Finite Group Databasebeta

← #27392285

Subgroups — #27392285

Order 13568.

738 conjugacy classes of subgroups.

The subgroup lattice has 536 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285058C211yesmaximal, characteristic
2#20283653C211yesmaximal, characteristic, complemented
3#20283431C211yesmaximal, characteristic
4#20275690C211yesmaximal, characteristic, complemented
5#20275541C211yesmaximal, characteristic
6#20275428C211yesmaximal, characteristic, complemented
7#20275424C211yesmaximal, characteristic, complemented
8#20275420C211yesmaximal, characteristic, complemented
9#20275416C211yesmaximal, characteristic, complemented
10#20274854C211yesmaximal, characteristic, complemented
11#20274561C211yesmaximal, characteristic, complemented
12#20274557C211yesmaximal, characteristic, complemented
13#20274552C211yesmaximal, characteristic, complemented
14C4×#13133127C211yesmaximal, characteristic, complemented
15C53×#1764C211yesmaximal, characteristic, complemented
16C2×#13133223C2211yescharacteristic
17#16811176C2211yescharacteristic
18#16811169C2211yescharacteristic
19#16810990C412yescomplemented
20#16810986C412yescomplemented
21#16810984C412yescomplemented
22#16810981C412yescomplemented
23#16810753—22no—
24#16810751—22no—
25#16810694—22no—
26#16810692—22no—
27#16810651C2211yescharacteristic
28#16810650C2211yescharacteristic
29#16810646C2211yescharacteristic
30#16810646C2211yescharacteristic
31C2×#13133145C2211yescharacteristic
32C2×#13133144C2211yescharacteristic
33C2×#13133143C2211yescharacteristic
34#16810620C2211yescharacteristic
35#16810619C2211yescharacteristic
36#16810617C2211yescharacteristic
37#16810591C2211yescharacteristic
38#16810590C2211yescharacteristic
39#16810578C2211yescharacteristic
40C2×#13133127C2211yescharacteristic
41C2×#13133127C2211yescharacteristic
42C2×#13133127C2211yescharacteristic
43C2×#13133127C2211yescharacteristic
44C2×C4×#88322C2211yescharacteristic
45C42×Dic53C2211yescharacteristic
46#16810388C2211yescharacteristic
47#16810387C2211yescharacteristic
48#16810387C2211yescharacteristic
49#16810384C2211yescharacteristic
50#16810383C2211yescharacteristic
51#16810382C2211yescharacteristic
52C2×#131×C53C2211yescharacteristic
53C2×#104×C53C2211yescharacteristic
54C53×#416C2211yescharacteristic
55C53×#405C2211yescharacteristic
56C2×C4×C8×C53C2211yescharacteristic
57C53×#341C2211yescharacteristic
58C53×#339C2211yescharacteristic
59C2×#11719945C2311yescharacteristic
60#13133223—22no—
61#13133223—22no—
62C2×#11719917C2311yescharacteristic
63C2×#11719917C2311yescharacteristic
64C2×#11719912—21no—
65C22×#88322C2311yescharacteristic
66C22×#88322C2311yescharacteristic
67C2×C8×D53—21no—
68#13133216C2×C412yes—
69#13133215C2×C411yescharacteristic
70#13133213C2×C411yescharacteristic
71#13133208C2×C412yes—
72#13133206C2×C411yescharacteristic
73#13133204C2×C411yescharacteristic
74#13133194—22no—
75#13133189—22no—
76C4×Dic106—22no—
77C4×C22⋊D53—22no—
78#13133176C2311yescharacteristic
79C2×C4×Dic53—22no—
80C2×C4×Dic53C2311yescharacteristic
81C2×C4×Dic53C2311yescharacteristic
82C4⋊#88322—22no—
83#13133146—22no—
84#13133146—22no—
85#13133146—22no—
86#13133145—22no—
87#13133144—22no—
88#13133143—22no—
89#13133142C2×C412yes—
90#13133141C2×C412yes—
91#13133140C2×C412yes—
92#13133139C2×C412yes—
