Finite Group Databasebeta

← #468×D107 #28346045

Subgroups — #468×D107 #28346045

Order 13696.

544 conjugacy classes of subgroups.

The subgroup lattice has 420 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×SD16×D107C211yesmaximal, characteristic, complemented
2C2×M16×D107C211yesmaximal, characteristic, complemented
3C22⋊Q8×D107C211yesmaximal, characteristic, complemented
4#121×D107C211yesmaximal, characteristic
5#20292876C211yesmaximal, characteristic, complemented
6#20292875C211yesmaximal, characteristic, complemented
7#20292874C211yesmaximal, characteristic, complemented
8#20292873C211yesmaximal, characteristic, complemented
9#20292872C211yesmaximal, characteristic, complemented
10#20292871C211yesmaximal, characteristic, complemented
11#20292870C211yesmaximal, characteristic, complemented
12#107×D107C211yesmaximal, characteristic, complemented
13#103×D107C211yesmaximal, characteristic, complemented
14#102×D107C211yesmaximal, characteristic, complemented
15#468×C107C211yesmaximal, characteristic, complemented
16C2×Q8×D107C2211yescharacteristic
17C2×D4×D107—21no—
18C2×D4×D107C2211yescharacteristic
19C22×C4×D107C2211yescharacteristic
20C2×#13133807C2211yescharacteristic, complemented
21C2×#13133806C2211yescharacteristic, complemented
22C2×#13133805C2211yescharacteristic, complemented
23SD16×D107—24no—
24C2×#13133802C2211yescharacteristic, complemented
25C2×#13133801C2211yescharacteristic, complemented
26C2×#13133801C2211yescharacteristic, complemented
27M16×D107—24no—
28C2×C8×D107C2211yescharacteristic
29C2×C8×D107C2211yescharacteristic
30#16812089C2211yescharacteristic, complemented
31#16812088C2211yescharacteristic, complemented
32#16812087—21no—
33#16812086—21no—
34#16812085C2211yescharacteristic, complemented
35#16812084C2211yescharacteristic
36#16812083C2211yescharacteristic
37#16812082—21no—
38#16812081—21no—
39#16812080C2211yescharacteristic
40C4⋊C4×D107—21no—
41C4⋊C4×D107C2211yescharacteristic
42C4⋊C4×D107C2211yescharacteristic
43C22⋊C4×D107—21no—
44C22⋊C4×D107—21no—
45#16812043C2211yescharacteristic, complemented
46#16812042C2211yescharacteristic, complemented
47#16812042C2211yescharacteristic, complemented
48#16812034C2211yescharacteristic, complemented
49#16812033C2211yescharacteristic, complemented
50#16812032C2211yescharacteristic, complemented
51#16812031C2211yescharacteristic, complemented
52#16812030C2211yescharacteristic, complemented
53#16812029C2211yescharacteristic, complemented
54C2×SD16×C107C2211yescharacteristic, complemented
55C2×M16×C107C2211yescharacteristic, complemented
56C22⋊Q8×C107C2211yescharacteristic, complemented
57#121×C107C2211yescharacteristic
58#107×C107C2211yescharacteristic, complemented
59#103×C107C2211yescharacteristic, complemented
60#102×C107C2211yescharacteristic, complemented
61C23×D107—21no—
62C23×D107D411yescharacteristic
63C2×Dic214—21no—
64C2×Dic214C2311yescharacteristic
65Q8×D107—22no—
66Q8×D107—22no—
67C2×D428—21no—
68C2×D428C2311yescharacteristic
69C2×C22⋊D107—21no—
70C2×C22⋊D107—21no—
71C2×C22⋊D107—21no—
72D4×D107—22no—
73D4×D107—22no—
74D4×D107—44no—
75C22×Dic107D411yescharacteristic
76C2×C4×D107—22no—
77C2×C4×D107—22no—
78C2×C4×D107—21no—
79C2×C4×D107—21no—
80C2×C4×D107—21no—
81C2×C4×D107D411yescharacteristic
82C2×C4×D107D411yescharacteristic
83C2×C4×D107C2311yescharacteristic
84C2×C4×D107C2311yescharacteristic
85#13133807—22no—
86#13133806—22no—
87#13133805—22no—
88#13133802—22no—
