Finite Group Databasebeta

← #27541132

Subgroups — #27541132

Order 13568.

562 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20286402C211yesmaximal, characteristic
2#20276954C211yesmaximal, characteristic, complemented
3#20276923C211yesmaximal, characteristic, complemented
4#20276906C211yesmaximal, characteristic, complemented
5#20276859C211yesmaximal, characteristic, complemented
6#20276821C211yesmaximal, characteristic, complemented
7#20276813C211yesmaximal, characteristic, complemented
8#20276777C211yesmaximal, characteristic, complemented
9#20276653C211yesmaximal, characteristic, complemented
10#20276643C211yesmaximal, characteristic, complemented
11#20276577C211yesmaximal, characteristic, complemented
12#20276541C211yesmaximal, characteristic, complemented
13#20276430C211yesmaximal, characteristic, complemented
14#20276420C211yesmaximal, characteristic, complemented
15C53×#3342C211yesmaximal, characteristic, complemented
16C2×#13133239C2211yescharacteristic
17C22×D424C2211yescharacteristic
18#16811269—21no—
19C2×#13133163C2211yescharacteristic, complemented
20C2×#13133160C2211yescharacteristic
21C2×#13133160C2211yescharacteristic
22C2×#13133157C2211yescharacteristic
23C22×#11719908C2211yescharacteristic, complemented
24#16811007C2211yescharacteristic
25#16810864C2211yescharacteristic
26#16810859C2211yescharacteristic
27#16810839—21no—
28#16810776—21no—
29#16810770—21no—
30C4×D424C2211yescharacteristic
31#16810450C2211yescharacteristic, complemented
32#16810450C2211yescharacteristic, complemented
33#16810448C2211yescharacteristic, complemented
34C4⋊#11719908C2211yescharacteristic, complemented
35#16810427C2211yescharacteristic, complemented
36#16810426C2211yescharacteristic, complemented
37#16810426C2211yescharacteristic, complemented
38#16810426C2211yescharacteristic, complemented
39#16810423C2211yescharacteristic
40#16810423C2211yescharacteristic
41#16810423C2211yescharacteristic
42#16810423C2211yescharacteristic
43#16810423C2211yescharacteristic
44#16810423C2211yescharacteristic
45C22⋊#11719908C2211yescharacteristic
46C22⋊#11719908C2211yescharacteristic, complemented
47#16810399C2211yescharacteristic, complemented
48C53×#556—21no—
49C53×#500C2211yescharacteristic, complemented
50C53×#485—21no—
51C53×#485—21no—
52C53×#475—21no—
53C53×#467C2211yescharacteristic
54C53×#467C2211yescharacteristic, complemented
55C4×Q16×C53C2211yescharacteristic, complemented
56C8×D4×C53C2211yescharacteristic
57C53×#427C2211yescharacteristic
58C53×#427C2211yescharacteristic, complemented
59C2×#11719945—21no—
60C22×D212—21no—
61#13133239—22no—
62#13133239—41no—
63C2×Dic212C2311yescharacteristic
64C2×#11719913—21no—
65C2×#11719913—21no—
66C2×D424—21no—
67C2×D424—22no—
68C2×D424—41no—
69C2×D424C2311yescharacteristic
70C2×D424C2311yescharacteristic
71#13133205—21no—
72#13133202—41no—
73#13133196—21no—
74#13133191—21no—
75#13133190—41no—
76#13133185—41no—
77C4×D212—21no—
78C22⋊C4×D53—41no—
79#13133163—41no—
80#13133160—22no—
81#13133160—41no—
82#13133157—41no—
83C2×#11719908—41no—
84C2×#11719908C2311yescharacteristic
85C2×#11719908C2311yescharacteristic
86C2×#11719908C2311yescharacteristic
87C2×#11719908C2311yescharacteristic
88#13133151C2311yescharacteristic
89#13133142—21no—
90#13133139—21no—
91#13133139—21no—
92C2×Q16×C53C2311yescharacteristic
93C22×C8×C53C2311yescharacteristic
94C22×C8×C53C2311yescharacteristic
95C4⋊Q8×C53—21no—
96#126×C53—41no—
97#126×C53—41no—
98#124×C53—41no—
99C22⋊Q8×C53—21no—
100C22⋊Q8×C53—21no—
101C4×Q8×C53—21no—
102C4×Q8×C53—41no—
103C4×D4×C53D411yescharacteristic
104#117×C53—21no—
105#117×C53—21no—
106#107×C53—21no—
107#107×C53C2311yescharacteristic
108#107×C53C2311yescharacteristic
109#107×C53C2311yescharacteristic
110#106×C53—21no—
111#106×C53—21no—
112C4⋊C8×C53D411yescharacteristic
113#103×C53—21no—
114#103×C53—21no—
115#103×C53—21no—
116C22⋊C8×C53D411yescharacteristic
117C22⋊C8×C53D411yescharacteristic
118C4×C8×C53C2311yescharacteristic
119C23×D53—41no—
120C2×Q8×C53—21no—
121C2×Q8×C53—41no—
122C2×D4×C53D811yescharacteristic
