Finite Group Databasebeta

← #28347958

Subgroups — #28347958

Order 13696.

546 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16812136C211yesmaximal, characteristic, complemented
2C2×SD16×D107C211yesmaximal, characteristic, complemented
3#20293289C211yesmaximal, characteristic, complemented
4D4×D428C211yesmaximal, characteristic, complemented
5#20293016C211yesmaximal, characteristic, complemented
6#20292874C211yesmaximal, characteristic, complemented
7#20292864C211yesmaximal, characteristic, complemented
8#20292842C211yesmaximal, characteristic, complemented
9#20292827C211yesmaximal, characteristic, complemented
10#20292779C211yesmaximal, characteristic, complemented
11#20292775C211yesmaximal, characteristic, complemented
12#20292767C211yesmaximal, characteristic, complemented
13#20292762C211yesmaximal, characteristic, complemented
14#20292700C211yesmaximal, characteristic, complemented
15#440×C107C211yesmaximal, characteristic, complemented
16C2×#13133819C2211yescharacteristic, complemented
17C2×Q8×D107C2211yescharacteristic, complemented
18C22×D428—21no—
19C2×D4×D107C2211yescharacteristic
20C2×D4×D107C2211yescharacteristic
21#16812136—24no—
22C2×#13133807C2211yescharacteristic, complemented
23C2×#13133807C2211yescharacteristic, complemented
24C2×#13133806C2211yescharacteristic
25C2×#13133805C2211yescharacteristic, complemented
26C2×#13133805C2211yescharacteristic, complemented
27SD16×D107—24no—
28C2×D856C2211yescharacteristic, complemented
29C2×D856C2211yescharacteristic, complemented
30C2×#13133803C2211yescharacteristic
31C2×#13133801C2211yescharacteristic, complemented
32C2×C8×D107C2211yescharacteristic, complemented
33#16812106C2211yescharacteristic
34#16812097—21no—
35#16812097C2211yescharacteristic, complemented
36#16812089—21no—
37#16812089C2211yescharacteristic, complemented
38#16812084—21no—
39#16812084C2211yescharacteristic
40#16812084C2211yescharacteristic
41#16812084C2211yescharacteristic, complemented
42#16812083—21no—
43#16812079—21no—
44#16812079—21no—
45C4×D428—21no—
46C4×D428C2211yescharacteristic
47C4⋊(C107⋊C8)C2211yescharacteristic, complemented
48#16812032C2211yescharacteristic, complemented
49#16812029C2211yescharacteristic, complemented
50#16812029C2211yescharacteristic, complemented
51#16812029C2211yescharacteristic, complemented
52(C2×D107)⋊C8C2211yescharacteristic, complemented
53(C2×D107)⋊C8C2211yescharacteristic, complemented
54C2×SD16×C107C2211yescharacteristic
55C4×Q8×C107C2211yescharacteristic, complemented
56C4×D4×C107C2211yescharacteristic
57#107×C107C2211yescharacteristic, complemented
58#103×C107C2211yescharacteristic, complemented
59#102×C107C2211yescharacteristic
60#97×C107C2211yescharacteristic, complemented
61C23×D107—21no—
62C23×D107—21no—
63#13133819—22no—
64#13133819—41no—
65C2×Dic214—21no—
66C2×Dic214C2311yescharacteristic
67Q8×D107—22no—
68Q8×D107—41no—
69C2×D428—21no—
70C2×D428—21no—
71C2×D428—21no—
72C2×D428—21no—
73C2×D428—21no—
74C2×D428—21no—
75C2×D428—44no—
76C2×D428D411yescharacteristic
77C2×D428C2311yescharacteristic
78C2×D428C2311yescharacteristic
79C2×D428C2311yescharacteristic
80C2×C22⋊D107—21no—
81C2×C22⋊D107—21no—
82D4×D107—22no—
83D4×D107—22no—
84D4×D107—41no—
85D4×D107—41no—
86C2×C4×D107—21no—
87C2×C4×D107—21no—
88C2×C4×D107C2311yescharacteristic
89C2×C4×D107C2311yescharacteristic
90#13133807—22no—
91#13133807—41no—
92#13133806—22no—
93#13133805—22no—
94#13133805—22no—
95D856—22no—
