Finite Group Databasebeta

← #28347942

Subgroups — #28347942

Order 13696.

568 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16812134C211yesmaximal, characteristic, complemented
2C22×#13133803C211yesmaximal, characteristic, complemented
3D4×D428C211yesmaximal, characteristic, complemented
4#20293017C211yesmaximal, characteristic, complemented
5#20292878C211yesmaximal, characteristic, complemented
6#20292858C211yesmaximal, characteristic, complemented
7#20292839C211yesmaximal, characteristic, complemented
8#20292822C211yesmaximal, characteristic, complemented
9#20292777C211yesmaximal, characteristic, complemented
10#20292772C211yesmaximal, characteristic, complemented
11#20292761C211yesmaximal, characteristic, complemented
12#20292760C211yesmaximal, characteristic, complemented
13#20292737C211yesmaximal, characteristic, complemented
14#20292699C211yesmaximal, characteristic, complemented
15#546×C107C211yesmaximal, characteristic, complemented
16C22×D428C2211yescharacteristic
17C22×D428C2211yescharacteristic, complemented
18C2×D4×D107—21no—
19#16812134—24no—
20C2×#13133805C2211yescharacteristic, complemented
21C2×#13133805C2211yescharacteristic, complemented
22C2×#13133805C2211yescharacteristic, complemented
23C2×#13133803—24no—
24C2×#13133803C2211yescharacteristic
25C2×#13133803C2211yescharacteristic
26C2×#13133803C2211yescharacteristic, complemented
27C2×#13133803C2211yescharacteristic, complemented
28C2×#13133803C2211yescharacteristic, complemented
29C2×#13133803C2211yescharacteristic, complemented
30C2×#13133802C2211yescharacteristic, complemented
31C22×C107⋊C8C2211yescharacteristic, complemented
32#16812106C2211yescharacteristic, complemented
33#16812084—21no—
34#16812083C2211yescharacteristic
35#16812083C2211yescharacteristic, complemented
36#16812079—21no—
37#16812079—21no—
38C4×D428C2211yescharacteristic, complemented
39#16812040C2211yescharacteristic, complemented
40C4⋊(C107⋊C8)C2211yescharacteristic, complemented
41#16812033C2211yescharacteristic, complemented
42#16812030C2211yescharacteristic, complemented
43#16812030C2211yescharacteristic, complemented
44#16812029C2211yescharacteristic, complemented
45#16812029C2211yescharacteristic, complemented
46#16812029C2211yescharacteristic, complemented
47C22⋊(C107⋊C8)C2211yescharacteristic, complemented
48C22⋊(C107⋊C8)C2211yescharacteristic, complemented
49#16812020C2211yescharacteristic, complemented
50C2×#41×C107C2211yescharacteristic, complemented
51C22×D4×C107C2211yescharacteristic, complemented
52#124×C107C2211yescharacteristic, complemented
53#123×C107—21no—
54C22⋊Q8×C107C2211yescharacteristic, complemented
55#121×C107—21no—
56#121×C107C2211yescharacteristic, complemented
57#120×C107—21no—
58C4×D4×C107C2211yescharacteristic
59C4×D4×C107C2211yescharacteristic, complemented
60C2×C22⋊C4×C107—21no—
61C23×D107—21no—
62C23×D107—21no—
63C2×D428—22no—
64C2×D428—22no—
65C2×D428—21no—
66C2×D428—21no—
67C2×D428—21no—
68C2×D428—21no—
69C2×D428—41no—
70C2×D428—41no—
71C2×D428C2311yescharacteristic
72C2×D428C2311yescharacteristic
73C2×D428C2311yescharacteristic
74C2×C22⋊D107—21no—
75C2×C22⋊D107—21no—
76D4×D107—44no—
77C2×C4×D107—21no—
78#13133805—22no—
79#13133805—22no—
80#13133805—41no—
81#13133803—22no—
82#13133803—22no—
83#13133803—22no—
84#13133803—22no—
85#13133803—41no—
86#13133803D414yescomplemented
87#13133802—22no—
88C2×C107⋊C8—22no—
89C2×C107⋊C8C2311yescharacteristic
90C2×C107⋊C8C2311yescharacteristic
91C2×C107⋊C8C2311yescharacteristic
92C2×C107⋊C8C2311yescharacteristic
93C4⋊Dic107C2311yescharacteristic
94(C2×D107)⋊C4—21no—
95(C2×D107)⋊C4—21no—
96(C2×D107)⋊C4—21no—
97C24×C107—21no—
