Finite Group Databasebeta

← #28347987

Subgroups — #28347987

Order 13696.

500 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×#16812134C211yesmaximal, characteristic, complemented
2C2×#16812129C211yesmaximal, characteristic, complemented
3#20293273C211yesmaximal, characteristic, complemented
4#20293019C211yesmaximal, characteristic, complemented
5#20292883C211yesmaximal, characteristic, complemented
6#20292858C211yesmaximal, characteristic, complemented
7#20292842C211yesmaximal, characteristic, complemented
8#20292829C211yesmaximal, characteristic, complemented
9#20292789C211yesmaximal, characteristic, complemented
10#20292787C211yesmaximal, characteristic, complemented
11#20292777C211yesmaximal, characteristic, complemented
12#20292772C211yesmaximal, characteristic, complemented
13#20292737C211yesmaximal, characteristic, complemented
14#20292699C211yesmaximal, characteristic, complemented
15#547×C107C211yesmaximal, characteristic, complemented
16C2×#13133818C2211yescharacteristic
17C22×D428C2211yescharacteristic
18#16812134—24no—
19#16812129—24no—
20C2×#13133808C2211yescharacteristic, complemented
21C2×#13133808C2211yescharacteristic, complemented
22C2×#13133806C2211yescharacteristic, complemented
23C2×#13133805C2211yescharacteristic
24C2×#13133805C2211yescharacteristic
25C2×#13133805C2211yescharacteristic
26C2×#13133805C2211yescharacteristic
27C2×#13133803C2211yescharacteristic
28C2×#13133803C2211yescharacteristic
29C2×#13133802C2211yescharacteristic
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
49#16812020C2211yescharacteristic, complemented
50C2×#41×C107C2211yescharacteristic
51C2×#41×C107C2211yescharacteristic, complemented
52C4⋊Q8×C107C2211yescharacteristic, complemented
53#123×C107—21no—
54#123×C107—21no—
55C22⋊Q8×C107C2211yescharacteristic
56C22⋊Q8×C107C2211yescharacteristic, complemented
57#121×C107—21no—
58C4×D4×C107C2211yescharacteristic
59C4×D4×C107C2211yescharacteristic, complemented
60C2×C4⋊C4×C107—21no—
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#13133805—41no—
81#13133803—22no—
82#13133803D414yescomplemented
83#13133802—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—22no—
98#41×C107—41no—
99#41×C107—41no—
100C2×Q8×C107C2311yescharacteristic
101C2×Q8×C107C2311yescharacteristic
102C2×D4×C107—21no—
103C2×D4×C107—21no—
104C2×D4×C107D411yescharacteristic
105C2×D4×C107C2311yescharacteristic
106C22×C4×C107—21no—
107C22×C4×C107—21no—
108C22×C4×C107—21no—
109C22×C4×C107C2311yescharacteristic
110C22×C4×C107C2311yescharacteristic
111C4⋊C4×C107—21no—
112C4⋊C4×C107—21no—
113C4⋊C4×C107—21no—
114C4⋊C4×C107—21no—
115C4⋊C4×C107D411yescharacteristic
116C4⋊C4×C107C2311yescharacteristic
117C22⋊C4×C107—21no—
118C22⋊C4×C107—21no—
119C22⋊C4×C107—21no—
120C22⋊C4×C107D411yescharacteristic
121C22⋊C4×C107D411yescharacteristic
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—21no—
135D4×C107—21no—
136D4×C107—41no—
137D4×C107—42no—
138D4×C107—42no—
139D4×C107C2×D412yes—
140C2×C4×C107—21no—
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—42no—
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—22no—
175C22×C107—21no—
176C22×C107—21no—
177C22×C107—21no—
178C22×C107—21no—
179C22×C107—41no—
180C22×C107—41no—
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—21no—
192C4×C107—21no—
193C4×C107—41no—
194C4×C107—41no—
195C4×C107—41no—
196C4×C107C22×D411yescharacteristic
197C4×C107H(5,𝔽2)11yescharacteristic
198Dic107—41no—
199Dic107—41no—
200C2×C107—41no—
201C2×C107—41no—
202C2×C107—42no—
203C2×C107D4211yescharacteristic
204C2×C107#57511yescharacteristic
205C2×C107#57711yescharacteristic
206D107—42no—
207D107—81no—
208#3267—1071nomaximal, Sylow
209C107#326711yescharacteristic, complemented, Sylow
210C2×AGL(1,ℤ/8)—1071no—
211C2×#135—1071no—
212#547D10711yescharacteristic, complemented
213#546—1071no—
214#497—1071no—
215#470—1071no—
216#465—1071no—
217#463—1071no—
218#461—1071no—
219#452—1071no—
220#452—1071no—
221#450—1071no—
222#449—1071no—
223#442—1071no—
224#435—1071no—
225C2×#41—1071no—
226C2×#41C2×D10711yescharacteristic
227C2×#41C2×D10711yescharacteristic, complemented
228C22×D4—1071no—
229AGL(1,ℤ/8)—2144no—
230#135—2144no—
231C2×Q16—1071no—
232C2×Q16—1071no—
233C2×SD16—1071no—
234C2×SD16—1071no—
235C2×SD16—1071no—
236C2×SD16—1071no—
237C2×SD16—1071no—
238C2×D8—1071no—
239C2×D8—1071no—
240C2×M16—1071no—
241C22×C8—1071no—
242C4⋊Q8C2×D10711yescharacteristic, complemented
243#124—1071no—
244#123—21no—
245#123—21no—
246#123—2141no—
247C22⋊Q8—1071no—
248C22⋊Q8C2×D10711yescharacteristic
249C22⋊Q8C2×D10711yescharacteristic, complemented
250#121—21no—
251#121—1071no—
252#121—2141no—
253#120—2141no—
254C4×D4—1071no—
255C4×D4C2×D10711yescharacteristic
256C4×D4C2×D10711yescharacteristic, complemented
257C2×C4⋊C4—21no—
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—
269#97—1071no—
270C24—2141no—
271#41—22no—
272#41—22no—
273#41—41no—
274#41—41no—
275#41—4281no—
276#41—2142no—
277C2×Q8—1071no—
278C2×Q8C22×D10711yescharacteristic
279C2×Q8C22×D10711yescharacteristic
280C2×D4—21no—
281C2×D4—21no—
282C2×D4—1071no—
283C2×D4—1071no—
284C2×D4—4281no—
285C2×D4—2142no—
286C2×D4—2141no—
287C2×D4—2141no—
288C2×D4—2141no—
289C2×D4C22⋊D10711yescharacteristic
290C2×D4C22×D10711yescharacteristic
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—2141no—
295C22×C4—2141no—
296C22×C4C22×D10711yescharacteristic
297C22×C4C22×D10711yescharacteristic
298Q16—4281no—
299Q16—2142no—
300SD16—4281no—
301SD16—2142no—
302SD16—2142no—
303SD16—2142no—
304SD16—2142no—
305D8—1074no—
306D8—2142no—
307M16—2142no—
308C2×C8—1071no—
309C2×C8—1071no—
310C2×C8—1071no—
311C2×C8—1071no—
312C2×C8—2142no—
313C4⋊C4—21no—
314C4⋊C4—21no—
315C4⋊C4—21no—
316C4⋊C4—21no—
317C4⋊C4—1071no—
318C4⋊C4—2141no—
319C4⋊C4C22⋊D10711yescharacteristic
320C4⋊C4C22×D10711yescharacteristic
321C22⋊C4—21no—
322C22⋊C4—21no—
323C22⋊C4—21no—
324C22⋊C4—2141no—
325C22⋊C4—2141no—
326C22⋊C4—2141no—
327C22⋊C4—2141no—
328C22⋊C4—2141no—
329C22⋊C4C22⋊D10711yescharacteristic
330C22⋊C4C22⋊D10711yescharacteristic
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—21no—
344Q8—2141no—
345Q8—2141no—
346D4—21no—
347D4—41no—
348D4—42no—
349D4—42no—
350D4—8561no—
351D4—1072no—
352D4—4282no—
353D4—4282no—
354D4—2141no—
355D4—2141no—
356D4—2141no—
357D4D4×D10712yes—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—42no—
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—21no—
387C22—22no—
388C22—41no—
389C22—41no—
390C22—8564no—
391C22—4281no—
392C22—4281no—
393C22—4281no—
394C22—4281no—
395C22—4281no—
396C22—2142no—
397C22C22×C22⋊D10711yescharacteristic
398C4—21no—
399C4—21no—
400C4—21no—
401C4—21no—
402C4—41no—
403C4—41no—
404C4—41no—
405C4—4281no—
406C4—4281no—
407C4C2×D4×D10711yescharacteristic
408C4#1681215411yescharacteristic
409C2—41no—
410C2—41no—
411C2—42no—
412C2—8561no—
413C2—4282no—
414C2D4×C22⋊D10711yescharacteristic
415C2#2029342511yescharacteristic
416C2#2029343511yescharacteristic
subgroupG/H
centerC22C22×C22⋊D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#547×C107C2
socleC22×C107C22×D4
hypercenter#547D107
hypocenterC107#3267
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C4×D4 ◁ #547

