Finite Group Databasebeta

← #27275960

Subgroups — #27275960

Order 13568.

758 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20274525C211yesmaximal, characteristic, complemented
2#20273322C211yesmaximal, characteristic, complemented
3#20273223C211yesmaximal, characteristic, complemented
4#20273201C211yesmaximal, characteristic, complemented
5#20273149C211yesmaximal, characteristic, complemented
6#20273132C211yesmaximal, characteristic, complemented
7C53×#3517C211yesmaximal, characteristic
8#16810680C2211yescharacteristic
9#16810676C2211yescharacteristic
10C2×#13133144—21no—
11C4×#11719917C2211yescharacteristic
12#16810469C2211yescharacteristic
13#16810382—21no—
14#16810365—21no—
15#16810363—21no—
16#16810360—21no—
17#16810357—21no—
18#16810357—22no—
19Q8⋊#88322—21no—
20D4⋊#88322—22no—
21D4⋊#88322—21no—
22C53×#557C411yescharacteristic, complemented
23C53×#553—22no—
24C53×#553—21no—
25C53×#550—21no—
26C53×#550—21no—
27C53×#550C411yescharacteristic, complemented
28C53×#550C411yescharacteristic, complemented
29C53×#547—21no—
30C53×#547C411yescharacteristic, complemented
31C53×#540—22no—
32C53×#540—21no—
33C53×#532—21no—
34C53×#532C2211yescharacteristic
35C53×#529—21no—
36C53×#529C2211yescharacteristic
37C53×#520—21no—
38C4×#41×C53C2211yescharacteristic
39C2×#11719917—21no—
40C2×C4×Dic53—21no—
41C4⋊#88322—21no—
42C4⋊#88322—21no—
43#13133144—24no—
44C22⋊#88322—21no—
45C22⋊#88322—21no—
46#13133128—21no—
47#13133128—21no—
48C4×#88322—21no—
49C4×#88322—21no—
50#13133124—21no—
51#13133124—21no—
52C2×#41×C53—21no—
53C2×#41×C53—21no—
54C2×#41×C53—21no—
55C2×#41×C53—21no—
56C2×#41×C53D411yescharacteristic
57C4⋊Q8×C53—21no—
58C4⋊Q8×C53C2×C411yescharacteristic
59#127×C53—22no—
60#127×C53—21no—
61#127×C53—21no—
62#127×C53C2×C411yescharacteristic
63#126×C53—22no—
64#126×C53—21no—
65#125×C53—22no—
66#125×C53—21no—
67#125×C53D411yescharacteristic
68#124×C53—21no—
69#124×C53D411yescharacteristic
70#123×C53—22no—
71#123×C53—22no—
72#123×C53—22no—
73#123×C53—22no—
74#123×C53—21no—
75C22⋊Q8×C53—21no—
76C22⋊Q8×C53D411yescharacteristic
77#121×C53—22no—
78#121×C53—22no—
79#121×C53—22no—
80#121×C53—22no—
81#121×C53—22no—
82#121×C53—21no—
83#121×C53—21no—
84#121×C53—21no—
85#121×C53—22no—
86#121×C53—22no—
87#121×C53—21no—
88#121×C53D411yescharacteristic
89#121×C53D412yes—
90C4×Q8×C53—21no—
91C4×Q8×C53—21no—
92C4×Q8×C53C2×C411yescharacteristic
93C4×Q8×C53C2×C411yescharacteristic
94C4×Q8×C53D411yescharacteristic
95C4×D4×C53—22no—
96C4×D4×C53—22no—
97C4×D4×C53—22no—
98C4×D4×C53—22no—
99C4×D4×C53—21no—
100C4×D4×C53—21no—
101C4×D4×C53—21no—
102C4×D4×C53—21no—
103C4×D4×C53—21no—
104C4×D4×C53—21no—
105C4×D4×C53C2×C411yescharacteristic
106C4×D4×C53C2×C411yescharacteristic
107C4×D4×C53D411yescharacteristic
108C4×D4×C53D412yes—
109#117×C53—22no—
110#117×C53—21no—
111#117×C53—21no—
112#117×C53D411yescharacteristic
113C2×C4⋊C4×C53—22no—
114C2×C4⋊C4×C53—21no—
115C2×C42×C53—21no—
116C2×C42×C53C2311yescharacteristic
117#41×C53—22no—
118#41×C53—41no—
119#41×C53—42no—
120#41×C53—42no—
121#41×C53—82no—
122#41×C53—82no—
123#41×C53—82no—
124C2×Q8×C53—21no—
125C2×Q8×C53C22⋊C411yescharacteristic
126C2×Q8×C53C22⋊C411yescharacteristic
