Finite Group Databasebeta

← #28347937

Subgroups — #28347937

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×#16812135C211yesmaximal, characteristic, complemented
2C22×#13133803C211yesmaximal, characteristic, complemented
3#20293273C211yesmaximal, characteristic, complemented
4#20293014C211yesmaximal, characteristic, complemented
5#20292878C211yesmaximal, characteristic, complemented
6#20292854C211yesmaximal, characteristic, complemented
7#20292842C211yesmaximal, characteristic, complemented
8#20292819C211yesmaximal, characteristic, complemented
9#20292778C211yesmaximal, characteristic, complemented
10#20292770C211yesmaximal, characteristic, complemented
11#20292761C211yesmaximal, characteristic, complemented
12#20292760C211yesmaximal, characteristic, complemented
13#20292737C211yesmaximal, characteristic, complemented
14#20292699C211yesmaximal, characteristic, complemented
15D42×C107C211yesmaximal, characteristic, complemented
16C2×#13133818C2211yescharacteristic, complemented
17C22×D428C2211yescharacteristic, complemented
18#16812135—24no—
19C2×#13133806C2211yescharacteristic, complemented
20C2×#13133806C2211yescharacteristic, complemented
21C2×#13133806C2211yescharacteristic, complemented
22C2×#13133803—24no—
23C2×#13133803C2211yescharacteristic
24C2×#13133803C2211yescharacteristic
25C2×#13133803C2211yescharacteristic, complemented
26C2×#13133803C2211yescharacteristic, complemented
27C2×#13133803C2211yescharacteristic, complemented
28C2×#13133803C2211yescharacteristic, complemented
29C2×#13133802C2211yescharacteristic, complemented
30C22×C107⋊C8C2211yescharacteristic, complemented
31#16812097C2211yescharacteristic, complemented
32#16812094—21no—
33#16812088C2211yescharacteristic, complemented
34#16812083C2211yescharacteristic, complemented
35#16812082—21no—
36#16812077—21no—
37C4×D428C2211yescharacteristic, complemented
38C22⋊C4×D107—21no—
39#16812040C2211yescharacteristic, complemented
40C4⋊(C107⋊C8)C2211yescharacteristic, complemented
41#16812032C2211yescharacteristic, complemented
42#16812030C2211yescharacteristic, complemented
43#16812030C2211yescharacteristic, complemented
44#16812030C2211yescharacteristic, complemented
45#16812029C2211yescharacteristic, complemented
46#16812029C2211yescharacteristic, complemented
47C22⋊(C107⋊C8)C2211yescharacteristic, complemented
48C22⋊(C107⋊C8)C2211yescharacteristic, complemented
49#16812020C2211yescharacteristic, complemented
50C22×D4×C107—21no—
51C22×D4×C107C2211yescharacteristic
52C22×D4×C107C2211yescharacteristic, complemented
53#127×C107C2211yescharacteristic, complemented
54#121×C107—21no—
55#121×C107C2211yescharacteristic
56#121×C107C2211yescharacteristic, complemented
57#120×C107—21no—
58#120×C107—21no—
59C4×D4×C107C2211yescharacteristic
60C4×D4×C107C2211yescharacteristic, complemented
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#13133806—22no—
75#13133806—22no—
76#13133806—41no—
77#13133803—22no—
78#13133803—22no—
79#13133803—22no—
80#13133803—22no—
81#13133803—41no—
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—
96C24×C107—21no—
97C24×C107—21no—
98C2×D4×C107—22no—
99C2×D4×C107—22no—
100C2×D4×C107—21no—
101C2×D4×C107—21no—
102C2×D4×C107—21no—
103C2×D4×C107—21no—
104C2×D4×C107—21no—
105C2×D4×C107—21no—
106C2×D4×C107—41no—
107C2×D4×C107—41no—
108C2×D4×C107—44no—
109C2×D4×C107D411yescharacteristic
110C2×D4×C107C2311yescharacteristic
111C2×D4×C107C2311yescharacteristic
112C2×D4×C107C2311yescharacteristic
113C22×C4×C107—21no—
114C22×C4×C107C2311yescharacteristic
115C22×C4×C107C2311yescharacteristic
116C4⋊C4×C107D411yescharacteristic
117C4⋊C4×C107C2311yescharacteristic
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—21no—
127C23×C107—42no—
128C23×C107—42no—
129C23×C107—42no—
130C23×C107—42no—
131C23×C107—81no—
132C23×C107—81no—
133C23×C107C2×D411yescharacteristic
134C23×C107C2×D411yescharacteristic
135C22×D107—21no—
136C22×D107—21no—
137C22×D107—42no—
138C22×D107—42no—
139C22×D107—81no—
140D4×C107—22no—
141D4×C107—21no—
142D4×C107—21no—
143D4×C107—21no—
144D4×C107—21no—
145D4×C107—21no—
146D4×C107—42no—
147D4×C107—42no—
148D4×C107—42no—
149D4×C107—42no—
150D4×C107—42no—
151D4×C107—81no—
152D4×C107C2×D412yes—
153C2×C4×C107—21no—
154C2×C4×C107—21no—
155C2×C4×C107—21no—
156C2×C4×C107—21no—
157C2×C4×C107—21no—
158C2×C4×C107—42no—
159C2×C4×C107C2×D411yescharacteristic
160C2×C4×C107C2×D411yescharacteristic