98#41×C107—22no—
99#41×C107—41no—
100C2×Q8×C107C2311yescharacteristic
101C2×D4×C107—22no—
102C2×D4×C107—21no—
103C2×D4×C107—21no—
104C2×D4×C107—21no—
105C2×D4×C107—41no—
106C2×D4×C107D411yescharacteristic
107C2×D4×C107C2311yescharacteristic
108C2×D4×C107C2311yescharacteristic
109C22×C4×C107—21no—
110C22×C4×C107—21no—
111C22×C4×C107C2311yescharacteristic
112C22×C4×C107C2311yescharacteristic
113C4⋊C4×C107—21no—
114C4⋊C4×C107D411yescharacteristic
115C4⋊C4×C107C2311yescharacteristic
116C22⋊C4×C107—21no—
117C22⋊C4×C107—21no—
118C22⋊C4×C107—21no—
119C22⋊C4×C107—21no—
120C22⋊C4×C107—21no—
121C22⋊C4×C107D411yescharacteristic
122C22⋊C4×C107D411yescharacteristic
123C42×C107C2311yescharacteristic
124C23×C107—21no—
125C23×C107—21no—
126C23×C107—42no—
127C23×C107—42no—
128C23×C107—81no—
129C23×C107C2×D411yescharacteristic
130C23×C107C2×D411yescharacteristic
131C22×D107—21no—
132C22×D107—21no—
133C22×D107—21no—
134C22×D107—42no—
135C22×D107—42no—
136C22×D107—42no—
137C22×D107—42no—
138C22×D107—81no—
139C22×D107—81no—
140Q8×C107—21no—
141Q8×C107—21no—
142D4×C107—22no—
143D4×C107—21no—
144D4×C107—21no—
145D4×C107—21no—
146D4×C107—42no—
147D4×C107—42no—
148D4×C107—81no—
149D4×C107C2×D412yescomplemented
150C2×C4×C107—21no—
151C2×C4×C107—21no—
152C2×C4×C107—21no—
153C2×C4×C107—21no—
154C2×C4×C107—21no—
155C2×C4×C107—21no—
156C2×C4×C107—42no—
157C2×C4×C107—81no—
158C2×C4×C107C2×D411yescharacteristic
159C2×C4×C107C2×D411yescharacteristic
160C2×C4×C107C2×D411yescharacteristic
161C2×C4×C107C2×D411yescharacteristic
162C2×C4×C107C2411yescharacteristic
163C22⋊D107—42no—
164C22⋊D107—42no—
165C2×Dic107—21no—
166D428—21no—
167D428—21no—
168D428—21no—
169D428—21no—
170D428—21no—
171D428—42no—
172D428—42no—
173D428—42no—
174D428—81no—
175D428C2×D412yes—
176C4×D107—42no—
177C107⋊C8—21no—
178C107⋊C8—21no—
179C107⋊C8—21no—
180C107⋊C8C2×D412yescomplemented
181C22×C107—21no—
182C22×C107—21no—
183C22×C107—21no—
184C22×C107—22no—
185C22×C107—21no—
186C22×C107—21no—
187C22×C107—41no—
188C22×C107—41no—
189C22×C107—41no—
190C22×C107—41no—
191C22×C107—41no—
192C22×C107—84no—
193C22×C107AGL(1,ℤ/8)12yescomplemented
194C22×C107C22×D411yescharacteristic
195C2×D107—22no—
196C2×D107—41no—
197C2×D107—41no—
198C2×D107—41no—
199C2×D107—41no—
200C2×D107—41no—
201C2×D107—41no—
202C2×D107—41no—
203C2×D107—41no—
204C2×D107—84no—
205C2×D107—84no—
206C4×C107—21no—
207C4×C107—21no—
208C4×C107—21no—
209C4×C107—21no—
210C4×C107—41no—
211C4×C107—41no—
212C4×C107C22×D411yescharacteristic
213C4×C107H(5,𝔽2)11yescharacteristic
214Dic107—41no—
215C2×C107—22no—
216C2×C107—41no—
217C2×C107—42no—
218C2×C107—81no—
219C2×C107D4211yescharacteristic
220C2×C107C2×AGL(1,ℤ/8)11yescharacteristic
221C2×C107#57611yescharacteristic
222D107—42no—
223D107—81no—
224D107—81no—
225#3264—1071nomaximal, Sylow
226C107#326411yescharacteristic, complemented, Sylow
227C2×AGL(1,ℤ/8)—1071no—
228C22×D8—1071no—
229#546D10711yescharacteristic, complemented
230D42—1071no—
231#496—1071no—
232#469—1071no—
233#465—1071no—
234#463—1071no—
235#459—1071no—
236#450—1071no—
237#449—1071no—
238#447—1071no—
239#447—1071no—
240#442—1071no—
241#435—1071no—
242C2×#41C2×D10711yescharacteristic, complemented
243C22×D4—1071no—
244C22×D4—1071no—
245C22×D4—2141no—
246C22×D4C2×D10711yescharacteristic, complemented
247AGL(1,ℤ/8)—2144no—
248C2×SD16—1071no—
249C2×SD16—1071no—
250C2×SD16—1071no—
251C2×D8—1071no—
252C2×D8—1071no—
253C2×D8—1071no—
254C2×D8—1071no—
255C2×D8—1071no—
256C2×D8—1071no—
257C2×D8—2144no—
258C2×M16—1071no—
259C22×C8—1071no—
260#127—1071no—
261#124C2×D10711yescharacteristic, complemented
262#123—21no—
263C22⋊Q8C2×D10711yescharacteristic, complemented
