Finite Group Databasebeta

← #28345709

Subgroups — #28345709

Order 13696.

806 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#41×D107C211yesmaximal, characteristic, complemented
2C2×#16812135C211yesmaximal, characteristic, complemented
3C2×#16812080C211yesmaximal, characteristic, complemented
4#20292771C212yesmaximal, complemented
5#20292769C212yesmaximal, complemented
6#20292764C212yesmaximal, complemented
7#20292759C212yesmaximal, complemented
8#20292629C211yesmaximal, characteristic, complemented
9#20292628C211yesmaximal, characteristic, complemented
10#20292590C211yesmaximal, characteristic, complemented
11C2×AGL(1,ℤ/8)×C107C211yesmaximal, characteristic, complemented
12C2×#13133819C2211yescharacteristic, complemented
13C2×#13133818—21no—
14C2×#13133818C2211yescharacteristic
15C2×#13133817—21no—
16C2×#13133817C2211yescharacteristic, complemented
17#41×D107—24no—
18#41×D107—24no—
19C2×Q8×D107C2211yescharacteristic, complemented
20C22×C22⋊D107—21no—
21C2×D4×D107—21no—
22C2×D4×D107C2211yescharacteristic, complemented
23C22×C4×D107—21no—
24C22×C4×D107C2211yescharacteristic
25#16812135—24no—
26C2×#13133806C2212yescomplemented
27C2×#13133803C2212yescomplemented
28C2×#13133802C2211yescharacteristic, complemented
29#16812080—21no—
30#16812080—21no—
31#16812080C2212yes—
32#16812080C2212yescomplemented
33C2×C4⋊Dic107C2211yescharacteristic, complemented
34C2×C22⋊Dic107—21no—
35#16812034C2212yescomplemented
36#16812033C2212yescomplemented
37#16812031C2212yescomplemented
38#16812030C2212yescomplemented
39(C2×D107)⋊C8C2212yescomplemented
40(C2×D107)⋊C8C2212yescomplemented
41C2×#41×C107C2211yescharacteristic
42C22×D4×C107C2211yescharacteristic, complemented
43AGL(1,ℤ/8)×C107—24no—
44C2×SD16×C107C2212yescomplemented
45C2×D8×C107C2212yescomplemented
46C2×M16×C107C2211yescharacteristic, complemented
47C23×D107—21no—
48C23×D107D411yescharacteristic
49#13133819—22no—
50#13133819—22no—
51#13133818—28no—
52#13133818—22no—
53#13133818D414yescomplemented
54#13133817—28no—
55#13133817—22no—
56#13133817—22no—
57C2×Dic214—21no—
58C2×Dic214C2311yescharacteristic
59Q8×D107—22no—
60Q8×D107—22no—
61C2×D428—21no—
62C2×D428C2311yescharacteristic
63C2×C22⋊D107—21no—
64C2×C22⋊D107—21no—
65C2×C22⋊D107—22no—
66C2×C22⋊D107—22no—
67C2×C22⋊D107—21no—
68C2×C22⋊D107—44no—
69D4×D107—28no—
70D4×D107—22no—
71D4×D107—22no—
72C22×Dic107—21no—
73C22×Dic107—21no—
74C22×Dic107D411yescharacteristic
75C2×C4×D107—28no—
76C2×C4×D107—21no—
77C2×C4×D107—21no—
78C2×C4×D107—22no—
79C2×C4×D107—22no—
80C2×C4×D107—21no—
81C2×C4×D107D411yescharacteristic
82C2×C4×D107D411yescharacteristic
83C2×C4×D107C2311yescharacteristic
84C2×C4×D107C2311yescharacteristic
85#13133806—24no—
86#13133803—24no—
