Finite Group Databasebeta

← #28347947

Subgroups — #28347947

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
4D4×D428C211yesmaximal, characteristic, complemented
5#20293017C211yesmaximal, characteristic, complemented
6#20292879C211yesmaximal, characteristic, complemented
7#20292859C211yesmaximal, characteristic, complemented
8#20292823C211yesmaximal, characteristic, complemented
9#20292776C211yesmaximal, characteristic, complemented
10#20292775C211yesmaximal, characteristic, complemented
11#20292762C211yesmaximal, characteristic, complemented
12#20292759C211yesmaximal, characteristic, complemented
13#20292737C211yesmaximal, characteristic, complemented
14#20292697C211yesmaximal, characteristic, complemented
15#463×C107C211yesmaximal, characteristic, complemented
16C2×#13133819C2211yescharacteristic, complemented
17C22×D428—21no—
18C2×D4×D107C2211yescharacteristic
19C2×D4×D107C2211yescharacteristic, complemented
20C2×D4×D107C2211yescharacteristic, complemented
21#16812136—24no—
22C2×#13133805C2211yescharacteristic, complemented
23C2×#13133805C2211yescharacteristic, complemented
24C2×D856C2211yescharacteristic, complemented
25C2×D856C2211yescharacteristic, complemented
26C2×#13133803C2211yescharacteristic
27C2×#13133803C2211yescharacteristic, complemented
28C2×#13133803C2211yescharacteristic, complemented
29D8×D107—24no—
30C2×#13133801C2211yescharacteristic, complemented
31C2×C8×D107C2211yescharacteristic, complemented
32#16812106C2211yescharacteristic, complemented
33#16812091—21no—
34#16812085C2211yescharacteristic, complemented
35#16812084C2211yescharacteristic
36#16812084C2211yescharacteristic, complemented
37#16812083—21no—
38#16812082—21no—
39#16812080C2211yescharacteristic, complemented
40#16812079—21no—
41#16812079—21no—
42#16812079—21no—
43C4×D428C2211yescharacteristic
44C4×D428C2211yescharacteristic, complemented
45C22⋊C4×D107—21no—
46#16812040C2211yescharacteristic, complemented
47#16812031C2211yescharacteristic, complemented
48#16812031C2211yescharacteristic, complemented
49#16812030C2211yescharacteristic, complemented
50#16812029C2211yescharacteristic, complemented
51(C2×D107)⋊C8C2211yescharacteristic, complemented
52(C2×D107)⋊C8C2211yescharacteristic, complemented
53#16812020C2211yescharacteristic, complemented
54C2×SD16×C107C2211yescharacteristic, complemented
55C2×D8×C107C2211yescharacteristic
56#124×C107C2211yescharacteristic, complemented
57C4×D4×C107C2211yescharacteristic, complemented
58C4⋊C8×C107C2211yescharacteristic, complemented
59#103×C107C2211yescharacteristic, complemented
60#102×C107C2211yescharacteristic, complemented
61C23×D107—21no—
62C23×D107—21no—
63C23×D107—21no—
64#13133819—22no—
65#13133819—41no—
66C2×D428—21no—
67C2×D428—21no—
68C2×D428—21no—
69C2×D428—21no—
70C2×D428—21no—
71C2×D428—44no—
72C2×D428D411yescharacteristic
73C2×D428C2311yescharacteristic
74C2×D428C2311yescharacteristic
75C2×D428C2311yescharacteristic
76C2×C22⋊D107—21no—
77C2×C22⋊D107—21no—
78C2×C22⋊D107—21no—
79C2×C22⋊D107—21no—
80D4×D107—22no—
81D4×D107—22no—
82D4×D107—22no—
83D4×D107—41no—
84D4×D107—41no—
85D4×D107—41no—
86C2×C4×D107—21no—
87C2×C4×D107—21no—
88C2×C4×D107C2311yescharacteristic
89C2×C4×D107C2311yescharacteristic
90#13133805—22no—
91#13133805—41no—
92D856—22no—
93D856—22no—
94#13133803—22no—
95#13133803—41no—
96#13133803D414yescomplemented
97#13133801—22no—
98C2×C107⋊C8C2311yescharacteristic
99C2×C107⋊C8C2311yescharacteristic
100C8×D107—22no—
101C4⋊Dic107D411yescharacteristic
102C4⋊Dic107C2311yescharacteristic
103Dic107⋊C4—21no—
104C22⋊Dic107—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×Q8×C107C2311yescharacteristic
112C2×D4×C107C2311yescharacteristic
113C2×D4×C107C2311yescharacteristic
114C22×C4×C107—21no—
115SD16×C107—22no—
116D8×C107—22no—
117C2×C8×C107C2311yescharacteristic
118C2×C8×C107C2311yescharacteristic
119C4⋊C4×C107C2311yescharacteristic
120C22⋊C4×C107—21no—
121C22⋊C4×C107—21no—
122C22⋊C4×C107—21no—
123C42×C107C2311yescharacteristic
124C23×C107—21no—
125C23×C107—21no—
126C22×D107—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×C107C2×D412yes—
146C2×C4×C107—21no—
147C2×C4×C107—21no—
