Finite Group Databasebeta

← #20286348

Subgroups — #20286348

Order 6784.

498 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#13133246C211yesmaximal, characteristic, complemented
2C2×#13133244C211yesmaximal, characteristic, complemented
3#16811285C211yesmaximal, characteristic, complemented
4#16811268C211yesmaximal, characteristic, complemented
5#16811007C211yesmaximal, characteristic, complemented
6#16810882C211yesmaximal, characteristic, complemented
7#16810877C211yesmaximal, characteristic, complemented
8#16810845C211yesmaximal, characteristic, complemented
9#16810838C211yesmaximal, characteristic, complemented
10#16810788C211yesmaximal, characteristic, complemented
11#16810775C211yesmaximal, characteristic, complemented
12#16810769C211yesmaximal, characteristic, complemented
13#16810763C211yesmaximal, characteristic, complemented
14#16810702C211yesmaximal, characteristic, complemented
15C4×SD16×C53C211yesmaximal, characteristic, complemented
16C2×#11719949C2211yescharacteristic
17C2×#11719947C2211yescharacteristic
18C2×Q8×D53C2211yescharacteristic, complemented
19C2×D4×D53C2211yescharacteristic
20#13133246—24no—
21#13133244—24no—
22C2×Dic212C2211yescharacteristic, complemented
23C2×#11719925C2211yescharacteristic, complemented
24C2×#11719913C2211yescharacteristic
25C2×#11719913C2211yescharacteristic, complemented
26C2×#11719923C2211yescharacteristic
27C2×#11719924C2211yescharacteristic
28C2×D424C2211yescharacteristic
29C2×#11719922C2211yescharacteristic
30C2×#11719912C2211yescharacteristic
31C2×#11719912C2211yescharacteristic, complemented
32#13133205—21no—
33#13133205C2211yescharacteristic
34#13133205C2211yescharacteristic, complemented
35#13133201—21no—
36#13133197—21no—
37#13133197C2211yescharacteristic
38#13133197C2211yescharacteristic
39#13133192—21no—
40#13133192C2211yescharacteristic
41#13133192C2211yescharacteristic, complemented
42#13133191—21no—
43#13133187—21no—
44C4×D212—21no—
45C4×D212C2211yescharacteristic
46C2×#11719921—21no—
47C4⋊#88322C2211yescharacteristic, complemented
48#13133140C2211yescharacteristic, complemented
49#13133140C2211yescharacteristic, complemented
50#13133137C2211yescharacteristic
51#13133137C2211yescharacteristic, complemented
52#13133129C2211yescharacteristic
53#13133129C2211yescharacteristic, complemented
54C2×SD16×C53C2211yescharacteristic
55C4×Q8×C53C2211yescharacteristic, complemented
56C4×D4×C53C2211yescharacteristic
57#106×C53C2211yescharacteristic, complemented
58#103×C53C2211yescharacteristic, complemented
59#102×C53C2211yescharacteristic
60C4×C8×C53C2211yescharacteristic, complemented
61C23×D53—21no—
62C2×Q8×C53C2311yescharacteristic
63C2×D4×C53C2311yescharacteristic
64C22×C4×C53—21no—
65C2×C22⋊D53—21no—
66C2×C22⋊D53—21no—
67C22×Dic53—21no—
68#11719949—22no—
69#11719949—41no—
70Q8×D53—22no—
71Q8×D53—41no—
72#11719947—22no—
73#11719947—41no—
74D4×D53—22no—
75D4×D53—41no—
76C2×D212—21no—
77C2×D212—21no—
78C2×D212—21no—
79C2×D212D411yescharacteristic
80C2×D212C2311yescharacteristic
81C2×D212C2311yescharacteristic
82C2×C4×D53—21no—
83C2×C4×D53—21no—
84C2×C4×D53C2311yescharacteristic
85C2×C4×D53C2311yescharacteristic
86C2×Dic106—21no—
87C2×Dic106C2311yescharacteristic
88C2×Dic106C2311yescharacteristic
89SD16×C53D414yescomplemented
90C2×C8×C53C2311yescharacteristic
91C2×C8×C53C2311yescharacteristic
92C4⋊C4×C53—21no—
93C4⋊C4×C53C2311yescharacteristic
94C4⋊C4×C53C2311yescharacteristic
95C22⋊C4×C53—21no—
96C42×C53—21no—
97C42×C53C2311yescharacteristic
98#11719925—22no—
99#11719924—22no—
100#11719923—22no—
101#11719922—22no—
102#11719921—21no—
103#11719921—21no—
104#11719921—21no—
105#11719921—21no—
106#11719921—21no—
107#11719921—21no—
108#11719921—21no—
109#11719921—21no—
110#11719921D411yescharacteristic
111#11719921D411yescharacteristic
112C4⋊Dic53—21no—
113C4⋊Dic53—21no—
114C4⋊Dic53D411yescharacteristic
115C2×#88322C2311yescharacteristic
116C2×#88322C2311yescharacteristic
117Dic212—22no—
118D424—22no—
119#11719913—41no—
120#11719913—41no—
121#11719912—22no—
122#11719912—22no—
123C23×C53—21no—
124C22×D53—21no—
125C22×D53—21no—
126C22×D53—42no—
127C22×D53—42no—
128C22×D53—81no—
129C22×D53C2×D411yescharacteristic
130C22×D53C2×D411yescharacteristic
131Q8×C53—21no—
132Q8×C53C2×D412yes—
133D4×C53—21no—
134D4×C53C2×D412yes—
135C2×C4×C53—21no—
136C2×C4×C53—21no—
137C2×C4×C53—21no—
138C2×C4×C53—81no—
139C2×C4×C53C2×D411yescharacteristic
140C2×C4×C53C2×D411yescharacteristic
141C2×C4×C53C2411yescharacteristic
142C22⋊D53—42no—
143C22⋊D53—42no—
144C2×Dic53—21no—
145C2×Dic53—21no—
146C2×Dic53—42no—
147C2×Dic53C2×D411yescharacteristic
148C2×Dic53C2×D411yescharacteristic
149D212—21no—
150D212—21no—
151D212—21no—
152D212—21no—
153D212—41no—
154D212—42no—
155D212—42no—
156D212—81no—
157C4×D53—21no—
158C4×D53—21no—
159C4×D53—21no—
160C4×D53—21no—
161C4×D53—42no—
162C4×D53—42no—
163Dic106—21no—