87#13133802—22no—
88C2×C107⋊C8C2312yes—
89C4⋊Dic107—21no—
90C4⋊Dic107C2312yes—
91C22⋊Dic107—22no—
92C22⋊Dic107—22no—
93C24×C107—21no—
94#41×C107—22no—
95#41×C107D414yescomplemented
96C2×Q8×C107C2311yescharacteristic
97C2×D4×C107—21no—
98C2×D4×C107—22no—
99C2×D4×C107—22no—
100C2×D4×C107—21no—
101C2×D4×C107C2312yes—
102C2×D4×C107C2311yescharacteristic
103C22×C4×C107—21no—
104C22×C4×C107C2311yescharacteristic
105SD16×C107—24no—
106D8×C107—24no—
107M16×C107—22no—
108C2×C8×C107C2312yes—
109C23×C107—21no—
110C23×C107—22no—
111C23×C107—42no—
112C23×C107—42no—
113C23×C107—42no—
114C23×C107C2×D411yescharacteristic
115C22×D107—24no—
116C22×D107—22no—
117C22×D107—21no—
118C22×D107—22no—
119C22×D107—41no—
120C22×D107—42no—
121C22×D107—42no—
122C22×D107C2×D411yescharacteristic
123C22×D107C2×D411yescharacteristic
124Q8×C107—21no—
125Q8×C107C2×D412yescomplemented
126D4×C107—24no—
127D4×C107—21no—
128D4×C107—22no—
129D4×C107—22no—
130D4×C107—42no—
131D4×C107C2×D412yescomplemented
132C2×C4×C107—21no—
133C2×C4×C107—24no—
134C2×C4×C107C2×D411yescharacteristic
135C2×C4×C107C2×D412yes—
136C2×C4×C107C2×D412yes—
137C2×C4×C107C2411yescharacteristic
138C22⋊D107—24no—
139C22⋊D107—24no—
140C22⋊D107—24no—
141C22⋊D107—44no—
142C22⋊D107—44no—
143C2×Dic107—24no—
144C2×Dic107—22no—
145C2×Dic107—21no—
146C2×Dic107—22no—
147C2×Dic107—42no—
148C2×Dic107C2×D411yescharacteristic
149C2×Dic107C2×D411yescharacteristic
150D428—24no—
151D428—21no—
152D428C2×D412yescomplemented
153C4×D107—24no—
154C4×D107—24no—
155C4×D107—22no—
156C4×D107—24no—
157C4×D107—22no—
158C4×D107—21no—
159C4×D107—21no—
160C4×D107—21no—
161C4×D107—21no—
162Dic214—24no—
163Dic214—21no—
164Dic214C2×D412yescomplemented
165C8×C107—22no—
166C107⋊C8—22no—
167C22×C107—21no—
168C22×C107—21no—
169C22×C107—22no—
170C22×C107—41no—
171C22×C107—41no—
172C22×C107—42no—
173C22×C107—42no—
174C22×C107—42no—
175C22×C107—84no—
176C22×C107#12012yes—
177C22×C107C22×D411yescharacteristic
178C2×D107—21no—
179C2×D107—21no—
180C2×D107—21no—
181C2×D107—22no—
182C2×D107—21no—
183C2×D107—21no—
184C2×D107—44no—
185C2×D107—48no—
186C2×D107—41no—
187C2×D107—41no—
188C2×D107AGL(1,ℤ/8)12yescomplemented
189C4×C107—22no—
190C4×C107#12012yes—
191C4×C107C22×D411yescharacteristic
192C4×C107C22×D411yescharacteristic
193Dic107—21no—
194Dic107—22no—
195Dic107—21no—
196Dic107—42no—
197C2×C107—22no—
198C2×C107—42no—
199C2×C107—82no—
200C2×C107C2×#12011yescharacteristic
201C2×C107C2×AGL(1,ℤ/8)11yescharacteristic
202C2×C107#57511yescharacteristic
203D107—22no—
204D107—41no—
205D107—42no—
206#3006—1071nomaximal, Sylow
207C107#300611yescharacteristic, complemented, Sylow
208C22×#41—1071no—
209C2×AGL(1,ℤ/8)—1071no—
210C2×AGL(1,ℤ/8)D10711yescharacteristic, complemented