93#13133138C2×C412yes—
94#13133137C2×C412yes—
95C22⋊#88322—22no—
96#13133129—22no—
97#13133127—21no—
98#13133127—22no—
99#13133127—22no—
100#13133127C2×C412yes—
101#13133127C2×C412yes—
102#13133127D412yes—
103#13133127D412yes—
104C4×#88322—22no—
105C8×Dic53—21no—
106C2×#41×C53C2311yescharacteristic
107#131×C53—22no—
108#131×C53—22no—
109C2×M16×C53—21no—
110C2×M16×C53C2311yescharacteristic
111C2×M16×C53C2311yescharacteristic
112C22×C8×C53—21no—
113C22×C8×C53C2311yescharacteristic
114C22×C8×C53C2311yescharacteristic
115#117×C53C2311yescharacteristic
116C2×C42×C53C2311yescharacteristic
117#104×C53—22no—
118#103×C53C2×C412yes—
119#102×C53C2×C412yes—
120#97×C53—21no—
121C4×C8×C53—21no—
122C4×C8×C53—22no—
123#41×C53—21no—
124#41×C53—21no—
125#41×C53—21no—
126#41×C53—21no—
127C2×Q8×C53C22×C411yescharacteristic
128C2×D4×C53—21no—
129C2×D4×C53C22×C411yescharacteristic
130C22×C4×C53—21no—
131C22×C4×C53—21no—
132C22×C4×C53C2411yescharacteristic
133C2×C22⋊D53—21no—
134C2×C22⋊D53—22no—
135C22×Dic53—21no—
136C22×Dic53#4112yes—
137#11719945—21no—
138#11719945—21no—
139#11719945—21no—
140#11719945—21no—
141C2×D212C22×C411yescharacteristic
142C2×C4×D53—21no—
143C2×Dic106—22no—
144C2×Dic106C22×C411yescharacteristic
145M16×C53—21no—
146M16×C53—21no—
147M16×C53—21no—
148M16×C53—21no—
149M16×C53—42no—
150C2×C8×C53—21no—
151C2×C8×C53—21no—
152C2×C8×C53—21no—
153C2×C8×C53—21no—
154C2×C8×C53—42no—
155C2×C8×C53C22×C412yes—
156C2×C8×C53C2×D411yescharacteristic
157C2×C8×C53C2×D411yescharacteristic
158C2×C8×C53#4112yes—
159C2×C8×C53#4112yes—
160C4⋊C4×C53C22×C412yes—
161C22⋊C4×C53—21no—
162C42×C53—21no—
163C42×C53—21no—
164C42×C53#4112yes—
165C22⋊Dic53—21no—
166C22⋊Dic53—22no—
167#11719921—22no—
168C4⋊Dic53—22no—
169C4⋊Dic53C22×C412yes—
170#11719919—22no—
171#11719919—22no—
172#11719919—22no—
173C4×Dic53—21no—
174C4×Dic53—21no—
175C4×Dic53—44no—
176C4×Dic53C22×C411yescharacteristic
177C4×Dic53C22×C411yescharacteristic
178C4×Dic53C2×D412yes—
179C4×Dic53#4112yes—
180#11719917—21no—
181#11719917—21no—
182#11719917—21no—
183#11719917—21no—
184C2×#88322—21no—
185C2×#88322—21no—
186C2×#88322—21no—
187C2×#88322—21no—
188C2×#88322—21no—
189C2×#88322—21no—
190C2×#88322C22×C412yes—
191C2×#88322C2×D412yes—
192#11719912—42no—
193C8×D53—42no—
194C23×C53—21no—
195C23×C53C2×#4111yescharacteristic
196C22×D53—21no—
197Q8×C53—21no—
198Q8×C53—21no—
199D4×C53—21no—
200D4×C53—21no—
201D4×C53—41no—
202D4×C53—41no—
203C2×C4×C53—24no—
204C2×C4×C53—22no—
205C2×C4×C53—22no—
206C2×C4×C53—21no—
207C2×C4×C53—41no—
208C2×C4×C53—41no—
209C2×C4×C53C23×C411yescharacteristic
210C2×C4×C53C22×D411yescharacteristic
211C2×C4×C53C2×#4111yescharacteristic
212C2×C4×C53C2×#4111yescharacteristic
213C2×C4×C53C2×#4111yescharacteristic
214C2×C4×C53C2×#4111yescharacteristic
215C22⋊D53—41no—
216C22⋊D53—41no—
217C22⋊D53—82no—
218C2×Dic53—22no—
219C2×Dic53—21no—
220C2×Dic53—22no—
221C2×Dic53—42no—
222C2×Dic53—42no—
223C2×Dic53C4×D412yes—
224C2×Dic53C4×D412yes—
225D212—21no—
226D212—21no—
227C4×D53—41no—
228C4×D53—41no—
229Dic106—21no—
230Dic106—21no—
231Dic106—82no—
232C8×C53—22no—
233C8×C53—41no—
234C8×C53—41no—
235C8×C53#12412yes—
236C8×C53#12412yes—
237#88322—22no—
238#88322—22no—
239#88322—41no—
240#88322—41no—
241#21662—531nomaximal, Sylow
242C22×C53—22no—
243C22×C53—42no—
244C22×C53—41no—
