Finite Group Databasebeta

← #28347944

Subgroups — #28347944

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
2C2×D8×D107C211yesmaximal, characteristic, complemented
3#20293265C211yesmaximal, characteristic, complemented
4#20293261C211yesmaximal, characteristic, complemented
5#20293013C211yesmaximal, characteristic, complemented
6#20292878C211yesmaximal, characteristic, complemented
7#20292859C211yesmaximal, characteristic, complemented
8#20292845C211yesmaximal, characteristic, complemented
9#20292820C211yesmaximal, characteristic, complemented
10#20292776C211yesmaximal, characteristic, complemented
11#20292763C211yesmaximal, characteristic, complemented
12#20292762C211yesmaximal, characteristic, complemented
13#20292736C211yesmaximal, characteristic, complemented
14#20292701C211yesmaximal, characteristic, complemented
15#449×C107C211yesmaximal, characteristic, complemented
16C22×D428C2211yescharacteristic, complemented
17C2×D4×D107—21no—
18C2×D4×D107C2211yescharacteristic
19C2×D4×D107C2211yescharacteristic, complemented
20#16812134—24no—
21C2×#13133805C2211yescharacteristic, complemented
22C2×#13133805C2211yescharacteristic, complemented
23C2×D856C2211yescharacteristic
24C2×D856C2211yescharacteristic, complemented
25C2×#13133803C2211yescharacteristic
26C2×#13133803C2211yescharacteristic, complemented
27C2×#13133803C2211yescharacteristic, complemented
28D8×D107—24no—
29C2×#13133802C2211yescharacteristic, complemented
30C2×C8×D107C2211yescharacteristic, complemented
31#16812105C2211yescharacteristic, complemented
32#16812098C2211yescharacteristic, complemented
33#16812092—21no—
34#16812085C2211yescharacteristic, complemented
35#16812084C2211yescharacteristic
36#16812083C2211yescharacteristic, complemented
37#16812082—21no—
38#16812081—21no—
39#16812079—21no—
40#16812077—21no—
41#16812077—21no—
42D4×Dic107C2211yescharacteristic, complemented
43#16812071C2211yescharacteristic, complemented
44C4×C22⋊D107C2211yescharacteristic
45C2×(C2×D107)⋊C4—21no—
46#16812040C2211yescharacteristic, complemented
47Dic107⋊C8C2211yescharacteristic, complemented
48#16812031C2211yescharacteristic, complemented
49#16812031C2211yescharacteristic, complemented
50#16812030C2211yescharacteristic, complemented
51#16812029C2211yescharacteristic, complemented
52(C2×D107)⋊C8C2211yescharacteristic, complemented
53#16812019C2211yescharacteristic, complemented
54C2×#41×C107C2211yescharacteristic, complemented
55C2×SD16×C107C2211yescharacteristic, complemented
56C2×D8×C107C2211yescharacteristic, complemented
57#121×C107C2211yescharacteristic, complemented
58#103×C107C2211yescharacteristic, complemented
59#102×C107C2211yescharacteristic, complemented
60C22⋊C8×C107C2211yescharacteristic, complemented
61C23×D107—21no—
62C23×D107—21no—
63C23×D107—21no—
64C2×D428—22no—
65C2×D428—21no—
66C2×D428—21no—
67C2×D428—41no—
68C2×D428C2311yescharacteristic
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⋊D107D411yescharacteristic
77D4×D107—22no—
78D4×D107—22no—
79D4×D107—41no—
80D4×D107—41no—
81D4×D107—44no—
82C22×Dic107—21no—
83C2×C4×D107—21no—
84C2×C4×D107C2311yescharacteristic
85#13133805—22no—
86#13133805—22no—
87D856—22no—
88D856—41no—
89#13133803—22no—
90#13133803—22no—
91#13133803D414yescomplemented
92#13133802—22no—
93C2×C107⋊C8C2311yescharacteristic
94C2×C107⋊C8C2311yescharacteristic
95C8×D107—22no—
96C4⋊Dic107C2311yescharacteristic
97Dic107⋊C4—21no—
98Dic107⋊C4D411yescharacteristic
99C22⋊Dic107—21no—
100C22⋊Dic107D411yescharacteristic
101(C2×D107)⋊C4—21no—
102(C2×D107)⋊C4—21no—
103(C2×D107)⋊C4—21no—
104(C2×D107)⋊C4—21no—
105(C2×D107)⋊C4—21no—
106(C2×D107)⋊C4—21no—
107(C2×D107)⋊C4D411yescharacteristic
108C4×Dic107C2311yescharacteristic
109#41×C107—22no—
110#41×C107—41no—
111C2×Q8×C107C2311yescharacteristic
112C2×D4×C107—21no—
113C2×D4×C107—21no—
114C2×D4×C107C2311yescharacteristic
115C2×D4×C107C2311yescharacteristic
116C22×C4×C107—21no—
117C22×C4×C107C2311yescharacteristic
118SD16×C107—41no—
119D8×C107—22no—
120C2×C8×C107C2311yescharacteristic
121C2×C8×C107C2311yescharacteristic
122C4⋊C4×C107C2311yescharacteristic
123C22⋊C4×C107—21no—
124C23×C107—21no—
125C23×C107—21no—
126C23×C107C2×D411yescharacteristic
127C22×D107—21no—
128C22×D107—21no—
129C22×D107—21no—
130C22×D107—42no—
131C22×D107—42no—
132C22×D107—42no—
133C22×D107—42no—
134C22×D107—42no—
135C22×D107—42no—
136C22×D107—81no—
137C22×D107—81no—
138C22×D107—81no—
139C22×D107C2×D411yescharacteristic
140Q8×C107—21no—
141Q8×C107—21no—
142D4×C107—21no—
143D4×C107—21no—
144D4×C107—21no—
145D4×C107—42no—
146D4×C107—42no—
147D4×C107C2×D412yescomplemented
148C2×C4×C107—21no—
149C2×C4×C107—21no—
150C2×C4×C107—21no—
151C2×C4×C107—21no—
152C2×C4×C107—42no—
153C2×C4×C107C2×D411yescharacteristic