161C2×C4×C107C2×D411yescharacteristic
162C2×C4×C107C2×D411yescharacteristic
163C2×C4×C107C2411yescharacteristic
164C22⋊D107—42no—
165C22⋊D107—81no—
166C2×Dic107—21no—
167C2×Dic107—21no—
168D428—21no—
169D428—21no—
170D428—21no—
171D428—42no—
172D428C2×D412yescomplemented
173C4×D107—42no—
174C4×D107—81no—
175Dic214—21no—
176Dic214—21no—
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—41no—
193C22×C107—41no—
194C22×C107—41no—
195C22×C107—84no—
196C22×C107—84no—
197C22×C107AGL(1,ℤ/8)12yescomplemented
198C22×C107C22×D411yescharacteristic
199C2×D107—22no—
200C2×D107—41no—
201C2×D107—41no—
202C2×D107—41no—
203C2×D107—41no—
204C2×D107—41no—
205C2×D107—84no—
206C4×C107—21no—
207C4×C107—21no—
208C4×C107—21no—
209C4×C107—21no—
210C4×C107—41no—
211C4×C107C22×D411yescharacteristic
212C4×C107H(5,𝔽2)11yescharacteristic
213Dic107—41no—
214Dic107—41no—
215C2×C107—22no—
216C2×C107—41no—
217C2×C107—42no—
218C2×C107—81no—
219C2×C107—81no—
220C2×C107D4211yescharacteristic
221C2×C107C2×AGL(1,ℤ/8)11yescharacteristic
222C2×C107#57611yescharacteristic
223D107—42no—
224D107—81no—
225#3264—1071nomaximal, Sylow
226C107#326411yescharacteristic, complemented, Sylow
227C2×AGL(1,ℤ/8)—1071no—
228C22×D8—1071no—
229#546—1071no—
230D42D10711yescharacteristic, complemented
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×#41—1071no—
243C22×D4—21no—
244C22×D4—1071no—
245C22×D4C2×D10711yescharacteristic
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#127C2×D10711yescharacteristic, complemented
261#124—1071no—
262#123—2141no—
263C22⋊Q8—1071no—
264#121—21no—
265#121—1071no—
266#121—2141no—
267#121C2×D10711yescharacteristic
268#121C2×D10711yescharacteristic, complemented
269#120—21no—
270#120—21no—
271#120—2141no—
272C4×D4—1071no—
273C4×D4C2×D10711yescharacteristic
274C4×D4C2×D10711yescharacteristic, complemented
275C2×C22⋊C4—2141no—
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—21no—
289C24—2141no—
290#41—4281no—
291#41—2142no—
292C2×Q8—1071no—
293C2×D4—22no—
294C2×D4—22no—
295C2×D4—21no—
296C2×D4—21no—
297C2×D4—21no—
298C2×D4—21no—
299C2×D4—21no—
300C2×D4—21no—
301C2×D4—41no—
302C2×D4—41no—
303C2×D4—44no—
304C2×D4—1071no—
305C2×D4—1071no—
306C2×D4—4281no—
307C2×D4—2142no—
308C2×D4—2141no—
309C2×D4—2141no—
310C2×D4—2141no—
311C2×D4C22⋊D10711yescharacteristic
312C2×D4C22×D10711yescharacteristic
313C2×D4C22×D10711yescharacteristic
314C2×D4C22×D10711yescharacteristic
315C22×C4—21no—
316C22×C4—2141no—
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—1071no—
336C4⋊C4—2141no—
337C4⋊C4C22⋊D10711yescharacteristic
338C4⋊C4C22×D10711yescharacteristic
339C22⋊C4—21no—
340C22⋊C4—21no—
341C22⋊C4—21no—
342C22⋊C4—2141no—
343C22⋊C4—2141no—
344C22⋊C4—2141no—
345C22⋊C4—2141no—
346C22⋊C4—2141no—
347C22⋊C4C22⋊D10711yescharacteristic
348C22⋊C4C22⋊D10711yescharacteristic
349C42C22×D10711yescharacteristic
350C23—21no—
351C23—21no—
352C23—21no—
353C23—42no—
354C23—42no—
355C23—42no—
356C23—42no—
357C23—81no—
358C23—81no—
359C23—8561no—
360C23—4282no—
361C23—4282no—
362C23—2141no—
363C23—2141no—
364C23C2×C22⋊D10711yescharacteristic
365C23C2×C22⋊D10711yescharacteristic
366Q8—2141no—
367Q8—2141no—
368D4—22no—
369D4—21no—
370D4—21no—
371D4—21no—
372D4—21no—
373D4—21no—
374D4—42no—
375D4—42no—
376D4—42no—
377D4—42no—
378D4—42no—
379D4—81no—
380D4—8561no—
381D4—1072no—
382D4—4282no—
383D4—4282no—
384D4—2141no—
385D4—2141no—
386D4—2141no—
387D4D4×D10712yes—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—42no—
394C2×C4—8561no—
395C2×C4—4282no—
396C2×C4—2141no—
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—41no—
419C22—41no—
420C22—41no—
421C22—84no—
422C22—84no—
423C22—8564no—
424C22—4281no—
425C22—4281no—
426C22—4281no—
427C22—4281no—
428C22—4281no—
429C22—2142no—
430C22#1681213512yescomplemented
431C22C22×C22⋊D10711yescharacteristic
432C4—21no—
433C4—21no—
434C4—21no—
435C4—21no—
436C4—41no—
437C4—4281no—
438C4—4281no—
439C4C2×D4×D10711yescharacteristic
440C4#1681215411yescharacteristic
441C2—22no—
442C2—41no—
443C2—42no—
444C2—81no—
445C2—81no—
446C2—8561no—
447C2—4282no—
448C2D4×C22⋊D10711yescharacteristic
449C2C2×#1681213511yescharacteristic
450C2#2029343011yescharacteristic
subgroupG/H
centerC22C22×C22⋊D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
FittingD42×C107C2
socleC22×C107C22×D4
hypercenterD42D107
hypocenterC107#3264
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C4×D4 ◁ D42

