Finite Group Databasebeta

← #27392294

Subgroups — #27392294

Order 13568.

730 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#20285058C211yesmaximal, characteristic
2#20283647C211yesmaximal, characteristic, complemented
3#20283425C211yesmaximal, characteristic
4#20275694C211yesmaximal, characteristic, complemented
5#20275545C211yesmaximal, characteristic
6#20275421C211yesmaximal, characteristic, complemented
7#20275420C211yesmaximal, characteristic, complemented
8#20275417C211yesmaximal, characteristic, complemented
9#20275416C211yesmaximal, characteristic, complemented
10#20274854C211yesmaximal, characteristic, complemented
11#20274561C211yesmaximal, characteristic, complemented
12#20274559C211yesmaximal, characteristic, complemented
13#20274557C211yesmaximal, characteristic, complemented
14#20274403C211yesmaximal, characteristic, complemented
15C53×#1765C211yesmaximal, characteristic, complemented
16C2×#13133223C2211yescharacteristic
17#16811176C2211yescharacteristic
18#16811169C2211yescharacteristic
19#16810990C412yescomplemented
20#16810986C412yescomplemented
21#16810984C412yescomplemented
22#16810981C412yescomplemented
23#16810751—22no—
24C8×Dic106—22no—
25#16810699—22no—
26C8×C22⋊D53—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×C4×#88322C2211yescharacteristic
43C2×C8×Dic53C2211yescharacteristic
44C2×C8×Dic53C2211yescharacteristic
45#16810469C2211yescharacteristic
46#16810388C2211yescharacteristic
47#16810387C2211yescharacteristic
48#16810387C2211yescharacteristic
49#16810384C2211yescharacteristic
50#16810383C2211yescharacteristic
51#16810382C2211yescharacteristic
52C2×#131×C53C2211yescharacteristic
53C2×#104×C53C2211yescharacteristic
54C53×#416C2211yescharacteristic
55C53×#405C2211yescharacteristic
56C2×#97×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×Dic53C2311yescharacteristic
80C2×C4×Dic53C2311yescharacteristic
81C4⋊#88322—22no—
82#13133146—22no—
83#13133146—22no—
84#13133146—22no—
85#13133145—22no—
86#13133144—22no—
87#13133143—22no—
88#13133142C2×C412yes—
89#13133141C2×C412yes—
90#13133140C2×C412yes—
91#13133139C2×C412yes—
92#13133138C2×C412yes—
93#13133137C2×C412yes—
94C22⋊#88322—22no—
95#13133129—22no—
96#13133127—21no—
97#13133127—22no—
98#13133127C2×C412yes—
99#13133127C2×C412yes—
100C4×#88322—22no—
101C8×Dic53—21no—
102C8×Dic53—22no—
103C8×Dic53D412yes—
104C8×Dic53D412yes—
105#13133124—22no—
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—
121#97×C53—22no—
122C4×C8×C53—21no—
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×Dic53C22×C411yescharacteristic
176C4×Dic53C22×C411yescharacteristic
177C4×Dic53C2×D412yes—
178C4×Dic53#4112yes—
179#11719917—21no—
180#11719917—21no—
181#11719917—21no—
182#11719917—21no—
183C2×#88322—21no—
184C2×#88322—21no—
185C2×#88322—21no—
186C2×#88322—21no—
187C2×#88322—21no—
188C2×#88322—21no—
189C2×#88322C22×C412yes—
190C2×#88322C2×D412yes—
191#11719912—42no—
192C8×D53—42no—
193C23×C53—21no—
194C23×C53C2×#4111yescharacteristic
195C22×D53—21no—
196Q8×C53—21no—
197Q8×C53—21no—
198D4×C53—21no—
199D4×C53—21no—
200D4×C53—41no—
201D4×C53—41no—
202C2×C4×C53—22no—
203C2×C4×C53—21no—
204C2×C4×C53—22no—
205C2×C4×C53—42no—
206C2×C4×C53—41no—
207C2×C4×C53—41no—
208C2×C4×C53C23×C411yescharacteristic
209C2×C4×C53C22×D411yescharacteristic
210C2×C4×C53C2×#4111yescharacteristic
211C2×C4×C53C2×#4111yescharacteristic
212C2×C4×C53C2×#4111yescharacteristic
213C2×C4×C53C2×#4111yescharacteristic
214C22⋊D53—44no—
215C22⋊D53—41no—
216C22⋊D53—41no—
217C2×Dic53—22no—
