Finite Group Databasebeta

← #20289487

Subgroups — #20289487

Order 6784.

1424 conjugacy classes of subgroups.

The subgroup lattice has 506 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16811460C211yesmaximal, characteristic, complemented
2#16811458C211yesmaximal, characteristic, complemented
3#16811439C213yesmaximal, complemented
4#16811438C211yesmaximal, characteristic, complemented
5#16811435C211yesmaximal, characteristic, complemented
6#16811430C213yesmaximal, complemented
7#16811429C213yesmaximal, complemented
8#16811427C213yesmaximal, complemented
9#16811426C211yesmaximal, characteristic, complemented
10#16811422C213yesmaximal, complemented
11#16811419C213yesmaximal, complemented
12C2×#13133238C213yesmaximal, complemented
13#135×D53C213yesmaximal, complemented
14#131×D53C211yesmaximal, characteristic, complemented
15C53×#577C211yesmaximal, characteristic, complemented
16#13133265—21no—
17#13133265C2213yescomplemented
18#13133264—21no—
19#13133264C2213yescomplemented
20#13133264C2211yescharacteristic, complemented
21#13133263—23no—
22#13133263C2213yescomplemented
23#13133263C2211yescharacteristic, complemented
24#13133262—23no—
25#13133262C2213yescomplemented
26C2×#11719949—21no—
27C2×#11719949C2213yescomplemented
28C2×#11719947—23no—
29C2×#11719947C2213yescomplemented
30#41×D53—21no—
31#41×D53—23no—
32#41×D53C2211yescharacteristic
33#41×D53C2213yescomplemented
34#41×D53C2213yescomplemented
35#13133250C2213yescomplemented
36#13133249C2216yescomplemented
37#13133248C2213yescomplemented
38#13133247C2213yescomplemented
39#13133246C2216yescomplemented
40#13133245C2213yescomplemented
41#13133244C2216yescomplemented
42#13133243C2213yescomplemented
43#13133242C2213yescomplemented
44#13133241C2216yescomplemented
45#13133240C2213yescomplemented
46#13133239C2213yescomplemented
47#13133238C2211yescharacteristic, complemented
48#13133238C2213yescomplemented
49#13133238C2216yescomplemented
50#13133238C2216yescomplemented
51#13133238C2216yescomplemented
52#13133237C2213yescomplemented
53#13133237C2213yescomplemented
54#13133236C2213yescomplemented
55C2×#11719925C2213yescomplemented
56Q16×D53C2213yescomplemented
57C2×#11719913C2213yescomplemented
58SD16×D53C2213yescomplemented
59SD16×D53C2213yescomplemented
60C2×#11719922C2213yescomplemented
61D8×D53C2213yescomplemented
62#13133225C2213yescomplemented
63#13133224C2211yescharacteristic, complemented
64#13133223C2213yescomplemented
65M16×D53C2213yescomplemented
66C2×C8×D53C2213yescomplemented
672−1+4×C53C2211yescharacteristic, complemented
68H(5,𝔽2)×C53C2211yescharacteristic, complemented
69#137×C53C2213yescomplemented
70AGL(1,ℤ/8)×C53C2213yescomplemented
71#135×C53C2213yescomplemented
72C2×SD16×C53C2213yescomplemented
73#131×C53C2211yescharacteristic, complemented
74#41×C53—21no—
75#41×C53—23no—
76#41×C53C2313yes—
77#41×C53C2313yes—
78#41×C53C2311yescharacteristic
79C2×Q8×C53—21no—
80C2×Q8×C53C2313yes—
81C2×D4×C53—23no—
82C2×D4×C53C2313yes—
83C2×C22⋊D53—23no—
84C2×C22⋊D53—26no—
85C22×Dic53—23no—
86#11719949—21no—
87#11719949—23no—
88#11719949—23no—
89#11719949—23no—
90#11719949D411yescharacteristic
91#11719949C2313yes—
92#11719949C2316yes—
93#11719949C2311yescharacteristic
94Q8×D53—21no—
95Q8×D53—23no—
96Q8×D53D411yescharacteristic
97Q8×D53C2313yes—
98Q8×D53C2313yes—
99#11719947—23no—
100#11719947—26no—
101#11719947—23no—
102#11719947—23no—
103#11719947—23no—
104#11719947—23no—
105#11719947—23no—
106#11719947D413yes—
107#11719947C2311yescharacteristic
108#11719947C2313yes—
109#11719947C2316yes—
110D4×D53—23no—
111D4×D53—23no—
112D4×D53—23no—
113D4×D53—26no—
114D4×D53—23no—
115D4×D53D413yes—
116D4×D53C2313yes—
117D4×D53C2313yes—
