Finite Group Databasebeta

← #28347950

Subgroups — #28347950

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×SD16×D107C211yesmaximal, characteristic, complemented
2#20293284C211yesmaximal, characteristic, complemented
3D4×C22⋊D107C211yesmaximal, characteristic, complemented
4#20293015C211yesmaximal, characteristic, complemented
5#20292873C211yesmaximal, characteristic, complemented
6#20292860C211yesmaximal, characteristic, complemented
7#20292840C211yesmaximal, characteristic, complemented
8#20292825C211yesmaximal, characteristic, complemented
9#20292778C211yesmaximal, characteristic, complemented
10#20292772C211yesmaximal, characteristic, complemented
11#20292764C211yesmaximal, characteristic, complemented
12#20292759C211yesmaximal, characteristic, complemented
13#20292728C211yesmaximal, characteristic, complemented
14#20292696C211yesmaximal, characteristic, complemented
15C2×AGL(1,ℤ/8)×C107C211yesmaximal, characteristic, complemented
16C2×Q8×D107C2211yescharacteristic, complemented
17C22×C22⋊D107—21no—
18C2×D4×D107C2211yescharacteristic
19C2×#13133807C2211yescharacteristic, complemented
20C2×#13133806C2211yescharacteristic
21C2×#13133806C2211yescharacteristic, complemented
22C2×#13133805C2211yescharacteristic, complemented
23SD16×D107—24no—
24C2×#13133803C2211yescharacteristic, complemented
25C2×C8×D107C2211yescharacteristic, complemented
26#16812105C2211yescharacteristic
27#16812096—21no—
28#16812096C2211yescharacteristic, complemented
29#16812088—21no—
30#16812088C2211yescharacteristic, complemented
31#16812083C2211yescharacteristic
32#16812081—21no—
33#16812080—21no—
34#16812080C2211yescharacteristic
35#16812080C2211yescharacteristic, complemented
36#16812078—21no—
37#16812077—21no—
38Q8×Dic107C2211yescharacteristic, complemented
39D4×Dic107C2211yescharacteristic
40C4×C22⋊D107—21no—
41C4×C22⋊D107C2211yescharacteristic
42#16812043C2211yescharacteristic, complemented
43Dic107⋊C8C2211yescharacteristic, complemented
44#16812034C2211yescharacteristic, complemented
45#16812033C2211yescharacteristic, complemented
46#16812031C2211yescharacteristic, complemented
47#16812030C2211yescharacteristic
48#16812030C2211yescharacteristic, complemented
49#16812030C2211yescharacteristic, complemented
50C22⋊(C107⋊C8)C2211yescharacteristic, complemented
51(C2×D107)⋊C8C2211yescharacteristic, complemented
52#16812019C2211yescharacteristic, complemented
53C2×#41×C107C2211yescharacteristic, complemented
54C22×D4×C107C2211yescharacteristic
55AGL(1,ℤ/8)×C107—24no—
56C2×SD16×C107C2211yescharacteristic
57C2×SD16×C107C2211yescharacteristic, complemented
58C2×D8×C107C2211yescharacteristic
59C2×D8×C107C2211yescharacteristic, complemented
60C2×M16×C107C2211yescharacteristic, complemented
61C23×D107—21no—
62C2×Dic214—21no—
63C2×Dic214C2311yescharacteristic
64Q8×D107—22no—
65Q8×D107—41no—
66C2×D428C2311yescharacteristic
67C2×C22⋊D107—21no—
68C2×C22⋊D107—21no—
