Finite Group Databasebeta

← #28347936

Subgroups — #28347936

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×#16812133C211yesmaximal, characteristic, complemented
2C2×D8×D107C211yesmaximal, characteristic, complemented
3#20293272C211yesmaximal, characteristic, complemented
4#20293262C211yesmaximal, characteristic, complemented
5#20293014C211yesmaximal, characteristic, complemented
6#20292879C211yesmaximal, characteristic, complemented
7#20292855C211yesmaximal, characteristic, complemented
8#20292843C211yesmaximal, characteristic, complemented
9#20292779C211yesmaximal, characteristic, complemented
10#20292769C211yesmaximal, characteristic, complemented
11#20292762C211yesmaximal, characteristic, complemented
12#20292759C211yesmaximal, characteristic, complemented
13#20292737C211yesmaximal, characteristic, complemented
14#20292697C211yesmaximal, characteristic, complemented
15#459×C107C211yesmaximal, characteristic, complemented
16C2×#13133817C2211yescharacteristic, complemented
17C2×D4×D107—21no—
18C2×D4×D107C2211yescharacteristic
19C2×D4×D107C2211yescharacteristic, complemented
20C2×D4×D107C2211yescharacteristic, complemented
21#16812133—24no—
22C2×#13133807C2211yescharacteristic, complemented
23C2×#13133806C2211yescharacteristic, complemented
24C2×#13133806C2211yescharacteristic, complemented
25C2×D856C2211yescharacteristic
26C2×#13133803C2211yescharacteristic
27C2×#13133803C2211yescharacteristic, complemented
28C2×#13133803C2211yescharacteristic, complemented
29D8×D107—24no—
30C2×#13133801C2211yescharacteristic, complemented
31C2×C8×D107C2211yescharacteristic, complemented
32#16812097C2211yescharacteristic, complemented
33#16812093—21no—
34#16812089C2211yescharacteristic, complemented
35#16812084C2211yescharacteristic, complemented
36#16812083—21no—
37#16812080—21no—
38#16812080C2211yescharacteristic
39#16812080C2211yescharacteristic, complemented
40#16812079—21no—
41#16812077—21no—
42#16812077—21no—
43C4×D428C2211yescharacteristic
44C4×D428C2211yescharacteristic, complemented
45C2×(C2×D107)⋊C4—21no—
46#16812040C2211yescharacteristic, complemented
47#16812034C2211yescharacteristic, complemented
48#16812031C2211yescharacteristic, complemented
49#16812030C2211yescharacteristic, complemented
50#16812029C2211yescharacteristic, complemented
51(C2×D107)⋊C8C2211yescharacteristic, complemented
52(C2×D107)⋊C8C2211yescharacteristic, complemented
53#16812020C2211yescharacteristic, complemented
54C2×D8×C107C2211yescharacteristic, complemented
55C2×D8×C107C2211yescharacteristic, complemented
56#127×C107C2211yescharacteristic, complemented
57C4×D4×C107C2211yescharacteristic, complemented
58C4⋊C8×C107C2211yescharacteristic, complemented
59#102×C107C2211yescharacteristic, complemented
60#102×C107C2211yescharacteristic, complemented
61C23×D107—21no—
62C23×D107—21no—
63C23×D107—21no—
64#13133817—22no—
65#13133817—41no—
66C2×Dic214C2311yescharacteristic
67C2×D428—21no—
68C2×D428D411yescharacteristic
69C2×D428C2311yescharacteristic
70C2×D428C2311yescharacteristic
71C2×C22⋊D107—21no—
72C2×C22⋊D107—21no—
73C2×C22⋊D107—21no—