118#11719945—26no—
119#11719945—23no—
120#11719945—23no—
121#11719945—23no—
122#11719945—23no—
123#11719945C2313yes—
124#11719945C2313yes—
125#11719945C2313yes—
126C2×D212—23no—
127C2×D212—23no—
128C2×D212C2313yes—
129C2×C4×D53—23no—
130C2×C4×D53—23no—
131C2×C4×D53—23no—
132C2×C4×D53C2313yes—
133C2×Dic106—23no—
134C2×Dic106C2313yes—
135Q16×C53C2313yes—
136SD16×C53C2313yes—
137SD16×C53C2316yes—
138SD16×C53C2311yescharacteristic
139D8×C53C2313yes—
140M16×C53C2313yes—
141C2×C8×C53C2313yes—
142#11719925C2311yescharacteristic
143#11719925C2313yes—
144#11719925C2316yes—
145#11719924C2313yes—
146#11719924C2313yes—
147#11719923C2313yes—
148#11719923C2313yes—
149#11719922C2311yescharacteristic
150#11719922C2313yes—
151#11719922C2316yes—
152#11719917C2313yes—
153C2×#88322C2313yes—
154Dic212C2313yes—
155D424C2313yes—
156#11719913C2311yescharacteristic
157#11719913C2313yes—
158#11719913C2316yes—
159#11719912C2316yes—
160C8×D53C2311yescharacteristic
161C8×D53C2313yes—
162C8×D53C2313yes—
163C8×D53C2313yes—
164C23×C53—23no—
165C22×D53—23no—
166C22×D53—23no—
167C22×D53—23no—
168C22×D53C2×D413yes—
169Q8×C53—23no—
170Q8×C53C2×D411yescharacteristic
171Q8×C53C2411yescharacteristic
172Q8×C53C2413yes—
173D4×C53—23no—
174D4×C53—23no—
175D4×C53—23no—
176D4×C53C2×D413yes—
177D4×C53C2411yescharacteristic
178D4×C53C2413yes—
179C2×C4×C53—23no—
180C2×C4×C53—23no—
181C2×C4×C53—23no—
182C2×C4×C53C2413yes—
183C22⋊D53—23no—
184C22⋊D53—23no—
185C22⋊D53—21no—
186C22⋊D53—26no—
187C22⋊D53—23no—
188C22⋊D53—23no—
189C22⋊D53—23no—
190C22⋊D53C2×D416yes—
191C2×Dic53—23no—
192C2×Dic53—26no—
193C2×Dic53—21no—
194C2×Dic53—23no—
195C2×Dic53C2×D413yes—
196D212—26no—
197D212—23no—
198D212—21no—
199D212—23no—
200D212C2×D413yes—
201D212C2413yes—
202D212C2411yescharacteristic
203C4×D53—23no—
204C4×D53—23no—
205C4×D53—23no—
206C4×D53—23no—
207C4×D53—21no—
208C4×D53—23no—
209C4×D53—26no—
210C4×D53C2×D413yes—
211C4×D53C2×D413yes—
212C4×D53C2411yescharacteristic
213C4×D53C2413yes—
214Dic106—23no—
215Dic106—23no—
216Dic106—23no—
217Dic106C2×D413yes—
218Dic106C2413yes—
219Dic106C2411yescharacteristic
220C8×C53C2411yescharacteristic
221C8×C53C2413yes—
222#88322C2411yescharacteristic
223#88322C2413yes—
224C22×C53—23no—
225C22×C53—21no—
226C22×C53—83no—
227C22×C53C22×D413yes—
228C2×D53—23no—
229C2×D53—21no—
230C2×D53—43no—
231C2×D53—83no—
232C2×D53—83no—
233C2×D53—83no—
234C2×D53C22×D413yes—
235C2×D53C22×D411yescharacteristic
236C4×C53—21no—
237C4×C53—23no—
238C4×C53C22×D413yes—
239C4×C53C2511yescharacteristic
240Dic53—21no—
241Dic53—23no—
242Dic53C22×D413yes—
243Dic53C22×D411yescharacteristic
244#3584—531nomaximal, Sylow
245C2×C53—23no—
246C2×C53—41no—
247C2×C53—43no—
248C2×C53C23×D411yescharacteristic
249D53—21no—
250D53—23no—
251D53—43no—
252D53—41no—
253#585—531no—
254#585—531no—
255#578—531no—
256#578—533no—
257#577—533no—
258#577—533no—
259#577—531no—
260#577D5311yescharacteristic, complemented
261#576—531no—
262#576—533no—
263#575—533no—
264#575—533no—
265C2×#135—533no—
266C2×#135—533no—
267C2×#131—531no—
268C53#358411yescharacteristic, complemented, Sylow
2692−1+4—533no—
2702−1+4—531no—
2712−1+4—533no—
2722−1+4—1061no—
2732−1+4—1061no—
2742−1+4C2×D5311yescharacteristic, complemented
275H(5,𝔽2)—533no—
276H(5,𝔽2)—531no—
277H(5,𝔽2)—533no—
278H(5,𝔽2)—1063no—
279H(5,𝔽2)—1063no—
280H(5,𝔽2)C2×D5311yescharacteristic, complemented
281C2×#41—531no—
282C2×#41—533no—
283C2×#41—533no—
284C2×#41—533no—
285C2×#41—533no—
286C2×#41—1063no—
287C2×#41—1061no—
288C2×#41—1061no—
289C2×#41—1063no—
290#137—533no—
291#137—533no—
292#137—536no—
293#137—536no—
294#137—533no—
295#137C2×D5313yescomplemented
296AGL(1,ℤ/8)—533no—
297AGL(1,ℤ/8)—533no—
