Finite Group Databasebeta

← #28345748

Subgroups — #28345748

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×#16812157C211yesmaximal, characteristic, complemented
2C2×#16812137C211yesmaximal, characteristic, complemented
3C22×#13133805C211yesmaximal, characteristic, complemented
4#20293196C211yesmaximal, characteristic, complemented
5#20292775C214yesmaximal, complemented
6#20292767C214yesmaximal, complemented
7#20292621C211yesmaximal, characteristic, complemented
8#20292595C211yesmaximal, characteristic, complemented
9#419×C107C211yesmaximal, characteristic, complemented
10#16812157—24no—
11#16812157—24no—
12C2×#13133819—21no—
13C2×#13133819C2212yescomplemented
14C2×#13133818—22no—
15C2×#13133818C2211yescharacteristic
16C2×#13133818C2211yescharacteristic
17C2×Q8×D107—21no—
18C2×Q8×D107C2212yescomplemented
19C22×D428C2211yescharacteristic
20#16812137—24no—
21C2×#13133807C2212yes—
22C2×#13133805—24no—
23C2×#13133805C2214yes—
24C2×D856C2212yes—
25C22×C107⋊C8C2211yescharacteristic
26#16812106—21no—
27#16812097—21no—
28#16812084C2214yes—
29#16812079—22no—
30#16812032C2212yescomplemented
31#16812029C2212yescomplemented
32(C2×D107)⋊C8C2214yescomplemented
33C22×Q8×C107C2211yescharacteristic
34C2×M16×C107C2211yescharacteristic
35#117×C107C2211yescharacteristic
36#103×C107C2214yescomplemented
37C23×D107—21no—
38#13133819—28no—
39#13133819—24no—
40#13133819—24no—
41#13133818—216no—
42#13133818—22no—
43#13133818—22no—
44#13133818—22no—
45#13133818D414yescomplemented
46C2×Dic214—22no—
47C2×Dic214D411yescharacteristic
48C2×Dic214C2311yescharacteristic
49Q8×D107—28no—
50Q8×D107—24no—
51Q8×D107—24no—
52C2×D428—22no—
53C2×D428—22no—
54C2×D428—22no—
55C2×D428—21no—
56C2×D428—24no—
57C2×D428D411yescharacteristic
58C2×D428C2312yes—
59C2×D428C2311yescharacteristic
60C2×C22⋊D107—21no—
61C2×C22⋊D107D412yes—
62C2×C4×D107—22no—
63C2×C4×D107—21no—
64C2×C4×D107—22no—
65C2×C4×D107C2312yes—
66#13133807—24no—
67#13133805—28no—
68D856—24no—
69C2×C107⋊C8—22no—
70C2×C107⋊C8C2312yes—
71(C2×D107)⋊C4—24no—
72C2×Q8×C107—21no—
73C2×Q8×C107—22no—
74C2×Q8×C107D414yescomplemented
75C2×Q8×C107C2312yes—
76C22×C4×C107—21no—
77C22×C4×C107C2311yescharacteristic
78M16×C107—22no—
79C2×C8×C107C2312yes—
80C4⋊C4×C107C2312yes—
81C22⋊C4×C107—21no—
82C42×C107—21no—
83C23×C107C2×D411yescharacteristic
84C22×D107—21no—
85C22×D107—22no—
86C22×D107—42no—
87C22×D107—82no—
88C22×D107C2×D412yes—
89Q8×C107—24no—
90Q8×C107—22no—
91Q8×C107C2×D414yes—
92C2×C4×C107—24no—
93C2×C4×C107—22no—
94C2×C4×C107—22no—
95C2×C4×C107C2×D411yescharacteristic
96C2×C4×C107C2×D412yes—
97C2×C4×C107C2×D412yes—
98C2×C4×C107C2411yescharacteristic
99C22⋊D107—24no—
100C22⋊D107—24no—
101C2×Dic107—21no—
102C2×Dic107C2×D412yes—
103D428—28no—
104D428—22no—
105D428—21no—
106D428—22no—
107D428—22no—
108D428—42no—
109D428—84no—
110D428C2×D412yes—
111C4×D107—28no—
112C4×D107—28no—
113C4×D107—24no—
114C4×D107—22no—
115C4×D107—22no—
116Dic214—28no—
117Dic214—22no—
118Dic214—21no—
119Dic214C2×D412yes—
120C8×C107—22no—
121C107⋊C8—22no—
122C22×C107—22no—
123C22×C107#12012yes—
124C22×C107C22×D411yescharacteristic
125C2×D107—22no—
126C2×D107—22no—
127C2×D107—24no—
128C2×D107—42no—
129C2×D107—42no—
130C2×D107—44no—
131C2×D107—84no—
132C4×C107—24no—
133C4×C107—42no—
134C4×C107#12012yes—
135C4×C107C22×D411yescharacteristic
136C4×C107C22×D411yescharacteristic
137Dic107—22no—
138Dic107—22no—
139C2×C107—22no—
140C2×C107C2×#12011yescharacteristic
141C2×C107#57712yes—
142D107—42no—
143D107—42no—
144D107—82no—
145#3010—1071nomaximal, Sylow
146C107#301011yescharacteristic, complemented, Sylow
147C2×2−1+4—1071no—
148C2×AGL(1,ℤ/8)—1071no—
