Finite Group Databasebeta

← #28348083

Subgroups — #28348083

Order 13696.

556 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 556 conjugacy classes of subgroups — too many to draw. Only its 133 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×#16812137C212yesmaximal, complemented
2#20293298C211yesmaximal, characteristic, complemented
3D4×D428C211yesmaximal, characteristic, complemented
4#20293004C211yesmaximal, characteristic, complemented
5#20292880C212yesmaximal, complemented
6#20292828C211yesmaximal, characteristic, complemented
7#20292823C211yesmaximal, characteristic, complemented
8#20292774C212yesmaximal, complemented
9#20292763C212yesmaximal, complemented
10C4×#13133807C211yesmaximal, characteristic, complemented
11#436×C107C211yesmaximal, characteristic, complemented
12C2×#13133818C2212yescomplemented
13C22×D428C2212yes—
14C2×D4×D107—21no—
15#16812137—28no—
16C2×#13133807C2211yescharacteristic
17C2×#13133807C2212yescomplemented
18C2×D856C2212yes—
19C2×D856C2212yescomplemented
20C2×D856C2212yescomplemented
21#16812106C2211yescharacteristic
22#16812106C2211yescharacteristic, complemented
23#16812105—21no—
24#16812084—21no—
25#16812083C2212yes—
26#16812083C2212yescomplemented
27#16812082—22no—
28#16812079—22no—
29C4×Dic214C2211yescharacteristic, complemented
30C4×D428C2211yescharacteristic
31#16812071—21no—
32#16812041C2211yescharacteristic, complemented
33#16812034C2211yescharacteristic, complemented
34#16812031C2211yescharacteristic
35#16812031C2212yescomplemented
36#16812031C2212yescomplemented
37C2×M16×C107C2212yescomplemented
38C4×D4×C107C2211yescharacteristic
39C4⋊C8×C107C2211yescharacteristic, complemented
40C22⋊C8×C107C2212yescomplemented
41C4×C8×C107C2211yescharacteristic, complemented
42C23×D107—22no—
43#13133818—24no—
44#13133818—42no—
45C2×Dic214C2311yescharacteristic
46C2×D428—24no—
47C2×D428—21no—
48C2×D428—22no—
49C2×D428—21no—
50C2×D428—21no—
51C2×D428—21no—
52C2×D428—42no—
53C2×D428C2312yes—
54C2×D428C2312yes—
55C2×D428C2311yescharacteristic
56C2×C22⋊D107—22no—
57C2×C22⋊D107—22no—
58C2×C22⋊D107—22no—
59D4×D107—44no—
60C2×C4×D107—22no—
61C2×C4×D107—21no—
62#13133807—24no—
63#13133807D414yescomplemented
64D856—24no—
65D856—24no—
66D856—42no—
67C4⋊Dic107C2311yescharacteristic
68C4⋊Dic107C2311yescharacteristic
69Dic107⋊C4—21no—
70(C2×D107)⋊C4—22no—
71(C2×D107)⋊C4—21no—
72(C2×D107)⋊C4—22no—
73C4×Dic107—21no—
74C2×D4×C107D411yescharacteristic
75C22×C4×C107C2312yes—
76M16×C107—24no—
77C2×C8×C107C2311yescharacteristic
78C2×C8×C107C2312yes—
79C2×C8×C107C2311yescharacteristic
80C4⋊C4×C107D411yescharacteristic
81C22⋊C4×C107D412yes—
82C42×C107C2311yescharacteristic
83C23×C107C2×D412yes—
84C22×D107—21no—
85C22×D107—22no—
86C22×D107—22no—
87C22×D107—44no—
88C22×D107—44no—
89C22×D107—82no—
90D4×C107—22no—
91C2×C4×C107—22no—
92C2×C4×C107—22no—
93C2×C4×C107C2×D411yescharacteristic
94C2×C4×C107C2×D412yes—
95C2×C4×C107C2×D411yescharacteristic
96C2×C4×C107C2411yescharacteristic
97C22⋊D107—44no—
98C22⋊D107—44no—
99C22⋊D107—82no—
100C2×Dic107—21no—
101C2×Dic107—21no—
102C2×Dic107—21no—
103D428—22no—
104D428—22no—
105D428—21no—
106D428—22no—
107D428—22no—
108D428—44no—
109D428—42no—
110D428—42no—
111D428—81no—
112D428—81no—
113D428C2×D412yes—
114C4×D107—44no—
115C4×D107—42no—
116Dic214—21no—
117Dic214C2×D412yescomplemented
118C8×C107—22no—
119C8×C107—21no—
120C8×C107C2×D412yescomplemented
121C22×C107—22no—
122C22×C107—22no—
123C22×C107—22no—
124C22×C107C22×D411yescharacteristic
125C2×D107—22no—
126C2×D107—41no—
127C2×D107—41no—
128C2×D107—42no—
129C2×D107—42no—
130C2×D107—42no—