298AGL(1,ℤ/8)—536no—
299AGL(1,ℤ/8)—536no—
300AGL(1,ℤ/8)—533no—
301AGL(1,ℤ/8)C2×D5313yescomplemented
302#135—533no—
303#135—533no—
304#135—533no—
305#135—531no—
306#135—533no—
307#135—536no—
308#135—536no—
309#135—536no—
310#135—533no—
311#135—533no—
312#135C2×D5313yescomplemented
313C2×Q16—533no—
314C2×Q16—533no—
315C2×SD16—533no—
316C2×SD16—533no—
317C2×SD16—533no—
318C2×SD16C2×D5313yescomplemented
319C2×D8—533no—
320C2×D8—533no—
321#131—531no—
322#131—533no—
323#131—533no—
324#131C2×D5311yescharacteristic, complemented
325C2×M16—533no—
326C22×C8—533no—
327#41—21no—
328#41—23no—
329#41—531no—
330#41—533no—
331#41—533no—
332#41—531no—
333#41—533no—
334#41—536no—
335#41—533no—
336#41—531no—
337#41—536no—
338#41—533no—
339#41—533no—
340#41—1063no—
341#41—1063no—
342#41—1066no—
343#41—1063no—
344#41—1063no—
345#41—1063no—
346#41—1063no—
347#41—1063no—
348#41—1063no—
349#41—1063no—
350#41—1063no—
351#41—1066no—
352#41—1063no—
353#41—1063no—
354#41—1061no—
355#41—1063no—
356#41C22×D5313yes—
357#41C22×D5313yes—
358#41C22×D5311yescharacteristic
359C2×Q8—21no—
360C2×Q8—531no—
361C2×Q8—533no—
362C2×Q8—533no—
363C2×Q8—533no—
364C2×Q8—1063no—
365C2×Q8—1063no—
366C2×Q8—1061no—
367C2×Q8C22×D5313yes—
368C2×D4—23no—
369C2×D4—533no—
370C2×D4—533no—
371C2×D4—533no—
372C2×D4—533no—
373C2×D4—1063no—
374C2×D4—1063no—
375C2×D4—1066no—
376C2×D4—1063no—
377C2×D4—1063no—
378C2×D4—1063no—
379C2×D4—1066no—
380C2×D4—1063no—
381C2×D4—1063no—
382C2×D4C22×D5313yes—
383C22×C4—533no—
384C22×C4—1063no—
385C22×C4—1063no—
386C22×C4—1063no—
387C22×C4—1063no—
388Q16—531no—
389Q16—533no—
390Q16—536no—
391Q16—533no—
392Q16C22×D5313yes—
393SD16—533no—
394SD16—533no—
395SD16—533no—
396SD16—533no—
397SD16—531no—
398SD16—533no—
399SD16—536no—
400SD16C22×D5311yescharacteristic
401SD16C22×D5313yes—
402SD16C22×D5316yes—
403D8—533no—
404D8—531no—
405D8—536no—
406D8—533no—
407D8C22×D5313yes—
408M16—533no—
409M16—536no—
410M16C22×D5313yes—
411C2×C8—533no—
412C2×C8—531no—
413C2×C8—533no—
414C2×C8—533no—
415C2×C8—533no—
416C2×C8C22×D5313yes—
417C23—23no—
418C23—533no—
419C23—1063no—
420C23—1063no—
421C23—1063no—
422Q8—23no—
423Q8—533no—
424Q8—533no—
425Q8—531no—
426Q8—1063no—
427Q8—1063no—
428Q8—1063no—
429Q8D4×D5311yescharacteristic
430Q8C23×D5311yescharacteristic
431Q8C23×D5313yes—
432D4—23no—
433D4—23no—
434D4—23no—
435D4—536no—
436D4—533no—
437D4—533no—
438D4—531no—
439D4—1061no—
440D4—1063no—
441D4—1066no—
442D4—1063no—
443D4—1063no—
444D4—1061no—
445D4—1063no—
446D4—1066no—
447D4—1063no—
448D4—1063no—
449D4—1063no—
450D4D4×D5313yes—
451D4C23×D5313yes—
452D4C23×D5311yescharacteristic
453C2×C4—23no—
454C2×C4—23no—
455C2×C4—23no—
456C2×C4—533no—
457C2×C4—533no—
458C2×C4—533no—
459C2×C4—531no—
460C2×C4—533no—
461C2×C4—1061no—
462C2×C4—1063no—
463C2×C4—1066no—
464C2×C4—1063no—
465C2×C4—1063no—
466C2×C4—1061no—
467C2×C4—1063no—
468C2×C4—1066no—
469C2×C4—1063no—
470C2×C4—1063no—
471C2×C4—1063no—
472C2×C4C23×D5313yes—
473C8—531no—
474C8—533no—
475C8C23×D5311yescharacteristic
476C8C23×D5313yes—
477C22—21no—
478C22—23no—
479C22—83no—
480C22—531no—
481C22—533no—
482C22—1063no—
483C22—1061no—
484C22—4243no—
485C22—4243no—
486C22—4243no—
487C22—2123no—
488C22C2×D4×D5313yes—
489C4—21no—
490C4—23no—
491C4—531no—
492C4—533no—
493C4—1061no—
494C4—1063no—
495C4C2×D4×D5313yes—
496C4C24×D5311yescharacteristic
497C2—23no—
498C2—43no—
499C2—41no—
500C2—1063no—
501C2—1061no—
502C2—2123no—
503C2—2121no—
504C2C22×D4×D5311yescharacteristic
subgroupG/H
centerC2C22×D4×D53
commutatorC4×C53C25
FrattiniC4C24×D53
FittingC53×#577C2
socleC2×C53C23×D4
hypercenter#577D53
hypocenterC53#3584
LCSG ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ #41 ◁ #577