211C2×#121—1071no—
212#449—1072no—
213#449—1072no—
214#448—1072no—
215#447—1072no—
216#417—1071no—
217#417—1071no—
218#407—1071no—
219C2×#41—1071no—
220C2×#41—1071no—
221C2×#41—1071no—
222C2×#41—2141no—
223C2×#41—2141no—
224C2×#41—2144no—
225C2×#41—2144no—
226C2×#41C2×D10711yescharacteristic
227C22×Q8—1071no—
228C22×D4—1071no—
229C22×D4—2141no—
230C22×D4—2141no—
231C22×D4C2×D10711yescharacteristic, complemented
232C23×C4—1071no—
233C23×C4—2141no—
234AGL(1,ℤ/8)—24no—
235AGL(1,ℤ/8)—2144no—
236C2×SD16—1072no—
237C2×SD16C2×D10712yescomplemented
238C2×D8—1072no—
239C2×D8C2×D10712yescomplemented
240C2×M16—1071no—
241C2×M16C2×D10711yescharacteristic, complemented
242#121—1072no—
243#121—1072no—
244#121—2141no—
245#121—2141no—
246C2×C4⋊C4—1071no—
247C2×C22⋊C4—2141no—
248#103—1072no—
249#103—1072no—
250#102—1072no—
251#102—1072no—
252C22⋊C8—1072no—
253C22⋊C8—1072no—
254C24—21no—
255C24—1071no—
256C24—2141no—
257#41—22no—
258#41—1074no—
259#41—2148no—
260#41—2142no—
261#41—2148no—
262#41—2142no—
263#41—2142no—
264#41—2142no—
265#41—2142no—
266#41C22⋊D10714yescomplemented
267C2×Q8—1071no—
268C2×Q8—2141no—
269C2×Q8—2142no—
270C2×Q8—2142no—
271C2×Q8C22×D10711yescharacteristic
272C2×D4—21no—
273C2×D4—22no—
274C2×D4—21no—
275C2×D4—22no—
276C2×D4—1071no—
277C2×D4—4284no—
278C2×D4—2141no—
279C2×D4—2141no—
280C2×D4—2142no—
281C2×D4—2148no—
282C2×D4—2142no—
283C2×D4—2142no—
284C2×D4—2141no—
285C2×D4—2141no—
286C2×D4—2142no—
287C2×D4C22×D10712yes—
288C2×D4C22×D10711yescharacteristic
289C22×C4—21no—
290C22×C4—1071no—
291C22×C4—1071no—
292C22×C4—1071no—
293C22×C4—1071no—
294C22×C4—1071no—
295C22×C4—2141no—
296C22×C4—2141no—
297C22×C4—2148no—
298C22×C4—2141no—
299C22×C4—2141no—
300C22×C4—2142no—
301C22×C4—2141no—
302C22×C4—2142no—
303C22×C4C22×D10711yescharacteristic
304SD16—24no—
305SD16—2144no—
306D8—24no—
307D8—2144no—
308M16—22no—
309M16—2142no—
310C2×C8—1072no—
311C2×C8C22×D10712yes—
312C4⋊C4—1072no—
313C4⋊C4—2141no—
314C22⋊C4—2142no—
315C22⋊C4—2142no—
316C23—21no—
317C23—22no—
318C23—42no—
319C23—42no—
320C23—42no—
321C23—1071no—
322C23—1071no—
323C23—4281no—
324C23—4282no—
325C23—4282no—
326C23—2142no—
327C23—2144no—
328C23—2142no—
329C23—2141no—
330C23D4×D10711yescharacteristic
331Q8—21no—
332Q8—1072no—
333Q8—2144no—
334Q8—2141no—
335Q8C2×C22⋊D10712yescomplemented
336D4—24no—
337D4—21no—
338D4—22no—
339D4—22no—
340D4—42no—
341D4—1072no—
342D4—4284no—
343D4—4284no—
344D4—2144no—
345D4—2144no—
346D4—2144no—
347D4—2144no—
348D4—2141no—
349D4C2×C22⋊D10712yescomplemented
350C2×C4—21no—
351C2×C4—24no—
352C2×C4—1071no—
353C2×C4—1071no—
354C2×C4—4282no—
355C2×C4—2144no—
356C2×C4—2144no—
357C2×C4—2142no—
358C2×C4—2144no—
359C2×C4—2142no—
