Finite Group Databasebeta

← #27540946

Subgroups — #27540946

Order 13568.

580 conjugacy classes of subgroups.

The subgroup lattice has 534 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20286402C211yesmaximal, characteristic, complemented
2#20276949C211yesmaximal, characteristic, complemented
3#20276917C211yesmaximal, characteristic, complemented
4#20276906C211yesmaximal, characteristic, complemented
5#20276869C211yesmaximal, characteristic, complemented
6#20276809C211yesmaximal, characteristic, complemented
7#20276799C211yesmaximal, characteristic, complemented
8#20276770C211yesmaximal, characteristic, complemented
9#20276636C211yesmaximal, characteristic, complemented
10#20276626C211yesmaximal, characteristic, complemented
11C4×#13133158C211yesmaximal, characteristic, complemented
12D4×#11719908C211yesmaximal, characteristic, complemented
13#20276420C211yesmaximal, characteristic, complemented
14C2×#16810423C211yesmaximal, characteristic, complemented
15C53×#3339C211yesmaximal, characteristic, complemented
16C2×#13133239C2211yescharacteristic
17C22×D424C2211yescharacteristic
18#16811269—21no—
19C2×#13133161C2211yescharacteristic, complemented
20C2×#13133158C2211yescharacteristic
21C2×#13133158C2211yescharacteristic
22C22×#11719908C2211yescharacteristic
23C22×#11719908C2211yescharacteristic, complemented
24#16811007C2211yescharacteristic
25#16810864C2211yescharacteristic
26#16810859C2211yescharacteristic
27#16810839—21no—
28#16810776—21no—
29#16810770—21no—
30C4×D424C2211yescharacteristic
31#16810450C2211yescharacteristic, complemented
32#16810448C2211yescharacteristic, complemented
33#16810448C2211yescharacteristic, complemented
34C4⋊#11719908C2211yescharacteristic, complemented
35#16810426C2211yescharacteristic, complemented
36#16810426C2211yescharacteristic, complemented
37#16810426C2211yescharacteristic, complemented
38#16810424C2211yescharacteristic, complemented
39#16810423C2211yescharacteristic
40#16810423C2211yescharacteristic
41#16810423C2211yescharacteristic
42#16810423C2211yescharacteristic
43#16810423C2211yescharacteristic
44#16810423C2211yescharacteristic
45C22⋊#11719908C2211yescharacteristic
46C22⋊#11719908C2211yescharacteristic, complemented
47C4×#11719908C2211yescharacteristic, complemented
48C53×#553—21no—
49C53×#499C2211yescharacteristic, complemented
50C53×#482—21no—
51C53×#481—21no—
52C53×#478—21no—
53C53×#466C2211yescharacteristic
54C53×#466C2211yescharacteristic, complemented
55C4×D8×C53C2211yescharacteristic, complemented
56C8×D4×C53C2211yescharacteristic
57C53×#427C2211yescharacteristic, complemented
58C2×#107×C53C2211yescharacteristic
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#13133161—41no—
80#13133158—22no—
81#13133158—41no—
82C2×#11719908—41no—
83C2×#11719908—41no—
84C2×#11719908C2311yescharacteristic
85C2×#11719908C2311yescharacteristic
86C2×#11719908C2311yescharacteristic
87C2×#11719908C2311yescharacteristic
88#13133151C2311yescharacteristic
89#13133142—21no—
90#13133139—21no—
91#13133139—21no—
92C2×D8×C53C2311yescharacteristic
93C22×C8×C53C2311yescharacteristic
94C22×C8×C53C2311yescharacteristic
95#126×C53—41no—
96#125×C53—21no—
97#123×C53—41no—
98#121×C53—21no—
99#121×C53—21no—
100#121×C53—41no—
101C4×D4×C53—21no—
102C4×D4×C53—41no—
103C4×D4×C53D411yescharacteristic
104#117×C53—21no—
105C2×C4⋊C4×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#102×C53—21no—
114#102×C53—21no—
115#102×C53—21no—
116C22⋊C8×C53D411yescharacteristic
117C22⋊C8×C53D411yescharacteristic
118C4×C8×C53C2311yescharacteristic
119C23×D53—41no—
120C2×D4×C53—21no—
121C2×D4×C53—41no—
122C2×D4×C53—41no—
123C2×D4×C53D811yescharacteristic
124C22×C4×C53—41no—
125C22×C4×C53—41no—
126C22×C4×C53C2×D411yescharacteristic
127C22×C4×C53C2×D411yescharacteristic
128C2×C22⋊D53—41no—
129C2×C22⋊D53—41no—
130#11719945—42no—
131#11719945—81no—
132C2×D212—21no—
