Finite Group Databasebeta

← #28345745

Subgroups — #28345745

Order 13696.

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

normal subgroups only

This group has 732 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×#16812156C211yesmaximal, characteristic, complemented
2C2×#16812135C211yesmaximal, characteristic, complemented
3C22×#13133806C211yesmaximal, characteristic, complemented
4#20292770C214yesmaximal, complemented
5#20292765C214yesmaximal, complemented
6#20292635C211yesmaximal, characteristic, complemented
7#20292621C211yesmaximal, characteristic, complemented
8#20292594C211yesmaximal, characteristic, complemented
9#535×C107C211yesmaximal, characteristic, complemented
10#16812156—24no—
11#16812156—24no—
12C2×#13133818C2211yescharacteristic
13C2×#13133818C2212yescomplemented
14C2×#13133817—22no—
15C2×#13133817—21no—
16C22×Dic214C2211yescharacteristic
17C22×Dic214C2212yescomplemented
18C2×Q8×D107—21no—
19#16812135—24no—
20C2×#13133806—24no—
21C2×#13133806C2214yes—
22C2×#13133806C2212yes—
23C2×#13133803C2212yes—
24C2×#13133802C2211yescharacteristic
25C22×C107⋊C8C2211yescharacteristic
26#16812032C2212yescomplemented
27#16812032C2214yescomplemented
28#16812029C2212yescomplemented
29C22⋊(C107⋊C8)C2214yescomplemented
30C2×#41×C107C2211yescharacteristic
31C22×D4×C107C2211yescharacteristic
32#127×C107—21no—
33#124×C107—21no—
34#121×C107C2214yes—
35#120×C107—22no—
36#117×C107C2211yescharacteristic
37#13133818—22no—
38#13133818—24no—
39#13133818—24no—
40#13133818D414yescomplemented
41#13133817—216no—
42#13133817—28no—
43C2×Dic214—21no—
44C2×Dic214—22no—
45C2×Dic214—22no—
46C2×Dic214—24no—
47C2×Dic214—24no—
48C2×Dic214D414yescomplemented
49C2×Dic214C2312yes—
50C2×Dic214C2311yescharacteristic
51Q8×D107—28no—
52C2×D428C2311yescharacteristic
53C2×C22⋊D107—21no—
54C2×C22⋊D107—22no—
55C22×Dic107—21no—
56C22×Dic107—22no—
57C2×C4×D107—21no—
58C2×C4×D107—22no—
59#13133806—28no—
60#13133806—24no—
61#13133803—24no—
62#13133802—22no—
63C2×C107⋊C8—22no—
64C2×C107⋊C8C2312yes—
65C2×C107⋊C8C2312yes—
66C24×C107—21no—
67#41×C107—22no—
68#41×C107—22no—
69C2×Q8×C107D411yescharacteristic
70C2×D4×C107—24no—
71C2×D4×C107—22no—
72C2×D4×C107—22no—
73C2×D4×C107—21no—
74C2×D4×C107D412yes—
75C2×D4×C107D411yescharacteristic
76C2×D4×C107C2312yes—
77C22×C4×C107C2312yes—
78C22×C4×C107C2311yescharacteristic
79C4⋊C4×C107C2312yes—
80C22⋊C4×C107—21no—
81C22⋊C4×C107—24no—
82C42×C107—21no—
83C23×C107—22no—
84C23×C107—42no—
85C23×C107—82no—
86C23×C107C2×D412yes—
87C23×C107C2×D411yescharacteristic
88C22×D107—21no—
89Q8×C107—22no—
90D4×C107—24no—
91D4×C107—22no—
92D4×C107—22no—
93D4×C107—22no—
94D4×C107—42no—
95D4×C107—84no—
96C2×C4×C107—22no—
97C2×C4×C107—22no—
98C2×C4×C107—22no—
99C2×C4×C107C2×D412yes—
100C2×C4×C107C2×D412yes—
101C2×C4×C107C2×D412yes—
102C2×C4×C107C2×D411yescharacteristic
103C2×C4×C107C2411yescharacteristic
104C22⋊D107—28no—
105C22⋊D107—24no—
106C2×Dic107—28no—
107C2×Dic107—24no—
108C2×Dic107—21no—
109C2×Dic107—22no—
110D428—21no—
111D428C2×D412yes—
112C4×D107—28no—
113C4×D107—24no—
114Dic214—28no—
115Dic214—24no—
116Dic214—22no—
117Dic214—21no—
118Dic214C2×D412yes—
119Dic214C2×D414yes—
120C107⋊C8—22no—
121C107⋊C8—22no—
122C22×C107—22no—
123C22×C107—22no—
124C22×C107—24no—
125C22×C107—42no—
126C22×C107—44no—
127C22×C107—84no—
128C22×C107#12012yes—
129C22×C107C22×D411yescharacteristic
130C2×D107—22no—
131C2×D107—42no—
132C4×C107—22no—
133C4×C107—42no—
134C4×C107#12012yes—
135C4×C107C22×D411yescharacteristic
136C4×C107C22×D411yescharacteristic
137Dic107—22no—
138Dic107—24no—
139C2×C107—22no—
140C2×C107—42no—
141C2×C107—82no—
142C2×C107C2×#12011yescharacteristic
143C2×C107#57712yes—
144D107—42no—