360C2×C4—2141no—
361C2×C4—2142no—
362C2×C4—2142no—
363C2×C4—2144no—
364C2×C4—2141no—
365C2×C4—2141no—
366C2×C4—2141no—
367C2×C4—2141no—
368C2×C4D4×D10711yescharacteristic
369C2×C4D4×D10712yes—
370C2×C4C2×C22⋊D10712yes—
371C2×C4C23×D10711yescharacteristic
372C8—22no—
373C8—2142no—
374C22—21no—
375C22—21no—
376C22—22no—
377C22—41no—
378C22—41no—
379C22—42no—
380C22—42no—
381C22—42no—
382C22—84no—
383C22—1072no—
384C22—4284no—
385C22—4288no—
386C22—4281no—
387C22—4281no—
388C22—2141no—
389C22—2141no—
390C22—2142no—
391C22—2141no—
392C22—2141no—
393C22—2141no—
394C22#1681207712yes—
395C22C2×D4×D10711yescharacteristic
396C4—22no—
397C4—4282no—
398C4—2141no—
399C4—2141no—
400C4—2142no—
401C4#1681207712yes—
402C4C2×D4×D10711yescharacteristic
403C4C22×C22⋊D10711yescharacteristic
404C2—22no—
405C2—42no—
406C2—82no—
407C2—4282no—
408C2—4281no—
409C2—2142no—
410C2C2×#1681207711yescharacteristic
411C2AGL(1,ℤ/8)×D10711yescharacteristic
412C2#2029342311yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
FittingC2×AGL(1,ℤ/8)×C107C2
socleC22×C107C22×D4
hypercenterC2×AGL(1,ℤ/8)D107
hypocenterC107#3006
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×#41×D107C2semidirect
2. C2×#16812135C2semidirect
3. C2×#16812080C2semidirect
4. #20292771C2semidirect
5. #20292769C2semidirect
6. #20292764C2semidirect
7. #20292759C2semidirect
8. #20292629C2semidirect
9. #20292628C2semidirect
10. #20292590C2semidirect
11. C2×AGL(1,ℤ/8)×C107C2semidirect
12. C2×#13133819C22semidirect
16. C2×#13133817C22semidirect
19. C2×Q8×D107C22semidirect
22. C2×D4×D107C22semidirect
26. C2×#13133806C22semidirect
27. C2×#13133803C22semidirect
28. C2×#13133802C22semidirect
32. #16812080C22semidirect
33. C2×C4⋊Dic107C22semidirect
35. #16812034C22semidirect
36. #16812033C22semidirect
37. #16812031C22semidirect
38. #16812030C22semidirect
39. (C2×D107)⋊C8C22semidirect
40. (C2×D107)⋊C8C22semidirect
42. C22×D4×C107C22semidirect
44. C2×SD16×C107C22semidirect
45. C2×D8×C107C22semidirect
46. C2×M16×C107C22semidirect
53. #13133818D4semidirect
95. #41×C107D4semidirect
125. Q8×C107C2×D4semidirect
131. D4×C107C2×D4semidirect
152. D428C2×D4semidirect
164. Dic214C2×D4semidirect
188. C2×D107AGL(1,ℤ/8)semidirect
207. C107#3006semidirect
210. C2×AGL(1,ℤ/8)D107semidirect
231. C22×D4C2×D107semidirect
237. C2×SD16C2×D107semidirect
239. C2×D8C2×D107semidirect
241. C2×M16C2×D107semidirect
266. #41C22⋊D107semidirect
335. Q8C2×C22⋊D107semidirect
349. D4C2×C22⋊D107semidirect

Maximal subgroups

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

subgroupindex
C2×#41×D1072
C2×#168121352
C2×#168120802
#202927712
#202927692
#202927642
#202927592
#202926292
#202926282
#202925902
C2×AGL(1,ℤ/8)×C1072
#3006107