149C22×SD16—1071no—
150#535—1071no—
151#449—1074no—
152#448—1074no—
153#419D10711yescharacteristic, complemented
154#416—1071no—
155#408—1071no—
1562−1+4—2144no—
1572−1+4—2144no—
158C2×#41—1071no—
159C2×#41—1072no—
160C2×#41—1071no—
161C2×#41—2142no—
162C2×#41—2141no—
163C22×Q8—1072no—
164C22×Q8—2141no—
165C22×Q8C2×D10711yescharacteristic
166C22×D4—1071no—
167AGL(1,ℤ/8)—2144no—
168C2×SD16—1074no—
169C2×SD16—1072no—
170C2×SD16—2144no—
171C2×D8—1072no—
172C2×M16C2×D10711yescharacteristic
173C22×C8—1071no—
174#127—2141no—
175#124—2141no—
176#121—1074no—
177#120—2142no—
178#117C2×D10711yescharacteristic
179#103—1072no—
180#103C2×D10714yescomplemented
181#102—1072no—
182C22⋊C8—1074no—
183C24—2141no—
184#41—1074no—
185#41—21416no—
186#41—2144no—
187#41—2148no—
188#41—2144no—
189#41—2142no—
190#41—2142no—
191#41—2142no—
192C2×Q8—21no—
193C2×Q8—22no—
194C2×Q8—1071no—
195C2×Q8—1071no—
196C2×Q8—2142no—
197C2×Q8—2144no—
198C2×Q8—2148no—
199C2×Q8—2144no—
200C2×Q8D42814yescomplemented
201C2×Q8C22×D10712yes—
202C2×D4—1072no—
203C2×D4—1071no—
204C2×D4—1071no—
205C2×D4—1072no—
206C2×D4—2144no—
207C2×D4—2142no—
208C2×D4—2141no—
209C2×D4—2141no—
210C2×D4—2142no—
211C2×D4—2142no—
212C22×C4—21no—
213C22×C4—1072no—
214C22×C4—2142no—
215C22×C4—2142no—
216C22×C4—2141no—
217C22×C4C22×D10711yescharacteristic
218SD16—2144no—
219SD16—2148no—
220D8—2144no—
221M16—22no—
222C2×C8—1072no—
223C2×C8—2142no—
224C2×C8C22×D10712yes—
225C4⋊C4C22×D10712yes—
226C22⋊C4—21no—
227C22⋊C4—2144no—
228C42—21no—
229C23—8562no—
230C23—1072no—
231C23—4282no—
232C23—2142no—
233C23—2141no—
234C23D4×D10711yescharacteristic
235Q8—24no—
236Q8—22no—
237Q8—1072no—
238Q8—2148no—
239Q8—2142no—
240Q8—2141no—
241Q8C2×D42814yes—
242D4—8564no—
243D4—1072no—
244D4—4282no—
245D4—2148no—
246D4—2144no—
247D4—2144no—
248D4—2142no—
249D4—2142no—
250D4—2142no—
251D4—2141no—
252C2×C4—24no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—1072no—
256C2×C4—2148no—
257C2×C4—2148no—
258C2×C4—2141no—
259C2×C4—2144no—
260C2×C4—2142no—
261C2×C4—2142no—
262C2×C4D4×D10711yescharacteristic
263C2×C4D4×D10712yes—
264C2×C4C2×D42812yes—
265C2×C4C23×D10711yescharacteristic
266C8—22no—
267C8—2142no—
268C22—22no—
269C22—8564no—
270C22—4284no—
271C22—4282no—
272C22—4282no—
273C22—2144no—
274C22—2142no—
275C22—2142no—
276C22#1681207912yes—
277C22C2×D4×D10711yescharacteristic
278C4—24no—
279C4—42no—
280C4—2142no—
281C4—2142no—
282C4#1681207912yes—
283C4C2×D4×D10711yescharacteristic
284C4C22×D42811yescharacteristic
285C2—22no—
286C2—8562no—
287C2—4282no—
288C2—4282no—
289C2C2×#1681207911yescharacteristic
290C2#2029343612yes—
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#419×C107C2
socleC22×C107C22×D4
hypercenter#419D107
hypocenterC107#3010
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #419

Complements

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

NH ≅ G/Ntype
1. C2×#16812157C2semidirect
2. C2×#16812137C2semidirect
3. C22×#13133805C2semidirect
4. #20293196C2semidirect
5. #20292775C2semidirect
6. #20292767C2semidirect
7. #20292621C2semidirect
8. #20292595C2semidirect
9. #419×C107C2semidirect
13. C2×#13133819C22semidirect
18. C2×Q8×D107C22semidirect
30. #16812032C22semidirect
31. #16812029C22semidirect
32. (C2×D107)⋊C8C22semidirect
36. #103×C107C22semidirect
45. #13133818D4semidirect
74. C2×Q8×C107D4semidirect
146. C107#3010semidirect
153. #419D107semidirect
180. #103C2×D107semidirect
200. C2×Q8D428semidirect

Maximal subgroups

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

subgroupindex
C2×#168121572
C2×#168121372
C22×#131338052
#202931962
#202927752
#202927672
#202926212
#202925952
#419×C1072
#3010107