69C2×C22⋊D107—21no—
70C2×C22⋊D107—21no—
71C2×C22⋊D107—21no—
72C2×C22⋊D107—21no—
73C2×C22⋊D107—44no—
74C2×C22⋊D107D411yescharacteristic
75D4×D107—22no—
76D4×D107—41no—
77C22×Dic107—21no—
78C2×C4×D107—21no—
79C2×C4×D107C2311yescharacteristic
80#13133807—22no—
81#13133806—22no—
82#13133806—41no—
83#13133805—22no—
84#13133803—41no—
85C2×C107⋊C8C2311yescharacteristic
86C2×C107⋊C8C2311yescharacteristic
87C8×D107—22no—
88C4⋊Dic107—21no—
89C4⋊Dic107C2311yescharacteristic
90C4⋊Dic107C2311yescharacteristic
91Dic107⋊C4—21no—
92Dic107⋊C4D411yescharacteristic
93C22⋊Dic107—21no—
94C22⋊Dic107—21no—
95C22⋊Dic107—21no—
96C22⋊Dic107—21no—
97C22⋊Dic107—21no—
98C22⋊Dic107—21no—
99C22⋊Dic107D411yescharacteristic
100(C2×D107)⋊C4—21no—
101(C2×D107)⋊C4D411yescharacteristic
102C4×Dic107—21no—
103C4×Dic107C2311yescharacteristic
104C24×C107—21no—
105#41×C107—22no—
106#41×C107—41no—
107C2×Q8×C107C2311yescharacteristic
108C2×D4×C107—22no—
109C2×D4×C107—21no—
110C2×D4×C107—21no—
111C2×D4×C107—41no—
112C2×D4×C107C2311yescharacteristic
113C2×D4×C107C2311yescharacteristic
114C2×D4×C107C2311yescharacteristic
115C22×C4×C107—21no—
116C22×C4×C107C2311yescharacteristic
117SD16×C107—22no—
118SD16×C107D414yescomplemented
119D8×C107—22no—
120D8×C107—22no—
121M16×C107—22no—
122C2×C8×C107C2311yescharacteristic
123C2×C8×C107C2311yescharacteristic
124C23×C107—21no—
125C23×C107—21no—
126C23×C107—21no—
127C23×C107—42no—
128C23×C107—42no—
129C23×C107—81no—
130C23×C107C2×D411yescharacteristic
131C22×D107—21no—
132C22×D107—42no—
133C22×D107—42no—
134C22×D107—81no—
135C22×D107C2×D411yescharacteristic
136Q8×C107—21no—
137Q8×C107C2×D412yescomplemented
138D4×C107—21no—
139D4×C107—21no—
140D4×C107—21no—
141D4×C107—21no—
142D4×C107—21no—
143D4×C107—42no—
144D4×C107—42no—
145D4×C107C2×D412yes—
146C2×C4×C107—21no—
147C2×C4×C107—21no—
148C2×C4×C107—21no—
149C2×C4×C107—42no—
150C2×C4×C107C2×D411yescharacteristic
151C2×C4×C107C2411yescharacteristic
152C22⋊D107—22no—
153C22⋊D107—42no—
154C22⋊D107—42no—
155C22⋊D107—42no—
156C22⋊D107—42no—
157C22⋊D107—42no—
158C22⋊D107—81no—
159C2×Dic107—21no—
160C2×Dic107—21no—
161C2×Dic107—21no—
162C2×Dic107—42no—
163C2×Dic107C2×D411yescharacteristic
164C2×Dic107C2×D411yescharacteristic
165C2×Dic107C2×D411yescharacteristic
166D428—21no—
167D428—21no—
168C4×D107—21no—
169C4×D107—21no—
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—21no—
180C22×C107—21no—
181C22×C107—21no—
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×C107—84no—
191C22×C107C22×D411yescharacteristic
192C2×D107—21no—
193C2×D107—21no—
194C2×D107—41no—
195C2×D107—41no—
196C2×D107—41no—
197C2×D107—84no—
198C2×D107AGL(1,ℤ/8)12yescomplemented
199C4×C107—22no—
200C4×C107—21no—
201C4×C107C22×D411yescharacteristic
202C4×C107H(5,𝔽2)11yescharacteristic
203Dic107—21no—
204Dic107—21no—
205Dic107—21no—
206Dic107—41no—
207Dic107—41no—
208Dic107—41no—
209C2×C107—42no—
210C2×C107—41no—
211C2×C107—81no—
212C2×C107—81no—