245C22×C53C2×C4×D411yescharacteristic
246C22×C53#52911yescharacteristic
247C22×C53#52911yescharacteristic
248C2×D53—42no—
249C2×D53—41no—
250C4×C53—24no—
251C4×C53—42no—
252C4×C53—41no—
253C4×C53C4×#4111yescharacteristic
254C4×C53C4×#4111yescharacteristic
255C4×C53C2×#12411yescharacteristic
256C4×C53#53211yescharacteristic
257Dic53—22no—
258Dic53—22no—
259Dic53—42no—
260Dic53—42no—
261Dic53—41no—
262#3135—531no—
263#2961—531no—
264#2932—531no—
265#1994—531no—
266#1962—531no—
267#1938—531no—
268#1938—531no—
269#1937—531no—
270#1937—531no—
271#1838—531no—
272#1765—531no—
273#1765—531no—
274#1764—531no—
275#1764D5311yescharacteristic, complemented
276C4×#97—531no—
277C2×C53—21no—
278C2×C53—21no—
279C2×C53—81no—
280C2×C53C4×#12411yescharacteristic
281C2×C53#295612yes—
282D53—81no—
283C2×#131—531no—
284C2×#131C2×D5311yescharacteristic
285#533—531no—
286#532—531no—
287#491—532no—
288#490—532no—
289#489—532no—
290#489—532no—
291#446—1062no—
292#446—1062no—
293#435—1062no—
294#435—1062no—
295#424—531no—
296#424—531no—
297#423—531no—
298#423—531no—
299C2×#104—531no—
300C2×#104—531no—
301C2×#104—531no—
302C2×#104C2×D5311yescharacteristic
303#416—531no—
304#416—531no—
305#416—531no—
306#416C2×D5311yescharacteristic
307#408—531no—
308#408—531no—
309#405—531no—
310#405C2×D5311yescharacteristic
311C2×#97—531no—
312C2×#97—531no—
313C2×#97—531no—
314C2×#97—531no—
315C2×C4×C8—531no—
316C2×C4×C8C2×D5311yescharacteristic
317C43—531no—
318#341—531no—
319#341—531no—
320#341—531no—
321#341C2×D5311yescharacteristic
322#339—531no—
323#339—531no—
324#339—531no—
325#339C2×D5311yescharacteristic
326C53#2166211yescharacteristic, complemented, Sylow
327C2×#41—531no—
328C2×#41C22×D5311yescharacteristic
329#131—22no—
330#131—22no—
331#131—1062no—
332#131—1062no—
333C2×M16—21no—
334C2×M16—531no—
335C2×M16—531no—
336C2×M16—1061no—
337C2×M16C22×D5311yescharacteristic
338C2×M16C22×D5311yescharacteristic
339C22×C8—21no—
340C22×C8—531no—
341C22×C8—531no—
342C22×C8—1061no—
343C22×C8C22×D5311yescharacteristic
344C22×C8C22×D5311yescharacteristic
345C4⋊Q8—532no—
346C4⋊Q8—531no—
347#127—531no—
348#125—532no—
349#124—531no—
350#124—531no—
351C22⋊Q8—1062no—
352#121—1062no—
353C4×Q8—1062no—
354C4×D4—1062no—
355#117—531no—
356#117C22×D5311yescharacteristic
357C2×C42—531no—
358C2×C42—531no—
359C2×C42—1062no—
360C2×C42C22×D5311yescharacteristic
361C4⋊C8—1062no—
362C4⋊C8—1062no—
363C4⋊C8—1062no—
364C4⋊C8—1062no—
365#104—22no—
366#104—1062no—
367#104—1062no—
368#104—1062no—
369#103—532no—
370#103—532no—
371#103—532no—
372#103C4×D5312yes—
373#102—532no—
374#102—532no—
375#102—532no—
376#102C4×D5312yes—
377C22⋊C8—1062no—
378C22⋊C8—1062no—
379#97—21no—
380#97—532no—
381#97—532no—
382#97—532no—
383#97—532no—
384#97—1062no—
385#97—1062no—
386#97—1061no—
387C4×C8—21no—
388C4×C8—22no—
389C4×C8—1061no—
390C4×C8—1062no—
391#41—21no—
392#41—21no—
393#41—21no—
394#41—21no—
395#41—1061no—
396#41—1061no—
397#41—1061no—
398#41—1061no—
399C2×Q8—531no—
400C2×Q8—1062no—
401C2×Q8C2×C4×D5311yescharacteristic
402C2×D4—21no—
403C2×D4—531no—
404C2×D4—1062no—
405C2×D4—1061no—
406C2×D4C2×C4×D5311yescharacteristic
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—532no—
410C22×C4—1061no—
411C22×C4—1061no—
412C22×C4C23×D5311yescharacteristic
413M16—21no—
414M16—21no—
415M16—21no—