96D856—41no—
97#13133803—22no—
98#13133801—22no—
99C2×C107⋊C8C2311yescharacteristic
100C2×C107⋊C8C2311yescharacteristic
101C8×D107—22no—
102C4⋊Dic107—21no—
103C4⋊Dic107D411yescharacteristic
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)⋊C4—21no—
110(C2×D107)⋊C4D411yescharacteristic
111(C2×D107)⋊C4D411yescharacteristic
112C2×Q8×C107C2311yescharacteristic
113C2×D4×C107C2311yescharacteristic
114C22×C4×C107—21no—
115SD16×C107D414yescomplemented
116C2×C8×C107C2311yescharacteristic
117C2×C8×C107C2311yescharacteristic
118C4⋊C4×C107—21no—
119C4⋊C4×C107C2311yescharacteristic
120C4⋊C4×C107C2311yescharacteristic
121C22⋊C4×C107—21no—
122C42×C107—21no—
123C42×C107C2311yescharacteristic
124C23×C107—21no—
125C22×D107—21no—
126C22×D107—21no—
127C22×D107—21no—
128C22×D107—42no—
129C22×D107—42no—
130C22×D107—42no—
131C22×D107—42no—
132C22×D107—81no—
133C22×D107—81no—
134C22×D107C2×D411yescharacteristic
135C22×D107C2×D411yescharacteristic
136Q8×C107—21no—
137Q8×C107C2×D412yescomplemented
138D4×C107—21no—
139D4×C107C2×D412yes—
140C2×C4×C107—21no—
141C2×C4×C107—21no—
142C2×C4×C107—21no—
143C2×C4×C107—42no—
144C2×C4×C107C2×D411yescharacteristic
145C2×C4×C107C2×D411yescharacteristic
146C2×C4×C107C2411yescharacteristic
147C22⋊D107—42no—
148C22⋊D107—42no—
149C2×Dic107—21no—
150C2×Dic107C2×D411yescharacteristic
151C2×Dic107C2×D411yescharacteristic
152D428—22no—
153D428—21no—
154D428—21no—
155D428—21no—
156D428—21no—
157D428—21no—
158D428—21no—
159D428—42no—
160D428—42no—
161D428—42no—
162D428—42no—
163D428—42no—
164D428—81no—
165C4×D107—21no—
166C4×D107—21no—
167C4×D107—21no—
168C4×D107—21no—
169C4×D107—42no—
170C4×D107—42no—
171Dic214—21no—
172Dic214—21no—
173Dic214—42no—
174C8×C107—21no—
175C8×C107C2×D412yescomplemented
176C107⋊C8—21no—
177C107⋊C8—21no—
178C22×C107—22no—
179C22×C107—41no—
180C22×C107—41no—
181C22×C107C22×D411yescharacteristic
182C2×D107—21no—
183C2×D107—21no—
184C2×D107—21no—
185C2×D107—21no—
186C2×D107—21no—
187C2×D107—41no—
188C2×D107—41no—
189C2×D107—41no—
190C2×D107—41no—
191C2×D107—41no—
192C2×D107—41no—
193C2×D107—41no—
194C2×D107—41no—
195C2×D107—41no—
196C2×D107—84no—
197C2×D107—84no—
198C2×D107AGL(1,ℤ/8)12yescomplemented
199C4×C107—22no—
200C4×C107—21no—
201C4×C107—21no—
202C4×C107—41no—
203C4×C107—41no—
204C4×C107C22×D411yescharacteristic
205C4×C107H(5,𝔽2)11yescharacteristic
206Dic107—21no—
207Dic107—21no—
208Dic107—41no—
209C2×C107—42no—
210C2×C107D4211yescharacteristic
211C2×C107C2×AGL(1,ℤ/8)11yescharacteristic
212C2×C107#57711yescharacteristic
213D107—22no—
214D107—41no—
215D107—81no—
216D107—81no—
217D107—81no—
218#3265—1071nomaximal, Sylow
219C107#326511yescharacteristic, complemented, Sylow
220C2×AGL(1,ℤ/8)—1071no—
221C22×SD16—1071no—
222#548—1071no—
223D42—1071no—
224#496—1071no—
225#468—1071no—
226#466—1071no—
227#463—1071no—
228#460—1071no—
229#450—1071no—
230#449—1071no—
231#448—1071no—
232#447—1071no—
233#440D10711yescharacteristic, complemented
234#435—1071no—
235C2×#41—1071no—
236C22×Q8—1071no—
237C22×D4—1071no—
238C22×D4—1071no—
239C22×D4—2141no—
240AGL(1,ℤ/8)—2144no—
241C2×SD16—1071no—
242C2×SD16—1071no—
243C2×SD16—1071no—
244C2×SD16—1071no—
245C2×SD16—1071no—
246C2×SD16—2144no—
247C2×SD16C2×D10711yescharacteristic
248C2×D8—1071no—
249C2×D8—1071no—
250C2×D8—1071no—
251C2×M16—1071no—
252C22×C8—1071no—
253#127—1071no—
254#124—1071no—
255#124—2141no—
256C22⋊Q8—1071no—
257C22⋊Q8—2141no—
258#121—1071no—
259#121—1071no—
260#121—1071no—