89#13133801—22no—
90#13133801—22no—
91C2×C107⋊C8C2311yescharacteristic
92C2×C107⋊C8C2311yescharacteristic
93C8×D107—22no—
94C8×D107D414yescomplemented
95C4⋊Dic107—21no—
96C4⋊Dic107C2311yescharacteristic
97C4⋊Dic107C2311yescharacteristic
98Dic107⋊C4—21no—
99Dic107⋊C4—21no—
100Dic107⋊C4—21no—
101Dic107⋊C4—21no—
102C22⋊Dic107—21no—
103C22⋊Dic107—21no—
104(C2×D107)⋊C4—21no—
105(C2×D107)⋊C4—21no—
106(C2×D107)⋊C4—21no—
107(C2×D107)⋊C4—21no—
108C2×Q8×C107C2311yescharacteristic
109C2×D4×C107—21no—
110C2×D4×C107C2311yescharacteristic
111C22×C4×C107C2311yescharacteristic
112SD16×C107—22no—
113M16×C107—22no—
114C2×C8×C107C2311yescharacteristic
115C2×C8×C107C2311yescharacteristic
116C4⋊C4×C107—21no—
117C4⋊C4×C107C2311yescharacteristic
118C4⋊C4×C107C2311yescharacteristic
119C22⋊C4×C107—21no—
120C22⋊C4×C107—21no—
121C23×C107—21no—
122C23×C107C2×D411yescharacteristic
123C22×D107—22no—
124C22×D107—22no—
125C22×D107—22no—
126C22×D107—21no—
127C22×D107—42no—
128C22×D107—42no—
129C22×D107—42no—
130C22×D107C2×D411yescharacteristic
131C22×D107C2×D411yescharacteristic
132Q8×C107—21no—
133Q8×C107—21no—
134D4×C107—21no—
135D4×C107—21no—
136D4×C107—42no—
137C2×C4×C107—21no—
138C2×C4×C107—21no—
139C2×C4×C107—21no—
140C2×C4×C107—21no—
141C2×C4×C107—21no—
142C2×C4×C107C2×D411yescharacteristic
143C2×C4×C107C2411yescharacteristic
144C22⋊D107—42no—
145C22⋊D107—42no—
146C22⋊D107—42no—
147C2×Dic107—22no—
148C2×Dic107—21no—
149C2×Dic107—21no—
150C2×Dic107—21no—
151C2×Dic107C2×D411yescharacteristic
152C2×Dic107C2×D411yescharacteristic
153D428—21no—
154D428—21no—
155D428—42no—
156C4×D107—22no—
157C4×D107—21no—
158C4×D107—21no—
159C4×D107—22no—
160C4×D107—42no—
161C4×D107—42no—
162C4×D107—42no—
163C4×D107C2×D412yes—
164C4×D107#4112yes—
165Dic214—21no—
166Dic214—21no—
167Dic214—42no—
168C8×C107—21no—
169C8×C107C2×D412yescomplemented
170C107⋊C8—21no—
171C107⋊C8C2×D412yescomplemented
172C22×C107—21no—
173C22×C107—21no—
174C22×C107—21no—
175C22×C107—41no—
176C22×C107—41no—
177C22×C107—41no—
178C22×C107C22×D411yescharacteristic
179C2×D107—21no—
180C2×D107—21no—
181C2×D107—21no—
182C2×D107—44no—
183C2×D107—41no—
184C2×D107—41no—
185C2×D107—41no—
186C2×D107—84no—
187C2×D107#12112yes—
188C2×D107AGL(1,ℤ/8)12yes—
189C2×D107#13712yes—
190C4×C107—21no—
191C4×C107—41no—
192C4×C107—41no—
193C4×C107—41no—
194C4×C107C22×D411yescharacteristic
195C4×C107C2×#4111yescharacteristic
196Dic107—21no—
197Dic107—41no—
198Dic107—41no—
199Dic107—41no—
200Dic107#12112yes—
201C2×C107—41no—
202C2×C107—81no—
203C2×C107C2×#12111yescharacteristic
204C2×C107C2×AGL(1,ℤ/8)11yescharacteristic
205C2×C107C2×#13711yescharacteristic
206D107—41no—
207D107—81no—
208D107#46814yesdirect product factor
209C2×#468—1071nomaximal, Sylow
210C107C2×#46811yescharacteristic, complemented, Sylow
211C22×SD16—1071no—
212C22×M16—1071no—
213C2×C22⋊Q8—1071no—
214C2×#121—1071no—
215#468—1071no—
216#468—1071no—
217#468—1071no—
218#468—1071no—
219#468—1071no—
220#468—1071no—
221#468—1071no—
222#468D10711yescharacteristic, direct product factor
223C2×#107—1071no—
224C2×#103—1071no—
225C2×#102—1071no—
226C22×Q8—1071no—
227C22×D4—1071no—
228C22×D4—2141no—
229C23×C4—1071no—
230C2×SD16—1071no—
231C2×SD16—1071no—
232C2×SD16—1071no—
233C2×SD16—2144no—
234C2×SD16C2×D10711yescharacteristic, complemented
235C2×M16—1071no—
236C2×M16—1071no—
237C2×M16—1071no—
238C2×M16—2144no—
239C2×M16C2×D10711yescharacteristic, complemented
240C22×C8—1071no—
241C22×C8—1071no—
242C22⋊Q8—1071no—