133C2×D212—21no—
134C2×D212—42no—
135C2×D212—41no—
136C2×D212—81no—
137C2×C4×D53—41no—
138C2×C4×D53—41no—
139C2×Dic106—21no—
140D8×C53—21no—
141D8×C53—21no—
142C2×C8×C53—21no—
143C2×C8×C53—21no—
144C2×C8×C53—21no—
145C2×C8×C53—21no—
146C2×C8×C53C2×D411yescharacteristic
147C2×C8×C53C2×D411yescharacteristic
148C2×C8×C53C2×D411yescharacteristic
149C2×C8×C53C2411yescharacteristic
150C4⋊C4×C53—21no—
151C4⋊C4×C53—21no—
152C4⋊C4×C53—21no—
153C4⋊C4×C53—41no—
154C4⋊C4×C53—41no—
155C4⋊C4×C53—41no—
156C4⋊C4×C53D811yescharacteristic
157C22⋊C4×C53—41no—
158C22⋊C4×C53—41no—
159C22⋊C4×C53—41no—
160C22⋊C4×C53—41no—
161C22⋊C4×C53D811yescharacteristic
162C22⋊C4×C53D811yescharacteristic
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×C53—41no—
186C23×C53C2×D811yescharacteristic
187C23×C53C2×D811yescharacteristic
188C22×D53—41no—
189C22×D53—41no—
190C22×D53—82no—
191C22×D53—82no—
192C22×D53—161no—
193D4×C53—22no—
194D4×C53—41no—
195D4×C53—41no—
196D4×C53—82no—
197D4×C53—161no—
198C2×C4×C53—21no—
199C2×C4×C53—21no—
200C2×C4×C53—21no—
201C2×C4×C53—21no—
202C2×C4×C53—41no—
203C2×C4×C53—41no—
204C2×C4×C53—41no—
205C2×C4×C53—161no—
206C2×C4×C53—161no—
207C2×C4×C53C2×D811yescharacteristic
208C2×C4×C53C2×D811yescharacteristic
209C2×C4×C53C2×D811yescharacteristic
210C2×C4×C53C2×D811yescharacteristic
211C2×C4×C53C22×D411yescharacteristic
212C22⋊D53—82no—
213C22⋊D53—161no—
214C2×Dic53—41no—
215C2×Dic53—41no—
216D212—22no—
217D212—41no—
218D212—41no—
219D212—41no—
220D212—82no—
221C4×D53—82no—
222C4×D53—161no—
223Dic106—41no—
224Dic106—41no—
225C8×C53—21no—
226C8×C53—21no—
227C8×C53—21no—
228C8×C53C2×#4111yescharacteristic
229C8×C53H(5,𝔽2)11yescharacteristic
230#33864—531nomaximal, Sylow
231C22×C53—21no—
232C22×C53—21no—
233C22×C53—21no—
234C22×C53—21no—
235C22×C53—21no—
236C22×C53—81no—
237C22×C53—81no—
238C22×C53—81no—
239C22×C53#50812yes—
240C22×C53C22×D811yescharacteristic
241C2×D53—42no—
242C2×D53—81no—
243C2×D53—81no—
244C2×D53—81no—
245C2×D53—81no—
246C2×D53—81no—
247C2×D53—164no—
248C4×C53—21no—
249C4×C53—21no—
250C4×C53—21no—
251C4×C53—21no—
252C4×C53—81no—
253C4×C53—81no—
254C4×C53—81no—
255C4×C53#54611yescharacteristic
256C4×C53#57711yescharacteristic
257Dic53—81no—
258Dic53—81no—
259#3339D5311yescharacteristic, complemented
260#3280—531no—
261#2242—531no—
262#2236—531no—
263#2234—531no—
264#2228—531no—
265#2217—531no—
266#2215—531no—
267#2210—531no—
268#2186—531no—
269#2184—531no—
270C4×D16—531no—
271D4×C16—531no—
272#2138—531no—
273C2×#357—531no—
274C2×C53—22no—
275C2×C53—41no—
276C2×C53—161no—
277C2×C53#329411yescharacteristic
278C2×C53C2×#50811yescharacteristic
279C2×C53#341511yescharacteristic
280D53—82no—
281D53—161no—
282C2×#135—531no—
283C22×D8—531no—
284#553—21no—
285#546—1061no—
286C2×SD32—531no—
287C2×D16—531no—
288C2×D16—531no—
289C22×C16—531no—
290C22×C16—531no—
291#499C2×D5311yescharacteristic, complemented
292#495—531no—
293#482—21no—
294#481—21no—
295#478—21no—
296#467—531no—
297#466—531no—
298#466C2×D5311yescharacteristic
299#466C2×D5311yescharacteristic, complemented
300#463—1061no—
301#450—1061no—
302#449—1061no—
303C4×D8—531no—
304C4×D8C2×D5311yescharacteristic, complemented
305C8×D4C2×D5311yescharacteristic
306#427C2×D5311yescharacteristic, complemented
307C2×#107C2×D5311yescharacteristic
308#367—531no—
309#366—531no—
310#366—531no—
311C4⋊C16—531no—
312#358—531no—
313#358—531no—
314#358—531no—
315#357—531no—
316#357—531no—
317#357—531no—
318#357—531no—
319#357—531no—
320#357—531no—
321#357—531no—
322C22⋊C16—531no—
323C22⋊C16—531no—
324C4×C16—531no—
325C53#3386411yescharacteristic, complemented, Sylow
326C2×#41—1061no—
327C22×D4—1061no—
328#135—1062no—
329#135—2121no—
330C2×Q16—531no—
331C2×SD16—1061no—
332C2×SD16—1061no—
333C2×D8—531no—
334C2×D8—531no—
335C2×D8—1062no—
336C2×D8—1061no—
337C2×D8—2121no—
338C2×D8C22×D5311yescharacteristic