131C2×D107—42no—
132C2×D107—42no—
133C2×D107—42no—
134C2×D107—88no—
135C4×C107—22no—
136C4×C107—21no—
137C4×C107AGL(1,ℤ/8)12yescomplemented
138C4×C107C22×D411yescharacteristic
139C4×C107H(5,𝔽2)11yescharacteristic
140Dic107—22no—
141Dic107—41no—
142Dic107—41no—
143C2×C107—42no—
144C2×C107D4211yescharacteristic
145C2×C107C2×AGL(1,ℤ/8)11yescharacteristic
146C2×C107#57611yescharacteristic
147D107—42no—
148D107—82no—
149D107—82no—
150#3274—1071nomaximal, Sylow
151C107#327411yescharacteristic, complemented, Sylow
152C2×AGL(1,ℤ/8)—1072no—
153#550—1071no—
154D42—1071no—
155#493—1071no—
156#469—1072no—
157#460—1071no—
158#459—1071no—
159#449—1072no—
160#447—1072no—
161C4×SD16—1071no—
162#436D10711yescharacteristic, complemented
163C2×#41—1072no—
164C22×D4—1072no—
165C22×D4—2141no—
166AGL(1,ℤ/8)—2148no—
167C2×SD16—1071no—
168C2×SD16—1072no—
169C2×D8—1072no—
170C2×D8—1072no—
171C2×D8—1072no—
172C2×M16C2×D10712yescomplemented
173#127—1071no—
174#127—1071no—
175#127—2141no—
176#121—1072no—
177#121—1072no—
178#121—2141no—
179#121—2142no—
180#120—2142no—
181C4×Q8—1071no—
182C4×D4—1071no—
183C4×D4—2141no—
184C4×D4C2×D10711yescharacteristic
185#106—1071no—
186C4⋊C8C2×D10711yescharacteristic, complemented
187#103—1071no—
188#102—1071no—
189#102—1072no—
190#102—1072no—
191C22⋊C8C2×D10712yescomplemented
192C4×C8C2×D10711yescharacteristic, complemented
193C24—2142no—
194#41—4282no—
195#41—2144no—
196C2×Q8—1071no—
197C2×D4—1071no—
198C2×D4—1072no—
199C2×D4—1072no—
200C2×D4—4282no—
201C2×D4—4284no—
202C2×D4—2144no—
203C2×D4—2142no—
204C2×D4—2142no—
205C2×D4—2142no—
206C2×D4—2141no—
207C2×D4—2141no—
208C2×D4—2141no—
209C2×D4—2142no—
210C2×D4—2141no—
211C2×D4D42811yescharacteristic
212C22×C4—2141no—
213C22×C4—2142no—
214C22×C4C22×D10712yes—
215SD16—1074no—
216SD16—2144no—
217D8—4282no—
218D8—2144no—
219D8—2144no—
220M16—24no—
221C2×C8C22×D10711yescharacteristic
222C2×C8C22×D10712yes—
223C2×C8C22×D10711yescharacteristic
224C4⋊C4—1071no—
225C4⋊C4—1071no—
226C4⋊C4—2141no—
227C4⋊C4D42811yescharacteristic
228C22⋊C4—2141no—
229C22⋊C4—2142no—
230C22⋊C4—2142no—
231C22⋊C4D42812yes—
232C42—2141no—
233C42C22×D10711yescharacteristic
234C23—8562no—
235C23—4284no—
236C23—4284no—
237C23—2142no—
238C23—2142no—
239C23—2141no—
240C23C2×D42812yes—
241Q8—1072no—
242Q8—2141no—
243D4—22no—
244D4—8561no—
245D4—8561no—
246D4—8562no—
247D4—1072no—
248D4—4284no—
249D4—4284no—
250D4—4284no—
251D4—4282no—
252D4—4282no—
253D4—2142no—
254D4—2142no—
255D4—2142no—
256D4—2142no—
257D4—2141no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—4284no—
261C2×C4—4282no—
262C2×C4—2141no—
263C2×C4—2141no—
264C2×C4—2141no—
265C2×C4C2×D42811yescharacteristic
266C2×C4C2×D42812yes—
267C2×C4C2×D42811yescharacteristic
268C2×C4C23×D10711yescharacteristic
269C8—22no—
270C8—21no—
271C8D4×D10712yescomplemented
272C22—22no—
273C22—22no—
274C22—22no—
275C22—8568no—
276C22—4282no—
277C22—4282no—
278C22—4282no—
279C22—4282no—
280C22—4282no—
281C22—4282no—
282C22—4281no—
283C22—4281no—
284C22—2142no—
285C22C22×D42811yescharacteristic
286C4—21no—
287C4—22no—
288C4—4281no—
289C4—4281no—
290C4—2142no—
291C4#1681213712yescomplemented
292C4C2×D4×D10711yescharacteristic
293C4#1681215511yescharacteristic
294C2—42no—
295C2—8562no—
296C2—8562no—
297C2—4282no—
298C2D4×D42811yescharacteristic
299C2C2×#1681213711yescharacteristic
300C2#2029343211yescharacteristic
subgroupG/H
centerC22C22×D428
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#436×C107C2
socleC22×C107C22×D4
hypercenter#436D107
hypocenterC107#3274
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C4×D4 ◁ #436