127C2×D4×C53—22no—
128C2×D4×C53—21no—
129C2×D4×C53—22no—
130C2×D4×C53—22no—
131C2×D4×C53—22no—
132C2×D4×C53—22no—
133C2×D4×C53—21no—
134C2×D4×C53—22no—
135C2×D4×C53—21no—
136C2×D4×C53—21no—
137C2×D4×C53C22⋊C411yescharacteristic
138C2×D4×C53C22⋊C411yescharacteristic
139C2×D4×C53C22⋊C412yes—
140C22×C4×C53—22no—
141C22×C4×C53—22no—
142C22×C4×C53—22no—
143C22×C4×C53—21no—
144C22×C4×C53—21no—
145C22×C4×C53—21no—
146C22×C4×C53—21no—
147C22×C4×C53—22no—
148C22×C4×C53C2×D411yescharacteristic
149C22×C4×C53C2×D411yescharacteristic
150C22×C4×C53C2×D411yescharacteristic
151C22×Dic53—41no—
152C4⋊C4×C53—22no—
153C4⋊C4×C53—22no—
154C4⋊C4×C53—22no—
155C4⋊C4×C53—22no—
156C4⋊C4×C53—21no—
157C4⋊C4×C53—21no—
158C4⋊C4×C53—21no—
159C4⋊C4×C53—21no—
160C4⋊C4×C53—21no—
161C4⋊C4×C53—21no—
162C4⋊C4×C53—21no—
163C4⋊C4×C53C22⋊C411yescharacteristic
164C4⋊C4×C53C22⋊C411yescharacteristic
165C4⋊C4×C53C22⋊C411yescharacteristic
166C4⋊C4×C53C22⋊C412yes—
167C4⋊C4×C53C22⋊C411yescharacteristic
168C22⋊C4×C53—22no—
169C22⋊C4×C53—22no—
170C22⋊C4×C53—22no—
171C22⋊C4×C53—21no—
172C22⋊C4×C53—21no—
173C22⋊C4×C53—21no—
174C22⋊C4×C53—21no—
175C22⋊C4×C53—21no—
176C22⋊C4×C53—21no—
177C22⋊C4×C53—22no—
178C22⋊C4×C53—21no—
179C22⋊C4×C53—21no—
180C22⋊C4×C53—22no—
181C42×C53—22no—
182C42×C53—21no—
183C42×C53—21no—
184C42×C53—21no—
185C42×C53—21no—
186C42×C53C22×C411yescharacteristic
187C42×C53C2×D411yescharacteristic
188C42×C53C2×D411yescharacteristic
189C42×C53C2×D411yescharacteristic
190C4×Dic53—22no—
191C4×Dic53—41no—
192#11719917—22no—
193#11719917—41no—
194C2×#88322—41no—
195C2×#88322—41no—
196C2×#88322—41no—
197C23×C53—21no—
198C23×C53—22no—
199C23×C53—21no—
200C23×C53#12011yescharacteristic
201Q8×C53—21no—
202Q8×C53—42no—
203Q8×C53—41no—
204Q8×C53#10412yes—
205D4×C53—21no—
206D4×C53—21no—
207D4×C53—42no—
208D4×C53—42no—
209D4×C53—41no—
210D4×C53—44no—
211D4×C53—42no—
212D4×C53—41no—
213D4×C53—41no—
214D4×C53—81no—
215D4×C53—82no—
216D4×C53—82no—
217D4×C53—82no—
218D4×C53—82no—
219D4×C53—82no—
220C2×C4×C53—21no—
221C2×C4×C53—21no—
222C2×C4×C53—21no—
223C2×C4×C53—21no—
224C2×C4×C53—21no—
225C2×C4×C53—22no—
226C2×C4×C53—21no—
227C2×C4×C53—21no—
228C2×C4×C53—21no—
229C2×C4×C53—41no—
230C2×C4×C53—41no—
231C2×C4×C53—42no—
232C2×C4×C53—44no—
233C2×C4×C53—42no—
234C2×C4×C53—41no—
235C2×C4×C53—41no—
236C2×C4×C53—81no—
237C2×C4×C53—82no—
238C2×C4×C53—82no—
239C2×C4×C53—82no—
240C2×C4×C53C2×C22⋊C411yescharacteristic
241C2×C4×C53C2×C22⋊C411yescharacteristic
242C2×C4×C53C2×C22⋊C411yescharacteristic
243C2×C4×C53#12011yescharacteristic
244C2×C4×C53#12011yescharacteristic
245C2×C4×C53#12011yescharacteristic
246C2×Dic53—42no—
247C2×Dic53—81no—
248C2×Dic53—81no—
249#88322—42no—
250#88322—81no—
251#88322—81no—
252#8414—531nomaximal, Sylow
253C22×C53—21no—
254C22×C53—21no—
255C22×C53—21no—
256C22×C53—41no—
257C22×C53—42no—
258C22×C53—41no—
259C22×C53—41no—
260C22×C53—42no—
261C22×C53—41no—
262C22×C53—41no—
263C22×C53—42no—
264C22×C53—41no—
265C22×C53#37911yescharacteristic
266C4×C53—21no—
267C4×C53—21no—
268C4×C53—21no—
269C4×C53—21no—
270C4×C53—21no—
271C4×C53—22no—
272C4×C53—41no—
273C4×C53—41no—
274C4×C53—41no—
275C4×C53—41no—
276C4×C53—42no—
277C4×C53—41no—
