Finite Group Databasebeta

← #28318617

Subgroups — #28318617

Order 13568.

998 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811666C212yesmaximal
2#20291287C211yesmaximal, characteristic, complemented
3#20291286C211yesmaximal, characteristic, complemented
4C4×#13133287C211yesmaximal, characteristic, complemented
5#20290518C212yesmaximal, complemented
6#20290496C212yesmaximal, complemented
7#20290494C212yesmaximal, complemented
8#20290479C211yesmaximal, characteristic, complemented
9#20290478C211yesmaximal, characteristic, complemented
10#20290371C211yesmaximal, characteristic, complemented
11C4×#13133238C211yesmaximal, characteristic
12C2×#13133305C2212yes—
13C2×#13133304C2212yes—
14#16811666C4116yescomplemented
15C2×#13133287C2212yes—
16C2×#13133282C2212yes—
17C2×#13133281C2212yes—
18#16811588—22no—
19#16811588C2212yes—
20#16811588C2212yes—
21#16811587—22no—
22C4×#11719936—21no—
23C4×#11719936C2211yescharacteristic
24C4×#11719936C2211yescharacteristic
25C4×#11719940—21no—
26#16811582—21no—
27#16811582C2211yescharacteristic
28C2×#13133272—21no—
29C2×#13133271C2212yes—
30C2×C8×#6157C2212yes—
31#16811559C2211yescharacteristic
32#16811499C2212yes—
33#16811493C2212yes—
34#16811493C2212yes—
35C2×#13133238C2211yescharacteristic
36C4×#11719949C2211yescharacteristic
37C4×#11719947C2211yescharacteristic
38C4×#11719925C411yescharacteristic, complemented
39#16810715C411yescharacteristic, complemented
40C4×#11719913C411yescharacteristic, complemented
41SD16×Dic53C411yescharacteristic, complemented
42#16810710C411yescharacteristic, complemented
43C4×#11719922C411yescharacteristic, complemented
44#16810703C411yescharacteristic, complemented
45#16810658C2211yescharacteristic
46#16810618C2211yescharacteristic
47#16810616C2211yescharacteristic
48C4×C8×D53C2211yescharacteristic
49C4×SD16×C53C411yescharacteristic, complemented
50#13133305—24no—
51#13133305C2×C418yescomplemented
52#13133304—24no—
53#13133304C2×C418yescomplemented
54C2×#11719936—22no—
55C2×#11719936C2312yes—
56C2×#11719936C2312yes—
57C2×#11719940—22no—
58C2×D53⋊C8—22no—
59C2×D53⋊C8C2312yes—
60C22×#88324—22no—
61C2×C4×#6157C2312yes—
62#13133287C2×C418yescomplemented
63#13133282C2×C418yescomplemented
64#13133281C2×C418yescomplemented
65#13133272—21no—
66#13133272—21no—
67#13133272—22no—
68#13133272—22no—
69#13133272D411yescharacteristic
70#13133272D411yescharacteristic
71#13133272D412yes—
72#13133271—24no—
73C4×#88324—21no—
74C4×#88324—21no—
75C4×#88324—22no—
76C4×#88324—22no—
77C8×#6157C2×C418yescomplemented
78(C2×#6157)⋊C4—21no—
79(C2×#6157)⋊C4—21no—
80C2×#11719949C2311yescharacteristic
81C2×#11719947C2311yescharacteristic
82#13133238C2×C418yes—
83C2×#11719925C2×C411yescharacteristic