164Dic106—21no—
165Dic106—21no—
166Dic106—21no—
167Dic106—42no—
168C8×C53—21no—
169C8×C53C2×D412yes—
170#88322—21no—
171#88322—21no—
172C22×C53—22no—
173C22×C53—41no—
174C22×C53—41no—
175C22×C53C22×D411yescharacteristic
176C2×D53—21no—
177C2×D53—21no—
178C2×D53—21no—
179C2×D53—21no—
180C2×D53—21no—
181C2×D53—21no—
182C2×D53—41no—
183C2×D53—41no—
184C2×D53—41no—
185C2×D53—41no—
186C2×D53—41no—
187C2×D53—41no—
188C2×D53—84no—
189C4×C53—22no—
190C4×C53—21no—
191C4×C53—21no—
192C4×C53—41no—
193C4×C53—41no—
194C4×C53C22×D411yescharacteristic
195C4×C53H(5,𝔽2)11yescharacteristic
196Dic53—21no—
197Dic53—21no—
198Dic53—41no—
199Dic53—41no—
200#3275—531nomaximal, Sylow
201C2×C53—42no—
202C2×C53D4211yescharacteristic
203C2×C53#57511yescharacteristic
204C2×C53#57711yescharacteristic
205D53—41no—
206D53—41no—
207D53—81no—
208D53—81no—
209C2×#137—531no—
210C2×AGL(1,ℤ/8)—531no—
211#548—531no—
212#546—531no—
213#495—531no—
214#470—531no—
215#469—531no—
216#464—531no—
217#463—531no—
218#452—531no—
219#450—531no—
220#449—531no—
221#448—531no—
222C4×SD16D5311yescharacteristic, complemented
223#436—531no—
224C53#327511yescharacteristic, complemented, Sylow
225C2×#41—531no—
226C2×#41—531no—
227C22×Q8—531no—
228C22×D4—531no—
229#137—1064no—
230AGL(1,ℤ/8)—1064no—
231C2×Q16—531no—
232C2×Q16—531no—
233C2×SD16—531no—
234C2×SD16—531no—
235C2×SD16—531no—
236C2×SD16—531no—
237C2×SD16C2×D5311yescharacteristic
238C2×D8—531no—
239C2×D8—531no—
240C2×M16—531no—
241C2×M16—531no—
242#124—531no—
243#124—531no—
244#124—1061no—
245#123—1061no—
246C22⋊Q8—531no—
247C22⋊Q8—531no—
248C22⋊Q8—1061no—
249#121—531no—
250#121—531no—
251#121—1061no—
252#121—1061no—
253#120—1061no—
254C4×Q8C2×D5311yescharacteristic, complemented
255C4×D4—531no—
256C4×D4—1061no—
257C4×D4C2×D5311yescharacteristic
258C2×C22⋊C4—1061no—
259#106C2×D5311yescharacteristic, complemented
260C4⋊C8—531no—
261#103—531no—
262#103—531no—
263#103C2×D5311yescharacteristic, complemented
264#102—531no—
265#102—531no—
266#102C2×D5311yescharacteristic
267C22⋊C8—531no—
268C22⋊C8—531no—
269C4×C8C2×D5311yescharacteristic, complemented
270C24—1061no—
271#41—1062no—
272#41—1062no—
273#41—2121no—
274#41—2121no—
275C2×Q8—531no—
276C2×Q8—531no—
277C2×Q8—1061no—
278C2×Q8—1062no—
279C2×Q8—2121no—
280C2×Q8C22×D5311yescharacteristic
281C2×D4—531no—
282C2×D4—531no—
283C2×D4—531no—
284C2×D4—1061no—
285C2×D4—1061no—
286C2×D4—1061no—
287C2×D4—1061no—
288C2×D4—1062no—
289C2×D4—1061no—
290C2×D4—2121no—
291C2×D4C22×D5311yescharacteristic
292C22×C4—21no—
293C22×C4—531no—
294C22×C4—531no—
295C22×C4—1061no—
296C22×C4—1061no—
297C22×C4—1061no—
298Q16—1062no—
299Q16—1062no—
300SD16—1062no—
301SD16—1062no—
302SD16—2121no—
303SD16—2121no—
304SD16D21214yescomplemented
305D8—1062no—
306D8—1062no—
307M16—1062no—
308M16—1062no—
309C2×C8—531no—
310C2×C8—531no—
311C2×C8C22×D5311yescharacteristic
312C2×C8C22×D5311yescharacteristic
313C4⋊C4—21no—
314C4⋊C4—531no—
315C4⋊C4—1061no—
316C4⋊C4—1061no—
317C4⋊C4C22×D5311yescharacteristic
318C4⋊C4C22×D5311yescharacteristic
319C22⋊C4—21no—
320C22⋊C4—531no—
321C22⋊C4—531no—
322C22⋊C4—1061no—
323C22⋊C4—1061no—
324C22⋊C4—1061no—
325C22⋊C4—1061no—
326C22⋊C4—1061no—
327C22⋊C4—1061no—
328C22⋊C4—1061no—
329C22⋊C4—1061no—
330C42—21no—
331C42C22×D5311yescharacteristic
332C23—21no—
333C23—531no—
334C23—531no—
335C23—1061no—
336C23—1061no—
337C23—4241no—
338C23—2122no—
339C23—2122no—
340Q8—21no—
341Q8—1061no—
342Q8—1061no—
343Q8—1061no—
344Q8—1061no—
345Q8—2122no—
346Q8C2×D21212yes—
347D4—21no—
348D4—1061no—
349D4—1061no—
350D4—1061no—
351D4—1061no—
352D4—4241no—
353D4—2121no—
354D4—2122no—
355D4—2122no—
356D4—2122no—
357D4—2122no—
358D4C2×D21212yes—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—81no—
363C2×C4—531no—
364C2×C4—531no—
365C2×C4—1061no—
366C2×C4—1061no—
367C2×C4—1061no—
368C2×C4—1061no—
369C2×C4—1061no—
370C2×C4—1061no—
371C2×C4—2122no—
372C2×C4—2122no—
373C2×C4—2122no—
374C2×C4D4×D5311yescharacteristic
375C2×C4D4×D5311yescharacteristic
376C2×C4C23×D5311yescharacteristic
377C8—21no—
378C8—1061no—
379C8—1061no—
380C8C2×D21212yes—
381C22—22no—
382C22—41no—
383C22—41no—
384C22—1061no—
385C22—1061no—
386C22—1061no—
387C22—1061no—
388C22—1061no—
389C22—1061no—
390C22—4244no—
391C22—2121no—
392C22—2121no—
393C22—2121no—
394C22—2121no—
395C22—2121no—
396C22—2121no—
397C22C2×D4×D5311yescharacteristic
398C4—21no—
399C4—21no—
400C4—22no—
401C4—41no—
402C4—41no—
403C4—1061no—
404C4—1061no—
405C4—2121no—
406C4—2121no—
407C4C22×D21211yescharacteristic
408C4#1313326311yescharacteristic
409C2—42no—
410C2—4241no—
411C2—4241no—
412C2—2121no—
413C2—2121no—
414C2D4×D21211yescharacteristic
415C2#1681142011yescharacteristic
416C2#1681142911yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC4×SD16×C53C2
socleC22×C53C22×D4
hypercenterC4×SD16D53
hypocenterC53#3275
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ C4×SD16