339C22×C8C22×D5311yescharacteristic
340C22×C8C22×D5311yescharacteristic
341#126—41no—
342#125—21no—
343#124—1061no—
344#123—41no—
345#123—2121no—
346C22⋊Q8—1061no—
347#121—21no—
348#121—21no—
349#121—41no—
350#121—1061no—
351#121—2121no—
352#120—2121no—
353C4×D4—21no—
354C4×D4—41no—
355C4×D4—1061no—
356C4×D4C22⋊D5311yescharacteristic
357#117—21no—
358C2×C4⋊C4—21no—
359C2×C22⋊C4—2121no—
360SD32—2121no—
361D16—1062no—
362D16—2121no—
363C2×C16—531no—
364C2×C16—531no—
365C2×C16—531no—
366C2×C16—531no—
367C2×C16—2121no—
368C2×C16—2121no—
369#107—21no—
370#107—531no—
371#107C22×D5311yescharacteristic
372#107C22×D5311yescharacteristic
373#107C22×D5311yescharacteristic
374#106—21no—
375#106—21no—
376C4⋊C8C22⋊D5311yescharacteristic
377#103—1061no—
378#102—21no—
379#102—21no—
380#102—21no—
381#102—1061no—
382#102—1061no—
383C22⋊C8C22⋊D5311yescharacteristic
384C22⋊C8C22⋊D5311yescharacteristic
385C4×C8C22×D5311yescharacteristic
386C24—2121no—
387#41—4241no—
388#41—2122no—
389C2×Q8—1061no—
390C2×D4—21no—
391C2×D4—41no—
392C2×D4—41no—
393C2×D4—1061no—
394C2×D4—1061no—
395C2×D4—4241no—
396C2×D4—2122no—
397C2×D4—2121no—
398C2×D4—2121no—
399C2×D4—2121no—
400C2×D4#1171992211yescharacteristic
401C22×C4—41no—
402C22×C4—41no—
403C22×C4—2121no—
404C22×C4—2121no—
405C22×C4C2×C22⋊D5311yescharacteristic
406C22×C4C2×C22⋊D5311yescharacteristic
407Q16—1061no—
408Q16—1061no—
409SD16—4241no—
410SD16—2122no—
411D8—21no—
412D8—21no—
413D8—532no—
414D8—1061no—
415D8—1061no—
416D8—1061no—
417D8—2122no—
418C2×C8—21no—
419C2×C8—21no—
420C2×C8—21no—
421C2×C8—21no—
422C2×C8C2×C22⋊D5311yescharacteristic
423C2×C8C2×C22⋊D5311yescharacteristic
424C2×C8C2×C22⋊D5311yescharacteristic
425C2×C8C23×D5311yescharacteristic
426C4⋊C4—21no—
427C4⋊C4—21no—
428C4⋊C4—21no—
429C4⋊C4—41no—
430C4⋊C4—41no—
431C4⋊C4—41no—
432C4⋊C4—1061no—
433C4⋊C4—2121no—
434C4⋊C4#1171992211yescharacteristic
435C22⋊C4—41no—
436C22⋊C4—41no—
437C22⋊C4—41no—
438C22⋊C4—41no—
439C22⋊C4—2121no—
440C22⋊C4—2121no—
441C22⋊C4—2121no—
442C22⋊C4—2121no—
443C22⋊C4—2121no—
444C22⋊C4#1171992211yescharacteristic
445C22⋊C4#1171992211yescharacteristic
446C42—41no—
447C42C2×C22⋊D5311yescharacteristic
448C16—1061no—
449C16—1061no—
450C16—1061no—
451C16—1061no—
452C23—41no—
453C23—8481no—
454C23—4242no—
455C23—4242no—
456C23—2121no—
457C23—2121no—
458C23C2×#1171992211yescharacteristic
459C23C2×#1171992211yescharacteristic
460Q8—2121no—
461Q8—2121no—
462D4—22no—
463D4—41no—
464D4—41no—
465D4—82no—
466D4—161no—
467D4—8481no—
468D4—1062no—
469D4—4242no—
470D4—4242no—
471D4—2121no—
472D4—2121no—
473D4—2121no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—161no—
482C2×C4—161no—
483C2×C4—8481no—
484C2×C4—4242no—
485C2×C4—2121no—
486C2×C4—2121no—
487C2×C4C2×#1171992211yescharacteristic
488C2×C4C2×#1171992211yescharacteristic
489C2×C4C2×#1171992211yescharacteristic
490C2×C4C2×#1171992211yescharacteristic
491C2×C4C22×C22⋊D5311yescharacteristic
492C8—21no—
493C8—21no—
494C8—21no—
495C8#41×D5311yescharacteristic
496C8#1313326311yescharacteristic
497C22—21no—
498C22—21no—
499C22—21no—
500C22—21no—
501C22—21no—
502C22—81no—
503C22—81no—
504C22—81no—
505C22—8484no—
506C22—4241no—
507C22—4241no—
508C22—4241no—
509C22—4241no—
510C22—4241no—
511C22—2122no—
512C22#1681105312yes—
513C22C22×#1171992211yescharacteristic
514C4—21no—
515C4—21no—
516C4—21no—
517C4—21no—
518C4—81no—
519C4—81no—
520C4—81no—
521C4—4241no—
522C4—4241no—
523C4#1681126111yescharacteristic
524C4#1681143111yescharacteristic
525C2—22no—
526C2—41no—
527C2—161no—
528C2—8481no—
529C2—4242no—
530C2#2028652211yescharacteristic
531C2C2×#1681105311yescharacteristic
532C2#2028785411yescharacteristic
subgroupG/H
centerC22C22×#11719922
commutator, cosocleC2×C8×C53C24
FrattiniC2×C8C23×D53
FittingC53×#3339C2
socleC22×C53C22×D8
hypercenter#3339D53
hypocenterC53#33864
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ #3339