264#121—21no—
265#121—1071no—
266#121—1071no—
267#121—2141no—
268#121C2×D10711yescharacteristic, complemented
269#120—21no—
270#120—2141no—
271#120—2141no—
272C4×D4—1071no—
273C4×D4C2×D10711yescharacteristic
274C4×D4C2×D10711yescharacteristic, complemented
275C2×C22⋊C4—21no—
276#106—1071no—
277C4⋊C8—1071no—
278#103—1071no—
279#102—1071no—
280#102—1071no—
281#102—1071no—
282#102—1071no—
283#102—1071no—
284C22⋊C8—1071no—
285C22⋊C8—1071no—
286#97—1071no—
287C24—21no—
288C24—2141no—
289C24—2141no—
290#41—22no—
291#41—41no—
292C2×Q8C22×D10711yescharacteristic
293C2×D4—22no—
294C2×D4—21no—
295C2×D4—21no—
296C2×D4—21no—
297C2×D4—41no—
298C2×D4—1071no—
299C2×D4—1071no—
300C2×D4—1071no—
301C2×D4—4281no—
302C2×D4—4281no—
303C2×D4—4284no—
304C2×D4—2142no—
305C2×D4—2142no—
306C2×D4—2141no—
307C2×D4—2141no—
308C2×D4—2141no—
309C2×D4—2141no—
310C2×D4—2141no—
311C2×D4—2141no—
312C2×D4C22⋊D10711yescharacteristic
313C2×D4C22×D10711yescharacteristic
314C2×D4C22×D10711yescharacteristic
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—2141no—
318C22×C4C22×D10711yescharacteristic
319C22×C4C22×D10711yescharacteristic
320SD16—4281no—
321SD16—2142no—
322SD16—2142no—
323D8—1074no—
324D8—4281no—
325D8—2142no—
326D8—2142no—
327D8—2142no—
328D8—2142no—
329M16—2142no—
330C2×C8—1071no—
331C2×C8—1071no—
332C2×C8—1071no—
333C2×C8—1071no—
334C2×C8—2142no—
335C4⋊C4—21no—
336C4⋊C4—1071no—
337C4⋊C4C22⋊D10711yescharacteristic
338C4⋊C4C22×D10711yescharacteristic
339C22⋊C4—21no—
340C22⋊C4—21no—
341C22⋊C4—21no—
342C22⋊C4—21no—
343C22⋊C4—21no—
344C22⋊C4—2141no—
345C22⋊C4—2141no—
346C22⋊C4—2141no—
347C22⋊C4C22⋊D10711yescharacteristic
348C22⋊C4C22⋊D10711yescharacteristic
349C42C22×D10711yescharacteristic
350C23—21no—
351C23—21no—
352C23—42no—
353C23—42no—
354C23—81no—
355C23—8561no—
356C23—8561no—
357C23—4282no—
358C23—4282no—
359C23—4282no—
360C23—4282no—
361C23—2141no—
362C23—2141no—
363C23—2141no—
364C23C2×C22⋊D10711yescharacteristic
365C23C2×C22⋊D10711yescharacteristic
366Q8—21no—
367Q8—21no—
368D4—22no—
369D4—21no—
370D4—21no—
371D4—21no—
372D4—42no—
373D4—42no—
374D4—81no—
375D4—8561no—
376D4—1072no—
377D4—4282no—
378D4—4282no—
379D4—4282no—
380D4—4282no—
381D4—4282no—
382D4—2141no—
383D4—2141no—
384D4—2141no—
385D4—2141no—
386D4—2141no—
387D4D4×D10712yescomplemented
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—42no—
395C2×C4—81no—
396C2×C4—4282no—
397C2×C4—2141no—
398C2×C4C2×C22⋊D10711yescharacteristic
399C2×C4C2×C22⋊D10711yescharacteristic
400C2×C4C2×C22⋊D10711yescharacteristic
401C2×C4C2×C22⋊D10711yescharacteristic
402C2×C4C23×D10711yescharacteristic
403C8—1072no—
404C8—2141no—
405C8—2141no—
406C8—2141no—
407C22—21no—
408C22—21no—
409C22—21no—
410C22—21no—
411C22—21no—
412C22—22no—
413C22—41no—
414C22—41no—
415C22—41no—
416C22—41no—
417C22—41no—
418C22—84no—
419C22—8564no—
420C22—8564no—
421C22—4281no—
422C22—4281no—
423C22—4281no—
424C22—4281no—
425C22—4281no—
426C22—4281no—
427C22—4281no—
428C22—4281no—
429C22—2142no—
430C22#1681213412yescomplemented
431C22C22×C22⋊D10711yescharacteristic
432C4—21no—
433C4—21no—
434C4—21no—
435C4—21no—
436C4—41no—
437C4—41no—
438C4—4281no—
439C4C2×D4×D10711yescharacteristic
440C4#1681215411yescharacteristic
441C2—22no—
442C2—41no—
443C2—42no—
444C2—81no—
445C2—8561no—
446C2—8561no—
447C2—4282no—
448C2D4×C22⋊D10711yescharacteristic
449C2C2×#1681213411yescharacteristic
450C2#2029343011yescharacteristic
subgroupG/H
centerC22C22×C22⋊D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#546×C107C2
socleC22×C107C22×D4
hypercenter#546D107
hypocenterC107#3264
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C4×D4 ◁ #546