243C22⋊Q8—1071no—
244C22⋊Q8—1071no—
245C22⋊Q8—2141no—
246C22⋊Q8—2141no—
247C22⋊Q8C2×D10711yescharacteristic, complemented
248#121—1071no—
249#121—1071no—
250#121—1071no—
251#121—2141no—
252#121—2141no—
253#121C2×D10711yescharacteristic
254C2×C4⋊C4—1071no—
255C2×C4⋊C4—1071no—
256C2×C4⋊C4—2141no—
257C2×C22⋊C4—2141no—
258C2×C22⋊C4—2141no—
259#107—1071no—
260#107—1071no—
261#107—1071no—
262#107C2×D10711yescharacteristic, complemented
263#103—1071no—
264#103—1071no—
265#103—1071no—
266#103C2×D10711yescharacteristic, complemented
267#102—1071no—
268#102—1071no—
269#102—1071no—
270#102C2×D10711yescharacteristic, complemented
271C24—1071no—
272C24—2141no—
273C2×Q8—1071no—
274C2×Q8—2141no—
275C2×Q8—2142no—
276C2×Q8—2142no—
277C2×Q8C22×D10711yescharacteristic
278C2×D4—21no—
279C2×D4—1071no—
280C2×D4—4284no—
281C2×D4—2141no—
282C2×D4—2142no—
283C2×D4—2142no—
284C2×D4—2141no—
285C2×D4—2141no—
286C2×D4—2141no—
287C2×D4C22×D10711yescharacteristic
288C22×C4—1071no—
289C22×C4—1071no—
290C22×C4—1071no—
291C22×C4—1071no—
292C22×C4—1071no—
293C22×C4—2142no—
294C22×C4—2142no—
295C22×C4—2141no—
296C22×C4—2141no—
297C22×C4—2141no—
298C22×C4C22×D10711yescharacteristic
299SD16—22no—
300SD16—2142no—
301SD16—2142no—
302SD16—2142no—
303M16—22no—
304M16—2142no—
305M16—2142no—
306M16—2142no—
307C2×C8—1071no—
308C2×C8—1074no—
309C2×C8—1071no—
310C2×C8—2142no—
311C2×C8C22×D10711yescharacteristic
312C2×C8C22×D10711yescharacteristic
313C4⋊C4—21no—
314C4⋊C4—1071no—
315C4⋊C4—1071no—
316C4⋊C4—2141no—
317C4⋊C4—2141no—
318C4⋊C4—2141no—
319C4⋊C4—2141no—
320C4⋊C4—2141no—
321C4⋊C4C22×D10711yescharacteristic
322C4⋊C4C22×D10711yescharacteristic
323C22⋊C4—21no—
324C22⋊C4—21no—
325C22⋊C4—2141no—
326C22⋊C4—2141no—
327C22⋊C4—2141no—
328C22⋊C4—2141no—
329C22⋊C4—2141no—
330C22⋊C4—2141no—
331C23—21no—
332C23—1071no—
333C23—1071no—
334C23—4282no—
335C23—4282no—
336C23—4282no—
337C23—2142no—
338C23—2142no—
339C23—2142no—
340C23—2141no—
341C23D4×D10711yescharacteristic
342Q8—21no—
343Q8—21no—
344Q8—4282no—
345Q8—2141no—
346Q8—2141no—
347D4—21no—
348D4—21no—
349D4—42no—
350D4—4282no—
351D4—4282no—
352D4—4282no—
353D4—4282no—
354D4—2141no—
355D4—2141no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—1071no—
362C2×C4—1072no—
363C2×C4—1072no—
364C2×C4—1071no—
365C2×C4—4282no—
366C2×C4—4282no—
367C2×C4—4282no—
368C2×C4—2142no—
369C2×C4—2142no—
370C2×C4—2142no—
371C2×C4—2141no—
372C2×C4—2141no—
373C2×C4—2141no—
374C2×C4—2141no—
375C2×C4—2141no—
376C2×C4D4×D10711yescharacteristic
377C2×C4C23×D10711yescharacteristic
378C8—21no—
379C8—1072no—
380C8—2141no—
381C8D4×D10712yescomplemented
382C22—21no—
383C22—21no—
384C22—21no—
385C22—41no—
386C22—41no—
387C22—41no—
388C22—8564no—
389C22—1072no—
390C22—1072no—
391C22—1072no—
392C22—4284no—
393C22—4281no—
394C22—4281no—
395C22—4281no—
396C22—2141no—
397C22—2141no—
398C22—2141no—
399C22C2×D4×D10711yescharacteristic
400C4—21no—
401C4—41no—
402C4—41no—
403C4—41no—
404C4—1072no—
405C4—4281no—
406C4—4281no—
407C4—4281no—
408C4—2141no—
409C4C2×D4×D10711yescharacteristic
410C4#41×D10711yescharacteristic
411C2—41no—
412C2—81no—
413C2—8561no—
414C2—1074no—
415C2—4281no—
416C2#121×D10711yescharacteristic
417C2AGL(1,ℤ/8)×D10711yescharacteristic
418C2#137×D10711yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#468×C107C2
socleC22×C107C22×D4
prime radical#121×D107C2
hypercenter#468D107
hypocenterC107C2×#468
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #468

