Finite Group Databasebeta

← #28345747

Subgroups — #28345747

Order 13696.

732 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 732 conjugacy classes of subgroups — too many to draw. Only its 165 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×#16812134C211yesmaximal, characteristic, complemented
2C22×#13133805C211yesmaximal, characteristic, complemented
3#20293191C211yesmaximal, characteristic, complemented
4#20292772C214yesmaximal, complemented
5#20292766C214yesmaximal, complemented
6#20292636C211yesmaximal, characteristic, complemented
7#20292623C211yesmaximal, characteristic, complemented
8#20292594C211yesmaximal, characteristic, complemented
9C2×2−1+4×C107C211yesmaximal, characteristic, complemented
10C22×D428C2211yescharacteristic
11#16812134—24no—
12C2×#13133805—24no—
13C2×#13133805C2214yes—
14C2×#13133805C2212yes—
15C2×#13133803C2212yes—
16C2×#13133802C2211yescharacteristic
17C22×C107⋊C8C2211yescharacteristic
18#16812105—21no—
19#16812098—21no—
20#16812083C2214yes—
21#16812077—22no—
22#16812068C2211yescharacteristic
23#16812033C2212yescomplemented
24#16812033C2214yescomplemented
25#16812030C2212yescomplemented
26C22⋊(C107⋊C8)C2214yescomplemented
272−1+4×C107—24no—
282−1+4×C107—24no—
29C2×#41×C107—22no—
30C2×#41×C107—21no—
31C2×#41×C107C2211yescharacteristic
32C2×#41×C107C2211yescharacteristic
33C2×#41×C107C2212yescomplemented
34C22×Q8×C107—21no—
35C22×Q8×C107C2211yescharacteristic
36C22×Q8×C107C2212yescomplemented
37C23×D107—21no—
38C2×D428—22no—
39C2×D428—22no—
40C2×D428—21no—
41C2×D428C2312yes—
42C2×C22⋊D107—24no—
43#13133805—28no—
44#13133805—24no—
45#13133803—24no—
46#13133802—22no—
47C2×C107⋊C8—22no—
48C2×C107⋊C8C2312yes—
49C2×C107⋊C8C2312yes—
50C4⋊Dic107C2312yes—
51C22⋊Dic107—21no—
52(C2×D107)⋊C4—24no—
53C4×Dic107—21no—
54#41×C107—216no—
55#41×C107—28no—
56#41×C107—24no—
57#41×C107—24no—
58#41×C107—22no—
59#41×C107—22no—
60#41×C107—22no—
61#41×C107D414yescomplemented
62C2×Q8×C107—28no—
63C2×Q8×C107—24no—
64C2×Q8×C107—24no—
65C2×Q8×C107—21no—
66C2×Q8×C107—22no—
67C2×Q8×C107—22no—
68C2×Q8×C107D411yescharacteristic
69C2×Q8×C107D414yescomplemented
70C2×Q8×C107C2312yes—
71C2×Q8×C107C2311yescharacteristic
72C2×D4×C107—21no—
73C2×D4×C107—22no—
74C2×D4×C107D412yes—
75C2×D4×C107D411yescharacteristic
76C2×D4×C107C2311yescharacteristic
77C22×C4×C107—21no—
78C22×C4×C107—22no—
79C22×C4×C107—21no—
80C22×C4×C107—22no—
81C22×C4×C107C2312yes—
82C22×C4×C107C2311yescharacteristic
83C23×C107—21no—
84C23×C107C2×D412yes—
85C23×C107C2×D411yescharacteristic
86C22×D107—22no—
87C22×D107—42no—
88C22×D107—82no—
89Q8×C107—28no—
90Q8×C107—22no—
91Q8×C107—24no—
92Q8×C107—21no—
93Q8×C107—22no—
94Q8×C107C2×D412yes—
95Q8×C107C2×D414yes—
96D4×C107—28no—
97D4×C107—24no—
98D4×C107—24no—
99D4×C107—22no—
100D4×C107—21no—
101D4×C107C2×D412yes—
102C2×C4×C107—28no—
103C2×C4×C107—28no—
104C2×C4×C107—24no—
105C2×C4×C107—22no—
106C2×C4×C107—21no—
107C2×C4×C107—24no—
108C2×C4×C107—22no—
109C2×C4×C107—22no—
110C2×C4×C107C2×D412yes—
111C2×C4×C107C2×D411yescharacteristic
112C2×C4×C107C2×D412yes—
113C2×C4×C107C2×D412yes—
114C2×C4×C107C2411yescharacteristic
115C22⋊D107—84no—
116C2×Dic107—22no—
117D428—22no—
118D428—22no—
119D428—42no—
120C107⋊C8—22no—
121C107⋊C8—22no—
122C22×C107—22no—
123C22×C107—22no—
124C22×C107—22no—
125C22×C107—24no—
126C22×C107—42no—
127C22×C107#12012yes—
128C22×C107C22×D411yescharacteristic
129C2×D107—42no—
130C2×D107—44no—
131C2×D107—84no—
132C4×C107—24no—
133C4×C107—22no—
134C4×C107—22no—
135C4×C107#12012yes—
136C4×C107C22×D411yescharacteristic
137C4×C107C22×D411yescharacteristic
138Dic107—42no—
139C2×C107—22no—
140C2×C107—42no—
141C2×C107—42no—
142C2×C107C2×#12011yescharacteristic
143C2×C107#57712yes—
144D107—82no—
145#3010—1071nomaximal, Sylow
146C107#301011yescharacteristic, complemented, Sylow