74C2×C22⋊D107—21no—
75C2×C22⋊D107—21no—
76C2×C22⋊D107—21no—
77D4×D107—22no—
78D4×D107—22no—
79D4×D107—22no—
80D4×D107—41no—
81D4×D107—41no—
82D4×D107—41no—
83D4×D107—44no—
84C22×Dic107—21no—
85C2×C4×D107—21no—
86C2×C4×D107C2311yescharacteristic
87C2×C4×D107C2311yescharacteristic
88#13133807—22no—
89#13133806—22no—
90#13133806—41no—
91D856—22no—
92#13133803—22no—
93#13133803—41no—
94#13133803D414yescomplemented
95#13133801—22no—
96C2×C107⋊C8C2311yescharacteristic
97C2×C107⋊C8C2311yescharacteristic
98C8×D107—22no—
99C4⋊Dic107—21no—
100C4⋊Dic107D411yescharacteristic
101C4⋊Dic107C2311yescharacteristic
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—
108(C2×D107)⋊C4—21no—
109(C2×D107)⋊C4D411yescharacteristic
110(C2×D107)⋊C4D411yescharacteristic
111C2×D4×C107—21no—
112C2×D4×C107—21no—
113C2×D4×C107C2311yescharacteristic
114C2×D4×C107C2311yescharacteristic
115C2×D4×C107C2311yescharacteristic
116C22×C4×C107—21no—
117D8×C107—22no—
118D8×C107—22no—
119C2×C8×C107C2311yescharacteristic
120C2×C8×C107C2311yescharacteristic
121C4⋊C4×C107C2311yescharacteristic
122C22⋊C4×C107—21no—
123C42×C107C2311yescharacteristic
124C23×C107—21no—
125C23×C107—21no—
126C23×C107—21no—
127C22×D107—21no—
128C22×D107—21no—
129C22×D107—42no—
130C22×D107—42no—
131C22×D107—42no—
132C22×D107—42no—
133C22×D107—42no—
134C22×D107—42no—
135C22×D107—81no—
136C22×D107—81no—
137C22×D107—81no—
138C22×D107C2×D411yescharacteristic
139C22×D107C2×D411yescharacteristic
140D4×C107—21no—
141D4×C107—21no—
142D4×C107—21no—
143D4×C107—21no—
144D4×C107—21no—
145D4×C107—42no—
146D4×C107—81no—
147D4×C107C2×D412yescomplemented
148C2×C4×C107—21no—
149C2×C4×C107—81no—
150C2×C4×C107C2×D411yescharacteristic
151C2×C4×C107C2×D411yescharacteristic
152C2×C4×C107C2411yescharacteristic
153C22⋊D107—42no—
154C22⋊D107—42no—
155C22⋊D107—42no—
156C22⋊D107—42no—
157C22⋊D107—42no—
158C22⋊D107—42no—
159C2×Dic107—21no—
160C2×Dic107—21no—
161C2×Dic107—42no—
162C2×Dic107C2×D411yescharacteristic
163C2×Dic107C2×D411yescharacteristic
164D428—22no—
165D428—21no—
166D428—21no—
167D428—21no—
168D428—81no—
169D428C2×D412yes—
170C4×D107—21no—
171C4×D107—21no—
172C4×D107—21no—
173C4×D107—21no—
174C4×D107—42no—
175Dic214—21no—
176Dic214—21no—
177C8×C107—21no—
178C8×C107—21no—
179C107⋊C8—21no—
180C107⋊C8C2×D412yescomplemented
181C22×C107—22no—
182C22×C107—41no—
183C22×C107—41no—
184C22×C107—41no—
185C22×C107—41no—
186C22×C107—41no—
187C22×C107—41no—
188C22×C107—41no—
189C22×C107—41no—
190C22×C107C22×D411yescharacteristic
191C2×D107—21no—
192C2×D107—21no—
193C2×D107—21no—
194C2×D107—22no—
195C2×D107—21no—
196C2×D107—21no—
197C2×D107—41no—
198C2×D107—41no—
199C2×D107—41no—
200C2×D107—41no—
201C2×D107—41no—
202C2×D107—84no—
203C2×D107—84no—
204C2×D107—84no—
205C2×D107AGL(1,ℤ/8)12yescomplemented
206C4×C107—21no—
207C4×C107—21no—
208C4×C107—41no—
209C4×C107C22×D411yescharacteristic
210C4×C107H(5,𝔽2)11yescharacteristic
211Dic107—21no—
212Dic107—21no—
213Dic107—41no—
214Dic107—41no—
215C2×C107—42no—
216C2×C107—81no—
217C2×C107—81no—
218C2×C107D4211yescharacteristic
219C2×C107C2×AGL(1,ℤ/8)11yescharacteristic