Complements

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

NH ≅ G/Ntype
1. #16811460C2semidirect
2. #16811458C2semidirect
3. #16811439C2semidirect
4. #16811438C2semidirect
5. #16811435C2semidirect
6. #16811430C2semidirect
7. #16811429C2semidirect
8. #16811427C2semidirect
9. #16811426C2semidirect
10. #16811422C2semidirect
11. #16811419C2semidirect
12. C2×#13133238C2semidirect
13. #135×D53C2semidirect
14. #131×D53C2semidirect
15. C53×#577C2semidirect
17. #13133265C22semidirect
19. #13133264C22semidirect
20. #13133264C22semidirect
22. #13133263C22semidirect
23. #13133263C22semidirect
25. #13133262C22semidirect
27. C2×#11719949C22semidirect
29. C2×#11719947C22semidirect
33. #41×D53C22semidirect
34. #41×D53C22semidirect
35. #13133250C22semidirect
36. #13133249C22semidirect
37. #13133248C22semidirect
38. #13133247C22semidirect
39. #13133246C22semidirect
40. #13133245C22semidirect
41. #13133244C22semidirect
42. #13133243C22semidirect
43. #13133242C22semidirect
44. #13133241C22semidirect
45. #13133240C22semidirect
46. #13133239C22semidirect
47. #13133238C22semidirect
48. #13133238C22semidirect
49. #13133238C22semidirect
50. #13133238C22semidirect
51. #13133238C22semidirect
52. #13133237C22semidirect
53. #13133237C22semidirect
54. #13133236C22semidirect
55. C2×#11719925C22semidirect
56. Q16×D53C22semidirect
57. C2×#11719913C22semidirect
58. SD16×D53C22semidirect
59. SD16×D53C22semidirect
60. C2×#11719922C22semidirect
61. D8×D53C22semidirect
62. #13133225C22semidirect
63. #13133224C22semidirect
64. #13133223C22semidirect
65. M16×D53C22semidirect
66. C2×C8×D53C22semidirect
67. 2−1+4×C53C22semidirect
68. H(5,𝔽2)×C53C22semidirect
69. #137×C53C22semidirect
70. AGL(1,ℤ/8)×C53C22semidirect
71. #135×C53C22semidirect
72. C2×SD16×C53C22semidirect
73. #131×C53C22semidirect
260. #577D53semidirect
268. C53#3584semidirect
274. 2−1+4C2×D53semidirect
280. H(5,𝔽2)C2×D53semidirect
295. #137C2×D53semidirect
301. AGL(1,ℤ/8)C2×D53semidirect
312. #135C2×D53semidirect
318. C2×SD16C2×D53semidirect
324. #131C2×D53semidirect