Finite Group Databasebeta

← #28345701

Subgroups — #28345701

Order 13696.

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

normal subgroups only

This group has 758 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
1C22×#13133818C211yesmaximal, characteristic, complemented
2C2×#16812139C211yesmaximal, characteristic, complemented
3C2×#16812134C211yesmaximal, characteristic, complemented
4#20292787C214yesmaximal, complemented
5#20292770C214yesmaximal, complemented
6#20292635C211yesmaximal, characteristic, complemented
7#20292631C211yesmaximal, characteristic, complemented
8#20292591C211yesmaximal, characteristic, complemented
9#525×C107C211yesmaximal, characteristic, complemented
10C2×#13133818—24no—
11C2×#13133818—24no—
12C2×#13133818—22no—
13C2×#13133818C2214yes—
14C22×Dic214C2211yescharacteristic
15C22×D428C2211yescharacteristic
16C22×C22⋊D107—21no—
17C22×C4×D107—21no—
18#16812139—24no—
19#16812134—24no—
20C2×#13133808C2212yescomplemented
21C2×#13133806C2212yescomplemented
22C2×#13133805C2212yescomplemented
23C2×#13133803C2212yescomplemented
24C2×#13133802C2211yescharacteristic
25C2×#13133802C2211yescharacteristic
26#16812032C2214yescomplemented
27#16812029C2214yescomplemented
28C22⋊(C107⋊C8)C2214yescomplemented
29C2×#41×C107C2211yescharacteristic
30C23×C4×C107C2211yescharacteristic
31#123×C107—21no—
32C22⋊Q8×C107C2212yescomplemented
33#121×C107C2212yescomplemented
34#120×C107—21no—
35C4×D4×C107—22no—
36#117×C107C2211yescharacteristic
37C23×D107—21no—
38#13133818—216no—
39#13133818—28no—
40#13133818—28no—
41C2×Dic214—22no—
42C2×Dic214—21no—
43C2×Dic214D414yescomplemented
44C2×Dic214C2312yes—
45C2×D428—22no—
46C2×D428—21no—
47C2×D428D414yescomplemented
48C2×D428C2312yes—
49C2×C22⋊D107—28no—
50C2×C22⋊D107—24no—
51C22×Dic107—21no—
52C2×C4×D107—28no—
53C2×C4×D107—24no—
54#13133808—24no—
55#13133806—24no—
56#13133805—24no—
57#13133803—24no—
58#13133802—22no—
59#13133802—22no—
60C2×C107⋊C8C2312yes—
61C2×C107⋊C8C2312yes—
62C24×C107D411yescharacteristic
63#41×C107—22no—
64#41×C107—22no—
65C2×Q8×C107C2311yescharacteristic
66C2×D4×C107—22no—
67C2×D4×C107—21no—
68C2×D4×C107C2311yescharacteristic
69C22×C4×C107—22no—
70C22×C4×C107—22no—
71C22×C4×C107—21no—
72C22×C4×C107D411yescharacteristic
73C22×C4×C107D412yes—
74C22×C4×C107C2312yes—
75C22×C4×C107C2311yescharacteristic
76C4⋊C4×C107—22no—
77C4⋊C4×C107C2312yes—
78C22⋊C4×C107—21no—
79C22⋊C4×C107—22no—
80C22⋊C4×C107—22no—
81C42×C107—21no—
82C23×C107—22no—
83C23×C107—21no—
84C23×C107—42no—
85C23×C107C2×D412yes—
86C23×C107C2×D411yescharacteristic
87C22×D107—24no—
88C22×D107—22no—
89C22×D107—44no—
90Q8×C107—21no—
91Q8×C107—21no—
92D4×C107—21no—
93D4×C107—21no—
94D4×C107—44no—
95D4×C107—42no—
96C2×C4×C107—22no—
97C2×C4×C107—24no—
98C2×C4×C107—22no—
99C2×C4×C107—22no—
100C2×C4×C107—22no—
101C2×C4×C107—21no—
102C2×C4×C107—42no—
103C2×C4×C107C2×D412yes—
104C2×C4×C107C2×D412yes—
105C2×C4×C107C2×D412yes—
106C2×C4×C107C2×D411yescharacteristic
107C2×C4×C107C2411yescharacteristic
108C22⋊D107—216no—
109C2×Dic107—24no—
110C2×Dic107—22no—
111D428—24no—
112D428—22no—
113D428C2×D414yes—
114C4×D107—216no—
115Dic214—24no—
116Dic214—22no—
117Dic214C2×D414yes—
118C107⋊C8—22no—
119C107⋊C8—22no—
120C22×C107—22no—
121C22×C107—22no—
122C22×C107—22no—
123C22×C107—22no—
124C22×C107—44no—
125C22×C107—41no—
126C22×C107—42no—
127C22×C107#12012yes—
128C22×C107C22×D411yescharacteristic
129C2×D107—24no—
130C2×D107—48no—
131C2×D107—44no—
132C4×C107—22no—
133C4×C107—41no—
134C4×C107—42no—
135C4×C107#12012yes—
136C4×C107C22×D411yescharacteristic
137C4×C107C22×D411yescharacteristic
138Dic107—24no—
139C2×C107—22no—
140C2×C107—42no—
141C2×C107—81no—
142C2×C107C2×#12011yescharacteristic
143C2×C107#57512yes—
144D107—44no—
145#3005—1071nomaximal, Sylow
146C107#300511yescharacteristic, complemented, Sylow
147C22×#41—1071no—
148C2×#137—1071no—