Complements

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

NH ≅ G/Ntype
1. C2×#16812134C2semidirect
2. C22×#13133803C2semidirect
3. D4×D428C2semidirect
4. #20293017C2semidirect
5. #20292878C2semidirect
6. #20292858C2semidirect
7. #20292839C2semidirect
8. #20292822C2semidirect
9. #20292777C2semidirect
10. #20292772C2semidirect
11. #20292761C2semidirect
12. #20292760C2semidirect
13. #20292737C2semidirect
14. #20292699C2semidirect
15. #546×C107C2semidirect
17. C22×D428C22semidirect
20. C2×#13133805C22semidirect
21. C2×#13133805C22semidirect
22. C2×#13133805C22semidirect
26. C2×#13133803C22semidirect
27. C2×#13133803C22semidirect
28. C2×#13133803C22semidirect
29. C2×#13133803C22semidirect
30. C2×#13133802C22semidirect
31. C22×C107⋊C8C22semidirect
32. #16812106C22semidirect
35. #16812083C22semidirect
38. C4×D428C22semidirect
39. #16812040C22semidirect
40. C4⋊(C107⋊C8)C22semidirect
41. #16812033C22semidirect
42. #16812030C22semidirect
43. #16812030C22semidirect
44. #16812029C22semidirect
45. #16812029C22semidirect
46. #16812029C22semidirect
47. C22⋊(C107⋊C8)C22semidirect
48. C22⋊(C107⋊C8)C22semidirect
49. #16812020C22semidirect
50. C2×#41×C107C22semidirect
51. C22×D4×C107C22semidirect
52. #124×C107C22semidirect
54. C22⋊Q8×C107C22semidirect
56. #121×C107C22semidirect
59. C4×D4×C107C22semidirect
86. #13133803D4semidirect
149. D4×C107C2×D4semidirect
180. C107⋊C8C2×D4semidirect
193. C22×C107AGL(1,ℤ/8)semidirect
226. C107#3264semidirect
229. #546D107semidirect
242. C2×#41C2×D107semidirect
246. C22×D4C2×D107semidirect
261. #124C2×D107semidirect
263. C22⋊Q8C2×D107semidirect
268. #121C2×D107semidirect
274. C4×D4C2×D107semidirect
387. D4D4×D107semidirect
430. C22#16812134semidirect

Maximal subgroups

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

subgroupindex
C2×#168121342
C22×#131338032
D4×D4282
#202930172
#202928782
#202928582
#202928392
#202928222
#202927772
#202927722
#202927612
#202927602
#202927372
#202926992
#546×C1072
#3264107