154C2×C4×C107C2411yescharacteristic
155C22⋊D107—22no—
156C22⋊D107—42no—
157C22⋊D107—42no—
158C22⋊D107—42no—
159C22⋊D107—81no—
160C22⋊D107—81no—
161C2×Dic107—21no—
162C2×Dic107—81no—
163C2×Dic107C2×D411yescharacteristic
164C2×Dic107C2×D411yescharacteristic
165C2×Dic107C2×D411yescharacteristic
166D428—21no—
167D428—21no—
168D428—21no—
169D428—21no—
170D428—21no—
171D428—42no—
172D428—42no—
173D428C2×D412yes—
174C4×D107—21no—
175C4×D107—21no—
176C4×D107—42no—
177C8×C107—21no—
178C8×C107—21no—
179C107⋊C8—21no—
180C107⋊C8C2×D412yescomplemented
181C22×C107—22no—
182C22×C107—21no—
183C22×C107—21no—
184C22×C107—21no—
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—22no—
194C2×D107—41no—
195C2×D107—41no—
196C2×D107—41no—
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—41no—
208C4×C107—41no—
209C4×C107C22×D411yescharacteristic
210C4×C107H(5,𝔽2)11yescharacteristic
211Dic107—21no—
212Dic107—21no—
213Dic107—21no—
214Dic107—41no—
215C2×C107—41no—
216C2×C107—42no—
217C2×C107—81no—
218C2×C107D4211yescharacteristic
219C2×C107C2×AGL(1,ℤ/8)11yescharacteristic
220C2×C107#57611yescharacteristic
221D107—22no—
222D107—42no—
223D107—81no—
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#459—1071no—
236#450—1071no—
237#449D10711yescharacteristic, complemented
238#447—1071no—
239#447—1071no—
240#442—1071no—
241#435—1071no—
242C2×#41C2×D10711yescharacteristic, complemented
243C22×D4—1071no—
244C22×D4—1071no—
245C22×D4—1071no—
246C22×D4—2141no—
247AGL(1,ℤ/8)—2144no—
248C2×SD16—1071no—
249C2×SD16—1071no—
250C2×SD16C2×D10711yescharacteristic, complemented
251C2×D8—1071no—
252C2×D8—1071no—
253C2×D8—1071no—
254C2×D8—1071no—
255C2×D8—1071no—
256C2×D8—2144no—
257C2×D8C2×D10711yescharacteristic, complemented
258C2×M16—1071no—
259C22×C8—1071no—
260#127—1071no—
261#124—1071no—
262#123—2141no—
263C22⋊Q8—1071no—
264#121—1071no—
265#121—1071no—
266#121—2141no—
267#121—2141no—
268#121C2×D10711yescharacteristic, complemented
269#120—2141no—
270#120—2141no—
271#120—2141no—
272C4×D4—1071no—
273C4×D4—1071no—
274C4×D4—1071no—
275C2×C22⋊C4—2141no—
276#106—1071no—
277C4⋊C8—1071no—
278#103C2×D10711yescharacteristic, complemented
279#102—1071no—
280#102—1071no—
281#102—1071no—
282#102—1071no—
283#102C2×D10711yescharacteristic, complemented
284C22⋊C8—1071no—
285C22⋊C8C2×D10711yescharacteristic, complemented
286#97—1071no—
287C24—2141no—
288C24—2141no—
289C24—2141no—
290#41—22no—
291#41—41no—
292C2×Q8C22×D10711yescharacteristic
293C2×D4—21no—
294C2×D4—21no—
295C2×D4—1071no—
296C2×D4—1071no—
297C2×D4—1071no—
298C2×D4—1071no—
299C2×D4—4281no—
300C2×D4—4284no—
301C2×D4—4281no—
302C2×D4—4281no—
303C2×D4—2142no—
304C2×D4—2141no—
305C2×D4—2141no—
306C2×D4—2141no—
307C2×D4—2141no—
308C2×D4—2141no—
309C2×D4—2142no—
310C2×D4—2141no—
311C2×D4—2142no—
312C2×D4—2141no—
313C2×D4C22×D10711yescharacteristic
314C2×D4C22×D10711yescharacteristic
315C22×C4—21no—
316C22×C4—1071no—
317C22×C4—2141no—
318C22×C4—2141no—
319C22×C4C22×D10711yescharacteristic
320SD16—41no—
321SD16—2142no—
322SD16—2142no—
323D8—22no—
324D8—1074no—
325D8—4281no—
326D8—2142no—
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—
349C42—1071no—
350C23—21no—
351C23—21no—
352C23—8561no—
353C23—8561no—
354C23—8561no—
355C23—1071no—
356C23—4282no—
357C23—4282no—
358C23—4282no—
359C23—4282no—
360C23—4282no—
361C23—4282no—
362C23—2141no—
363C23—2141no—
364C23—2141no—
365C23D4×D10711yescharacteristic
366Q8—21no—
367Q8—21no—
368D4—21no—
369D4—21no—
370D4—21no—
371D4—42no—
372D4—42no—
373D4—8561no—
374D4—8561no—
375D4—1072no—
376D4—4282no—
377D4—4282no—
378D4—4282no—
379D4—4282no—
380D4—4282no—
381D4—2142no—
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—42no—
393C2×C4—8561no—
394C2×C4—1071no—
395C2×C4—1071no—
396C2×C4—1071no—
397C2×C4—4282no—
398C2×C4—2141no—
399C2×C4—2141no—
400C2×C4—2141no—
401C2×C4D4×D10711yescharacteristic
402C2×C4C23×D10711yescharacteristic
403C8—21no—
404C8—21no—
405C8—1072no—
406C8—2141no—
407C22—21no—
408C22—21no—
409C22—21no—
410C22—22no—
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—4281no—
426C22—4281no—
427C22—4281no—
428C22—2142no—
429C22—2141no—
430C22—2141no—
431C22C2×D4×D10711yescharacteristic
432C4—21no—
433C4—41no—
434C4—41no—
435C4—4281no—
436C4—2141no—
437C4—2141no—
438C4—2141no—
439C4C2×D4×D10711yescharacteristic
440C4#1681215411yescharacteristic
441C2—41no—
442C2—42no—
443C2—81no—
444C2—8561no—
445C2—8561no—
446C2—4282no—
447C2—2142no—
448C2#2029325911yescharacteristic
449C2AGL(1,ℤ/8)×D10711yescharacteristic
450C2#2029342911yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#449×C107C2
socleC22×C107C22×D4
hypercenter#449D107
hypocenterC107#3264
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #449