149C2×AGL(1,ℤ/8)—1071no—
150#525D10711yescharacteristic, complemented
151#452—1074no—
152#449—1074no—
153#419—1071no—
154#418—1071no—
155#407—1071no—
156C2×#41—1074no—
157C2×#41—2144no—
158C2×#41—2144no—
159C2×#41—2142no—
160C2×#41C2×D10711yescharacteristic
161C22×Q8—1071no—
162C22×D4—1071no—
163C22×D4—2141no—
164C23×C4—2141no—
165C23×C4C2×D10711yescharacteristic
166#137—2144no—
167AGL(1,ℤ/8)—2144no—
168C2×Q16—1072no—
169C2×SD16—1072no—
170C2×SD16—1072no—
171C2×D8—1072no—
172C2×M16—1071no—
173C2×M16—1071no—
174#123—21no—
175C22⋊Q8C2×D10712yescomplemented
176#121C2×D10712yescomplemented
177#120—21no—
178C4×D4—22no—
179#117C2×D10711yescharacteristic
180#103—1074no—
181#102—1074no—
182C22⋊C8—1074no—
183C24—2141no—
184C24C22⋊D10711yescharacteristic
185#41—22no—
186#41—22no—
187#41—2148no—
188#41—2148no—
189#41—21416no—
190C2×Q8—1074no—
191C2×Q8—1072no—
192C2×Q8—2142no—
193C2×Q8—2141no—
194C2×Q8C22×D10711yescharacteristic
195C2×D4—22no—
196C2×D4—21no—
197C2×D4—1074no—
198C2×D4—1072no—
199C2×D4—2148no—
200C2×D4—2144no—
201C2×D4—2142no—
202C2×D4—2141no—
203C2×D4C22×D10711yescharacteristic
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—21no—
207C22×C4—2148no—
208C22×C4—2141no—
209C22×C4—2144no—
210C22×C4C22⋊D10712yes—
211C22×C4C22⋊D10711yescharacteristic
212C22×C4C22×D10712yes—
213C22×C4C22×D10711yescharacteristic
214Q16—2144no—
215SD16—2144no—
216SD16—2144no—
217D8—2144no—
218M16—2142no—
219M16—2142no—
220C2×C8—1072no—
221C2×C8—1072no—
222C4⋊C4—22no—
223C4⋊C4C22×D10712yes—
224C22⋊C4—21no—
225C22⋊C4—22no—
226C22⋊C4—22no—
227C42—21no—
228C23—22no—
229C23—21no—
230C23—42no—
231C23—4284no—
232C23—2144no—
233C23—2142no—
234C23C2×C22⋊D10712yes—
235C23C2×C22⋊D10711yescharacteristic
236Q8—21no—
237Q8—21no—
238Q8—1074no—
239Q8—2144no—
240Q8—2142no—
241D4—21no—
242D4—21no—
243D4—44no—
244D4—42no—
245D4—1074no—
246D4—21416no—
247D4—2144no—
248D4—2142no—
249C2×C4—22no—
250C2×C4—24no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—21no—
255C2×C4—42no—
256C2×C4—21416no—
257C2×C4—2144no—
258C2×C4—2142no—
259C2×C4D4×D10712yes—
260C2×C4D4×D10712yes—
261C2×C4C2×C22⋊D10712yes—
262C2×C4C2×C22⋊D10711yescharacteristic
263C2×C4C23×D10711yescharacteristic
264C8—2142no—
265C8—2142no—
266C22—22no—
267C22—22no—
268C22—22no—
269C22—22no—
270C22—44no—
271C22—42no—
272C22—41no—
273C22—4288no—
274C22—4284no—
275C22—2144no—
276C22#1681207712yes—
277C22C22×C22⋊D10711yescharacteristic
278C4—22no—
279C4—42no—
280C4—41no—
281C4—2144no—
282C4#1681207712yes—
283C4C2×D4×D10711yescharacteristic
284C4C2×D4×D10711yescharacteristic
285C2—22no—
286C2—42no—
287C2—81no—
288C2—4284no—
289C2C2×#1681207711yescharacteristic
290C2#2029342512yes—
subgroupG/H
centerC22C22×C22⋊D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#525×C107C2
socleC22×C107C22×D4
hypercenter#525D107
hypocenterC107#3005
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C23×C4 ◁ #525

Complements

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

NH ≅ G/Ntype
1. C22×#13133818C2semidirect
2. C2×#16812139C2semidirect
3. C2×#16812134C2semidirect
4. #20292787C2semidirect
5. #20292770C2semidirect
6. #20292635C2semidirect
7. #20292631C2semidirect
8. #20292591C2semidirect
9. #525×C107C2semidirect
20. C2×#13133808C22semidirect
21. C2×#13133806C22semidirect
22. C2×#13133805C22semidirect
23. C2×#13133803C22semidirect
26. #16812032C22semidirect
27. #16812029C22semidirect
28. C22⋊(C107⋊C8)C22semidirect
32. C22⋊Q8×C107C22semidirect
33. #121×C107C22semidirect
43. C2×Dic214D4semidirect
47. C2×D428D4semidirect
146. C107#3005semidirect
150. #525D107semidirect
175. C22⋊Q8C2×D107semidirect
176. #121C2×D107semidirect

Maximal subgroups

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

subgroupindex
C22×#131338182
C2×#168121392
C2×#168121342
#202927872
#202927702
#202926352
#202926312
#202925912
#525×C1072
#3005107