Complements

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

NH ≅ G/Ntype
1. C2×#13133246C2semidirect
2. C2×#13133244C2semidirect
3. #16811285C2semidirect
4. #16811268C2semidirect
5. #16811007C2semidirect
6. #16810882C2semidirect
7. #16810877C2semidirect
8. #16810845C2semidirect
9. #16810838C2semidirect
10. #16810788C2semidirect
11. #16810775C2semidirect
12. #16810769C2semidirect
13. #16810763C2semidirect
14. #16810702C2semidirect
15. C4×SD16×C53C2semidirect
18. C2×Q8×D53C22semidirect
22. C2×Dic212C22semidirect
23. C2×#11719925C22semidirect
25. C2×#11719913C22semidirect
31. C2×#11719912C22semidirect
34. #13133205C22semidirect
41. #13133192C22semidirect
47. C4⋊#88322C22semidirect
48. #13133140C22semidirect
49. #13133140C22semidirect
51. #13133137C22semidirect
53. #13133129C22semidirect
55. C4×Q8×C53C22semidirect
57. #106×C53C22semidirect
58. #103×C53C22semidirect
60. C4×C8×C53C22semidirect
89. SD16×C53D4semidirect
222. C4×SD16D53semidirect
224. C53#3275semidirect
254. C4×Q8C2×D53semidirect
259. #106C2×D53semidirect
263. #103C2×D53semidirect
269. C4×C8C2×D53semidirect
304. SD16D212semidirect