Complements

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

NH ≅ G/Ntype
1. C2×#16812135C2semidirect
2. C22×#13133803C2semidirect
3. #20293273C2semidirect
4. #20293014C2semidirect
5. #20292878C2semidirect
6. #20292854C2semidirect
7. #20292842C2semidirect
8. #20292819C2semidirect
9. #20292778C2semidirect
10. #20292770C2semidirect
11. #20292761C2semidirect
12. #20292760C2semidirect
13. #20292737C2semidirect
14. #20292699C2semidirect
15. D42×C107C2semidirect
16. C2×#13133818C22semidirect
17. C22×D428C22semidirect
19. C2×#13133806C22semidirect
20. C2×#13133806C22semidirect
21. C2×#13133806C22semidirect
25. C2×#13133803C22semidirect
26. C2×#13133803C22semidirect
27. C2×#13133803C22semidirect
28. C2×#13133803C22semidirect
29. C2×#13133802C22semidirect
30. C22×C107⋊C8C22semidirect
31. #16812097C22semidirect
33. #16812088C22semidirect
34. #16812083C22semidirect
37. C4×D428C22semidirect
39. #16812040C22semidirect
40. C4⋊(C107⋊C8)C22semidirect
41. #16812032C22semidirect
42. #16812030C22semidirect
43. #16812030C22semidirect
44. #16812030C22semidirect
45. #16812029C22semidirect
46. #16812029C22semidirect
47. C22⋊(C107⋊C8)C22semidirect
48. C22⋊(C107⋊C8)C22semidirect
49. #16812020C22semidirect
52. C22×D4×C107C22semidirect
53. #127×C107C22semidirect
56. #121×C107C22semidirect
60. C4×D4×C107C22semidirect
82. #13133803D4semidirect
172. D428C2×D4semidirect
180. C107⋊C8C2×D4semidirect
197. C22×C107AGL(1,ℤ/8)semidirect
226. C107#3264semidirect
230. D42D107semidirect
246. C22×D4C2×D107semidirect
260. #127C2×D107semidirect
268. #121C2×D107semidirect
274. C4×D4C2×D107semidirect
430. C22#16812135semidirect

Maximal subgroups

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

subgroupindex
C2×#168121352
C22×#131338032
#202932732
#202930142
#202928782
#202928542
#202928422
#202928192
#202927782
#202927702
#202927612
#202927602
#202927372
#202926992
D42×C1072
#3264107