148C2×C4×C107—42no—
149C2×C4×C107C2×D411yescharacteristic
150C2×C4×C107C2×D411yescharacteristic
151C2×C4×C107C2411yescharacteristic
152C22⋊D107—42no—
153C22⋊D107—42no—
154C22⋊D107—42no—
155C22⋊D107—81no—
156C2×Dic107—21no—
157C2×Dic107C2×D411yescharacteristic
158C2×Dic107C2×D411yescharacteristic
159D428—22no—
160D428—21no—
161D428—21no—
162D428—21no—
163D428—21no—
164D428—21no—
165D428—42no—
166D428—42no—
167D428—42no—
168D428—42no—
169D428—81no—
170D428C2×D412yescomplemented
171C4×D107—21no—
172C4×D107—21no—
173C4×D107—21no—
174C4×D107—21no—
175C4×D107—42no—
176C4×D107—81no—
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×C107C22×D411yescharacteristic
188C2×D107—21no—
189C2×D107—21no—
190C2×D107—21no—
191C2×D107—22no—
192C2×D107—21no—
193C2×D107—21no—
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—21no—
208C4×C107—41no—
209C4×C107—41no—
210C4×C107C22×D411yescharacteristic
211C4×C107H(5,𝔽2)11yescharacteristic
212Dic107—21no—
213Dic107—21no—
214Dic107—41no—
215C2×C107—42no—
216C2×C107—81no—
217C2×C107D4211yescharacteristic
218C2×C107C2×AGL(1,ℤ/8)11yescharacteristic
219C2×C107#57611yescharacteristic
220D107—22no—
221D107—41no—
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#463D10711yescharacteristic, complemented
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×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
258C2×M16—1071no—
259C22×C8—1071no—
260#127—1071no—
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×D4C2×D10711yescharacteristic, complemented
275C2×C22⋊C4—2141no—
276#106—1071no—
277C4⋊C8C2×D10711yescharacteristic, complemented
278#103C2×D10711yescharacteristic, complemented
279#102—1071no—
280#102—1071no—
281#102—1071no—
282#102—1071no—
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×Q8C22×D10711yescharacteristic
293C2×D4—1071no—
294C2×D4—1071no—
295C2×D4—1071no—
296C2×D4—1071no—
297C2×D4—4284no—
298C2×D4—4281no—
299C2×D4—4281no—
300C2×D4—4281no—
301C2×D4—2141no—
302C2×D4—2141no—
303C2×D4—2141no—
304C2×D4—2141no—
305C2×D4—2141no—
306C2×D4—2141no—
307C2×D4—2142no—
308C2×D4—2141no—
309C2×D4—2142no—
310C2×D4—2142no—
311C2×D4—2141no—
312C2×D4—2141no—
313C2×D4C22×D10711yescharacteristic
314C2×D4C22×D10711yescharacteristic
315C22×C4—21no—
316C22×C4—1071no—
317C22×C4—1071no—
318C22×C4—2141no—
319C22×C4—2141no—
320SD16—22no—
321SD16—4281no—
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—21no—
341C22⋊C4—21no—
342C22⋊C4—1071no—
343C22⋊C4—1071no—
344C22⋊C4—2141no—
345C22⋊C4—2141no—
346C22⋊C4—2141no—
347C22⋊C4—2141no—
348C22⋊C4—2141no—
349C42C22×D10711yescharacteristic
350C23—21no—
351C23—21no—
352C23—8561no—
353C23—8561no—
354C23—8561no—
355C23—1071no—
356C23—1071no—
357C23—4282no—
358C23—4282no—
359C23—4282no—
360C23—4282no—
361C23—4282no—
362C23—4282no—
363C23—2141no—
364C23—2141no—
365C23—2141no—
366Q8—21no—
367Q8—21no—
368D4—21no—
369D4—21no—
370D4—21no—
371D4—8561no—
372D4—8561no—
373D4—1072no—
374D4—4282no—
375D4—4282no—
376D4—4282no—
377D4—4282no—
378D4—4282no—
379D4—4282no—
380D4—4282no—
381D4—2142no—
382D4—2141no—
383D4—2141no—
384D4—2141no—
385D4—2141no—
386D4—2141no—
387D4D4×D10712yes—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—42no—
391C2×C4—8561no—
392C2×C4—1071no—
393C2×C4—1071no—
394C2×C4—4282no—
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—8564no—
414C22—8564no—
415C22—8564no—
416C22—1072no—
417C22—4281no—
418C22—4281no—
419C22—4281no—
420C22—4281no—
421C22—4281no—
422C22—4281no—
423C22—4281no—
424C22—4281no—
425C22—2141no—
426C22—2141no—
427C22—2142no—
428C22—2141no—
429C22—2141no—
430C22—2141no—
431C22C2×D4×D10711yescharacteristic
432C4—21no—
433C4—21no—
434C4—41no—
435C4—41no—
436C4—4281no—
437C4—2141no—
438C4—2141no—
439C4C2×D4×D10711yescharacteristic
440C4#1681215511yescharacteristic
441C2—42no—
442C2—81no—
443C2—8561no—
444C2—8561no—
445C2—4281no—
446C2—4282no—
447C2—2142no—
448C2#2029326111yescharacteristic
449C2AGL(1,ℤ/8)×D10711yescharacteristic
450C2#2029342911yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#463×C107C2
socleC22×C107C22×D4
hypercenter#463D107
hypocenterC107#3264
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #463

