Finite Group Databasebeta

← #20286476

Subgroups — #20286476

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#16811434C211yesmaximal, characteristic, complemented
2#16811433C211yesmaximal, characteristic, complemented
3#16811427C211yesmaximal, characteristic, complemented
4#16811426C211yesmaximal, characteristic, complemented
5#16811010C211yesmaximal, characteristic, complemented
6#16810892C211yesmaximal, characteristic, complemented
7#16810891C211yesmaximal, characteristic, complemented
8#16810885C211yesmaximal, characteristic, complemented
9#16810884C211yesmaximal, characteristic, complemented
10#16810802C211yesmaximal, characteristic, complemented
11#16810797C211yesmaximal, characteristic, complemented
12#16810790C211yesmaximal, characteristic, complemented
13#16810742C211yesmaximal, characteristic, complemented
14#16810742C211yesmaximal, characteristic, complemented
15C53×#453C211yesmaximal, characteristic, complemented
16#13133264C2211yescharacteristic, complemented
17#13133263C2211yescharacteristic
18#13133262C2211yescharacteristic, complemented
19#13133250C2211yescharacteristic, complemented
20#13133247—21no—
21#13133247C2211yescharacteristic, complemented
22#13133246—21no—
23#13133245C2211yescharacteristic, complemented
24#13133244—21no—
25#13133243—21no—
26#13133243—21no—
27#13133243C2211yescharacteristic
28#13133243C2211yescharacteristic, complemented
29#13133242—21no—
30#13133242C2211yescharacteristic, complemented
31#13133241—21no—
32#13133241—21no—
33#13133238—21no—
34#13133237—21no—
35#13133237—21no—
36#13133237C2211yescharacteristic
37#13133237C2211yescharacteristic
38#13133236—21no—
39C2×#11719923—21no—
40C2×#11719923C2211yescharacteristic, complemented
41C2×#11719923C2211yescharacteristic, complemented
42SD16×D53—21no—
43C2×#11719922—21no—
44C2×#11719922C2211yescharacteristic, complemented
45C2×#11719922C2211yescharacteristic, complemented
46D8×D53—21no—
47#13133225C2211yescharacteristic, complemented
48#13133225C2211yescharacteristic, complemented
49#13133224C2211yescharacteristic, complemented
50#13133224C2211yescharacteristic, complemented
51#13133205C2211yescharacteristic, complemented
52#13133152C2211yescharacteristic, complemented
53#13133152C2211yescharacteristic, complemented
54#13133145C2211yescharacteristic, complemented
55#13133145C2211yescharacteristic, complemented
56#13133135C2211yescharacteristic, complemented
57#13133134C2211yescharacteristic, complemented
58#13133133C2211yescharacteristic, complemented
59#13133128C2211yescharacteristic, complemented
60H(5,𝔽2)×C53C2211yescharacteristic, complemented
61AGL(1,ℤ/8)×C53C2211yescharacteristic
62AGL(1,ℤ/8)×C53C2211yescharacteristic, complemented
63#127×C53C2211yescharacteristic, complemented
64#104×C53C2211yescharacteristic, complemented
65#104×C53C2211yescharacteristic, complemented
66#100×C53C2211yescharacteristic, complemented
67#41×C53—21no—
68#41×C53—21no—
69#41×C53C2311yescharacteristic
70#41×C53C2311yescharacteristic
71C2×D4×C53—21no—
72C2×D4×C53—21no—
73C2×D4×C53—41no—
74C2×D4×C53—41no—
75C2×D4×C53C2311yescharacteristic
76C2×D4×C53C2311yescharacteristic
77C2×C22⋊D53—41no—
78#11719949—21no—
79Q8×D53—21no—
80#11719947—21no—
81#11719947—21no—
82#11719947—21no—
83#11719947—41no—
84D4×D53—21no—
85D4×D53—21no—
86D4×D53—21no—
87D4×D53—41no—
88#11719945—21no—
89#11719945—21no—
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—
100M16×C53C2311yescharacteristic
101M16×C53C2311yescharacteristic
102C42×C53C2311yescharacteristic
103#11719925—21no—
104#11719925D411yescharacteristic, complemented
105#11719925D411yescharacteristic, complemented
106#11719924—21no—
107#11719924D411yescharacteristic, complemented
108#11719924D411yescharacteristic, complemented
109#11719923—21no—
110#11719923—21no—
111#11719923—21no—
112#11719923—21no—
113#11719923—21no—
114#11719923—21no—
115#11719923—21no—
116#11719923D411yescharacteristic, complemented
117#11719923D411yescharacteristic, complemented
118#11719922—21no—
119#11719922—21no—
120#11719922—21no—
121#11719922—21no—
122#11719922—21no—
123#11719922—21no—
124#11719922—21no—
125#11719922D411yescharacteristic, complemented
126#11719922D411yescharacteristic, complemented
127#11719921—41no—
128#11719917—21no—
129#11719917—21no—
130#11719917C2311yescharacteristic
131#11719917C2311yescharacteristic
132C2×#88322—21no—
133C2×#88322—21no—
134C2×#88322C2311yescharacteristic
135C2×#88322C2311yescharacteristic
136Dic212—21no—
137D424—21no—
138#11719913—21no—
139#11719913—21no—
140#11719912—21no—
141#11719912—21no—
142C8×D53—21no—
143C8×D53—21no—
144C23×C53—21no—
145C23×C53—21no—
146C23×C53—41no—
147C22×D53—21no—
148C22×D53—41no—
149C22×D53—41no—
150Q8×C53C2×D411yescharacteristic
151Q8×C53C2×D411yescharacteristic
152D4×C53—21no—
153D4×C53—21no—
154D4×C53—21no—