220C2×C107#57611yescharacteristic
221D107—22no—
222D107—41no—
223D107—42no—
224D107—81no—
225#3264—1071nomaximal, Sylow
226C107#326411yescharacteristic, complemented, Sylow
227C2×AGL(1,ℤ/8)—1071no—
228C22×D8—1071no—
229#546—1071no—
230D42—1071no—
231#496—1071no—
232#469—1071no—
233#465—1071no—
234#463—1071no—
235#459D10711yescharacteristic, complemented
236#450—1071no—
237#449—1071no—
238#447—1071no—
239#447—1071no—
240#442—1071no—
241#435—1071no—
242C2×#41—1071no—
243C22×D4—1071no—
244C22×D4—1071no—
245C22×D4—1071no—
246C22×D4—2141no—
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—2144no—
256C2×D8C2×D10711yescharacteristic, complemented
257C2×D8C2×D10711yescharacteristic, complemented
258C2×M16—1071no—
259C22×C8—1071no—
260#127C2×D10711yescharacteristic, complemented
261#124—1071no—
262#123—2141no—
263C22⋊Q8—1071no—
264#121—1071no—
265#121—1071no—
266#121—1071no—
267#121—2141no—
268#121—2141no—
269#120—2141no—
270#120—2141no—
271#120—2141no—
272C4×D4—1071no—
273C4×D4—1071no—
274C4×D4C2×D10711yescharacteristic, complemented
275C2×C22⋊C4—2141no—
276#106—1071no—
277C4⋊C8C2×D10711yescharacteristic, complemented
278#103—1071no—
279#102—1071no—
280#102—1071no—
281#102—1071no—
282#102C2×D10711yescharacteristic, complemented
283#102C2×D10711yescharacteristic, complemented
284C22⋊C8—1071no—
285C22⋊C8—1071no—
286#97—1071no—
287C24—2141no—
288C24—2141no—
289C24—2141no—
290#41—4281no—
291#41—2142no—
292C2×Q8—1071no—
293C2×D4—21no—
294C2×D4—21no—
295C2×D4—1071no—
296C2×D4—1071no—
297C2×D4—1071no—
298C2×D4—4284no—
299C2×D4—4281no—
300C2×D4—4281no—
301C2×D4—4281no—
302C2×D4—2141no—
303C2×D4—2141no—
304C2×D4—2141no—
305C2×D4—2141no—
306C2×D4—2142no—
307C2×D4—2142no—
308C2×D4—2141no—
309C2×D4—2141no—
310C2×D4—2141no—
311C2×D4—2142no—
312C2×D4C22×D10711yescharacteristic
313C2×D4C22×D10711yescharacteristic
314C2×D4C22×D10711yescharacteristic
315C22×C4—21no—
316C22×C4—1071no—
317C22×C4—1071no—
318C22×C4—2141no—
319C22×C4—2141no—
320SD16—4281no—
321SD16—2142no—
322SD16—2142no—
323D8—22no—
324D8—22no—
325D8—1074no—
326D8—4281no—
327D8—2142no—
328D8—2142no—
329M16—2142no—
330C2×C8—1071no—
331C2×C8—1071no—
332C2×C8—2142no—
333C2×C8C22×D10711yescharacteristic
334C2×C8C22×D10711yescharacteristic
335C4⋊C4—1071no—
336C4⋊C4—1071no—
337C4⋊C4—2141no—
338C4⋊C4C22×D10711yescharacteristic
339C22⋊C4—21no—
340C22⋊C4—1071no—
341C22⋊C4—1071no—
342C22⋊C4—2141no—
343C22⋊C4—2141no—
344C22⋊C4—2141no—
345C22⋊C4—2141no—
346C22⋊C4—2141no—
347C22⋊C4—2141no—
348C22⋊C4—2141no—
349C42C22×D10711yescharacteristic
350C23—21no—
351C23—21no—
352C23—21no—
353C23—8561no—
354C23—8561no—
355C23—8561no—
356C23—1071no—
357C23—1071no—
358C23—4282no—
359C23—4282no—
360C23—4282no—
361C23—4282no—
362C23—4282no—
363C23—4282no—
364C23—2141no—
365C23—2141no—
366Q8—2141no—
367Q8—2141no—
368D4—21no—
369D4—21no—
370D4—21no—
371D4—21no—
372D4—21no—
373D4—42no—
374D4—81no—
375D4—8561no—
376D4—1072no—
377D4—4282no—
378D4—4282no—
379D4—4282no—
380D4—4282no—
381D4—4282no—
382D4—4282no—
383D4—2142no—
384D4—2141no—
385D4—2141no—
386D4—2141no—
387D4D4×D10712yescomplemented
388C2×C4—21no—
