Finite Group Databasebeta

← #28347940

Subgroups — #28347940

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×#16812136C211yesmaximal, characteristic, complemented
2C2×D8×D107C211yesmaximal, characteristic, complemented
3#20293270C211yesmaximal, characteristic, complemented
4#20293262C211yesmaximal, characteristic, complemented
5#20292877C211yesmaximal, characteristic, complemented
6#20292853C211yesmaximal, characteristic, complemented
7#20292839C211yesmaximal, characteristic, complemented
8#20292819C211yesmaximal, characteristic, complemented
9#20292776C211yesmaximal, characteristic, complemented
10#20292771C211yesmaximal, characteristic, complemented
11#20292759C211yesmaximal, characteristic, complemented
12#20292759C211yesmaximal, characteristic, complemented
13#20292740C211yesmaximal, characteristic, complemented
14#20292700C211yesmaximal, characteristic, complemented
15#496×C107C211yesmaximal, characteristic, complemented
16C2×#13133819C2211yescharacteristic, complemented
17C2×D4×D107—21no—
18C2×D4×D107C2211yescharacteristic
19C2×D4×D107C2211yescharacteristic, complemented
20C2×D4×D107C2211yescharacteristic, complemented
21#16812136—24no—
22C2×#13133805C2211yescharacteristic, complemented
23C2×D856C2211yescharacteristic
24C2×#13133803C2211yescharacteristic
25C2×#13133803C2211yescharacteristic, complemented
26C2×#13133803C2211yescharacteristic, complemented
27D8×D107—24no—
28C2×#13133801C2211yescharacteristic, complemented
29C2×C8×D107C2211yescharacteristic, complemented
30#16812091—21no—
31#16812085C2211yescharacteristic, complemented
32#16812082—21no—
33#16812080—21no—
34#16812080C2211yescharacteristic
35#16812080C2211yescharacteristic, complemented
36#16812080C2211yescharacteristic, complemented
37#16812079—21no—
38#16812077—21no—
39#16812077—21no—
40C4×D428C2211yescharacteristic
41C4×D428C2211yescharacteristic, complemented
42C4×D428C2211yescharacteristic, complemented
43C22⋊C4×D107—21no—
44#16812041C2211yescharacteristic, complemented
45C4⋊(C107⋊C8)C2211yescharacteristic, complemented
46#16812033C2211yescharacteristic, complemented
47#16812031C2211yescharacteristic, complemented
48#16812031C2211yescharacteristic, complemented
49#16812030C2211yescharacteristic, complemented
50#16812030C2211yescharacteristic, complemented
51#16812030C2211yescharacteristic, complemented
52(C2×D107)⋊C8C2211yescharacteristic, complemented
53(C2×D107)⋊C8C2211yescharacteristic, complemented
54C2×SD16×C107C2211yescharacteristic, complemented
55C2×SD16×C107C2211yescharacteristic, complemented
56C2×D8×C107C2211yescharacteristic, complemented
57C2×D8×C107C2211yescharacteristic, complemented
58#127×C107C2211yescharacteristic, complemented
59#124×C107C2211yescharacteristic, complemented
60#97×C107C2211yescharacteristic, complemented
61C23×D107—21no—
62C23×D107—21no—
63C23×D107—21no—
64#13133819—22no—
65#13133819—41no—
66C2×D428—21no—
67C2×D428D411yescharacteristic
68C2×D428C2311yescharacteristic
69C2×D428C2311yescharacteristic
70C2×C22⋊D107—21no—
71C2×C22⋊D107—21no—
72C2×C22⋊D107—21no—
73C2×C22⋊D107—21no—
74C2×C22⋊D107—21no—
75C2×C22⋊D107—21no—
76D4×D107—22no—
77D4×D107—22no—
78D4×D107—22no—
79D4×D107—41no—
80D4×D107—41no—
81D4×D107—41no—
82D4×D107—44no—
83C2×C4×D107—21no—
84C2×C4×D107—21no—
85C2×C4×D107—21no—
86C2×C4×D107C2311yescharacteristic
87C2×C4×D107C2311yescharacteristic
88#13133805—22no—
89D856D414yescomplemented
90#13133803—22no—
91#13133803—22no—
92#13133803—22no—
93#13133801—22no—
94C2×C107⋊C8C2311yescharacteristic
95C2×C107⋊C8C2311yescharacteristic
96C8×D107—22no—
97C4⋊Dic107D411yescharacteristic
98C4⋊Dic107C2311yescharacteristic
99C4⋊Dic107C2311yescharacteristic
100Dic107⋊C4—21no—
101C22⋊Dic107—21no—
102C22⋊Dic107—21no—
103C22⋊Dic107—21no—
104(C2×D107)⋊C4—21no—
105(C2×D107)⋊C4—21no—
106(C2×D107)⋊C4—21no—
107(C2×D107)⋊C4D411yescharacteristic
108(C2×D107)⋊C4D411yescharacteristic