Complements

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

NH ≅ G/Ntype
1. #20286402C2semidirect
2. #20276949C2semidirect
3. #20276917C2semidirect
4. #20276906C2semidirect
5. #20276869C2semidirect
6. #20276809C2semidirect
7. #20276799C2semidirect
8. #20276770C2semidirect
9. #20276636C2semidirect
10. #20276626C2semidirect
11. C4×#13133158C2semidirect
12. D4×#11719908C2semidirect
13. #20276420C2semidirect
14. C2×#16810423C2semidirect
15. C53×#3339C2semidirect
19. C2×#13133161C22semidirect
23. C22×#11719908C22semidirect
31. #16810450C22semidirect
32. #16810448C22semidirect
33. #16810448C22semidirect
34. C4⋊#11719908C22semidirect
35. #16810426C22semidirect
36. #16810426C22semidirect
37. #16810426C22semidirect
38. #16810424C22semidirect
46. C22⋊#11719908C22semidirect
47. C4×#11719908C22semidirect
49. C53×#499C22semidirect
54. C53×#466C22semidirect
55. C4×D8×C53C22semidirect
57. C53×#427C22semidirect
259. #3339D53semidirect
291. #499C2×D53semidirect
299. #466C2×D53semidirect
304. C4×D8C2×D53semidirect
306. #427C2×D53semidirect
325. C53#33864semidirect

Maximal subgroups

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

subgroupindex
#202864022
#202769492
#202769172
#202769062
#202768692
#202768092
#202767992
#202767702
#202766362
#202766262
C4×#131331582
D4×#117199082
#202764202
C2×#168104232
C53×#33392
#3386453