213C2×C107D4211yescharacteristic
214C2×C107C2×AGL(1,ℤ/8)11yescharacteristic
215C2×C107#57711yescharacteristic
216D107—22no—
217D107—81no—
218#3265—1071nomaximal, Sylow
219C107#326511yescharacteristic, complemented, Sylow
220C2×AGL(1,ℤ/8)D10711yescharacteristic, complemented
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#440—1071no—
234#435—1071no—
235C2×#41C2×D10711yescharacteristic, complemented
236C22×Q8—1071no—
237C22×D4—1071no—
238C22×D4—2141no—
239C22×D4C2×D10711yescharacteristic
240AGL(1,ℤ/8)—24no—
241C2×SD16—1071no—
242C2×SD16—1071no—
243C2×SD16—1071no—
244C2×SD16—1071no—
245C2×SD16—2144no—
246C2×SD16C2×D10711yescharacteristic
247C2×SD16C2×D10711yescharacteristic, complemented
248C2×D8—1071no—
249C2×D8C2×D10711yescharacteristic
250C2×D8C2×D10711yescharacteristic, complemented
251C2×M16C2×D10711yescharacteristic, complemented
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×Q8—1071no—
266C4×D4—1071no—
267C4×D4—1071no—
268C4×D4—2141no—
269#107—1071no—
270C4⋊C8—1071no—
271#103—1071no—
272#103—1071no—
273#102—1071no—
274#102—1071no—
275#102—1071no—
276#102—1071no—
277C22⋊C8—1071no—
278C22⋊C8—1071no—
279#97—1071no—
280C24—21no—
281C24—2141no—
282#41—22no—
283#41—41no—
284C2×Q8—1071no—
285C2×Q8—4281no—
286C2×Q8—2141no—
287C2×Q8—2142no—
288C2×Q8C22×D10711yescharacteristic
289C2×D4—22no—
290C2×D4—21no—
291C2×D4—21no—
292C2×D4—41no—
293C2×D4—1071no—
294C2×D4—1071no—
295C2×D4—4284no—
296C2×D4—4281no—
297C2×D4—2141no—
298C2×D4—2141no—
299C2×D4—2141no—
300C2×D4—2141no—
301C2×D4—2141no—
302C2×D4—2141no—
303C2×D4—2142no—
304C2×D4C22×D10711yescharacteristic
305C2×D4C22×D10711yescharacteristic
306C2×D4C22×D10711yescharacteristic
307C22×C4—21no—
308C22×C4—1071no—
309C22×C4—2141no—
310C22×C4—2141no—
311C22×C4C22×D10711yescharacteristic
312SD16—22no—
313SD16—4281no—
314SD16—2142no—
315SD16—2142no—
316SD16—2142no—
317SD16C22⋊D10714yescomplemented
318D8—22no—
319D8—22no—
320D8—4281no—
321M16—22no—
322C2×C8—1071no—
323C2×C8—1071no—
324C2×C8—2142no—
325C2×C8C22×D10711yescharacteristic
326C2×C8C22×D10711yescharacteristic
327C4⋊C4—1071no—
328C4⋊C4—1071no—
329C4⋊C4—1071no—
330C4⋊C4—2141no—
331C4⋊C4—2141no—
332C22⋊C4—1071no—
333C22⋊C4—1071no—
334C22⋊C4—2141no—
335C22⋊C4—2141no—
336C22⋊C4—2141no—
337C22⋊C4—2141no—
338C22⋊C4—2141no—
339C22⋊C4—2141no—
340C22⋊C4—2141no—
341C42—1071no—
342C42—2141no—
343C23—21no—
344C23—21no—
345C23—21no—
346C23—42no—
347C23—42no—
348C23—81no—
349C23—8561no—
350C23—1071no—
351C23—4282no—
352C23—4282no—
353C23—2141no—
354C23D4×D10711yescharacteristic
355Q8—21no—
356Q8—4282no—
357Q8—2141no—
358Q8—2141no—
359Q8C2×C22⋊D10712yescomplemented
360D4—21no—
361D4—21no—
362D4—21no—
363D4—21no—
364D4—21no—
365D4—42no—
366D4—42no—
367D4—8561no—
368D4—4282no—
369D4—4282no—
370D4—4282no—
371D4—4282no—
372D4—4282no—
373D4—2142no—
374D4—2141no—
375D4—2141no—
376D4C2×C22⋊D10712yes—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—42no—