Complements

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

NH ≅ G/Ntype
1. C2×#16812136C2semidirect
2. C2×D8×D107C2semidirect
3. #20293270C2semidirect
4. D4×D428C2semidirect
5. #20293017C2semidirect
6. #20292879C2semidirect
7. #20292859C2semidirect
8. #20292823C2semidirect
9. #20292776C2semidirect
10. #20292775C2semidirect
11. #20292762C2semidirect
12. #20292759C2semidirect
13. #20292737C2semidirect
14. #20292697C2semidirect
15. #463×C107C2semidirect
16. C2×#13133819C22semidirect
19. C2×D4×D107C22semidirect
20. C2×D4×D107C22semidirect
22. C2×#13133805C22semidirect
23. C2×#13133805C22semidirect
24. C2×D856C22semidirect
25. C2×D856C22semidirect
27. C2×#13133803C22semidirect
28. C2×#13133803C22semidirect
30. C2×#13133801C22semidirect
31. C2×C8×D107C22semidirect
32. #16812106C22semidirect
34. #16812085C22semidirect
36. #16812084C22semidirect
39. #16812080C22semidirect
44. C4×D428C22semidirect
46. #16812040C22semidirect
47. #16812031C22semidirect
48. #16812031C22semidirect
49. #16812030C22semidirect
50. #16812029C22semidirect
51. (C2×D107)⋊C8C22semidirect
52. (C2×D107)⋊C8C22semidirect
53. #16812020C22semidirect
54. C2×SD16×C107C22semidirect
56. #124×C107C22semidirect
57. C4×D4×C107C22semidirect
58. C4⋊C8×C107C22semidirect
59. #103×C107C22semidirect
60. #102×C107C22semidirect
96. #13133803D4semidirect
170. D428C2×D4semidirect
180. C107⋊C8C2×D4semidirect
205. C2×D107AGL(1,ℤ/8)semidirect
226. C107#3264semidirect
234. #463D107semidirect
250. C2×SD16C2×D107semidirect
261. #124C2×D107semidirect
274. C4×D4C2×D107semidirect
277. C4⋊C8C2×D107semidirect
278. #103C2×D107semidirect
283. #102C2×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
D4×D4282
#202930172
#202928792
#202928592
#202928232
#202927762
#202927752
#202927622
#202927592
#202927372
#202926972
#463×C1072
#3264107