109C2×Q8×C107C2311yescharacteristic
110C2×D4×C107—21no—
111C2×D4×C107—21no—
112C2×D4×C107C2311yescharacteristic
113C2×D4×C107C2311yescharacteristic
114C2×D4×C107C2311yescharacteristic
115SD16×C107—22no—
116SD16×C107—41no—
117D8×C107—22no—
118D8×C107—41no—
119C2×C8×C107C2311yescharacteristic
120C2×C8×C107C2311yescharacteristic
121C22⋊C4×C107—21no—
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
140Q8×C107—21no—
141Q8×C107—21no—
142D4×C107—21no—
143D4×C107—21no—
144D4×C107—21no—
145D4×C107—21no—
146D4×C107—21no—
147D4×C107—21no—
148D4×C107—42no—
149D4×C107—81no—
150C2×C4×C107—21no—
151C2×C4×C107C2×D411yescharacteristic
152C2×C4×C107C2×D411yescharacteristic
153C2×C4×C107C2411yescharacteristic
154C22⋊D107—42no—
155C22⋊D107—42no—
156C22⋊D107—42no—
157C22⋊D107—42no—
158C22⋊D107—42no—
159C22⋊D107—81no—
160C2×Dic107—21no—
161C2×Dic107—21no—
162C2×Dic107C2×D411yescharacteristic
163C2×Dic107C2×D411yescharacteristic
164D428—22no—
165D428—21no—
166D428—21no—
167D428—42no—
168D428C2×D412yes—
169D428C2×D412yescomplemented
170C4×D107—21no—
171C4×D107—21no—
172C4×D107—21no—
173C4×D107—21no—
174C4×D107—42no—
175C4×D107—42no—
176C4×D107—81no—
177C8×C107—21no—
178C8×C107C2×D412yescomplemented
179C107⋊C8—21no—
180C107⋊C8—21no—
181C22×C107—41no—
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—22no—
196C2×D107—21no—
197C2×D107—21no—
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—81no—
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—42no—
225#3264—1071nomaximal, Sylow
226C107#326411yescharacteristic, complemented, Sylow
227C2×AGL(1,ℤ/8)—1071no—
228C22×D8—1071no—
229#546—1071no—
230D42—1071no—
231#496D10711yescharacteristic, complemented
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—1071no—
244C22×D4—1071no—
245C22×D4—1071no—
246C22×D4—2141no—
247AGL(1,ℤ/8)—2144no—
248C2×SD16—1071no—
249C2×SD16C2×D10711yescharacteristic, complemented
250C2×SD16C2×D10711yescharacteristic, complemented
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#124C2×D10711yescharacteristic, complemented
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×D4—1071no—
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#97C2×D10711yescharacteristic, complemented
287C24—2141no—
288C24—2141no—
289C24—2141no—
290#41—4281no—
291#41—2142no—
292C2×Q8C22×D10711yescharacteristic
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—2142no—
309C2×D4—2141no—
310C2×D4—2141no—
311C2×D4—2141no—
312C2×D4C22×D10711yescharacteristic
313C2×D4C22×D10711yescharacteristic
314C2×D4C22×D10711yescharacteristic
315C22×C4—1071no—
316C22×C4—1071no—
317C22×C4—2141no—
318C22×C4—2141no—
319C22×C4—2141no—
320SD16—22no—
321SD16—41no—
322SD16—2142no—
323D8—22no—
324D8—41no—
325D8—1074no—
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—1071no—
338C4⋊C4—2141no—
339C22⋊C4—21no—
340C22⋊C4—21no—
341C22⋊C4—1071no—
342C22⋊C4—1071no—
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—21no—
367Q8—21no—
368D4—21no—
369D4—21no—
370D4—21no—
371D4—21no—
372D4—21no—
373D4—21no—
374D4—42no—
375D4—81no—
376D4—8561no—
377D4—1072no—
378D4—1072no—
379D4—4282no—
380D4—4282no—
381D4—4282no—
382D4—4282no—
383D4—4282no—
384D4—4282no—
385D4—2142no—
386D4—2141no—
387D4—2141no—
388C2×C4—21no—
389C2×C4—8561no—
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—2141no—
405C8—2141no—
406C8D4×D10712yescomplemented
407C22—41no—
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—2141no—
425C22—2141no—
426C22—2142no—
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#1681215411yescharacteristic
441C2—81no—
442C2—81no—
443C2—81no—
444C2—4282no—
445C2—4281no—
446C2—4282no—
447C2—2142no—
448C2#2029326211yescharacteristic
449C2AGL(1,ℤ/8)×D10711yescharacteristic
450C2#2029343111yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#496×C107C2
socleC22×C107C22×D4
hypercenter#496D107
hypocenterC107#3264
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #496

