Finite Group Databasebeta

← #28346944

Subgroups — #28346944

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×#16812135C211yesmaximal, characteristic, complemented
2C2×#16812134C211yesmaximal, characteristic, complemented
3C2×#16812129C211yesmaximal, characteristic, complemented
4C22×#13133803C211yesmaximal, characteristic, complemented
5#20293196C211yesmaximal, characteristic, complemented
6#20293017C212yesmaximal, complemented
7#20293014C212yesmaximal, complemented
8#20293009C212yesmaximal, complemented
9#20293003C212yesmaximal, complemented
10#20292583C211yesmaximal, characteristic, complemented
11#535×C107C211yesmaximal, characteristic, complemented
12C2×#13133818C2211yescharacteristic
13C22×D428C2211yescharacteristic
14#16812135—24no—
15#16812134—24no—
16#16812129—24no—
17C2×#13133808C2211yescharacteristic, complemented
18C2×#13133806C2211yescharacteristic, complemented
19C2×#13133806C2212yescomplemented
20C2×#13133805C2211yescharacteristic, complemented
21C2×#13133805C2212yescomplemented
22C2×#13133803—24no—
23C2×#13133803C2212yes—
24C2×#13133803C2212yes—
25C2×#13133803C2211yescharacteristic, complemented
26C2×#13133803C2212yescomplemented
27C2×#13133803C2212yescomplemented
28C2×#13133802C2211yescharacteristic
29C22×C107⋊C8C2211yescharacteristic
30#16812106C2212yescomplemented
31#16812097C2212yescomplemented
32#16812084—22no—
33#16812079—22no—
34#16812020C2212yescomplemented
35C4×C107⋊C8C2212yescomplemented
36C2×#41×C107C2211yescharacteristic
37C22×D4×C107C2211yescharacteristic
38#127×C107C2212yescomplemented
39#124×C107C2212yescomplemented
40#121×C107—22no—
41#120×C107—22no—
42#117×C107C2211yescharacteristic
43C23×D107—21no—
44#13133818—22no—
45#13133818—22no—
46C2×Dic214C2311yescharacteristic
47C2×D428—22no—
48C2×D428—22no—
49C2×D428—21no—
50C2×D428—24no—
51C2×D428C2312yes—
52C2×D428C2311yescharacteristic
53C2×C22⋊D107—21no—
54C2×C4×D107—21no—
55#13133808—22no—
56#13133806—22no—
57#13133806—24no—
58#13133805—22no—
59#13133805—24no—
60#13133803—24no—
61#13133803—24no—
62#13133803—22no—
63#13133803—24no—
64#13133803—24no—
65#13133802D414yescomplemented
66C2×C107⋊C8D414yescomplemented
67C2×C107⋊C8C2311yescharacteristic
68C2×C107⋊C8C2311yescharacteristic
69C2×C107⋊C8C2312yes—
70(C2×D107)⋊C4—24no—
71C24×C107—21no—
72#41×C107—22no—
73#41×C107—22no—
74C2×Q8×C107C2311yescharacteristic
75C2×D4×C107—21no—
76C2×D4×C107—22no—
77C2×D4×C107—22no—
78C2×D4×C107—21no—
79C2×D4×C107—24no—
80C2×D4×C107C2312yes—
81C2×D4×C107C2311yescharacteristic
82C22×C4×C107—21no—
83C22×C4×C107C2311yescharacteristic
84C4⋊C4×C107D412yes—
85C22⋊C4×C107—24no—
86C22⋊C4×C107D412yes—
87C42×C107C2312yes—
88C23×C107—22no—
89C23×C107—21no—
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—22no—
100D4×C107—22no—
101D4×C107—21no—
102D4×C107—21no—
103D4×C107—42no—
104D4×C107—42no—
105D4×C107—84no—
106C2×C4×C107—21no—
107C2×C4×C107—42no—
108C2×C4×C107C2×D412yes—
109C2×C4×C107C2×D412yes—
110C2×C4×C107C2×D414yes—
111C2×C4×C107C2×D411yescharacteristic
112C2×C4×C107C2411yescharacteristic
113C22⋊D107—42no—
114C2×Dic107—21no—
115D428—22no—
116D428—21no—
117D428—22no—
118D428—21no—
119D428—42no—
120D428—84no—
121C4×D107—42no—
122Dic214—21no—
123Dic214—21no—
124C107⋊C8C2×D412yes—
125C107⋊C8C2×D412yes—
126C107⋊C8C2×D414yes—
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—24no—
141C4×C107—41no—
142C4×C107#12712yes—
143C4×C107C22×D411yescharacteristic
144C4×C107C22×D411yescharacteristic
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)—1071no—
157C2×#135—1071no—
158C22×D8—1071no—
159#535—1071no—
160#535D10711yescharacteristic, complemented
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)—2144no—
171AGL(1,ℤ/8)—2144no—
172#135—2144no—
173C2×Q16—1071no—
174C2×SD16—1071no—
175C2×SD16—1072no—
176C2×SD16—1071no—
177C2×SD16—1072no—
178C2×D8—1072no—
179C2×D8—1072no—
180C2×D8—1071no—
181C2×D8—1072no—
182C2×D8—1072no—
183C2×D8—2144no—
184C2×M16—1071no—
185C22×C8—1071no—
186#127—1072no—
187#127C2×D10712yescomplemented
188#124—1072no—
189#124C2×D10712yescomplemented
190#121—22no—
191#121—2142no—
192#120—22no—
193#120—2142no—
194#117C2×D10711yescharacteristic
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—22no—
208C2×D4—21no—
209C2×D4—24no—
210C2×D4—1072no—
211C2×D4—1071no—
212C2×D4—2141no—
213C2×D4—2142no—
214C2×D4—2142no—
215C2×D4—2144no—
216C2×D4—2141no—
217C2×D4C22×D10712yes—
218C2×D4C22×D10711yescharacteristic
219C22×C4—21no—
220C22×C4—2141no—
221C22×C4C22×D10711yescharacteristic
222Q16—2142no—
223SD16—2142no—
224SD16—2144no—
225SD16—2142no—
226SD16—2144no—
227D8—2144no—
228D8—2144no—
229D8—2142no—
230D8—2144no—
231D8—2144no—
232M16—1074no—
233C2×C8—1074no—
234C2×C8—1071no—
235C2×C8—1072no—
236C2×C8—1071no—
237C4⋊C4C22⋊D10712yes—
238C22⋊C4—24no—
239C22⋊C4—2144no—
240C22⋊C4C22⋊D10712yes—
241C42C22×D10712yes—
242C23—22no—
243C23—21no—
244C23—42no—
245C23—82no—
246C23—8562no—
247C23—4282no—
248C23—2142no—
249C23—2141no—
250C23C2×C22⋊D10711yescharacteristic
251Q8—21no—
252Q8—21no—
253Q8—2141no—
254Q8—2141no—
255D4—22no—
256D4—21no—
257D4—22no—
258D4—21no—
259D4—42no—
260D4—42no—
261D4—84no—
262D4—8564no—
263D4—4282no—
264D4—4282no—
265D4—2142no—
266D4—2141no—
267D4—2142no—
268D4—2141no—
269C2×C4—21no—
270C2×C4—42no—
271C2×C4—4282no—
272C2×C4—2141no—
273C2×C4D4×D10712yes—
274C2×C4D4×D10712yes—
275C2×C4C2×C22⋊D10714yes—
276C2×C4C2×C22⋊D10711yescharacteristic
277C2×C4C23×D10711yescharacteristic
278C8—1072no—
279C8—1072no—
280C8—1074no—
281C22—22no—
282C22—41no—
283C22—42no—
284C22—42no—
285C22—44no—
286C22—84no—
287C22—8564no—
288C22—4281no—
289C22—4282no—
290C22—4282no—
291C22—4284no—
292C22#1681210512yes—
293C22C22×C22⋊D10711yescharacteristic
294C4—24no—
295C4—41no—
296C4—4281no—
297C4#1681210512yes—
298C4C2×D4×D10711yescharacteristic
299C4C2×D4×D10711yescharacteristic
300C2—22no—
301C2—82no—
302C2—81no—
303C2—8562no—
304C2—8561no—
305C2C2×#1681210511yescharacteristic
306C2#2029343012yes—
subgroupG/H
centerC22C22×C22⋊D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#535×C107C2
socleC22×C107C22×D4
hypercenter#535D107
hypocenterC107#3155
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ #117 ◁ #535

