Finite Group Databasebeta

← #469×D107 #28346053

Subgroups — #469×D107 #28346053

Order 13696.

596 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 596 conjugacy classes of subgroups — too many to draw. Only its 135 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×D8×D107C211yesmaximal, characteristic, complemented
2C2×M16×D107C211yesmaximal, characteristic, complemented
3#121×D107C212yesmaximal, complemented
4#20292881C211yesmaximal, characteristic, complemented
5#20292880C211yesmaximal, characteristic, complemented
6#20292879C212yesmaximal, complemented
7#20292878C212yesmaximal, complemented
8#20292877C211yesmaximal, characteristic, complemented
9#106×D107C211yesmaximal, characteristic, complemented
10#102×D107C212yesmaximal, complemented
11#469×C107C211yesmaximal, characteristic, complemented
12C2×D4×D107—22no—
13C2×D4×D107C2212yes—
14C22×C4×D107C2211yescharacteristic
15C2×D856C2211yescharacteristic, complemented
16C2×#13133803C2212yescomplemented
17D8×D107—24no—
18C2×#13133802C2211yescharacteristic, complemented
19C2×#13133801C2211yescharacteristic, complemented
20C2×#13133801C2211yescharacteristic, complemented
21M16×D107—24no—
22C2×C8×D107C2211yescharacteristic
23C2×C8×D107C2211yescharacteristic
24#16812084C2212yescomplemented
25#16812083C2212yescomplemented
26#16812082—22no—
27#16812081—22no—
28#16812080C2212yescomplemented
29C4⋊C4×D107C2212yes—
30C22⋊C4×D107—22no—
31#16812041C2211yescharacteristic, complemented
32#16812040C2212yescomplemented
33#16812031C2212yescomplemented
34#16812030C2212yescomplemented
35#16812029C2212yescomplemented
36C2×D8×C107C2211yescharacteristic, complemented
37C2×M16×C107C2211yescharacteristic, complemented
38#121×C107C2212yescomplemented
39#106×C107C2211yescharacteristic, complemented
40#102×C107C2212yescomplemented
41C23×D107—22no—
42C23×D107D411yescharacteristic
43C2×D428—22no—
44C2×D428C2312yes—
45C2×C22⋊D107—22no—
46C2×C22⋊D107—22no—
47C2×C22⋊D107—22no—
48D4×D107—24no—
49D4×D107—24no—
50D4×D107—48no—
51C22×Dic107D411yescharacteristic
52C2×C4×D107—22no—
53C2×C4×D107—22no—
54C2×C4×D107—22no—
55C2×C4×D107D411yescharacteristic
56C2×C4×D107D411yescharacteristic
57C2×C4×D107C2311yescharacteristic
58C2×C4×D107C2311yescharacteristic
59D856—22no—
60#13133803—24no—
61#13133802—22no—
62#13133801—22no—
63#13133801—22no—
64C2×C107⋊C8C2311yescharacteristic
65C2×C107⋊C8C2311yescharacteristic
66C8×D107—22no—
67C8×D107D414yescomplemented
68C4⋊Dic107C2312yes—
69Dic107⋊C4—22no—
70C22⋊Dic107—22no—
71(C2×D107)⋊C4—22no—
72(C2×D107)⋊C4—22no—
73C2×D4×C107—22no—
74C2×D4×C107C2312yes—
75C22×C4×C107C2311yescharacteristic
76D8×C107—22no—
77M16×C107—22no—
78C2×C8×C107C2311yescharacteristic
79C2×C8×C107C2311yescharacteristic
80C4⋊C4×C107C2312yes—
81C22⋊C4×C107—22no—
82C23×C107—22no—
83C23×C107C2×D411yescharacteristic
84C22×D107—22no—
85C22×D107—24no—
86C22×D107—22no—
87C22×D107—44no—
88C22×D107—44no—
89C22×D107—44no—
90C22×D107C2×D411yescharacteristic
91C22×D107C2×D411yescharacteristic
92D4×C107—22no—
93D4×C107—22no—
94D4×C107—44no—
95C2×C4×C107—21no—
96C2×C4×C107—21no—
97C2×C4×C107—22no—
98C2×C4×C107C2×D411yescharacteristic
99C2×C4×C107C2411yescharacteristic
100C22⋊D107—44no—
101C22⋊D107—44no—
102C22⋊D107—44no—
103C2×Dic107—22no—
104C2×Dic107—22no—
105C2×Dic107C2×D411yescharacteristic
106C2×Dic107C2×D411yescharacteristic
107D428—22no—
108D428—22no—
109D428—44no—
110C4×D107—22no—
111C4×D107—21no—
112C4×D107—21no—
113C4×D107—22no—
114C4×D107—44no—
115C4×D107C2×D412yes—
116C4×D107#4112yes—
117C8×C107—21no—
118C8×C107C2×D412yescomplemented
119C107⋊C8—21no—
120C107⋊C8C2×D412yescomplemented
121C22×C107—21no—
122C22×C107—22no—
123C22×C107—42no—