84C2×#11719913C2×C411yescharacteristic
85C2×#11719922C2×C411yescharacteristic
86C2×C8×D53C2311yescharacteristic
87C2×C8×D53C2311yescharacteristic
88Q8×Dic53C2×C411yescharacteristic
89C4×Dic106C2×C411yescharacteristic
90#13133182C2×C411yescharacteristic
91C4×D212—21no—
92C4×D212C2×C411yescharacteristic
93D4×Dic53C2×C411yescharacteristic
94#13133179—21no—
95#13133179C2×C411yescharacteristic
96C4×C22⋊D53—21no—
97#13133177—21no—
98#13133175—21no—
99#13133175C2311yescharacteristic
100#13133175C2311yescharacteristic
101#13133173—21no—
102C2×C4×Dic53—21no—
103C42×D53—21no—
104C42×D53C2311yescharacteristic
105#13133150C2×C411yescharacteristic
106#13133150C2×C411yescharacteristic
107#13133149C2×C411yescharacteristic
108#13133142C2×C411yescharacteristic
109#13133141C2×C411yescharacteristic
110#13133140C2×C411yescharacteristic
111#13133139C2×C411yescharacteristic
112#13133138C2×C411yescharacteristic
113#13133137C2×C411yescharacteristic
114C4×#88322C2×C411yescharacteristic
115C8×Dic53C2×C411yescharacteristic
116C8×Dic53C2×C411yescharacteristic
117C2×SD16×C53C2×C411yescharacteristic
118C4×Q8×C53C2×C411yescharacteristic
119C4×D4×C53C2×C411yescharacteristic
120#106×C53C2×C411yescharacteristic
121#103×C53C2×C411yescharacteristic
122#102×C53C2×C411yescharacteristic
123C4×C8×C53C2×C411yescharacteristic
124C22×#6157—22no—
125C2×Q8×C53C22×C411yescharacteristic
126C2×D4×C53C22×C411yescharacteristic
127C22×C4×C53—21no—
128C2×C22⋊D53—21no—
129C22×Dic53—21no—
130C22×Dic53—21no—
131#11719949—22no—
132#11719949C22×C414yes—
133#11719947—22no—
134#11719947C22×C414yes—
135C2×D212—21no—
136C2×D212C22×C411yescharacteristic
137C2×C4×D53—21no—
138C2×C4×D53—21no—
139C2×C4×D53—21no—
140C2×C4×D53#4111yescharacteristic
141C2×C4×D53#4111yescharacteristic
142C2×C4×D53C2411yescharacteristic
143C2×Dic106C22×C411yescharacteristic
144#11719940—28no—
145C2×#88324—22no—
146C2×#88324—22no—
147C2×#88324—28no—
148C2×#88324C2×D412yes—
149C2×#88324C2×D412yes—
150C4×#6157—22no—
151C4×#6157C22×C414yes—
152#11719936—22no—
153#11719936—22no—
154#11719936—28no—
155#11719936C22×C414yes—
156#11719936C22×C414yes—
157D53⋊C8—28no—
158D53⋊C8C22×C414yes—
159D53⋊C8#4114yes—
160SD16×C53C4214yes—
161C2×C8×C53C22×C411yescharacteristic
162C2×C8×C53C22×C411yescharacteristic
163C4⋊C4×C53—21no—
164C4⋊C4×C53C22×C411yescharacteristic
165C4⋊C4×C53C22×C411yescharacteristic
166C22⋊C4×C53—21no—
167C42×C53—21no—
168C42×C53C22×C411yescharacteristic
169C22⋊Dic53—21no—
170#11719925C4214yes—
171#11719922C4214yes—
172#11719921—21no—
173#11719921—21no—
174#11719921—21no—
175#11719921—21no—
176C4⋊Dic53—21no—
177C4⋊Dic53C22×C411yescharacteristic
178C4⋊Dic53C22×C411yescharacteristic
179#11719919—21no—
180#11719919—21no—