261#121—2141no—
262#121—2141no—
263#120—2141no—
264#120—2141no—
265C4×Q8C2×D10711yescharacteristic, complemented
266C4×D4—1071no—
267C4×D4—2141no—
268C4×D4C2×D10711yescharacteristic
269#107C2×D10711yescharacteristic, complemented
270C4⋊C8—1071no—
271#103—1071no—
272#103C2×D10711yescharacteristic, complemented
273#102—1071no—
274#102—1071no—
275#102—1071no—
276#102C2×D10711yescharacteristic
277C22⋊C8—1071no—
278C22⋊C8—1071no—
279#97C2×D10711yescharacteristic, complemented
280C24—2141no—
281C24—2141no—
282#41—4281no—
283#41—2142no—
284C2×Q8—1071no—
285C2×Q8—4281no—
286C2×Q8—2141no—
287C2×Q8—2142no—
288C2×Q8C22×D10711yescharacteristic
289C2×D4—1071no—
290C2×D4—1071no—
291C2×D4—1071no—
292C2×D4—1071no—
293C2×D4—4284no—
294C2×D4—4281no—
295C2×D4—4281no—
296C2×D4—2141no—
297C2×D4—2141no—
298C2×D4—2141no—
299C2×D4—2141no—
300C2×D4—2141no—
301C2×D4—2141no—
302C2×D4—2142no—
303C2×D4—2142no—
304C2×D4—2141no—
305C2×D4—2141no—
306C2×D4C22×D10711yescharacteristic
307C22×C4—21no—
308C22×C4—1071no—
309C22×C4—1071no—
310C22×C4—2141no—
311C22×C4—2141no—
312SD16—4281no—
313SD16—2142no—
314SD16—2142no—
315SD16—2142no—
316SD16—2142no—
317SD16D42814yescomplemented
318D8—4281no—
319D8—2142no—
320D8—2142no—
321M16—2142no—
322C2×C8—1071no—
323C2×C8—1071no—
324C2×C8—2142no—
325C2×C8C22×D10711yescharacteristic
326C2×C8C22×D10711yescharacteristic
327C4⋊C4—21no—
328C4⋊C4—1071no—
329C4⋊C4—2141no—
330C4⋊C4C22×D10711yescharacteristic
331C4⋊C4C22×D10711yescharacteristic
332C22⋊C4—21no—
333C22⋊C4—1071no—
334C22⋊C4—1071no—
335C22⋊C4—2141no—
336C22⋊C4—2141no—
337C22⋊C4—2141no—
338C22⋊C4—2141no—
339C22⋊C4—2141no—
340C22⋊C4—2141no—
341C42—21no—
342C42C22×D10711yescharacteristic
343C23—21no—
344C23—8561no—
345C23—8561no—
346C23—1071no—
347C23—1071no—
348C23—4282no—
349C23—4282no—
350C23—4282no—
351C23—4282no—
352C23—2141no—
353C23—2141no—
354C23—2141no—
355Q8—21no—
356Q8—4282no—
357Q8—2141no—
358Q8—2141no—
359Q8C2×D42812yescomplemented
360D4—21no—
361D4—8561no—
362D4—4282no—
363D4—4282no—
364D4—4282no—
365D4—4282no—
366D4—4282no—
367D4—4282no—
368D4—4282no—
369D4—2142no—
370D4—2141no—
371D4—2141no—
372D4—2141no—
373D4—2141no—
374D4—2141no—
375D4—2141no—
376D4C2×D42812yes—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—42no—
381C2×C4—1071no—
382C2×C4—1071no—
383C2×C4—4282no—
384C2×C4—4282no—
385C2×C4—2141no—
386C2×C4—2141no—
387C2×C4—2141no—
388C2×C4—2141no—
389C2×C4—2141no—
390C2×C4D4×D10711yescharacteristic
391C2×C4D4×D10711yescharacteristic
392C2×C4C23×D10711yescharacteristic
393C8—21no—
394C8—2141no—
395C8—2141no—
396C8C2×D42812yescomplemented
397C22—22no—
398C22—41no—
399C22—41no—
400C22—8564no—
401C22—8564no—
402C22—1072no—
403C22—4281no—
404C22—4281no—
405C22—4281no—
406C22—4281no—
407C22—4281no—
408C22—4281no—
409C22—4281no—
410C22—4281no—
411C22—4281no—
412C22—2141no—
413C22—2141no—
414C22—2141no—
415C22—2141no—
416C22—2141no—
417C22C2×D4×D10711yescharacteristic
418C4—21no—
419C4—21no—
420C4—22no—
421C4—41no—
422C4—41no—
423C4—4281no—
424C4—2141no—
425C4—2141no—
426C4C22×D42811yescharacteristic
427C4#1681215511yescharacteristic
428C2—42no—
429C2—8561no—
430C2—8561no—
431C2—8561no—
432C2—4281no—
433C2—2142no—
434C2D4×D42811yescharacteristic
435C2AGL(1,ℤ/8)×D10711yescharacteristic
436C2#2029343611yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#440×C107C2
socleC22×C107C22×D4
hypercenter#440D107
hypocenterC107#3265
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #440