Complements

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

NH ≅ G/Ntype
1. C2×#16812137C2semidirect
2. #20293298C2semidirect
3. D4×D428C2semidirect
4. #20293004C2semidirect
5. #20292880C2semidirect
6. #20292828C2semidirect
7. #20292823C2semidirect
8. #20292774C2semidirect
9. #20292763C2semidirect
10. C4×#13133807C2semidirect
11. #436×C107C2semidirect
12. C2×#13133818C22semidirect
17. C2×#13133807C22semidirect
19. C2×D856C22semidirect
20. C2×D856C22semidirect
22. #16812106C22semidirect
26. #16812083C22semidirect
29. C4×Dic214C22semidirect
32. #16812041C22semidirect
33. #16812034C22semidirect
35. #16812031C22semidirect
36. #16812031C22semidirect
37. C2×M16×C107C22semidirect
39. C4⋊C8×C107C22semidirect
40. C22⋊C8×C107C22semidirect
41. C4×C8×C107C22semidirect
63. #13133807D4semidirect
117. Dic214C2×D4semidirect
120. C8×C107C2×D4semidirect
137. C4×C107AGL(1,ℤ/8)semidirect
151. C107#3274semidirect
162. #436D107semidirect
172. C2×M16C2×D107semidirect
186. C4⋊C8C2×D107semidirect
191. C22⋊C8C2×D107semidirect
192. C4×C8C2×D107semidirect
271. C8D4×D107semidirect
291. C4#16812137semidirect

Maximal subgroups

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

subgroupindex
C2×#168121372
#202932982
D4×D4282
#202930042
#202928802
#202928282
#202928232
#202927742
#202927632
C4×#131338072
#436×C1072
#3274107