Finite Group Databasebeta

← #20286478

Subgroups — #20286478

Order 6784.

464 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16811435C211yesmaximal, characteristic, complemented
2#16811433C211yesmaximal, characteristic, complemented
3#16811427C211yesmaximal, characteristic, complemented
4#16811424C211yesmaximal, characteristic, complemented
5#16811011C211yesmaximal, characteristic, complemented
6#16810892C211yesmaximal, characteristic, complemented
7#16810887C211yesmaximal, characteristic, complemented
8#16810885C211yesmaximal, characteristic, complemented
9#16810803C211yesmaximal, characteristic, complemented
10#16810794C211yesmaximal, characteristic, complemented
11#16810793C211yesmaximal, characteristic, complemented
12#16810789C211yesmaximal, characteristic, complemented
13#16810744C211yesmaximal, characteristic, complemented
14#16810743C211yesmaximal, characteristic, complemented
15C53×#472C211yesmaximal, characteristic, complemented
16#13133264C2211yescharacteristic, complemented
17#13133263C2211yescharacteristic, complemented
18#13133262C2211yescharacteristic
19#13133262C2211yescharacteristic, complemented
20#13133250—21no—
21#13133250C2211yescharacteristic, complemented
22#13133250C2211yescharacteristic, complemented
23#13133246—21no—
24#13133245—21no—
25#13133245C2211yescharacteristic
26#13133245C2211yescharacteristic, complemented
27#13133244—21no—
28#13133243C2211yescharacteristic
29#13133243C2211yescharacteristic, complemented
30#13133241—21no—
31#13133241—21no—
32#13133241—21no—
33#13133241—21no—
34#13133239—21no—
35#13133239C2211yescharacteristic
36#13133238—21no—
37#13133237C2211yescharacteristic
38#13133236—21no—
39#13133236—21no—
40C2×#11719913—21no—
41C2×#11719913C2211yescharacteristic, complemented
42C2×#11719923C2211yescharacteristic, complemented
43SD16×D53—21no—
44C2×#11719922C2211yescharacteristic, complemented
45D8×D53—21no—
46D8×D53—21no—
47#13133225C2211yescharacteristic, complemented
48#13133225C2211yescharacteristic, complemented
49#13133223C2211yescharacteristic, complemented
50#13133213C2211yescharacteristic, complemented
51#13133204C2211yescharacteristic, complemented
52#13133152C2211yescharacteristic, complemented
53#13133144C2211yescharacteristic, complemented
54#13133143C2211yescharacteristic, complemented
55#13133143C2211yescharacteristic, complemented
56#13133143C2211yescharacteristic, complemented
57#13133135C2211yescharacteristic, complemented
58#13133133C2211yescharacteristic, complemented
59#13133127C2211yescharacteristic, complemented
60AGL(1,ℤ/8)×C53C2211yescharacteristic, complemented
61AGL(1,ℤ/8)×C53C2211yescharacteristic, complemented
62C2×D8×C53C2211yescharacteristic, complemented
63#131×C53C2211yescharacteristic, complemented
64#108×C53C2211yescharacteristic, complemented
65#100×C53C2211yescharacteristic, complemented
66#100×C53C2211yescharacteristic, complemented
67#41×C53C2311yescharacteristic
68C2×D4×C53C2311yescharacteristic
69C2×D4×C53C2311yescharacteristic
70C2×C22⋊D53—41no—
71C2×C22⋊D53—41no—
72C2×C22⋊D53—41no—
73#11719949—21no—
74Q8×D53—21no—
75#11719947—21no—
76#11719947—21no—
77#11719947—21no—
78#11719947—21no—
79#11719947—21no—
80#11719947—41no—
81D4×D53—21no—
82D4×D53—21no—
83D4×D53—21no—
84D4×D53—21no—
85D4×D53—21no—
86D4×D53—41no—
87#11719945—21no—
88#11719945—21no—
89#11719945C2311yescharacteristic
90#11719945C2311yescharacteristic
91#11719945C2311yescharacteristic
92C2×D212—21no—
93C2×D212C2311yescharacteristic
94C2×Dic106—21no—
95C2×Dic106C2311yescharacteristic
96SD16×C53—21no—
97SD16×C53—21no—
98D8×C53—21no—
99D8×C53—21no—
100D8×C53—21no—