145#3010—1071nomaximal, Sylow
146C107#301011yescharacteristic, complemented, Sylow
147C2×2−1+4—1071no—
148C2×AGL(1,ℤ/8)—1071no—
149C22×SD16—1071no—
150#535D10711yescharacteristic, complemented
151#449—1074no—
152#448—1074no—
153#419—1071no—
154#416—1071no—
155#408—1071no—
1562−1+4—2144no—
1572−1+4—2144no—
158C2×#41—1071no—
159C2×#41—1072no—
160C2×#41—2142no—
161C2×#41—2141no—
162C2×#41C2×D10711yescharacteristic
163C22×Q8—1071no—
164C22×Q8—1072no—
165C22×Q8—2141no—
166C22×D4C2×D10711yescharacteristic
167AGL(1,ℤ/8)—2144no—
168C2×SD16—1074no—
169C2×SD16—1072no—
170C2×SD16—2144no—
171C2×D8—1072no—
172C2×M16—1071no—
173C22×C8—1071no—
174#127—21no—
175#124—21no—
176#121C2×D10714yes—
177#120—22no—
178#117C2×D10711yescharacteristic
179#103—1072no—
180#103—1074no—
181#102—1072no—
182C22⋊C8—1074no—
183C24—21no—
184#41—22no—
185#41—22no—
186#41—1074no—
187#41—2144no—
188#41—2144no—
189#41—21416no—
190#41—2148no—
191#41—2142no—
192C2×Q8—1074no—
193C2×Q8—1072no—
194C2×Q8—1071no—
195C2×Q8—2144no—
196C2×Q8—2144no—
197C2×Q8—2148no—
198C2×Q8—2142no—
199C2×Q8—2142no—
200C2×Q8—2141no—
201C2×Q8C22⋊D10711yescharacteristic
202C2×D4—24no—
203C2×D4—22no—
204C2×D4—22no—
205C2×D4—21no—
206C2×D4—1071no—
207C2×D4—2142no—
208C2×D4—2141no—
209C2×D4C22⋊D10712yes—
210C2×D4C22⋊D10711yescharacteristic
211C2×D4C22×D10712yes—
212C22×C4—2142no—
213C22×C4—2141no—
214C22×C4—2142no—
215C22×C4—2141no—
216C22×C4C22×D10712yes—
217C22×C4C22×D10711yescharacteristic
218SD16—2148no—
219SD16—2144no—
220D8—2144no—
221M16—2142no—
222C2×C8—1072no—
223C2×C8—1072no—
224C2×C8—2142no—
225C4⋊C4C22×D10712yes—
226C22⋊C4—21no—
227C22⋊C4—24no—
228C42—21no—
229C23—22no—
230C23—42no—
231C23—82no—
232C23—2141no—
233C23C2×C22⋊D10712yes—
234C23C2×C22⋊D10711yescharacteristic
235Q8—22no—
236Q8—1074no—
237Q8—1072no—
238Q8—2148no—
239Q8—2144no—
240Q8—2142no—
241Q8—2141no—
242D4—24no—
243D4—22no—
244D4—22no—
245D4—22no—
246D4—42no—
247D4—84no—
248D4—1072no—
249D4—2148no—
250D4—2144no—
251D4—2141no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—2148no—
256C2×C4—2148no—
257C2×C4—2144no—
258C2×C4—2142no—
259C2×C4—2141no—
260C2×C4—2144no—
261C2×C4D4×D10712yes—
262C2×C4D4×D10712yes—
263C2×C4C2×C22⋊D10712yes—
264C2×C4C2×C22⋊D10711yescharacteristic
265C2×C4C23×D10711yescharacteristic
266C8—2142no—
267C8—2142no—
268C22—22no—
269C22—24no—
270C22—22no—
271C22—44no—
272C22—42no—
273C22—84no—
274C22—4282no—
275C22—2142no—
276C22#1681207712yes—
277C22C22×C22⋊D10711yescharacteristic
278C4—22no—
279C4—42no—
280C4—2144no—
281C4—2142no—
282C4#1681207712yes—
283C4C2×D4×D10711yescharacteristic
284C4C2×D4×D10711yescharacteristic
285C2—22no—
286C2—42no—
287C2—82no—
288C2—4282no—
289C2C2×#1681207711yescharacteristic
290C2#2029343812yes—
subgroupG/H
centerC22C22×C22⋊D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#535×C107C2
socleC22×C107C22×D4
hypercenter#535D107
hypocenterC107#3010
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C2×#41 ◁ #535

Complements

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

NH ≅ G/Ntype
1. C2×#16812156C2semidirect
2. C2×#16812135C2semidirect
3. C22×#13133806C2semidirect
4. #20292770C2semidirect
5. #20292765C2semidirect
6. #20292635C2semidirect
7. #20292621C2semidirect
8. #20292594C2semidirect
9. #535×C107C2semidirect
13. C2×#13133818C22semidirect
17. C22×Dic214C22semidirect
26. #16812032C22semidirect
27. #16812032C22semidirect
28. #16812029C22semidirect
29. C22⋊(C107⋊C8)C22semidirect
40. #13133818D4semidirect
48. C2×Dic214D4semidirect
146. C107#3010semidirect
150. #535D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168121562
C2×#168121352
C22×#131338062
#202927702
#202927652
#202926352
#202926212
#202925942
#535×C1072
#3010107