181C4×Dic53—21no—
182C4×Dic53—21no—
183C4×Dic53—21no—
184C4×Dic53—21no—
185C4×Dic53—21no—
186C4×Dic53C22×C411yescharacteristic
187C4×Dic53C2×D411yescharacteristic
188C4×Dic53C2×D411yescharacteristic
189C2×#88322C22×C411yescharacteristic
190C2×#88322C22×C411yescharacteristic
191#11719913C4214yes—
192C8×D53—22no—
193C8×D53C22×C414yes—
194C23×C53—21no—
195C22×D53—21no—
196C22×D53C2×#4111yescharacteristic
197C2×#6157—24no—
198C2×#6157—42no—
199C2×#6157—42no—
200Q8×C53—21no—
201Q8×C53C2×C4212yes—
202D4×C53—21no—
203D4×C53C2×C4212yes—
204C2×C4×C53—21no—
205C2×C4×C53—21no—
206C2×C4×C53—21no—
207C2×C4×C53—81no—
208C2×C4×C53C4×D411yescharacteristic
209C2×C4×C53C4×D411yescharacteristic
210C2×C4×C53C23×C411yescharacteristic
211C22⋊D53—24no—
212C2×Dic53—21no—
213C2×Dic53—24no—
214C2×Dic53—21no—
215C2×Dic53—21no—
216C2×Dic53—81no—
217C2×Dic53C4×D411yescharacteristic
218C2×Dic53C4×D411yescharacteristic
219C2×Dic53C22×D411yescharacteristic
220D212—24no—
221D212—21no—
222D212C2×C4212yes—
223C4×D53—24no—
224C4×D53—41no—
225C4×D53—41no—
226C4×D53—81no—
227C4×D53—81no—
228C4×D53C23×C412yes—
229C4×D53C2×#4112yes—
230Dic106—21no—
231Dic106C2×C4212yes—
232#88324—24no—
233#88324—24no—
234#88324C4×D414yes—
235#88324C4×D414yes—
236C8×C53—21no—
237C8×C53C2×C4212yes—
238#88322—21no—
239#88322C2×C4212yes—
240#20514—531nomaximal, Sylow
241C22×C53—22no—
242C22×C53—41no—
243C22×C53—41no—
244C22×C53C2×C4×D411yescharacteristic
245C2×D53—21no—
246C2×D53—21no—
247C2×D53—22no—
248C2×D53—41no—
249C2×D53—41no—
250C2×D53C4×#4112yes—
251#6157—44no—
252C4×C53—21no—
253C4×C53—21no—
254C4×C53—22no—
255C4×C53—41no—
256C4×C53—41no—
257C4×C53C22×C4211yescharacteristic
258C4×C53C4×#4111yescharacteristic
259Dic53—21no—
260Dic53—21no—
261Dic53—22no—
262Dic53—41no—
263Dic53—41no—
264Dic53C2×C4×D412yes—
265C2×#443—532no—
266C4×#135—531no—
267#2875—531no—
268#2875—531no—
269C4×#108—531no—
270#1772—532no—
271#1765—532no—
272#1765—532no—
273#1759—531no—
274#1759—531no—
275#1726—531no—
276C2×C53—42no—
277C2×C53C42×D411yescharacteristic
278C2×C53#295711yescharacteristic
279C2×C53#295711yescharacteristic
280D53—22no—
281D53—42no—
282C2×#135—531no—
283C2×#131—532no—
284C2×#131—532no—
285C4×#41—531no—
286C4×#41—531no—
287#443—5316no—
288C4×Q16—531no—
289C4×Q16—531no—
290C4×SD16—531no—
291C4×SD16—531no—
292C4×SD16—531no—
293C4×SD16#615711yescharacteristic, complemented
294C4×D8—531no—
295C4×D8—531no—
296C2×#108—532no—
297#427—531no—
298C2×#104—532no—
299C2×#104—532no—
300#416—531no—
301#416—531no—
302#405—532no—
303#405—532no—
304#405—1062no—
305#405—1062no—
306C4×M16—531no—
307C4×M16—531no—
308C4×M16—1061no—
309C4×M16—1061no—
310C2×#97—531no—
311C2×#97—532no—
312C2×#97—1061no—