101D8×C53—21no—
102M16×C53—21no—
103M16×C53C2311yescharacteristic
104M16×C53C2311yescharacteristic
105M16×C53C2311yescharacteristic
106C2×C8×C53—21no—
107C2×C8×C53C2311yescharacteristic
108C22⋊Dic53—41no—
109#11719925D411yescharacteristic, complemented
110#11719924D411yescharacteristic, complemented
111#11719923—21no—
112#11719923—21no—
113#11719923—21no—
114#11719923—21no—
115#11719923D411yescharacteristic, complemented
116#11719922—21no—
117#11719922—21no—
118#11719922—21no—
119#11719922—21no—
120#11719922D411yescharacteristic, complemented
121C4×Dic53C2311yescharacteristic
122#11719917C2311yescharacteristic
123C2×#88322C2311yescharacteristic
124Dic212—21no—
125Dic212—21no—
126Dic212D411yescharacteristic, complemented
127D424—21no—
128D424—21no—
129D424D411yescharacteristic, complemented
130#11719913—21no—
131#11719913—21no—
132#11719913—21no—
133#11719913—21no—
134#11719913—21no—
135#11719913—21no—
136#11719913D411yescharacteristic, complemented
137#11719913D411yescharacteristic, complemented
138#11719912—21no—
139#11719912—21no—
140#11719912—21no—
141C8×D53—21no—
142C8×D53—21no—
143C8×D53—21no—
144C23×C53—21no—
145C23×C53—21no—
146C22×D53—21no—
147C22×D53—41no—
148C22×D53—41no—
149C22×D53—41no—
150Q8×C53C2×D411yescharacteristic
151D4×C53—21no—
152D4×C53—21no—
153D4×C53—21no—
154D4×C53—21no—
155D4×C53—21no—
156D4×C53C2×D411yescharacteristic
157C2×C4×C53—21no—
158C2×C4×C53C2411yescharacteristic
159C22⋊D53—21no—
160C22⋊D53—21no—
161C22⋊D53—21no—
162C22⋊D53—41no—
163C22⋊D53—41no—
164C22⋊D53—41no—
165C22⋊D53—41no—
166C22⋊D53—41no—
167C22⋊D53—41no—
168C22⋊D53—82no—
169C2×Dic53—21no—
170C2×Dic53—21no—
171C2×Dic53—41no—
172C2×Dic53—41no—
173C2×Dic53—41no—
174D212—21no—
175D212—21no—
176D212—41no—
177D212C2×D411yescharacteristic
178D212C2×D411yescharacteristic
179D212C2×D411yescharacteristic
180C4×D53—21no—
181C4×D53—21no—
182C4×D53—21no—
183C4×D53—41no—
184C4×D53—41no—
185Dic106—21no—
186Dic106—21no—
187Dic106—41no—
188Dic106C2×D411yescharacteristic
189Dic106C2×D411yescharacteristic
190Dic106C2×D411yescharacteristic
191C8×C53—21no—
192C8×C53—21no—
193C8×C53—21no—
194C8×C53C2×D411yescharacteristic
195C8×C53C2×D411yescharacteristic
196#88322—21no—
197#88322C2×D411yescharacteristic
198#88322C2×D411yescharacteristic
199C22×C53—21no—
200C22×C53—41no—
201C22×C53—41no—
202C22×C53—81no—
203C22×C53—81no—
204C22×C53H(5,𝔽2)11yescharacteristic
205C2×D53—21no—
206C2×D53—21no—
207C2×D53—21no—
208C2×D53—41no—
209C2×D53—81no—
210C2×D53—161no—
211C2×D53—161no—
212C2×D53—161no—
213C4×C53—21no—
214C4×C53C22×D411yescharacteristic
215C4×C53C22×D411yescharacteristic
216Dic53—21no—
217Dic53—21no—
218Dic53—21no—
219Dic53—41no—
220Dic53—41no—
221#3288—531nomaximal, Sylow
222C2×C53—21no—
223C2×C53—41no—
224C2×C53—81no—
225C2×C53—81no—
226C2×C53D4211yescharacteristic
227D53—41no—
228D53—41no—
229D53—41no—
230D53—81no—
231#577—531no—
232#577—531no—
233#576—531no—
234#576—531no—
235#496—531no—
236#472—531no—
237#472D5311yescharacteristic, complemented
238#471—531no—
239#471—531no—
240#455—531no—
241#454—531no—
242#453—531no—
243#453—531no—
244#444—531no—
245#444—531no—
246C53#328811yescharacteristic, complemented, Sylow
2472−1+4—531no—
248H(5,𝔽2)—531no—
249H(5,𝔽2)—531no—
250H(5,𝔽2)—531no—
251#137—531no—
252#137—531no—
253#137—1061no—
254#137—1061no—
255AGL(1,ℤ/8)—531no—
256AGL(1,ℤ/8)—531no—
257AGL(1,ℤ/8)—531no—
258AGL(1,ℤ/8)—531no—
259AGL(1,ℤ/8)—1061no—
260AGL(1,ℤ/8)—1061no—
261AGL(1,ℤ/8)—1061no—
262AGL(1,ℤ/8)—1061no—
263AGL(1,ℤ/8)—1061no—
264AGL(1,ℤ/8)—1061no—
265AGL(1,ℤ/8)C2×D5311yescharacteristic, complemented
