Finite Group Databasebeta

← #28348169

Subgroups — #28348169

Order 13696.

504 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16812129C211yesmaximal, characteristic, complemented
2C22×#13133805C211yesmaximal, characteristic, complemented
3#20293273C211yesmaximal, characteristic, complemented
4#20293009C211yesmaximal, characteristic, complemented
5#20292858C211yesmaximal, characteristic, complemented
6#20292854C211yesmaximal, characteristic, complemented
7#20292849C211yesmaximal, characteristic, complemented
8#20292839C211yesmaximal, characteristic, complemented
9#20292787C211yesmaximal, characteristic, complemented
10#20292783C211yesmaximal, characteristic, complemented
11#20292772C211yesmaximal, characteristic, complemented
12#20292760C211yesmaximal, characteristic, complemented
13C4×#13133805C211yesmaximal, characteristic, complemented
14D4×C107⋊C8C211yesmaximal, characteristic, complemented
15#548×C107C211yesmaximal, characteristic, complemented
16C2×#13133818C2211yescharacteristic
17C22×D428C2211yescharacteristic
18#16812129—24no—
19C2×#13133808C2211yescharacteristic, complemented
20C2×#13133808C2211yescharacteristic, complemented
21C2×#13133806C2211yescharacteristic, complemented
22C2×#13133805—24no—
23C2×#13133805C2211yescharacteristic
24C2×#13133805C2211yescharacteristic
25C2×#13133805C2211yescharacteristic
26C2×#13133805C2211yescharacteristic
27C2×#13133803C2211yescharacteristic
28C2×#13133803C2211yescharacteristic
29C22×C107⋊C8C2211yescharacteristic
30C22×C107⋊C8C2211yescharacteristic, complemented
31#16812097C2211yescharacteristic
32#16812094—21no—
33#16812088C2211yescharacteristic
34#16812083C2211yescharacteristic
35#16812082—21no—
36#16812077—21no—
37C4×D428C2211yescharacteristic
38C22⋊C4×D107—21no—
39#16812040C2211yescharacteristic, complemented
40C4⋊(C107⋊C8)C2211yescharacteristic, complemented
41#16812033C2211yescharacteristic, complemented
42#16812033C2211yescharacteristic, complemented
43#16812032C2211yescharacteristic, complemented
44#16812030C2211yescharacteristic, complemented
45#16812029C2211yescharacteristic
46#16812029C2211yescharacteristic
47C22⋊(C107⋊C8)C2211yescharacteristic
48C22⋊(C107⋊C8)C2211yescharacteristic, complemented
49C4×C107⋊C8C2211yescharacteristic, complemented
50C2×#41×C107C2211yescharacteristic, complemented
51C22×Q8×C107C2211yescharacteristic
52#124×C107—21no—
53#124×C107C2211yescharacteristic, complemented
54C22⋊Q8×C107—21no—
55C22⋊Q8×C107C2211yescharacteristic, complemented
56#121×C107—21no—
57#121×C107C2211yescharacteristic
58C4×Q8×C107C2211yescharacteristic, complemented
59C4×D4×C107—21no—
60C4×D4×C107C2211yescharacteristic
61C23×D107—21no—
62#13133818—22no—
63#13133818—41no—
64C2×Dic214C2311yescharacteristic
65C2×D428—22no—
66C2×D428—21no—
67C2×D428—41no—
68C2×D428C2311yescharacteristic
69C2×D428C2311yescharacteristic
70C2×C22⋊D107—21no—
71C2×C22⋊D107—21no—
72C2×C4×D107—21no—
73C2×C4×D107—21no—
74#13133808—22no—
75#13133808—41no—
76#13133806—22no—
77#13133805—22no—
78#13133805—22no—
79#13133805—22no—
80#13133805D414yescomplemented
81#13133803—22no—
82#13133803—41no—
83C2×C107⋊C8—22no—
84C2×C107⋊C8—22no—
85C2×C107⋊C8C2311yescharacteristic
86C2×C107⋊C8C2311yescharacteristic
87C2×C107⋊C8C2311yescharacteristic
88C2×C107⋊C8C2311yescharacteristic
89C4⋊Dic107C2311yescharacteristic
90Dic107⋊C4—21no—
91C22⋊Dic107—21no—
92(C2×D107)⋊C4—21no—