123C22×C4×C53C2×D411yescharacteristic
124C22×C4×C53C2×D411yescharacteristic
125C2×C22⋊D53—41no—
126C2×C22⋊D53—41no—
127#11719945—42no—
128#11719945—81no—
129C2×D212—21no—
130C2×D212—21no—
131C2×D212—42no—
132C2×D212—41no—
133C2×D212—81no—
134C2×C4×D53—41no—
135C2×C4×D53—41no—
136C2×Dic106—21no—
137Q16×C53—21no—
138Q16×C53—21no—
139C2×C8×C53—21no—
140C2×C8×C53—21no—
141C2×C8×C53—21no—
142C2×C8×C53—21no—
143C2×C8×C53C2×D411yescharacteristic
144C2×C8×C53C2×D411yescharacteristic
145C2×C8×C53C2×D411yescharacteristic
146C2×C8×C53C2411yescharacteristic
147C4⋊C4×C53—21no—
148C4⋊C4×C53—21no—
149C4⋊C4×C53—21no—
150C4⋊C4×C53—41no—
151C4⋊C4×C53—41no—
152C4⋊C4×C53—41no—
153C4⋊C4×C53—41no—
154C4⋊C4×C53D811yescharacteristic
155C22⋊C4×C53—41no—
156C22⋊C4×C53—41no—
157C22⋊C4×C53—41no—
158C22⋊C4×C53—41no—
159C22⋊C4×C53D811yescharacteristic
160C22⋊C4×C53D811yescharacteristic
161C42×C53—41no—
162C42×C53—41no—
163C42×C53—41no—
164C42×C53C2×D411yescharacteristic
165C22⋊Dic53—41no—
166#11719921—41no—
167#11719921—41no—
168#11719921—41no—
169#11719921—41no—
170C4⋊Dic53—21no—
171#11719919—41no—
172Dic212—21no—
173Dic212—21no—
174D424—21no—
175D424—21no—
176D424—21no—
177D424—42no—
178D424#4112yes—
179#11719913—42no—
180#11719913—81no—
181#11719908—21no—
182#11719908—21no—
183#11719908—21no—
184#11719908—21no—
185C23×C53C2×D811yescharacteristic
186C23×C53C2×D811yescharacteristic
187C22×D53—41no—
188C22×D53—41no—
189C22×D53—82no—
190C22×D53—82no—
191C22×D53—161no—
192Q8×C53—41no—
193Q8×C53—41no—
194Q8×C53—82no—
195D4×C53—41no—
196C2×C4×C53—21no—
197C2×C4×C53—21no—
198C2×C4×C53—21no—
199C2×C4×C53—21no—
200C2×C4×C53—41no—
201C2×C4×C53—41no—
202C2×C4×C53—41no—
203C2×C4×C53—41no—
204C2×C4×C53C2×D811yescharacteristic
205C2×C4×C53C2×D811yescharacteristic
206C2×C4×C53C2×D811yescharacteristic
207C2×C4×C53C2×D811yescharacteristic
208C2×C4×C53C22×D411yescharacteristic
209C22⋊D53—82no—
210C22⋊D53—161no—
211C2×Dic53—41no—
212C2×Dic53—41no—
213D212—22no—
214D212—41no—
215D212—41no—
216D212—41no—
217D212—82no—
218C4×D53—82no—
219C4×D53—161no—
220Dic106—41no—
221Dic106—41no—
222C8×C53—21no—
223C8×C53—21no—
224C8×C53—21no—
225C8×C53C2×#4111yescharacteristic
226C8×C53H(5,𝔽2)11yescharacteristic
227#33878—531nomaximal, Sylow
228C22×C53—21no—
229C22×C53—21no—
230C22×C53—21no—
231C22×C53—21no—
232C22×C53—21no—
233C22×C53—21no—
234C22×C53C22×D811yescharacteristic
235C2×D53—42no—
236C2×D53—81no—
237C2×D53—81no—
238C2×D53—81no—
239C2×D53—81no—
240C2×D53—81no—
241C2×D53—164no—
242C4×C53—21no—
243C4×C53—21no—
244C4×C53—21no—
245C4×C53—21no—
246C4×C53—81no—
247C4×C53—81no—
248C4×C53—81no—
249C4×C53—81no—
250C4×C53#54611yescharacteristic
251C4×C53#57711yescharacteristic
252Dic53—81no—
253Dic53—81no—
254#3342D5311yescharacteristic, complemented
255#3280—531no—
256#2243—531no—
257#2237—531no—
258#2234—531no—
259#2226—531no—
260#2219—531no—
261#2217—531no—
262#2211—531no—
263#2189—531no—
264#2187—531no—
265#2175—531no—
266#2168—531no—
267#2140—531no—
268#2138—531no—
269C2×C53—41no—
270C2×C53—41no—
271C2×C53#329411yescharacteristic
272C2×C53#341411yescharacteristic
273C2×C53#341611yescharacteristic
274D53—82no—
275D53—161no—
276C2×#135—531no—
277C22×D8—531no—
278#556—21no—
279#546—1061no—
280C2×Q32—531no—
281C2×SD32—531no—
282C2×SD32—531no—
283C2×M32—531no—
284C22×C16—531no—
285#500C2×D5311yescharacteristic, complemented
286#495—531no—
287#485—21no—
288#485—21no—
289#475—21no—
290#467—531no—
291#467C2×D5311yescharacteristic
292#467C2×D5311yescharacteristic, complemented
293#466—531no—
294#463—1061no—
295#450—1061no—
296#449—1061no—
297C4×Q16C2×D5311yescharacteristic, complemented
298C4×D8—531no—
299C8×D4C2×D5311yescharacteristic
300#427C2×D5311yescharacteristic
301#427C2×D5311yescharacteristic, complemented
302#367—531no—
303#367—531no—
304#366—531no—
305C4⋊C16—531no—
306#358—531no—
307#358—531no—
308#358—531no—
309#358—531no—
310#357—531no—
311#357—531no—
312#357—531no—
313#357—531no—
314#357—531no—
315#357—531no—
316C22⋊C16—531no—
317C22⋊C16—531no—
318#346—531no—
319C53#3387811yescharacteristic, complemented, Sylow