416M16—21no—
417M16—42no—
418M16—1061no—
419M16—1061no—
420M16—1061no—
421M16—1061no—
422M16—2122no—
423C2×C8—21no—
424C2×C8—21no—
425C2×C8—21no—
426C2×C8—21no—
427C2×C8—42no—
428C2×C8—532no—
429C2×C8—532no—
430C2×C8—1061no—
431C2×C8—1061no—
432C2×C8—1061no—
433C2×C8—1061no—
434C2×C8—1061no—
435C2×C8—1061no—
436C2×C8—2122no—
437C2×C8C2×C4×D5312yes—
438C2×C8#1171994512yes—
439C2×C8D4×D5311yescharacteristic
440C2×C8D4×D5311yescharacteristic
441C2×C8#1171994712yes—
442C4⋊C4—532no—
443C4⋊C4—1062no—
444C4⋊C4—1062no—
445C4⋊C4—1062no—
446C4⋊C4—1062no—
447C4⋊C4C2×C4×D5312yes—
448C22⋊C4—21no—
449C22⋊C4—1062no—
450C22⋊C4—1062no—
451C22⋊C4—1061no—
452C42—21no—
453C42—21no—
454C42—531no—
455C42—531no—
456C42—532no—
457C42—532no—
458C42—1061no—
459C42—1061no—
460C42—2124no—
461C42#1171994512yes—
462C23—21no—
463C23—1061no—
464C23#41×D5311yescharacteristic
465Q8—21no—
466Q8—21no—
467Q8—1061no—
468Q8—1061no—
469Q8—4242no—
470D4—21no—
471D4—21no—
472D4—41no—
473D4—41no—
474D4—1061no—
475D4—1061no—
476D4—4242no—
477D4—2121no—
478D4—2121no—
479C2×C4—24no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—21no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—532no—
486C2×C4—532no—
487C2×C4—1062no—
488C2×C4—1061no—
489C2×C4—1062no—
490C2×C4—2122no—
491C2×C4—2122no—
492C2×C4—2121no—
493C2×C4—2121no—
494C2×C4C22×C4×D5311yescharacteristic
495C2×C4C2×D4×D5311yescharacteristic
496C2×C4#41×D5311yescharacteristic
497C2×C4C2×#1171994711yescharacteristic
498C2×C4C2×#1171994511yescharacteristic
499C2×C4C2×#1171994511yescharacteristic
500C8—22no—
501C8—41no—
502C8—41no—
503C8—1062no—
504C8—1062no—
505C8—2121no—
506C8—2121no—
507C8#1313320312yes—
508C8#1313320312yes—
509C22—22no—
510C22—41no—
511C22—42no—
512C22—2121no—
513C22—2122no—
514C22C4×D4×D5311yescharacteristic
515C22#1681115111yescharacteristic
516C22#1681115311yescharacteristic
517C4—24no—
518C4—41no—
519C4—42no—
520C4—1062no—
521C4—1062no—
522C4—2122no—
523C4—2121no—
524C4—2122no—
525C4C4×#1171994711yescharacteristic
526C4C4×#1171994511yescharacteristic
527C4C2×#1313320311yescharacteristic
528C4#1681117011yescharacteristic
529C2—21no—
530C2—21no—
531C2—81no—
532C2—4241no—
533C2C4×#1313320311yescharacteristic
534C2#2028362012yes—
subgroupG/H
centerC2×C4C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1764C2
socleC22×C53C2×C4×D4
prime radical#16811169C22
cosocleC22×C4×C53C24
hypercenter#1764D53
hypocenterC53#21662
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ #1764

Complements

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

NH ≅ G/Ntype
2. #20283653C2semidirect
4. #20275690C2semidirect
6. #20275428C2semidirect
7. #20275424C2semidirect
8. #20275420C2semidirect
9. #20275416C2semidirect
10. #20274854C2semidirect
11. #20274561C2semidirect
12. #20274557C2semidirect
13. #20274552C2semidirect
14. C4×#13133127C2semidirect
15. C53×#1764C2semidirect
19. #16810990C4semidirect
20. #16810986C4semidirect
21. #16810984C4semidirect
22. #16810981C4semidirect
275. #1764D53semidirect
326. C53#21662semidirect

Maximal subgroups

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

subgroupindex
#202850582
#202836532
#202834312
#202756902
#202755412
#202754282
#202754242
#202754202
#202754162
#202748542
#202745612
#202745572
#202745522
C4×#131331272
C53×#17642
#2166253