Complements

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

NH ≅ G/Ntype
1. C2×#16812135C2semidirect
2. C2×#16812134C2semidirect
3. C2×#16812129C2semidirect
4. C22×#13133803C2semidirect
5. #20293196C2semidirect
6. #20293017C2semidirect
7. #20293014C2semidirect
8. #20293009C2semidirect
9. #20293003C2semidirect
10. #20292583C2semidirect
11. #535×C107C2semidirect
17. C2×#13133808C22semidirect
18. C2×#13133806C22semidirect
19. C2×#13133806C22semidirect
20. C2×#13133805C22semidirect
21. C2×#13133805C22semidirect
25. C2×#13133803C22semidirect
26. C2×#13133803C22semidirect
27. C2×#13133803C22semidirect
30. #16812106C22semidirect
31. #16812097C22semidirect
34. #16812020C22semidirect
35. C4×C107⋊C8C22semidirect
38. #127×C107C22semidirect
39. #124×C107C22semidirect
65. #13133802D4semidirect
66. C2×C107⋊C8D4semidirect
154. C107#3155semidirect
160. #535D107semidirect
187. #127C2×D107semidirect
189. #124C2×D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168121352
C2×#168121342
C2×#168121292
C22×#131338032
#202931962
#202930172
#202930142
#202930092
#202930032
#202925832
#535×C1072
#3155107