93(C2×D107)⋊C4—21no—
94(C2×D107)⋊C4—21no—
95(C2×D107)⋊C4—21no—
96#41×C107—22no—
97#41×C107—41no—
98C2×Q8×C107—21no—
99C2×Q8×C107—22no—
100C2×Q8×C107—41no—
101C2×Q8×C107C2311yescharacteristic
102C2×Q8×C107C2311yescharacteristic
103C2×D4×C107—21no—
104C2×D4×C107—21no—
105C2×D4×C107D411yescharacteristic
106C2×D4×C107C2311yescharacteristic
107C22×C4×C107—21no—
108C22×C4×C107—21no—
109C22×C4×C107C2311yescharacteristic
110C22×C4×C107C2311yescharacteristic
111C4⋊C4×C107—21no—
112C4⋊C4×C107—21no—
113C4⋊C4×C107D411yescharacteristic
114C4⋊C4×C107C2311yescharacteristic
115C22⋊C4×C107—21no—
116C22⋊C4×C107—21no—
117C22⋊C4×C107—21no—
118C22⋊C4×C107—21no—
119C22⋊C4×C107D411yescharacteristic
120C22⋊C4×C107D411yescharacteristic
121C42×C107—21no—
122C42×C107C2311yescharacteristic
123C23×C107—21no—
124C23×C107C2×D411yescharacteristic
125C23×C107C2×D411yescharacteristic
126C22×D107—21no—
127C22×D107—21no—
128C22×D107—42no—
129C22×D107—42no—
130C22×D107—81no—
131Q8×C107—21no—
132Q8×C107—21no—
133Q8×C107—21no—
134Q8×C107—42no—
135Q8×C107C2×D412yes—
136D4×C107—22no—
137D4×C107—21no—
138D4×C107—21no—
139D4×C107—42no—
140D4×C107—42no—
141C2×C4×C107—21no—
142C2×C4×C107—21no—
143C2×C4×C107—21no—
144C2×C4×C107—21no—
145C2×C4×C107—21no—
146C2×C4×C107—21no—
147C2×C4×C107—21no—
148C2×C4×C107—42no—
149C2×C4×C107—42no—
150C2×C4×C107C2×D411yescharacteristic
151C2×C4×C107C2×D411yescharacteristic
152C2×C4×C107C2×D411yescharacteristic
153C2×C4×C107C2×D411yescharacteristic
154C2×C4×C107C2411yescharacteristic
155C22⋊D107—42no—
156C22⋊D107—81no—
157C2×Dic107—21no—
158C2×Dic107—21no—
159D428—21no—
160D428—21no—
161D428—21no—
162D428—42no—
163D428C2×D412yes—
164C4×D107—42no—
165C4×D107—81no—
166Dic214—21no—
167Dic214—21no—
168C107⋊C8—21no—
169C107⋊C8—21no—
170C107⋊C8—21no—
171C107⋊C8C2×D412yes—
172C22×C107—21no—
173C22×C107—21no—
174C22×C107—21no—
175C22×C107—21no—
176C22×C107—21no—
177C22×C107—41no—
178C22×C107—41no—
179C22×C107—41no—
180C22×C107#13512yescomplemented
181C22×C107C22×D411yescharacteristic
182C2×D107—22no—
183C2×D107—41no—
184C2×D107—41no—
185C2×D107—41no—
186C2×D107—41no—
187C2×D107—41no—
188C2×D107—84no—
189C4×C107—21no—
190C4×C107—21no—
191C4×C107—22no—
192C4×C107—21no—
193C4×C107—21no—
194C4×C107—41no—
195C4×C107—41no—
196C4×C107C22×D411yescharacteristic
197C4×C107H(5,𝔽2)11yescharacteristic
198Dic107—41no—
199Dic107—41no—
200C2×C107—22no—
201C2×C107—41no—
202C2×C107—81no—
203C2×C107D4211yescharacteristic
204C2×C107C2×#13511yescharacteristic
205C2×C107#57711yescharacteristic
206D107—42no—
207D107—81no—
208#3282—1071nomaximal, Sylow
209C107#328211yescharacteristic, complemented, Sylow
210C2×#135—1071no—
211C22×SD16—1071no—
212#548D10711yescharacteristic, complemented
213#546—1071no—
214#495—1071no—
215#465—1071no—
216#465—1071no—
217#464—1071no—
218#463—1071no—
219#452—1071no—
220#451—1071no—
221#449—1071no—
222#447—1071no—
223C4×SD16—1071no—
224C8×D4—1071no—
225C2×#41—1071no—
226C2×#41C2×D10711yescharacteristic, complemented
227C22×Q8C2×D10711yescharacteristic
228C22×D4—1071no—
229#135—2144no—
230C2×Q16—1071no—
231C2×Q16—1071no—
232C2×SD16—1071no—
233C2×SD16—1071no—
234C2×SD16—1071no—
235C2×SD16—1071no—
236C2×SD16—1071no—
237C2×SD16—2144no—
238C2×D8—1071no—
239C2×D8—1071no—
240C22×C8—1071no—
241C22×C8—1071no—
242#124—21no—
243#124—1071no—
244#124C2×D10711yescharacteristic, complemented
245#123—2141no—