389C2×C4—81no—
390C2×C4—1071no—
391C2×C4—1071no—
392C2×C4—4282no—
393C2×C4—4282no—
394C2×C4—2141no—
395C2×C4—2141no—
396C2×C4—2141no—
397C2×C4—2141no—
398C2×C4—2141no—
399C2×C4—2141no—
400C2×C4D4×D10711yescharacteristic
401C2×C4D4×D10711yescharacteristic
402C2×C4C23×D10711yescharacteristic
403C8—21no—
404C8—21no—
405C8—1072no—
406C8—2141no—
407C22—22no—
408C22—41no—
409C22—41no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—41no—
414C22—41no—
415C22—41no—
416C22—8564no—
417C22—8564no—
418C22—8564no—
419C22—1072no—
420C22—4281no—
421C22—4281no—
422C22—4281no—
423C22—4281no—
424C22—4281no—
425C22—2141no—
426C22—2141no—
427C22—2141no—
428C22—2142no—
429C22—2141no—
430C22—2141no—
431C22C2×D4×D10711yescharacteristic
432C4—21no—
433C4—21no—
434C4—41no—
435C4—4281no—
436C4—4281no—
437C4—2141no—
438C4—2141no—
439C4C2×D4×D10711yescharacteristic
440C4#1681215511yescharacteristic
441C2—42no—
442C2—81no—
443C2—81no—
444C2—8561no—
445C2—4282no—
446C2—4281no—
447C2—2142no—
448C2#2029326111yescharacteristic
449C2AGL(1,ℤ/8)×D10711yescharacteristic
450C2#2029342811yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#459×C107C2
socleC22×C107C22×D4
hypercenter#459D107
hypocenterC107#3264
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. C2×#16812133C2semidirect
2. C2×D8×D107C2semidirect
3. #20293272C2semidirect
4. #20293262C2semidirect
5. #20293014C2semidirect
6. #20292879C2semidirect
7. #20292855C2semidirect
8. #20292843C2semidirect
9. #20292779C2semidirect
10. #20292769C2semidirect
11. #20292762C2semidirect
12. #20292759C2semidirect
13. #20292737C2semidirect
14. #20292697C2semidirect
15. #459×C107C2semidirect
16. C2×#13133817C22semidirect
19. C2×D4×D107C22semidirect
20. C2×D4×D107C22semidirect
22. C2×#13133807C22semidirect
23. C2×#13133806C22semidirect
24. C2×#13133806C22semidirect
27. C2×#13133803C22semidirect
28. C2×#13133803C22semidirect
30. C2×#13133801C22semidirect
31. C2×C8×D107C22semidirect
32. #16812097C22semidirect
34. #16812089C22semidirect
35. #16812084C22semidirect
39. #16812080C22semidirect
44. C4×D428C22semidirect
46. #16812040C22semidirect
47. #16812034C22semidirect
48. #16812031C22semidirect
49. #16812030C22semidirect
50. #16812029C22semidirect
51. (C2×D107)⋊C8C22semidirect
52. (C2×D107)⋊C8C22semidirect
53. #16812020C22semidirect
54. C2×D8×C107C22semidirect
55. C2×D8×C107C22semidirect
56. #127×C107C22semidirect
57. C4×D4×C107C22semidirect
58. C4⋊C8×C107C22semidirect
59. #102×C107C22semidirect
60. #102×C107C22semidirect
94. #13133803D4semidirect
147. D4×C107C2×D4semidirect
180. C107⋊C8C2×D4semidirect
205. C2×D107AGL(1,ℤ/8)semidirect
226. C107#3264semidirect
235. #459D107semidirect
256. C2×D8C2×D107semidirect
257. C2×D8C2×D107semidirect
260. #127C2×D107semidirect
274. C4×D4C2×D107semidirect
277. C4⋊C8C2×D107semidirect
282. #102C2×D107semidirect
283. #102C2×D107semidirect
387. D4D4×D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168121332
C2×D8×D1072
#202932722
#202932622
#202930142
#202928792
#202928552
#202928432
#202927792
#202927692
#202927622
#202927592
#202927372
#202926972
#459×C1072
#3264107