Complements

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

NH ≅ G/Ntype
1. C2×#16812134C2semidirect
2. C2×#16812129C2semidirect
3. #20293273C2semidirect
4. #20293019C2semidirect
5. #20292883C2semidirect
6. #20292858C2semidirect
7. #20292842C2semidirect
8. #20292829C2semidirect
9. #20292789C2semidirect
10. #20292787C2semidirect
11. #20292777C2semidirect
12. #20292772C2semidirect
13. #20292737C2semidirect
14. #20292699C2semidirect
15. #547×C107C2semidirect
20. C2×#13133808C22semidirect
21. C2×#13133808C22semidirect
22. 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. #16812020C22semidirect
51. C2×#41×C107C22semidirect
52. C4⋊Q8×C107C22semidirect
56. C22⋊Q8×C107C22semidirect
59. C4×D4×C107C22semidirect
82. #13133803D4semidirect
209. C107#3267semidirect
212. #547D107semidirect
227. C2×#41C2×D107semidirect
242. C4⋊Q8C2×D107semidirect
249. C22⋊Q8C2×D107semidirect
256. C4×D4C2×D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168121342
C2×#168121292
#202932732
#202930192
#202928832
#202928582
#202928422
#202928292
#202927892
#202927872
#202927772
#202927722
#202927372
#202926992
#547×C1072
#3267107