Complements

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

NH ≅ G/Ntype
1. C2×SD16×D107C2semidirect
2. C2×M16×D107C2semidirect
3. C22⋊Q8×D107C2semidirect
5. #20292876C2semidirect
6. #20292875C2semidirect
7. #20292874C2semidirect
8. #20292873C2semidirect
9. #20292872C2semidirect
10. #20292871C2semidirect
11. #20292870C2semidirect
12. #107×D107C2semidirect
13. #103×D107C2semidirect
14. #102×D107C2semidirect
15. #468×C107C2semidirect
20. C2×#13133807C22semidirect
21. C2×#13133806C22semidirect
22. C2×#13133805C22semidirect
24. C2×#13133802C22semidirect
25. C2×#13133801C22semidirect
26. C2×#13133801C22semidirect
30. #16812089C22semidirect
31. #16812088C22semidirect
34. #16812085C22semidirect
45. #16812043C22semidirect
46. #16812042C22semidirect
47. #16812042C22semidirect
48. #16812034C22semidirect
49. #16812033C22semidirect
50. #16812032C22semidirect
51. #16812031C22semidirect
52. #16812030C22semidirect
53. #16812029C22semidirect
54. C2×SD16×C107C22semidirect
55. C2×M16×C107C22semidirect
56. C22⋊Q8×C107C22semidirect
58. #107×C107C22semidirect
59. #103×C107C22semidirect
60. #102×C107C22semidirect
94. C8×D107D4semidirect
169. C8×C107C2×D4semidirect
171. C107⋊C8C2×D4semidirect
208. D107#468direct (N × H)
210. C107C2×#468semidirect
222. #468D107direct (N × H)
234. C2×SD16C2×D107semidirect
239. C2×M16C2×D107semidirect
247. C22⋊Q8C2×D107semidirect
262. #107C2×D107semidirect
266. #103C2×D107semidirect
270. #102C2×D107semidirect
381. C8D4×D107semidirect

Maximal subgroups

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

subgroupindex
C2×SD16×D1072
C2×M16×D1072
C22⋊Q8×D1072
#121×D1072
#202928762
#202928752
#202928742
#202928732
#202928722
#202928712
#202928702
#107×D1072
#103×D1072
#102×D1072
#468×C1072
C2×#468107