266AGL(1,ℤ/8)C2×D5311yescharacteristic, complemented
267#135—531no—
268#135—531no—
269#135—1061no—
270#135—1061no—
271#135—1061no—
272#135—1061no—
273C2×SD16—531no—
274C2×SD16—531no—
275C2×SD16—1061no—
276C2×SD16—1061no—
277C2×D8—531no—
278C2×D8—1061no—
279C2×D8—1061no—
280C2×D8C2×D5311yescharacteristic, complemented
281#131—531no—
282#131—531no—
283#131—531no—
284#131C2×D5311yescharacteristic, complemented
285#127—531no—
286#124—531no—
287#108—531no—
288#108C2×D5311yescharacteristic, complemented
289#104—531no—
290#104—531no—
291#104—531no—
292#104—531no—
293#101—531no—
294#100—531no—
295#100C2×D5311yescharacteristic, complemented
296#100C2×D5311yescharacteristic, complemented
297#97—531no—
298#41—531no—
299#41—531no—
300#41—531no—
301#41—1061no—
302#41—1061no—
303#41—1061no—
304#41—1061no—
305#41—1061no—
306#41—1061no—
307#41—1061no—
308#41—1061no—
309#41—2121no—
310#41C22×D5311yescharacteristic
311C2×Q8—531no—
312C2×Q8—1061no—
313C2×Q8—1061no—
314C2×D4—531no—
315C2×D4—1061no—
316C2×D4—1061no—
317C2×D4—1061no—
318C2×D4—1061no—
319C2×D4—1061no—
320C2×D4—1061no—
321C2×D4—2121no—
322C2×D4—2121no—
323C2×D4—2121no—
324C2×D4—2121no—
325C2×D4C22×D5311yescharacteristic
326C2×D4C22×D5311yescharacteristic
327Q16—531no—
328Q16—531no—
329Q16—1061no—
330Q16—1061no—
331SD16—21no—
332SD16—21no—
333SD16—531no—
334SD16—531no—
335SD16—531no—
336SD16—531no—
337SD16—1061no—
338SD16—1061no—
339SD16—1061no—
340SD16—1061no—
341SD16—1061no—
342SD16—1061no—
343SD16—1061no—
344SD16—1061no—
345SD16—1061no—
346SD16—1061no—
347D8—21no—
348D8—21no—
349D8—21no—
350D8—21no—
351D8—531no—
352D8—531no—
353D8—1061no—
354D8—1061no—
355D8—1061no—
356D8—1061no—
357D8—1061no—
358D8—1061no—
359M16—21no—
360M16—531no—
361M16—1061no—
362M16—1061no—
363M16—1061no—
364M16C22×D5311yescharacteristic
365M16C22×D5311yescharacteristic
366M16C22×D5311yescharacteristic
367C2×C8—21no—
368C2×C8—531no—
369C2×C8—1061no—
370C2×C8—1061no—
371C2×C8—1061no—
372C2×C8C22×D5311yescharacteristic
373C22⋊C4—2121no—
374C42—531no—
375C23—21no—
376C23—21no—
377C23—1061no—
378C23—2121no—
379C23—2121no—
380C23—2121no—
381Q8—531no—
382Q8—531no—
383Q8—531no—
384Q8—1061no—
385Q8—1061no—
386Q8—2121no—
387Q8D4×D5311yescharacteristic
388D4—21no—
389D4—21no—
390D4—21no—
391D4—21no—
392D4—21no—
393D4—531no—
394D4—531no—
395D4—531no—
396D4—1061no—
397D4—1061no—
398D4—1061no—
399D4—1061no—
400D4—1061no—
401D4—4242no—
402D4—2121no—
403D4—2121no—
404D4—2121no—
405D4—2121no—
406D4—2121no—
407D4—2121no—
408D4—2121no—
409D4D4×D5311yescharacteristic
410C2×C4—21no—
411C2×C4—1061no—
412C2×C4—1061no—
413C2×C4—1061no—
414C2×C4—1061no—
415C2×C4—1061no—
416C2×C4—2121no—
417C2×C4—2121no—
418C2×C4—2121no—
419C2×C4—2121no—
420C2×C4—2121no—
421C2×C4C23×D5311yescharacteristic
422C8—21no—
423C8—21no—
424C8—21no—
425C8—531no—
426C8—531no—
427C8—1061no—
428C8D4×D5311yescharacteristic
429C8D4×D5311yescharacteristic
430C22—21no—
431C22—41no—
432C22—41no—
433C22—81no—
434C22—81no—
435C22—8481no—
436C22—8481no—
437C22—8481no—
438C22—1061no—
439C22—1061no—
440C22—1061no—
441C22—4241no—
442C22—2121no—
443C22#1313326311yescharacteristic
444C4—21no—
445C4—1061no—
446C4—1061no—
447C4—1061no—
448C4—2121no—
449C4—2121no—
450C4C2×D4×D5311yescharacteristic
451C4C2×D4×D5311yescharacteristic
452C2—21no—
453C2—41no—
454C2—81no—
455C2—81no—
456C2—4241no—
457C2—2121no—
458C2—2121no—
459C2—2121no—
460C2#1681125711yescharacteristic
subgroupG/H
centerC2#16811257
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#472C2
socleC2×C53D42
hypercenter#472D53
hypocenterC53#3288
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C2×C4 ◁ #472