320C2×#41—1061no—
321C22×D4—1061no—
322#135—1062no—
323#135—2121no—
324C2×Q16—531no—
325C2×Q16C22×D5311yescharacteristic
326C2×SD16—1061no—
327C2×SD16—1061no—
328C2×D8—531no—
329C2×D8—531no—
330C2×D8—1062no—
331C2×D8—1061no—
332C2×D8—2121no—
333C22×C8C22×D5311yescharacteristic
334C22×C8C22×D5311yescharacteristic
335C4⋊Q8—21no—
336#126—41no—
337#126—41no—
338#124—41no—
339#124—1061no—
340#123—2121no—
341C22⋊Q8—21no—
342C22⋊Q8—21no—
343C22⋊Q8—1061no—
344#121—1061no—
345#121—2121no—
346#120—2121no—
347C4×Q8—21no—
348C4×Q8—41no—
349C4×D4—1061no—
350C4×D4C22⋊D5311yescharacteristic
351#117—21no—
352#117—21no—
353C2×C22⋊C4—2121no—
354Q32—2121no—
355SD32—1062no—
356SD32—2121no—
357M32—2121no—
358C2×C16—531no—
359C2×C16—531no—
360C2×C16—531no—
361C2×C16—531no—
362C2×C16—2121no—
363#107—21no—
364#107—531no—
365#107C22×D5311yescharacteristic
366#107C22×D5311yescharacteristic
367#107C22×D5311yescharacteristic
368#106—21no—
369#106—21no—
370C4⋊C8C22⋊D5311yescharacteristic
371#103—21no—
372#103—21no—
373#103—21no—
374#103—1061no—
375#102—1061no—
376#102—1061no—
377C22⋊C8C22⋊D5311yescharacteristic
378C22⋊C8C22⋊D5311yescharacteristic
379C4×C8C22×D5311yescharacteristic
380C24—2121no—
381#41—4241no—
382#41—2122no—
383C2×Q8—21no—
384C2×Q8—41no—
385C2×Q8—1061no—
386C2×D4—1061no—
387C2×D4—1061no—
388C2×D4—4241no—
389C2×D4—2122no—
390C2×D4—2121no—
391C2×D4—2121no—
392C2×D4—2121no—
393C2×D4#1171992211yescharacteristic
394C22×C4—2121no—
395C22×C4—2121no—
396C22×C4C2×C22⋊D5311yescharacteristic
397C22×C4C2×C22⋊D5311yescharacteristic
398Q16—21no—
399Q16—21no—
400Q16—1061no—
401Q16—1061no—
402SD16—4241no—
403SD16—2122no—
404D8—532no—
405D8—1061no—
406D8—1061no—
407D8—1061no—
408D8—2122no—
409C2×C8—21no—
410C2×C8—21no—
411C2×C8—21no—
412C2×C8—21no—
413C2×C8C2×C22⋊D5311yescharacteristic
414C2×C8C2×C22⋊D5311yescharacteristic
415C2×C8C2×C22⋊D5311yescharacteristic
416C2×C8C23×D5311yescharacteristic
417C4⋊C4—21no—
418C4⋊C4—21no—
419C4⋊C4—21no—
420C4⋊C4—41no—
421C4⋊C4—41no—
422C4⋊C4—41no—
423C4⋊C4—41no—
424C4⋊C4—1061no—
425C4⋊C4—2121no—
426C4⋊C4#1171992211yescharacteristic
427C22⋊C4—41no—
428C22⋊C4—41no—
429C22⋊C4—41no—
430C22⋊C4—41no—
431C22⋊C4—2121no—
432C22⋊C4—2121no—
433C22⋊C4—2121no—
434C22⋊C4—2121no—
435C22⋊C4—2121no—
436C22⋊C4#1171992211yescharacteristic
437C22⋊C4#1171992211yescharacteristic
438C42—41no—
439C42—41no—
440C42—41no—
441C42C2×C22⋊D5311yescharacteristic
442C16—1061no—
443C16—1061no—
444C16—1061no—
445C16—1061no—
446C23—8481no—
447C23—4242no—
448C23—4242no—
449C23—2121no—
450C23—2121no—
451C23C2×#1171992211yescharacteristic
452C23C2×#1171992211yescharacteristic
453Q8—41no—
454Q8—41no—
455Q8—82no—
456Q8—2121no—
457Q8—2121no—
458D4—41no—
459D4—8481no—
460D4—1062no—
461D4—4242no—
462D4—4242no—
463D4—2121no—
464D4—2121no—
465D4—2121no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—8481no—
475C2×C4—4242no—
476C2×C4—2121no—
477C2×C4—2121no—
478C2×C4C2×#1171992211yescharacteristic
479C2×C4C2×#1171992211yescharacteristic
480C2×C4C2×#1171992211yescharacteristic
481C2×C4C2×#1171992211yescharacteristic
482C2×C4C22×C22⋊D5311yescharacteristic
483C8—21no—
484C8—21no—
485C8—21no—
486C8#41×D5311yescharacteristic
487C8#1313326311yescharacteristic
488C22—21no—
489C22—21no—
490C22—21no—
491C22—21no—
492C22—21no—
493C22—21no—
494C22—8484no—
495C22—4241no—
496C22—4241no—
497C22—4241no—
498C22—4241no—
499C22—4241no—
500C22—2122no—
501C22C22×#1171992211yescharacteristic
502C4—21no—
503C4—21no—
504C4—21no—
505C4—21no—
506C4—81no—
507C4—81no—
508C4—81no—
509C4—81no—
510C4—4241no—
511C4—4241no—
512C4#1681126111yescharacteristic
513C4#1681143111yescharacteristic
514C2—41no—
515C2—41no—
516C2—8481no—
517C2—4242no—
518C2#2028652211yescharacteristic
519C2#2028784911yescharacteristic
520C2#2028785911yescharacteristic
subgroupG/H
centerC22C22×#11719922
commutator, cosocleC2×C8×C53C24
FrattiniC2×C8C23×D53
FittingC53×#3342C2
socleC22×C53C22×D8
prime radical#20286402C2
hypercenter#3342D53
hypocenterC53#33878
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ #3342