Complements

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

NH ≅ G/Ntype
1. C2×#16812136C2semidirect
2. C2×SD16×D107C2semidirect
3. #20293289C2semidirect
4. D4×D428C2semidirect
5. #20293016C2semidirect
6. #20292874C2semidirect
7. #20292864C2semidirect
8. #20292842C2semidirect
9. #20292827C2semidirect
10. #20292779C2semidirect
11. #20292775C2semidirect
12. #20292767C2semidirect
13. #20292762C2semidirect
14. #20292700C2semidirect
15. #440×C107C2semidirect
16. C2×#13133819C22semidirect
17. C2×Q8×D107C22semidirect
22. C2×#13133807C22semidirect
23. C2×#13133807C22semidirect
25. C2×#13133805C22semidirect
26. C2×#13133805C22semidirect
28. C2×D856C22semidirect
29. C2×D856C22semidirect
31. C2×#13133801C22semidirect
32. C2×C8×D107C22semidirect
35. #16812097C22semidirect
37. #16812089C22semidirect
41. #16812084C22semidirect
47. C4⋊(C107⋊C8)C22semidirect
48. #16812032C22semidirect
49. #16812029C22semidirect
50. #16812029C22semidirect
51. #16812029C22semidirect
52. (C2×D107)⋊C8C22semidirect
53. (C2×D107)⋊C8C22semidirect
55. C4×Q8×C107C22semidirect
57. #107×C107C22semidirect
58. #103×C107C22semidirect
60. #97×C107C22semidirect
115. SD16×C107D4semidirect
137. Q8×C107C2×D4semidirect
175. C8×C107C2×D4semidirect
198. C2×D107AGL(1,ℤ/8)semidirect
219. C107#3265semidirect
233. #440D107semidirect
265. C4×Q8C2×D107semidirect
269. #107C2×D107semidirect
272. #103C2×D107semidirect
279. #97C2×D107semidirect
317. SD16D428semidirect
359. Q8C2×D428semidirect
396. C8C2×D428semidirect

Maximal subgroups

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

subgroupindex
C2×#168121362
C2×SD16×D1072
#202932892
D4×D4282
#202930162
#202928742
#202928642
#202928422
#202928272
#202927792
#202927752
#202927672
#202927622
#202927002
#440×C1072
#3265107