155D4×C53—21no—
156D4×C53—21no—
157D4×C53—21no—
158D4×C53—41no—
159D4×C53—41no—
160D4×C53—82no—
161D4×C53C2×D411yescharacteristic
162D4×C53C2×D411yescharacteristic
163C2×C4×C53—21no—
164C2×C4×C53—21no—
165C2×C4×C53—21no—
166C2×C4×C53—41no—
167C2×C4×C53C2411yescharacteristic
168C22⋊D53—21no—
169C22⋊D53—21no—
170C22⋊D53—41no—
171C22⋊D53—41no—
172C22⋊D53—41no—
173C22⋊D53—41no—
174C2×Dic53—21no—
175C2×Dic53—41no—
176C2×Dic53—41no—
177D212—21no—
178D212—21no—
179D212—41no—
180D212C2×D411yescharacteristic
181D212C2×D411yescharacteristic
182C4×D53—21no—
183C4×D53—21no—
184C4×D53—41no—
185C4×D53—41no—
186Dic106—21no—
187Dic106—21no—
188Dic106—41no—
189Dic106C2×D411yescharacteristic
190Dic106C2×D411yescharacteristic
191C8×C53—21no—
192C8×C53—21no—
193#88322—21no—
194#88322—21no—
195#88322C2×D411yescharacteristic
196#88322C2×D411yescharacteristic
197#88322C2×D411yescharacteristic
198#88322C2×D411yescharacteristic
199C22×C53—21no—
200C22×C53—21no—
201C22×C53—41no—
202C22×C53—41no—
203C22×C53—81no—
204C22×C53—81no—
205C22×C53—161no—
206C22×C53H(5,𝔽2)11yescharacteristic
207C2×D53—21no—
208C2×D53—21no—
209C2×D53—41no—
210C2×D53—81no—
211C2×D53—161no—
212C2×D53—161no—
213C4×C53—21no—
214C4×C53—21no—
215C4×C53—41no—
216C4×C53C22×D411yescharacteristic
217C4×C53C22×D411yescharacteristic
218Dic53—21no—
219Dic53—21no—
220Dic53—41no—
221#3288—531nomaximal, Sylow
222C2×C53—21no—
223C2×C53—41no—
224C2×C53—41no—
225C2×C53—81no—
226C2×C53—81no—
227C2×C53D4211yescharacteristic
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#472—531no—
238#471—531no—
239#471—531no—
240#455—531no—
241#454—531no—
242#453—531no—
243#453D5311yescharacteristic, complemented
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)C2×D5311yescharacteristic, complemented
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
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—531no—
279C2×D8—1061no—
280C2×D8—1061no—
281#131—531no—
282#131—531no—
283#131—531no—
284#131—531no—
285#127C2×D5311yescharacteristic, complemented
286#124—531no—
287#108—531no—
288#108—531no—
289#104—531no—
290#104—531no—
291#104C2×D5311yescharacteristic, complemented
292#104C2×D5311yescharacteristic, complemented
293#101—531no—
294#100—531no—
295#100—531no—
296#100C2×D5311yescharacteristic, complemented
297#97—531no—
298#41—21no—
299#41—21no—
300#41—531no—
301#41—531no—
302#41—1061no—
303#41—1061no—
304#41—1061no—
305#41—1061no—
306#41—1061no—
307#41—1061no—
308#41—2121no—
309#41C22×D5311yescharacteristic
310#41C22×D5311yescharacteristic
311C2×Q8—531no—
312C2×Q8—1061no—
313C2×Q8—1061no—
314C2×D4—21no—
315C2×D4—21no—
316C2×D4—41no—
317C2×D4—41no—
318C2×D4—531no—
319C2×D4—1061no—
320C2×D4—1061no—
321C2×D4—1061no—
322C2×D4—1061no—
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—531no—
350D8—531no—
351D8—1061no—
352D8—1061no—
353D8—1061no—
354D8—1061no—
355D8—1061no—
356D8—1061no—
357D8—1061no—
358D8—1061no—
359M16—531no—
360M16—531no—
361M16—1061no—
362M16—1061no—
363M16—1061no—
364M16—1061no—
365M16C22×D5311yescharacteristic
366M16C22×D5311yescharacteristic
367C2×C8—531no—
368C2×C8—531no—
369C2×C8—1061no—
370C2×C8—1061no—
371C2×C8—1061no—
372C2×C8—1061no—
373C22⋊C4—2121no—
374C42C22×D5311yescharacteristic
375C23—21no—
376C23—21no—
377C23—41no—
378C23—1061no—
379C23—2121no—
380C23—2121no—
381Q8—531no—
382Q8—531no—
383Q8—1061no—
384Q8—1061no—
385Q8—2121no—
386Q8D4×D5311yescharacteristic
387Q8D4×D5311yescharacteristic
388D4—21no—
389D4—21no—
390D4—21no—
391D4—21no—
392D4—21no—
393D4—21no—
394D4—41no—
395D4—41no—
396D4—82no—
397D4—531no—
398D4—531no—
399D4—1061no—
400D4—1061no—
401D4—1061no—
402D4—1061no—
403D4—2121no—
404D4—2121no—
405D4—2121no—
406D4—2121no—
407D4—2121no—
408D4D4×D5311yescharacteristic
409D4D4×D5311yescharacteristic
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—41no—
414C2×C4—1061no—
415C2×C4—1061no—
416C2×C4—1061no—
417C2×C4—2121no—
418C2×C4—2121no—
419C2×C4—2121no—
420C2×C4—2121no—
421C2×C4C23×D5311yescharacteristic
422C8—21no—
423C8—21no—
424C8—531no—
425C8—531no—
426C8—531no—
427C8—531no—
428C8—1061no—
429C8—1061no—
430C22—21no—
431C22—21no—
432C22—41no—
433C22—41no—
434C22—81no—
435C22—81no—
436C22—161no—
437C22—8481no—
438C22—8481no—
439C22—1061no—
440C22—1061no—
441C22—4241no—
442C22—2121no—
443C22#1313326211yescharacteristic
444C4—21no—
445C4—21no—
446C4—41no—
447C4—1061no—
448C4—1061no—
449C4—2121no—
450C4C2×D4×D5311yescharacteristic
451C4C2×D4×D5311yescharacteristic
452C2—21no—
453C2—41no—
454C2—41no—
455C2—81no—
456C2—81no—
457C2—4241no—
458C2—2121no—
459C2—2121no—
460C2#1681125511yescharacteristic
subgroupG/H
centerC2#16811255
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#453C2
socleC2×C53D42
hypercenter#453D53
hypocenterC53#3288
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C2×C4 ◁ #453