Complements

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

NH ≅ G/Ntype
1. #16811435C2semidirect
2. #16811433C2semidirect
3. #16811427C2semidirect
4. #16811424C2semidirect
5. #16811011C2semidirect
6. #16810892C2semidirect
7. #16810887C2semidirect
8. #16810885C2semidirect
9. #16810803C2semidirect
10. #16810794C2semidirect
11. #16810793C2semidirect
12. #16810789C2semidirect
13. #16810744C2semidirect
14. #16810743C2semidirect
15. C53×#472C2semidirect
16. #13133264C22semidirect
17. #13133263C22semidirect
19. #13133262C22semidirect
21. #13133250C22semidirect
22. #13133250C22semidirect
26. #13133245C22semidirect
29. #13133243C22semidirect
41. C2×#11719913C22semidirect
42. C2×#11719923C22semidirect
44. C2×#11719922C22semidirect
47. #13133225C22semidirect
48. #13133225C22semidirect
49. #13133223C22semidirect
50. #13133213C22semidirect
51. #13133204C22semidirect
52. #13133152C22semidirect
53. #13133144C22semidirect
54. #13133143C22semidirect
55. #13133143C22semidirect
56. #13133143C22semidirect
57. #13133135C22semidirect
58. #13133133C22semidirect
59. #13133127C22semidirect
60. AGL(1,ℤ/8)×C53C22semidirect
61. AGL(1,ℤ/8)×C53C22semidirect
62. C2×D8×C53C22semidirect
63. #131×C53C22semidirect
64. #108×C53C22semidirect
65. #100×C53C22semidirect
66. #100×C53C22semidirect
109. #11719925D4semidirect
110. #11719924D4semidirect
115. #11719923D4semidirect
120. #11719922D4semidirect
126. Dic212D4semidirect
129. D424D4semidirect
136. #11719913D4semidirect
137. #11719913D4semidirect
237. #472D53semidirect
246. C53#3288semidirect
265. AGL(1,ℤ/8)C2×D53semidirect
266. AGL(1,ℤ/8)C2×D53semidirect
280. C2×D8C2×D53semidirect
284. #131C2×D53semidirect
288. #108C2×D53semidirect
295. #100C2×D53semidirect
296. #100C2×D53semidirect