246C22⋊Q8—21no—
247C22⋊Q8—1071no—
248C22⋊Q8C2×D10711yescharacteristic, complemented
249#121—21no—
250#121—1071no—
251#121—2141no—
252#121C2×D10711yescharacteristic
253#120—2141no—
254C4×Q8C2×D10711yescharacteristic, complemented
255C4×D4—21no—
256C4×D4—1071no—
257C4×D4C2×D10711yescharacteristic
258C2×C22⋊C4—2141no—
259#106—1071no—
260C4⋊C8—1071no—
261#103—1071no—
262#103—1071no—
263#103—1071no—
264#102—1071no—
265#102—1071no—
266#102—1071no—
267C22⋊C8—1071no—
268C22⋊C8—1071no—
269C4×C8—1071no—
270C24—2141no—
271#41—22no—
272#41—41no—
273#41—4281no—
274#41—2142no—
275C2×Q8—21no—
276C2×Q8—22no—
277C2×Q8—41no—
278C2×Q8—1071no—
279C2×Q8C22×D10711yescharacteristic
280C2×Q8C22×D10711yescharacteristic
281C2×D4—21no—
282C2×D4—21no—
283C2×D4—1071no—
284C2×D4—1071no—
285C2×D4—4281no—
286C2×D4—2142no—
287C2×D4—2141no—
288C2×D4—2141no—
289C2×D4—2141no—
290C2×D4C22⋊D10711yescharacteristic
291C2×D4C22×D10711yescharacteristic
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—2141no—
295C22×C4—2141no—
296C22×C4C22×D10711yescharacteristic
297C22×C4C22×D10711yescharacteristic
298Q16—4281no—
299Q16—2142no—
300SD16—1074no—
301SD16—2142no—
302SD16—2142no—
303SD16—2142no—
304SD16—2142no—
305D8—4281no—
306D8—2142no—
307C2×C8—1071no—
308C2×C8—1071no—
309C2×C8—1071no—
310C2×C8—1071no—
311C2×C8—2142no—
312C2×C8—2142no—
313C4⋊C4—21no—
314C4⋊C4—21no—
315C4⋊C4—1071no—
316C4⋊C4—2141no—
317C4⋊C4C22⋊D10711yescharacteristic
318C4⋊C4C22×D10711yescharacteristic
319C22⋊C4—21no—
320C22⋊C4—21no—
321C22⋊C4—21no—
322C22⋊C4—21no—
323C22⋊C4—2141no—
324C22⋊C4—2141no—
325C22⋊C4—2141no—
326C22⋊C4—2141no—
327C22⋊C4—2141no—
328C22⋊C4C22⋊D10711yescharacteristic
329C22⋊C4C22⋊D10711yescharacteristic
330C42—21no—
331C42C22×D10711yescharacteristic
332C23—21no—
333C23—8561no—
334C23—4282no—
335C23—4282no—
336C23—2141no—
337C23—2141no—
338C23C2×C22⋊D10711yescharacteristic
339C23C2×C22⋊D10711yescharacteristic
340Q8—21no—
341Q8—21no—
342Q8—21no—
343Q8—42no—
344Q8—2141no—
345Q8—2141no—
346Q8D4×D10712yes—
347D4—22no—
348D4—21no—
349D4—21no—
350D4—42no—
351D4—42no—
352D4—8561no—
353D4—1072no—
354D4—4282no—
355D4—4282no—
356D4—2141no—
357D4—2141no—
358D4—2141no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—8561no—
369C2×C4—4282no—
370C2×C4—2141no—
371C2×C4—2141no—
372C2×C4C2×C22⋊D10711yescharacteristic
373C2×C4C2×C22⋊D10711yescharacteristic
374C2×C4C2×C22⋊D10711yescharacteristic
375C2×C4C2×C22⋊D10711yescharacteristic
376C2×C4C23×D10711yescharacteristic
377C8—1072no—
378C8—2141no—
379C8—2141no—
380C8—2141no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—41no—
387C22—41no—
388C22—41no—
389C22—8564no—
390C22—4281no—
391C22—4281no—
392C22—4281no—
393C22—4281no—
394C22—4281no—
395C22—2142no—
396C22#1681212912yescomplemented
397C22C22×C22⋊D10711yescharacteristic
398C4—21no—
399C4—21no—
400C4—21no—
401C4—21no—
402C4—22no—
403C4—41no—
404C4—41no—
405C4—4281no—
406C4—4281no—
407C4C2×D4×D10711yescharacteristic
408C4#1681215411yescharacteristic
409C2—22no—
410C2—41no—
411C2—81no—
412C2—8561no—
413C2—4282no—
414C2D4×C22⋊D10711yescharacteristic
415C2C2×#1681212911yescharacteristic
416C2#2029343511yescharacteristic
subgroupG/H
centerC22C22×C22⋊D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#548×C107C2
socleC22×C107C22×D4
hypercenter#548D107
hypocenterC107#3282
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C4×D4 ◁ #548