313C2×#97—1061no—
314C2×C4×C8—532no—
315C2×C4×C8—531no—
316#376—531no—
317#341—532no—
318#339—532no—
319#339—532no—
320C53#2051411yescharacteristic, complemented, Sylow
321C2×#41—531no—
322C2×#41—531no—
323#135—538no—
324C2×Q16—531no—
325C2×SD16—531no—
326C2×SD16C2×#615711yescharacteristic
327C2×D8—531no—
328#131—538no—
329#131—538no—
330#131—1064no—
331#131—1064no—
332C2×M16—532no—
333C2×M16—532no—
334C2×M16—1062no—
335C2×M16—1062no—
336C22×C8—532no—
337C22×C8—531no—
338C22×C8—531no—
339C22×C8—1062no—
340C22×C8—1062no—
341C4×Q8—531no—
342C4×Q8—531no—
343C4×Q8—531no—
344C4×Q8C2×#615711yescharacteristic
345C4×D4—531no—
346C4×D4—531no—
347C4×D4—531no—
348C4×D4—1061no—
349C4×D4—1061no—
350C4×D4—1061no—
351C4×D4—1061no—
352C4×D4C2×#615711yescharacteristic
353#117—531no—
354#117—531no—
355#117—1061no—
356#117—1061no—
357C2×C42—531no—
358C2×C42—532no—
359C2×C42—1061no—
360C2×C42—1061no—
361#108—538no—
362#107—531no—
363#107—531no—
364#106—531no—
365#106C2×#615711yescharacteristic
366#104—538no—
367#104—538no—
368#103—531no—
369#103—531no—
370#103—531no—
371#103C2×#615711yescharacteristic
372#102—531no—
373#102—531no—
374#102—531no—
375#102C2×#615711yescharacteristic
376#97—531no—
377#97—531no—
378#97—532no—
379#97—1061no—
380#97—1061no—
381#97—1064no—
382#97—1062no—
383#97—1062no—
384C4×C8—538no—
385C4×C8—531no—
386C4×C8—531no—
387C4×C8—531no—
388C4×C8—1061no—
389C4×C8—1061no—
390C4×C8—1062no—
391C4×C8—1062no—
392C4×C8C2×#615711yescharacteristic
393#95—1061no—
394#95—1061no—
395#41—534no—
396#41—534no—
397#41—1062no—
398#41—1062no—
399C2×Q8—531no—
400C2×Q8C22×#615711yescharacteristic
401C2×D4—531no—
402C2×D4—1061no—
403C2×D4—1061no—
404C2×D4C22×#615711yescharacteristic
405C22×C4—21no—
406C22×C4—531no—
407C22×C4—531no—
408C22×C4—531no—
409C22×C4—1061no—
410C22×C4—1061no—
411C22×C4—1061no—
412C22×C4—1061no—
413C22×C4—1062no—
414C22×C4—1061no—
415Q16—534no—
416SD16—534no—
417SD16C4×#615714yes—
418D8—534no—
419M16—534no—
420M16—534no—
421M16—1068no—
422M16—1068no—
423M16—1062no—
424M16—1062no—
425C2×C8—532no—
426C2×C8—532no—
427C2×C8—534no—
428C2×C8—534no—
429C2×C8—531no—
430C2×C8—534no—
431C2×C8—531no—
432C2×C8—1062no—
433C2×C8—1068no—
434C2×C8—1068no—
435C2×C8—1062no—
436C2×C8—1062no—
437C2×C8C22×#615711yescharacteristic
438C2×C8C22×#615711yescharacteristic
439C4⋊C4—21no—
440C4⋊C4—531no—
441C4⋊C4—531no—
442C4⋊C4—1061no—
443C4⋊C4—1061no—
444C4⋊C4—1061no—
445C4⋊C4C22×#615711yescharacteristic
446C4⋊C4C22×#615711yescharacteristic
447C22⋊C4—21no—
448C22⋊C4—1061no—
449C22⋊C4—1061no—
450C22⋊C4—1061no—
451C22⋊C4—1061no—
452C22⋊C4—1061no—
453C42—21no—
454C42—531no—
455C42—531no—
456C42—531no—
457C42—534no—
458C42—1061no—
459C42—1061no—
460C42—1061no—