147C2×2−1+4D10711yescharacteristic, complemented
148C2×AGL(1,ℤ/8)—1071no—
149C22×SD16—1071no—
150#535—1071no—
151#449—1074no—
152#448—1074no—
153#419—1071no—
154#416—1071no—
155#408—1071no—
1562−1+4—24no—
1572−1+4—24no—
158C2×#41—22no—
159C2×#41—21no—
160C2×#41C2×D10711yescharacteristic
161C2×#41C2×D10711yescharacteristic
162C2×#41C2×D10712yescomplemented
163C22×Q8—21no—
164C22×Q8C2×D10711yescharacteristic
165C22×Q8C2×D10712yescomplemented
166C22×D4—1071no—
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—2141no—
175#124—2141no—
176#121—1074no—
177#120—2142no—
178#117—1071no—
179#103—1072no—
180#103—1074no—
181#102—1072no—
182C22⋊C8—1074no—
183C24—2141no—
184#41—216no—
185#41—28no—
186#41—24no—
187#41—24no—
188#41—22no—
189#41—22no—
190#41—22no—
191#41C22⋊D10714yescomplemented
192C2×Q8—28no—
193C2×Q8—24no—
194C2×Q8—24no—
195C2×Q8—21no—
196C2×Q8—22no—
197C2×Q8—22no—
198C2×Q8C22⋊D10711yescharacteristic
199C2×Q8C22⋊D10714yescomplemented
200C2×Q8C22×D10712yes—
201C2×Q8C22×D10711yescharacteristic
202C2×D4—21no—
203C2×D4—22no—
204C2×D4—1072no—
205C2×D4—2144no—
206C2×D4—2141no—
207C2×D4—2142no—
208C2×D4—2142no—
209C2×D4C22⋊D10712yes—
210C2×D4C22⋊D10711yescharacteristic
211C2×D4C22×D10711yescharacteristic
212C22×C4—21no—
213C22×C4—22no—
214C22×C4—21no—
215C22×C4—22no—
216C22×C4C22×D10712yes—
217C22×C4C22×D10711yescharacteristic
218SD16—2148no—
219SD16—2144no—
220D8—2144no—
221M16—2142no—
222C2×C8—1072no—
223C2×C8—1072no—
224C2×C8—2142no—
225C4⋊C4—1072no—
226C22⋊C4—2141no—
227C22⋊C4—2144no—
228C42—2141no—
229C23—21no—
230C23—8562no—
231C23—4282no—
232C23—2142no—
233C23C2×C22⋊D10712yes—
234C23C2×C22⋊D10711yescharacteristic
235Q8—28no—
236Q8—22no—
237Q8—24no—
238Q8—21no—
239Q8—22no—
240Q8C2×C22⋊D10712yes—
241Q8C2×C22⋊D10714yes—
242D4—28no—
243D4—24no—
244D4—24no—
245D4—22no—
246D4—21no—
247D4—8564no—
248D4—4282no—
249D4—2142no—
250D4—2142no—
251D4C2×C22⋊D10712yes—
252C2×C4—28no—
253C2×C4—28no—
254C2×C4—24no—
255C2×C4—22no—
256C2×C4—21no—
257C2×C4—24no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—2142no—
261C2×C4C2×C22⋊D10712yes—
262C2×C4C2×C22⋊D10711yescharacteristic
263C2×C4C2×C22⋊D10712yes—
264C2×C4C2×C22⋊D10712yes—
265C2×C4C23×D10711yescharacteristic
266C8—2142no—
267C8—2142no—
268C22—22no—
269C22—24no—
270C22—22no—
271C22—22no—
272C22—42no—
273C22—8564no—
274C22—4284no—
275C22—4282no—
276C22#1681207812yes—
277C22C22×C22⋊D10711yescharacteristic
278C4—24no—
279C4—22no—
280C4—22no—
281C4—4282no—
282C4#1681207812yes—
283C4C22×C22⋊D10711yescharacteristic
284C4C22×C22⋊D10711yescharacteristic
285C2—22no—
286C2—42no—
287C2—42no—
288C2—8562no—
289C2C2×#1681207811yescharacteristic
290C2#2029343512yes—
subgroupG/H
centerC22C22×C22⋊D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
FittingC2×2−1+4×C107C2
socleC22×C107C22×D4
hypercenterC2×2−1+4D107
hypocenterC107#3010
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C2×#41 ◁ C2×2−1+4

Complements

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

NH ≅ G/Ntype
1. C2×#16812134C2semidirect
2. C22×#13133805C2semidirect
3. #20293191C2semidirect
4. #20292772C2semidirect
5. #20292766C2semidirect
6. #20292636C2semidirect
7. #20292623C2semidirect
8. #20292594C2semidirect
9. C2×2−1+4×C107C2semidirect
23. #16812033C22semidirect
24. #16812033C22semidirect
25. #16812030C22semidirect
26. C22⋊(C107⋊C8)C22semidirect
33. C2×#41×C107C22semidirect
36. C22×Q8×C107C22semidirect
61. #41×C107D4semidirect
69. C2×Q8×C107D4semidirect
146. C107#3010semidirect
147. C2×2−1+4D107semidirect
162. C2×#41C2×D107semidirect
165. C22×Q8C2×D107semidirect
191. #41C22⋊D107semidirect
199. C2×Q8C22⋊D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168121342
C22×#131338052
#202931912
#202927722
#202927662
#202926362
#202926232
#202925942
C2×2−1+4×C1072
#3010107