Complements

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

NH ≅ G/Ntype
1. C2×#16812129C2semidirect
2. C22×#13133805C2semidirect
3. #20293273C2semidirect
4. #20293009C2semidirect
5. #20292858C2semidirect
6. #20292854C2semidirect
7. #20292849C2semidirect
8. #20292839C2semidirect
9. #20292787C2semidirect
10. #20292783C2semidirect
11. #20292772C2semidirect
12. #20292760C2semidirect
13. C4×#13133805C2semidirect
14. D4×C107⋊C8C2semidirect
15. #548×C107C2semidirect
19. C2×#13133808C22semidirect
20. C2×#13133808C22semidirect
21. C2×#13133806C22semidirect
30. C22×C107⋊C8C22semidirect
39. #16812040C22semidirect
40. C4⋊(C107⋊C8)C22semidirect
41. #16812033C22semidirect
42. #16812033C22semidirect
43. #16812032C22semidirect
44. #16812030C22semidirect
48. C22⋊(C107⋊C8)C22semidirect
49. C4×C107⋊C8C22semidirect
50. C2×#41×C107C22semidirect
53. #124×C107C22semidirect
55. C22⋊Q8×C107C22semidirect
58. C4×Q8×C107C22semidirect
80. #13133805D4semidirect
180. C22×C107#135semidirect
209. C107#3282semidirect
212. #548D107semidirect
226. C2×#41C2×D107semidirect
244. #124C2×D107semidirect
248. C22⋊Q8C2×D107semidirect
254. C4×Q8C2×D107semidirect
396. C22#16812129semidirect

Maximal subgroups

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

subgroupindex
C2×#168121292
C22×#131338052
#202932732
#202930092
#202928582
#202928542
#202928492
#202928392
#202927872
#202927832
#202927722
#202927602
C4×#131338052
D4×C107⋊C82
#548×C1072
#3282107