461C42—1061no—
462C42—1062no—
463C42—1061no—
464C42C22×#615711yescharacteristic
465C23—21no—
466C23—531no—
467C23—1061no—
468Q8—21no—
469Q8—532no—
470Q8—1061no—
471Q8C2×C4×#615712yes—
472D4—21no—
473D4—532no—
474D4—1064no—
475D4—1064no—
476D4—1061no—
477D4C2×C4×#615712yes—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—81no—
482C2×C4—531no—
483C2×C4—532no—
484C2×C4—532no—
485C2×C4—531no—
486C2×C4—531no—
487C2×C4—1064no—
488C2×C4—1064no—
489C2×C4—1061no—
490C2×C4—1064no—
491C2×C4—1061no—
492C2×C4—1061no—
493C2×C4—4241no—
494C2×C4—4241no—
495C2×C4—4241no—
496C2×C4—2121no—
497C2×C4—2121no—
498C2×C4—2122no—
499C2×C4—2122no—
500C2×C4D4×#615711yescharacteristic
501C2×C4D4×#615711yescharacteristic
502C2×C4C23×#615711yescharacteristic
503C8—21no—
504C8—534no—
505C8—534no—
506C8—532no—
507C8—1064no—
508C8—1064no—
509C8—1061no—
510C8C2×C4×#615712yes—
511C22—22no—
512C22—41no—
513C22—41no—
514C22—532no—
515C22—1061no—
516C22—1061no—
517C22—1062no—
518C22—2121no—
519C22—2121no—
520C22C2×D4×#615711yescharacteristic
521C4—22no—
522C4—21no—
523C4—21no—
524C4—41no—
525C4—41no—
526C4—532no—
527C4—1062no—
528C4—1061no—
529C4—1061no—
530C4—2124no—
531C4—2121no—
532C4—2121no—
533C4C22×C4×#615711yescharacteristic
534C4#41×#615711yescharacteristic
535C2—42no—
536C2—1062no—
537C2—2122no—
538C2C4×D4×#615711yescharacteristic
539C2#2029149311yescharacteristic
540C2#2029149411yescharacteristic
subgroupG/H
centerC22C2×D4×#6157
commutatorC4×C53C22×C42
FrattiniC2×C4C23×#6157
FittingC4×SD16×C53C4
socleC22×C53C2×C4×D4
prime radicalC2×#13133238C22
cosocleC2×C4×D53C24
hypercenterC4×SD16#6157
hypocenterC53#20514
LCSG ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ C4×SD16

Complements

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

NH ≅ G/Ntype
2. #20291287C2semidirect
3. #20291286C2semidirect
4. C4×#13133287C2semidirect
5. #20290518C2semidirect
6. #20290496C2semidirect
7. #20290494C2semidirect
8. #20290479C2semidirect
9. #20290478C2semidirect
10. #20290371C2semidirect
14. #16811666C4semidirect
38. C4×#11719925C4semidirect
39. #16810715C4semidirect
40. C4×#11719913C4semidirect
41. SD16×Dic53C4semidirect
42. #16810710C4semidirect
43. C4×#11719922C4semidirect
44. #16810703C4semidirect
49. C4×SD16×C53C4semidirect
51. #13133305C2×C4semidirect
53. #13133304C2×C4semidirect
62. #13133287C2×C4semidirect
63. #13133282C2×C4semidirect
64. #13133281C2×C4semidirect
77. C8×#6157C2×C4semidirect
293. C4×SD16#6157semidirect
320. C53#20514semidirect

Maximal subgroups

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

subgroupindex
C2×#168116662
#202912872
#202912862
C4×#131332872
#202905182
#202904962
#202904942
#202904792
#202904782
#202903712
C4×#131332382
#2051453