Complements

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

NH ≅ G/Ntype
2. #20276954C2semidirect
3. #20276923C2semidirect
4. #20276906C2semidirect
5. #20276859C2semidirect
6. #20276821C2semidirect
7. #20276813C2semidirect
8. #20276777C2semidirect
9. #20276653C2semidirect
10. #20276643C2semidirect
11. #20276577C2semidirect
12. #20276541C2semidirect
13. #20276430C2semidirect
14. #20276420C2semidirect
15. C53×#3342C2semidirect
19. C2×#13133163C22semidirect
23. C22×#11719908C22semidirect
31. #16810450C22semidirect
32. #16810450C22semidirect
33. #16810448C22semidirect
34. C4⋊#11719908C22semidirect
35. #16810427C22semidirect
36. #16810426C22semidirect
37. #16810426C22semidirect
38. #16810426C22semidirect
46. C22⋊#11719908C22semidirect
47. #16810399C22semidirect
49. C53×#500C22semidirect
54. C53×#467C22semidirect
55. C4×Q16×C53C22semidirect
58. C53×#427C22semidirect
254. #3342D53semidirect
285. #500C2×D53semidirect
292. #467C2×D53semidirect
297. C4×Q16C2×D53semidirect
301. #427C2×D53semidirect
319. C53#33878semidirect

Maximal subgroups

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

subgroupindex
#202864022
#202769542
#202769232
#202769062
#202768592
#202768212
#202768132
#202767772
#202766532
#202766432
#202765772
#202765412
#202764302
#202764202
C53×#33422
#3387853