381C2×C4—1071no—
382C2×C4—1071no—
383C2×C4—1071no—
384C2×C4—4282no—
385C2×C4—4282no—
386C2×C4—2141no—
387C2×C4—2141no—
388C2×C4—2141no—
389C2×C4—2141no—
390C2×C4—2141no—
391C2×C4D4×D10711yescharacteristic
392C2×C4C23×D10711yescharacteristic
393C8—21no—
394C8—2141no—
395C8—2141no—
396C8C2×C22⋊D10712yescomplemented
397C22—21no—
398C22—21no—
399C22—21no—
400C22—22no—
401C22—41no—
402C22—41no—
403C22—41no—
404C22—41no—
405C22—41no—
406C22—41no—
407C22—41no—
408C22—41no—
409C22—84no—
410C22—8564no—
411C22—1072no—
412C22—4281no—
413C22—4281no—
414C22—4281no—
415C22—2141no—
416C22—2141no—
417C22C2×D4×D10711yescharacteristic
418C4—21no—
419C4—22no—
420C4—4281no—
421C4—4281no—
422C4—4281no—
423C4—2141no—
424C4—2141no—
425C4—2141no—
426C4C22×C22⋊D10711yescharacteristic
427C4#1681215411yescharacteristic
428C2—41no—
429C2—42no—
430C2—81no—
431C2—81no—
432C2—8561no—
433C2—2142no—
434C2D4×C22⋊D10711yescharacteristic
435C2AGL(1,ℤ/8)×D10711yescharacteristic
436C2#2029343711yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
FittingC2×AGL(1,ℤ/8)×C107C2
socleC22×C107C22×D4
hypercenterC2×AGL(1,ℤ/8)D107
hypocenterC107#3265
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×SD16×D107C2semidirect
2. #20293284C2semidirect
3. D4×C22⋊D107C2semidirect
4. #20293015C2semidirect
5. #20292873C2semidirect
6. #20292860C2semidirect
7. #20292840C2semidirect
8. #20292825C2semidirect
9. #20292778C2semidirect
10. #20292772C2semidirect
11. #20292764C2semidirect
12. #20292759C2semidirect
13. #20292728C2semidirect
14. #20292696C2semidirect
15. C2×AGL(1,ℤ/8)×C107C2semidirect
16. C2×Q8×D107C22semidirect
19. C2×#13133807C22semidirect
21. C2×#13133806C22semidirect
22. C2×#13133805C22semidirect
24. C2×#13133803C22semidirect
25. C2×C8×D107C22semidirect
28. #16812096C22semidirect
30. #16812088C22semidirect
35. #16812080C22semidirect
38. Q8×Dic107C22semidirect
42. #16812043C22semidirect
43. Dic107⋊C8C22semidirect
44. #16812034C22semidirect
45. #16812033C22semidirect
46. #16812031C22semidirect
48. #16812030C22semidirect
49. #16812030C22semidirect
50. C22⋊(C107⋊C8)C22semidirect
51. (C2×D107)⋊C8C22semidirect
52. #16812019C22semidirect
53. C2×#41×C107C22semidirect
57. C2×SD16×C107C22semidirect
59. C2×D8×C107C22semidirect
60. C2×M16×C107C22semidirect
118. SD16×C107D4semidirect
137. Q8×C107C2×D4semidirect
175. C8×C107C2×D4semidirect
198. C2×D107AGL(1,ℤ/8)semidirect
219. C107#3265semidirect
220. C2×AGL(1,ℤ/8)D107semidirect
235. C2×#41C2×D107semidirect
247. C2×SD16C2×D107semidirect
250. C2×D8C2×D107semidirect
251. C2×M16C2×D107semidirect
317. SD16C22⋊D107semidirect
359. Q8C2×C22⋊D107semidirect
396. C8C2×C22⋊D107semidirect

Maximal subgroups

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

subgroupindex
C2×SD16×D1072
#202932842
D4×C22⋊D1072
#202930152
#202928732
#202928602
#202928402
#202928252
#202927782
#202927722
#202927642
#202927592
#202927282
#202926962
C2×AGL(1,ℤ/8)×C1072
#3265107