278C4×C53C2×#10411yescharacteristic
279C4×C53#42111yescharacteristic
280Dic53—82no—
281#3517D5311yescharacteristic
282#1759—531no—
283#1529—531no—
284#1505—531no—
285#1500—531no—
286#1488—531no—
287#1484—531no—
288C2×C53—41no—
289C2×C53—81no—
290C2×C53—82no—
291C2×C53—81no—
292C2×C53#178911yescharacteristic
293C2×C53#179411yescharacteristic
294C2×C53#179511yescharacteristic
295#557Dic5311yescharacteristic, complemented
296#553—22no—
297#553—21no—
298#550—21no—
299#550—21no—
300#550Dic5311yescharacteristic, complemented
301#550Dic5311yescharacteristic, complemented
302#547—21no—
303#547Dic5311yescharacteristic, complemented
304#540—22no—
305#540—21no—
306#532—21no—
307#532C2×D5311yescharacteristic
308#529—21no—
309#529C2×D5311yescharacteristic
310#520—21no—
311C4×#41C2×D5311yescharacteristic
312#432—531no—
313#431—531no—
314C2×#104—1061no—
315C4×M16—531no—
316#376—531no—
317#339—1061no—
318#330—1061no—
319#329—1061no—
320#328—1061no—
321#327—1061no—
322#327—1062no—
323Q8⋊C8—1061no—
324D4⋊C8—1062no—
325D4⋊C8—1061no—
326C53#841411yescharacteristic, complemented, Sylow
327C2×#41—21no—
328C2×#41—21no—
329C2×#41—21no—
330C2×#41—21no—
331C2×#41C22⋊D5311yescharacteristic
332C2×M16—1061no—
333C4⋊Q8—21no—
334C4⋊Q8C2×Dic5311yescharacteristic
335#127—22no—
336#127—21no—
337#127—21no—
338#127C2×Dic5311yescharacteristic
339#126—22no—
340#126—21no—
341#125—22no—
342#125—21no—
343#125C22⋊D5311yescharacteristic
344#124—21no—
345#124C22⋊D5311yescharacteristic
346#123—22no—
347#123—22no—
348#123—22no—
349#123—21no—
350#123—22no—
351C22⋊Q8—21no—
352C22⋊Q8C22⋊D5311yescharacteristic
353#121—22no—
354#121—22no—
355#121—22no—
356#121—22no—
357#121—22no—
358#121—21no—
359#121—22no—
360#121—22no—
361#121—21no—
362#121—21no—
363#121—21no—
364#121C22⋊D5311yescharacteristic
365#121C22⋊D5312yes—
366C4×Q8—21no—
367C4×Q8—21no—
368C4×Q8C2×Dic5311yescharacteristic
369C4×Q8C2×Dic5311yescharacteristic
370C4×Q8C22⋊D5311yescharacteristic
371C4×D4—22no—
372C4×D4—22no—
373C4×D4—22no—
374C4×D4—22no—
375C4×D4—21no—
376C4×D4—21no—
377C4×D4—21no—
378C4×D4—21no—
379C4×D4—21no—
380C4×D4—21no—
381C4×D4C2×Dic5311yescharacteristic
382C4×D4C2×Dic5311yescharacteristic
383C4×D4C22⋊D5311yescharacteristic
384C4×D4C22⋊D5312yes—
385#117—22no—
386#117—21no—
387#117—21no—
388#117C22⋊D5311yescharacteristic
389C2×C4⋊C4—22no—
390C2×C4⋊C4—21no—
391C2×C42—21no—
392C2×C42—1061no—
393C2×C42C22×D5311yescharacteristic
394C4⋊C8—1061no—
395C4⋊C8—1061no—
396#104—1064no—
397C22⋊C8—1061no—
398C22⋊C8—1061no—
399#97—1061no—
400#97—1061no—
401C4×C8—1061no—
402C4×C8—1061no—
403#95—1061no—
404#95—1061no—
405#41—22no—
406#41—41no—
407#41—42no—
408#41—42no—
409#41—82no—
410#41—82no—
411#41—82no—
412C2×Q8—21no—
413C2×Q8C22⋊Dic5311yescharacteristic
414C2×Q8C22⋊Dic5311yescharacteristic
415C2×D4—22no—
416C2×D4—21no—
417C2×D4—22no—
418C2×D4—22no—
419C2×D4—22no—
420C2×D4—22no—
421C2×D4—21no—
422C2×D4—22no—
423C2×D4—21no—
424C2×D4—21no—
425C2×D4C22⋊Dic5311yescharacteristic
426C2×D4C22⋊Dic5311yescharacteristic
427C2×D4C22⋊Dic5312yes—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—22no—
436C22×C4—2121no—
437C22×C4C2×C22⋊D5311yescharacteristic
438C22×C4C2×C22⋊D5311yescharacteristic
