138C4×C107C22×D411yescharacteristic
139Dic107—24no—
140Dic107—42no—
141Dic107—41no—
142C2×C107—22no—
143C2×C107—81no—
144C2×C107—82no—
145C2×C107C2×#12711yescharacteristic
146C2×C107#57712yes—
147D107—81no—
148#3154—1071nomaximal, Sylow
149C107#315411yescharacteristic, complemented, Sylow
150C2×#137—1071no—
151C2×AGL(1,ℤ/8)D10711yescharacteristic, complemented
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#405—1071no—
161C2×#41—1071no—
162C2×#41C2×D10711yescharacteristic
163C22×Q8—1071no—
164C22×D4C2×D10711yescharacteristic
165#137—2144no—
166AGL(1,ℤ/8)—24no—
167#135—2144no—
168C2×Q16—1071no—
169C2×Q16—1072no—
170C2×SD16—1071no—
171C2×SD16—1072no—
172C2×SD16—1072no—
173C2×SD16—1071no—
174C2×SD16—1072no—
175C2×SD16—2144no—
176C2×SD16C2×D10712yes—
177C2×D8—1071no—
178C2×D8C2×D10712yes—
179C2×M16C2×D10711yescharacteristic
180C22×C8—1071no—
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#117—1071no—
190#97—1072no—
191C4×C8—1072no—
192C24—21no—
193#41—22no—
194#41—22no—
195#41—2142no—
196#41—2142no—
197C2×Q8—1071no—
198C2×Q8—1072no—
199C2×Q8—2142no—
200C2×Q8—2142no—
201C2×Q8—2141no—
202C2×Q8C22×D10711yescharacteristic
203C2×D4—21no—
204C2×D4—22no—
205C2×D4—21no—
206C2×D4—22no—
207C2×D4—1071no—
208C2×D4—2144no—
209C2×D4—2141no—
210C2×D4C22×D10711yescharacteristic
211C2×D4C22×D10712yes—
212C22×C4—21no—
213C22×C4—2141no—
214C22×C4—2141no—
215C22×C4C22×D10711yescharacteristic
216Q16—2142no—
217Q16—2144no—
218SD16—24no—
219SD16—2144no—
220SD16—2144no—
221SD16—2142no—
222SD16—2142no—
223SD16—2144no—
224D8—24no—
225D8—2142no—
226M16C22⋊D10714yescomplemented
227C2×C8—1074no—
228C2×C8—1071no—
229C2×C8—1071no—
230C2×C8C22×D10712yes—
231C4⋊C4—1072no—
232C4⋊C4—2144no—
233C22⋊C4—1072no—
234C22⋊C4—2144no—
235C22⋊C4—2144no—
236C42—1072no—
237C23—21no—
238C23—22no—
239C23—42no—
240C23—82no—
241C23—2141no—
242C23D4×D10711yescharacteristic
243Q8—21no—
244Q8—21no—
245Q8—4282no—
246Q8—2142no—
247Q8—2141no—
248Q8—2142no—
249Q8—2141no—
250D4—21no—
251D4—22no—
252D4—21no—
253D4—22no—
254D4—42no—
255D4—42no—
256D4—8564no—
257D4—4282no—
258D4—2141no—
259D4—2141no—
260C2×C4—21no—
261C2×C4—42no—
262C2×C4—1074no—
263C2×C4—4282no—
264C2×C4—4282no—
265C2×C4—2141no—
266C2×C4—2142no—
267C2×C4D4×D10711yescharacteristic
268C2×C4D4×D10712yes—
269C2×C4C2×C22⋊D10712yes—
270C2×C4C23×D10711yescharacteristic
271C8—1072no—
272C8—1072no—
273C8C2×C22⋊D10714yes—
274C22—22no—
275C22—41no—
276C22—42no—
277C22—44no—
278C22—42no—
279C22—84no—
280C22—4281no—
281C22—4282no—
282C22#1681210512yes—
283C22C2×D4×D10711yescharacteristic
284C4—41no—
285C4—4281no—
286C4—4282no—
287C4—2144no—
288C4#1681210512yes—
289C4C2×D4×D10711yescharacteristic
290C4C22×C22⋊D10711yescharacteristic
291C2—22no—
292C2—82no—
293C2—81no—
294C2—8561no—
295C2C2×#1681210511yescharacteristic
296C2#2029343712yes—
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#3154
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×#16812142C2semidirect
2. C2×#16812129C2semidirect
3. C22×#13133806C2semidirect
4. #20293198C2semidirect
5. #20293194C2semidirect
6. #20293018C2semidirect
7. #20293015C2semidirect
8. #20293010C2semidirect
9. #20292998C2semidirect
10. #20292585C2semidirect
11. C2×AGL(1,ℤ/8)×C107C2semidirect
16. C2×Dic428C22semidirect
17. C2×#13133808C22semidirect
18. C2×#13133807C22semidirect
22. C2×#13133806C22semidirect
23. C2×#13133805C22semidirect
24. C2×#13133803C22semidirect
26. #16812107C22semidirect
29. #16812096C22semidirect
35. #16812019C22semidirect
36. C8×Dic107C22semidirect
63. C2×C107⋊C8D4semidirect
86. M16×C107D4semidirect
149. C107#3154semidirect
151. C2×AGL(1,ℤ/8)D107semidirect
226. M16C22⋊D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168121422
C2×#168121292
C22×#131338062
#202931982
#202931942
#202930182
#202930152
#202930102
#202929982
#202925852
C2×AGL(1,ℤ/8)×C1072
#3154107