218C2×Dic53—21no—
219C2×Dic53—22no—
220C2×Dic53—42no—
221C2×Dic53—42no—
222C2×Dic53C4×D412yes—
223C2×Dic53C4×D412yes—
224D212—21no—
225D212—21no—
226C4×D53—41no—
227C4×D53—41no—
228Dic106—21no—
229Dic106—21no—
230Dic106—44no—
231C8×C53—22no—
232C8×C53—41no—
233C8×C53—41no—
234C8×C53#12412yes—
235C8×C53#12412yes—
236#88322—22no—
237#88322—22no—
238#88322—41no—
239#88322—41no—
240#21663—531nomaximal, Sylow
241C22×C53—22no—
242C22×C53—42no—
243C22×C53—41no—
244C22×C53C2×C4×D411yescharacteristic
245C22×C53#52911yescharacteristic
246C22×C53#52911yescharacteristic
247C2×D53—42no—
248C2×D53—41no—
249C4×C53—42no—
250C4×C53—42no—
251C4×C53—41no—
252C4×C53C4×#4111yescharacteristic
253C4×C53C4×#4111yescharacteristic
254C4×C53C2×#12411yescharacteristic
255C4×C53#53211yescharacteristic
256Dic53—22no—
257Dic53—22no—
258Dic53—42no—
259Dic53—42no—
260Dic53—41no—
261#3135—531no—
262#2960—531no—
263#2931—531no—
264#1995—531no—
265#1963—531no—
266#1937—531no—
267#1937—531no—
268#1937—531no—
269#1937—531no—
270#1838—531no—
271#1765—531no—
272#1765—531no—
273#1765—531no—
274#1765D5311yescharacteristic, complemented
275#1726—531no—
276C2×C53—21no—
277C2×C53—21no—
278C2×C53—81no—
279C2×C53C4×#12411yescharacteristic
280C2×C53#295512yes—
281D53—81no—
282C2×#131—531no—
283C2×#131C2×D5311yescharacteristic
284#533—531no—
285#532—531no—
286#491—532no—
287#490—532no—
288#489—532no—
289#489—532no—
290#446—1062no—
291C8×Q8—1062no—
292#436—1062no—
293C8×D4—1062no—
294#424—531no—
295#424—531no—
296#423—531no—
297#423—531no—
298C2×#104—531no—
299C2×#104—531no—
300C2×#104—531no—
301C2×#104C2×D5311yescharacteristic
302#416—531no—
303#416—531no—
304#416—531no—
305#416C2×D5311yescharacteristic
306#408—531no—
307#408—531no—
308#405—531no—
309#405C2×D5311yescharacteristic
310C2×#97—531no—
311C2×#97—531no—
312C2×#97C2×D5311yescharacteristic
313C2×C4×C8—531no—
314C2×C4×C8—531no—
315C2×C4×C8—531no—
316#376—531no—
317#341—531no—
318#341—531no—
319#341—531no—
320#341C2×D5311yescharacteristic
321#339—531no—
322#339—531no—
323#339—531no—
324#339C2×D5311yescharacteristic
325C53#2166311yescharacteristic, complemented, Sylow
326C2×#41—531no—
327C2×#41C22×D5311yescharacteristic
328#131—22no—
329#131—22no—
330#131—1062no—
331#131—1062no—
332C2×M16—21no—
333C2×M16—531no—
334C2×M16—531no—
335C2×M16—1061no—
336C2×M16C22×D5311yescharacteristic
337C2×M16C22×D5311yescharacteristic
338C22×C8—21no—
339C22×C8—531no—
340C22×C8—531no—
341C22×C8—1061no—
342C22×C8C22×D5311yescharacteristic
343C22×C8C22×D5311yescharacteristic
344C4⋊Q8—532no—
345C4⋊Q8—531no—
346#127—531no—
347#125—532no—
348#124—531no—
349#124—531no—
350C22⋊Q8—1062no—
351#121—1062no—
352C4×Q8—1062no—
353C4×D4—1062no—
354#117—531no—
355#117C22×D5311yescharacteristic
356C2×C42—531no—
357C2×C42—531no—
358C2×C42C22×D5311yescharacteristic
359C4⋊C8—1062no—
360C4⋊C8—1062no—
361C4⋊C8—1062no—
362C4⋊C8—1062no—
363#104—22no—
364#104—1062no—
365#104—1062no—
366#104—1062no—
367#103—532no—
368#103—532no—
369#103—532no—
370#103C4×D5312yes—
371#102—532no—
372#102—532no—
373#102—532no—
374#102C4×D5312yes—
375C22⋊C8—1062no—
376C22⋊C8—1062no—
377#97—21no—
378#97—22no—
379#97—532no—
380#97—532no—
381#97—1062no—
382#97—1061no—
383C4×C8—21no—
384C4×C8—532no—
385C4×C8—532no—
386C4×C8—1062no—
387C4×C8—1061no—
388C4×C8—1062no—
389#95—1062no—
390#41—21no—
391#41—21no—
392#41—21no—
393#41—21no—
394#41—1061no—
395#41—1061no—
396#41—1061no—
397#41—1061no—
398C2×Q8—531no—
399C2×Q8—1062no—
400C2×Q8C2×C4×D5311yescharacteristic
401C2×D4—21no—