Complements

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

NH ≅ G/Ntype
1. C2×#16812136C2semidirect
2. C2×D8×D107C2semidirect
3. #20293270C2semidirect
4. #20293262C2semidirect
5. #20292877C2semidirect
6. #20292853C2semidirect
7. #20292839C2semidirect
8. #20292819C2semidirect
9. #20292776C2semidirect
10. #20292771C2semidirect
11. #20292759C2semidirect
12. #20292759C2semidirect
13. #20292740C2semidirect
14. #20292700C2semidirect
15. #496×C107C2semidirect
16. C2×#13133819C22semidirect
19. C2×D4×D107C22semidirect
20. C2×D4×D107C22semidirect
22. C2×#13133805C22semidirect
25. C2×#13133803C22semidirect
26. C2×#13133803C22semidirect
28. C2×#13133801C22semidirect
29. C2×C8×D107C22semidirect
31. #16812085C22semidirect
35. #16812080C22semidirect
36. #16812080C22semidirect
41. C4×D428C22semidirect
42. C4×D428C22semidirect
44. #16812041C22semidirect
45. C4⋊(C107⋊C8)C22semidirect
46. #16812033C22semidirect
47. #16812031C22semidirect
48. #16812031C22semidirect
49. #16812030C22semidirect
50. #16812030C22semidirect
51. #16812030C22semidirect
52. (C2×D107)⋊C8C22semidirect
53. (C2×D107)⋊C8C22semidirect
54. C2×SD16×C107C22semidirect
55. C2×SD16×C107C22semidirect
56. C2×D8×C107C22semidirect
57. C2×D8×C107C22semidirect
58. #127×C107C22semidirect
59. #124×C107C22semidirect
60. #97×C107C22semidirect
89. D856D4semidirect
169. D428C2×D4semidirect
178. C8×C107C2×D4semidirect
205. C2×D107AGL(1,ℤ/8)semidirect
226. C107#3264semidirect
231. #496D107semidirect
249. C2×SD16C2×D107semidirect
250. C2×SD16C2×D107semidirect
256. C2×D8C2×D107semidirect
257. C2×D8C2×D107semidirect
260. #127C2×D107semidirect
261. #124C2×D107semidirect
286. #97C2×D107semidirect
406. C8D4×D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168121362
C2×D8×D1072
#202932702
#202932622
#202928772
#202928532
#202928392
#202928192
#202927762
#202927712
#202927592
#202927402
#202927002
#496×C1072
#3264107