Complements

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

NH ≅ G/Ntype
1. C2×#16812134C2semidirect
2. C2×D8×D107C2semidirect
3. #20293265C2semidirect
4. #20293261C2semidirect
5. #20293013C2semidirect
6. #20292878C2semidirect
7. #20292859C2semidirect
8. #20292845C2semidirect
9. #20292820C2semidirect
10. #20292776C2semidirect
11. #20292763C2semidirect
12. #20292762C2semidirect
13. #20292736C2semidirect
14. #20292701C2semidirect
15. #449×C107C2semidirect
16. C22×D428C22semidirect
19. C2×D4×D107C22semidirect
21. C2×#13133805C22semidirect
22. C2×#13133805C22semidirect
24. C2×D856C22semidirect
26. C2×#13133803C22semidirect
27. C2×#13133803C22semidirect
29. C2×#13133802C22semidirect
30. C2×C8×D107C22semidirect
31. #16812105C22semidirect
32. #16812098C22semidirect
34. #16812085C22semidirect
36. #16812083C22semidirect
42. D4×Dic107C22semidirect
43. #16812071C22semidirect
46. #16812040C22semidirect
47. Dic107⋊C8C22semidirect
48. #16812031C22semidirect
49. #16812031C22semidirect
50. #16812030C22semidirect
51. #16812029C22semidirect
52. (C2×D107)⋊C8C22semidirect
53. #16812019C22semidirect
54. C2×#41×C107C22semidirect
55. C2×SD16×C107C22semidirect
56. C2×D8×C107C22semidirect
57. #121×C107C22semidirect
58. #103×C107C22semidirect
59. #102×C107C22semidirect
60. C22⋊C8×C107C22semidirect
91. #13133803D4semidirect
147. D4×C107C2×D4semidirect
180. C107⋊C8C2×D4semidirect
205. C2×D107AGL(1,ℤ/8)semidirect
226. C107#3264semidirect
237. #449D107semidirect
242. C2×#41C2×D107semidirect
250. C2×SD16C2×D107semidirect
257. C2×D8C2×D107semidirect
268. #121C2×D107semidirect
278. #103C2×D107semidirect
283. #102C2×D107semidirect
285. C22⋊C8C2×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×#168121342
C2×D8×D1072
#202932652
#202932612
#202930132
#202928782
#202928592
#202928452
#202928202
#202927762
#202927632
#202927622
#202927362
#202927012
#449×C1072
#3264107