124C22×C107—42no—
125C22×C107—42no—
126C22×C107C22×D411yescharacteristic
127C2×D107—21no—
128C2×D107—22no—
129C2×D107—44no—
130C2×D107—42no—
131C2×D107—42no—
132C2×D107—42no—
133C2×D107—88no—
134C2×D107#12112yes—
135C2×D107AGL(1,ℤ/8)14yes—
136C4×C107—21no—
137C4×C107—42no—
138C4×C107C22×D411yescharacteristic
139C4×C107C2×#4111yescharacteristic
140Dic107—21no—
141Dic107—42no—
142Dic107#12112yes—
143C2×C107—41no—
144C2×C107—82no—
145C2×C107C2×#12111yescharacteristic
146C2×C107C2×AGL(1,ℤ/8)12yes—
147D107—41no—
148D107—82no—
149D107#46914yesdirect product factor
150C2×#469—1071nomaximal, Sylow
151C107C2×#46911yescharacteristic, complemented, Sylow
152C22×D8—1071no—
153C22×M16—1071no—
154C2×#121—1072no—
155#469—1071no—
156#469—1071no—
157#469—1072no—
158#469—1071no—
159#469—1072no—
160#469D10711yescharacteristic, direct product factor
161C2×#106—1071no—
162C2×#102—1072no—
163C22×D4—1072no—
164C22×D4—2142no—
165C23×C4—1071no—
166C2×D8—1072no—
167C2×D8—1071no—
168C2×D8—2144no—
169C2×D8C2×D10711yescharacteristic, complemented
170C2×M16—1071no—
171C2×M16—1071no—
172C2×M16—1071no—
173C2×M16—2144no—
174C2×M16C2×D10711yescharacteristic, complemented
175C22×C8—1071no—
176C22×C8—1071no—
177#121—1072no—
178#121—1072no—
179#121—1072no—
180#121—2142no—
181#121—2142no—
182#121C2×D10712yescomplemented
183C2×C4⋊C4—1072no—
184C2×C22⋊C4—2142no—
185#106—1071no—
186#106—1072no—
187#106C2×D10711yescharacteristic, complemented
188#102—1072no—
189#102—1072no—
190#102—1072no—
191#102C2×D10712yescomplemented
192C24—1071no—
193C24—2142no—
194C2×D4—22no—
195C2×D4—1072no—
196C2×D4—4288no—
197C2×D4—2142no—
198C2×D4—2144no—
199C2×D4—2144no—
200C2×D4—2142no—
201C2×D4—2142no—
202C2×D4—2142no—
203C2×D4C22×D10712yes—
204C22×C4—1071no—
205C22×C4—1071no—
206C22×C4—1071no—
207C22×C4—1071no—
208C22×C4—1071no—
209C22×C4—2142no—
210C22×C4—2142no—
211C22×C4—2142no—
212C22×C4C22×D10711yescharacteristic
213D8—22no—
214D8—2144no—
215D8—2142no—
216M16—22no—
217M16—2142no—
218M16—2142no—
219M16—2142no—
220C2×C8—1071no—
221C2×C8—1074no—
222C2×C8—1071no—
223C2×C8—2142no—
224C2×C8C22×D10711yescharacteristic
225C2×C8C22×D10711yescharacteristic
226C4⋊C4—1072no—
227C4⋊C4—2142no—
228C4⋊C4C22×D10712yes—
229C22⋊C4—22no—
230C22⋊C4—2142no—
231C22⋊C4—2142no—
232C22⋊C4—2142no—
233C23—22no—
234C23—1071no—
235C23—1071no—
236C23—4284no—
237C23—4284no—
238C23—4284no—
239C23—2142no—
240C23—2144no—
241C23—2142no—
242C23D4×D10711yescharacteristic
243D4—22no—
244D4—22no—
245D4—44no—
246D4—4284no—
247D4—4284no—
248D4—4284no—
249D4—4284no—
250D4—2142no—
251D4—2142no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—22no—
255C2×C4—1071no—
256C2×C4—1072no—
257C2×C4—1072no—
258C2×C4—1071no—
259C2×C4—4284no—
260C2×C4—2142no—
261C2×C4—2142no—
262C2×C4—2142no—
263C2×C4—2142no—
264C2×C4—2141no—
265C2×C4—2141no—
266C2×C4D4×D10711yescharacteristic
267C2×C4C23×D10711yescharacteristic
268C8—21no—
269C8—1072no—
270C8—2141no—
271C8D4×D10712yescomplemented
272C22—22no—
273C22—21no—
274C22—42no—
275C22—42no—
276C22—42no—
277C22—8568no—
278C22—1074no—
279C22—1072no—
280C22—4284no—
281C22—4282no—
282C22—4282no—
283C22—4282no—
284C22—2142no—
285C22—2141no—
286C22C2×D4×D10711yescharacteristic
287C4—21no—
288C4—42no—
289C4—1072no—
290C4—4282no—
291C4—2141no—
292C4C2×D4×D10711yescharacteristic
293C4#41×D10711yescharacteristic
294C2—41no—
295C2—82no—
296C2—8562no—
297C2—1074no—
298C2—4281no—
299C2#121×D10711yescharacteristic
300C2AGL(1,ℤ/8)×D10712yes—
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#469×C107C2
socleC22×C107C22×D4
hypercenter#469D107
hypocenterC107C2×#469
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #469