Complements

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

NH ≅ G/Ntype
1. #16811434C2semidirect
2. #16811433C2semidirect
3. #16811427C2semidirect
4. #16811426C2semidirect
5. #16811010C2semidirect
6. #16810892C2semidirect
7. #16810891C2semidirect
8. #16810885C2semidirect
9. #16810884C2semidirect
10. #16810802C2semidirect
11. #16810797C2semidirect
12. #16810790C2semidirect
13. #16810742C2semidirect
14. #16810742C2semidirect
15. C53×#453C2semidirect
16. #13133264C22semidirect
18. #13133262C22semidirect
19. #13133250C22semidirect
21. #13133247C22semidirect
23. #13133245C22semidirect
28. #13133243C22semidirect
30. #13133242C22semidirect
40. C2×#11719923C22semidirect
41. C2×#11719923C22semidirect
44. C2×#11719922C22semidirect
45. C2×#11719922C22semidirect
47. #13133225C22semidirect
48. #13133225C22semidirect
49. #13133224C22semidirect
50. #13133224C22semidirect
51. #13133205C22semidirect
52. #13133152C22semidirect
53. #13133152C22semidirect
54. #13133145C22semidirect
55. #13133145C22semidirect
56. #13133135C22semidirect
57. #13133134C22semidirect
58. #13133133C22semidirect
59. #13133128C22semidirect
60. H(5,𝔽2)×C53C22semidirect
62. AGL(1,ℤ/8)×C53C22semidirect
63. #127×C53C22semidirect
64. #104×C53C22semidirect
65. #104×C53C22semidirect
66. #100×C53C22semidirect
104. #11719925D4semidirect
105. #11719925D4semidirect
107. #11719924D4semidirect
108. #11719924D4semidirect
116. #11719923D4semidirect
117. #11719923D4semidirect
125. #11719922D4semidirect
126. #11719922D4semidirect
243. #453D53semidirect
246. C53#3288semidirect
250. H(5,𝔽2)C2×D53semidirect
266. AGL(1,ℤ/8)C2×D53semidirect
285. #127C2×D53semidirect
291. #104C2×D53semidirect
292. #104C2×D53semidirect
296. #100C2×D53semidirect