439C22×C4C2×C22⋊D5311yescharacteristic
440M16—1062no—
441M16—2121no—
442C2×C8—2121no—
443C2×C8—2121no—
444C2×C8—2121no—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4C22⋊Dic5311yescharacteristic
457C4⋊C4C22⋊Dic5311yescharacteristic
458C4⋊C4C22⋊Dic5311yescharacteristic
459C4⋊C4C22⋊Dic5312yes—
460C4⋊C4C22⋊Dic5311yescharacteristic
461C22⋊C4—22no—
462C22⋊C4—22no—
463C22⋊C4—22no—
464C22⋊C4—21no—
465C22⋊C4—21no—
466C22⋊C4—21no—
467C22⋊C4—21no—
468C22⋊C4—21no—
469C22⋊C4—21no—
470C22⋊C4—22no—
471C22⋊C4—21no—
472C22⋊C4—21no—
473C22⋊C4—22no—
474C42—22no—
475C42—21no—
476C42—21no—
477C42—21no—
478C42—21no—
479C42—1062no—
480C42—2121no—
481C42C22×Dic5311yescharacteristic
482C42C2×C22⋊D5311yescharacteristic
483C42C2×C22⋊D5311yescharacteristic
484C42C2×C22⋊D5311yescharacteristic
485C23—21no—
486C23—22no—
487C23—21no—
488C23#1313318611yescharacteristic
489Q8—21no—
490Q8—41no—
491Q8—42no—
492Q8#1313314412yes—
493D4—21no—
494D4—21no—
495D4—42no—
496D4—42no—
497D4—41no—
498D4—44no—
499D4—42no—
500D4—41no—
501D4—41no—
502D4—81no—
503D4—82no—
504D4—82no—
505D4—82no—
506D4—82no—
507D4—82no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—22no—
516C2×C4—21no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—42no—
520C2×C4—44no—
521C2×C4—42no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—81no—
525C2×C4—82no—
526C2×C4—82no—
527C2×C4—82no—
528C2×C4—4241no—
529C2×C4—4241no—
530C2×C4—2122no—
531C2×C4C2×C22⋊Dic5311yescharacteristic
532C2×C4C2×C22⋊Dic5311yescharacteristic
533C2×C4C2×C22⋊Dic5311yescharacteristic
534C2×C4#1313318611yescharacteristic
535C2×C4#1313318611yescharacteristic
536C2×C4#1313318611yescharacteristic
537C8—4241no—
538C8—4241no—
539C8—2122no—
540C22—21no—
541C22—21no—
542C22—21no—
543C22—41no—
544C22—42no—
545C22—41no—
546C22—41no—
547C22—42no—
548C22—41no—
549C22—41no—
550C22—42no—
551C22—41no—
552C22#1681047911yescharacteristic
553C4—21no—
554C4—22no—
555C4—21no—
556C4—21no—
557C4—21no—
558C4—21no—
559C4—41no—
560C4—41no—
561C4—41no—
562C4—41no—
563C4—41no—
564C4—42no—
565C4—4242no—
566C4C2×#1313314411yescharacteristic
567C4#1681064011yescharacteristic
568C2—41no—
569C2—81no—
570C2—82no—
571C2—81no—
572C2#2027465911yescharacteristic
573C2#2027467911yescharacteristic
574C2#2027468511yescharacteristic
subgroupG/H
centerC22#16810479
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
Fitting, prime radicalC53×#3517C2
socleC22×C53#379
cosocleC2×C42×C53C23
hypercenter#3517D53
hypocenterC53#8414
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3517

Complements

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

NH ≅ G/Ntype
1. #20274525C2semidirect
2. #20273322C2semidirect
3. #20273223C2semidirect
4. #20273201C2semidirect
5. #20273149C2semidirect
6. #20273132C2semidirect
22. C53×#557C4semidirect
27. C53×#550C4semidirect
28. C53×#550C4semidirect
30. C53×#547C4semidirect
295. #557Dic53semidirect
300. #550Dic53semidirect
301. #550Dic53semidirect
303. #547Dic53semidirect
326. C53#8414semidirect

Maximal subgroups

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

subgroupindex
#202745252
#202733222
#202732232
#202732012
#202731492
#202731322
C53×#35172
#841453