Complements

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

NH ≅ G/Ntype
1. C2×D8×D107C2semidirect
2. C2×M16×D107C2semidirect
3. #121×D107C2semidirect
4. #20292881C2semidirect
5. #20292880C2semidirect
6. #20292879C2semidirect
7. #20292878C2semidirect
8. #20292877C2semidirect
9. #106×D107C2semidirect
10. #102×D107C2semidirect
11. #469×C107C2semidirect
15. C2×D856C22semidirect
16. C2×#13133803C22semidirect
18. C2×#13133802C22semidirect
19. C2×#13133801C22semidirect
20. C2×#13133801C22semidirect
24. #16812084C22semidirect
25. #16812083C22semidirect
28. #16812080C22semidirect
31. #16812041C22semidirect
32. #16812040C22semidirect
33. #16812031C22semidirect
34. #16812030C22semidirect
35. #16812029C22semidirect
36. C2×D8×C107C22semidirect
37. C2×M16×C107C22semidirect
38. #121×C107C22semidirect
39. #106×C107C22semidirect
40. #102×C107C22semidirect
67. C8×D107D4semidirect
118. C8×C107C2×D4semidirect
120. C107⋊C8C2×D4semidirect
149. D107#469direct (N × H)
151. C107C2×#469semidirect
160. #469D107direct (N × H)
169. C2×D8C2×D107semidirect
174. C2×M16C2×D107semidirect
182. #121C2×D107semidirect
187. #106C2×D107semidirect
191. #102C2×D107semidirect
271. C8D4×D107semidirect

Maximal subgroups

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

subgroupindex
C2×D8×D1072
C2×M16×D1072
#121×D1072
#202928812
#202928802
#202928792
#202928782
#202928772
#106×D1072
#102×D1072
#469×C1072
C2×#469107