402C2×D4—531no—
403C2×D4—1062no—
404C2×D4—1061no—
405C2×D4C2×C4×D5311yescharacteristic
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—532no—
409C22×C4—1061no—
410C22×C4—1061no—
411C22×C4C23×D5311yescharacteristic
412M16—21no—
413M16—21no—
414M16—21no—
415M16—21no—
416M16—42no—
417M16—1061no—
418M16—1061no—
419M16—1061no—
420M16—1061no—
421M16—2122no—
422C2×C8—21no—
423C2×C8—21no—
424C2×C8—21no—
425C2×C8—21no—
426C2×C8—42no—
427C2×C8—532no—
428C2×C8—532no—
429C2×C8—1061no—
430C2×C8—1061no—
431C2×C8—1061no—
432C2×C8—1061no—
433C2×C8—1061no—
434C2×C8—1061no—
435C2×C8—2122no—
436C2×C8C2×C4×D5312yes—
437C2×C8#1171994512yes—
438C2×C8D4×D5311yescharacteristic
439C2×C8D4×D5311yescharacteristic
440C2×C8#1171994712yes—
441C4⋊C4—532no—
442C4⋊C4—1062no—
443C4⋊C4—1062no—
444C4⋊C4—1062no—
445C4⋊C4—1062no—
446C4⋊C4C2×C4×D5312yes—
447C22⋊C4—21no—
448C22⋊C4—1062no—
449C22⋊C4—1062no—
450C22⋊C4—1061no—
451C42—21no—
452C42—21no—
453C42—531no—
454C42—531no—
455C42—532no—
456C42—532no—
457C42—1061no—
458C42—1061no—
459C42#1171994512yes—
460C23—21no—
461C23—1061no—
462C23#41×D5311yescharacteristic
463Q8—21no—
464Q8—21no—
465Q8—1061no—
466Q8—1061no—
467Q8—2124no—
468D4—21no—
469D4—21no—
470D4—41no—
471D4—41no—
472D4—1061no—
473D4—1061no—
474D4—2124no—
475D4—2121no—
476D4—2121no—
477C2×C4—22no—
478C2×C4—21no—
479C2×C4—22no—
480C2×C4—42no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—532no—
484C2×C4—532no—
485C2×C4—1062no—
486C2×C4—1061no—
487C2×C4—1062no—
488C2×C4—2122no—
489C2×C4—2122no—
490C2×C4—2121no—
491C2×C4—2121no—
492C2×C4C22×C4×D5311yescharacteristic
493C2×C4C2×D4×D5311yescharacteristic
494C2×C4#41×D5311yescharacteristic
495C2×C4C2×#1171994711yescharacteristic
496C2×C4C2×#1171994511yescharacteristic
497C2×C4C2×#1171994511yescharacteristic
498C8—22no—
499C8—41no—
500C8—41no—
501C8—1062no—
502C8—1062no—
503C8—2121no—
504C8—2121no—
505C8#1313320312yes—
506C8#1313320312yes—
507C22—22no—
508C22—41no—
509C22—42no—
510C22—2121no—
511C22—2122no—
512C22C4×D4×D5311yescharacteristic
513C22#1681115111yescharacteristic
514C22#1681115311yescharacteristic
515C4—42no—
516C4—41no—
517C4—42no—
518C4—1062no—
519C4—1062no—
520C4—2122no—
521C4—2121no—
522C4—2122no—
523C4C4×#1171994711yescharacteristic
524C4C4×#1171994511yescharacteristic
525C4C2×#1313320311yescharacteristic
526C4#1681117011yescharacteristic
527C2—21no—
528C2—21no—
529C2—81no—
530C2—4241no—
531C2C4×#1313320311yescharacteristic
532C2#2028361512yes—
subgroupG/H
centerC2×C4C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1765C2
socleC22×C53C2×C4×D4
prime radical#16811169C22
cosocleC22×C4×C53C24
hypercenter#1765D53
hypocenterC53#21663
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ #1765

Complements

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

NH ≅ G/Ntype
2. #20283647C2semidirect
4. #20275694C2semidirect
6. #20275421C2semidirect
7. #20275420C2semidirect
8. #20275417C2semidirect
9. #20275416C2semidirect
10. #20274854C2semidirect
11. #20274561C2semidirect
12. #20274559C2semidirect
13. #20274557C2semidirect
14. #20274403C2semidirect
15. C53×#1765C2semidirect
19. #16810990C4semidirect
20. #16810986C4semidirect
21. #16810984C4semidirect
22. #16810981C4semidirect
274. #1765D53semidirect
325. C53#21663semidirect

Maximal subgroups

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

subgroupindex
#202850582
#202836472
#202834252
#202756942
#202755452
#202754212
#202754202
#202754172
#202754162
#202748542
#202745612
#202745592
#202745572
#202744032
C53×#17652
#2166353