Finite Group Databasebeta

← #27469499

Subgroups — #27469499

Order 13568.

944 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810958C211yesmaximal, characteristic, complemented
2C2×#16810947C211yesmaximal, characteristic
3C2×#16810869C211yesmaximal, characteristic
4#20279182C211yesmaximal, characteristic
5#20279164C211yesmaximal, characteristic, complemented
6#20276255C211yesmaximal, characteristic, complemented
7#20276250C211yesmaximal, characteristic, complemented
8#20276240C211yesmaximal, characteristic, complemented
9#20276237C211yesmaximal, characteristic, complemented
10#20275594C211yesmaximal, characteristic, complemented
11#20275567C211yesmaximal, characteristic, complemented
12#20274880C211yesmaximal, characteristic, complemented
13#20274873C211yesmaximal, characteristic, complemented
14#20274496C211yesmaximal, characteristic, complemented
15C2×C53×#428C211yesmaximal, characteristic, complemented
16C22×#11719913C2211yescharacteristic
17C2×#13133200—22no—
18C2×#13133196C2211yescharacteristic
19C2×#13133191C2211yescharacteristic
20#16810958—24no—
21#16810947—24no—
22#16810869—24no—
23C2×#13133151C2211yescharacteristic
24C2×#13133150C2211yescharacteristic
25C2×#13133150C2211yescharacteristic
26C2×#13133150C2211yescharacteristic
27C2×#13133148C2211yescharacteristic
28C2×#13133142C2211yescharacteristic
29C2×#13133140C2211yescharacteristic
30C2×#13133140C2211yescharacteristic
31C2×#13133139C2211yescharacteristic
32C2×#13133137C2211yescharacteristic
33C2×#13133137C2211yescharacteristic
34C2×C22⋊#88322C2211yescharacteristic
35C2×#13133127C2211yescharacteristic
36C2×C8×Dic53C2211yescharacteristic
37#16810558—22no—
38#16810551—22no—
39#16810528C2211yescharacteristic
40#16810528C2211yescharacteristic
41#16810524C2211yescharacteristic
42#16810524C2211yescharacteristic
43#16810507—22no—
44#16810492C2211yescharacteristic
45#16810492C2211yescharacteristic
46#16810492C2211yescharacteristic
47#16810488C2211yescharacteristic
48C4×C22⋊Dic53C2211yescharacteristic
49#16810377C2211yescharacteristic
50#16810377C2211yescharacteristic
51C22×M16×C53C2211yescharacteristic
52C2×#117×C53C2211yescharacteristic
53C22×C4⋊C4×C53C2211yescharacteristic
54C53×#428—24no—
55C2×#107×C53C2211yescharacteristic
56C2×#107×C53C2211yescharacteristic
57C2×#106×C53C2211yescharacteristic
58C2×#106×C53C2211yescharacteristic
59C22×Dic106C2311yescharacteristic
60C22×D212C2311yescharacteristic
61C22×C22⋊D53—21no—
62C2×#11719913—22no—
63C2×#11719913—22no—
64C2×#11719913—22no—
65C22×#88322C2311yescharacteristic
66C22×#88322C2311yescharacteristic
67#13133200—48no—
68#13133196—22no—
69#13133196—22no—
70#13133191—22no—
71#13133191—22no—
72C2×C4⋊Dic53C2311yescharacteristic
73C2×C4⋊Dic53C2311yescharacteristic
74C2×#11719919—21no—
75C2×#11719919—22no—
76C2×#11719919—22no—
77C2×#11719919—22no—
78C2×C22⋊Dic53—21no—
79C2×C22⋊Dic53D412yes—
80C2×#11719921—21no—
81C2×#11719921—22no—
82C2×#11719921—22no—
83C2×C4×Dic53C2311yescharacteristic
84C2×C4×Dic53C2311yescharacteristic
85#13133151—22no—
86#13133150—22no—
87#13133150—22no—
88#13133150—22no—
89#13133148—22no—
90#13133142—22no—
91#13133140—22no—
92#13133140—22no—
93#13133139—22no—
94#13133137—22no—
95#13133137—22no—
96C22⋊#88322D414yes—
97#13133127—22no—
98C8×Dic53—22no—
99#13133124—22no—
100#13133124—22no—
101#13133124—22no—
102#13133124D412yes—
103C23×C4×C53—21no—
104C23×C4×C53C2311yescharacteristic
105C2×M16×C53—22no—
106C2×M16×C53—22no—
107C2×M16×C53D414yes—
108C22×C8×C53C2311yescharacteristic
109C22×C8×C53C2311yescharacteristic
110#117×C53—22no—
111#117×C53—22no—
112C2×C4⋊C4×C53—21no—
113C2×C4⋊C4×C53—22no—
114C2×C4⋊C4×C53—22no—
115C2×C4⋊C4×C53C2311yescharacteristic
116C2×C4⋊C4×C53C2311yescharacteristic
117C2×C4⋊C4×C53C2311yescharacteristic
118C2×C4⋊C4×C53C2311yescharacteristic
119C2×C22⋊C4×C53—21no—
120C2×C42×C53—21no—
121#107×C53—22no—
122#107×C53—22no—
123#106×C53—22no—
124#106×C53—22no—
125C24×C53C2×D411yescharacteristic
126C23×D53—21no—
127C22×C4×C53—21no—
128C22×C4×C53—21no—
129C22×C4×C53—21no—
130C22×C4×C53—21no—
131C22×C4×C53—21no—
132C22×C4×C53—21no—
133C22×C4×C53—21no—
134C22×C4×C53—21no—
135C22×C4×C53—44no—
136C22×C4×C53—42no—
137C22×C4×C53C2×D411yescharacteristic
138C22×C4×C53C2×D412yes—
139C22×C4×C53C2×D412yes—
140C22×C4×C53C2411yescharacteristic
141C2×C22⋊D53—44no—
142C2×C22⋊D53—42no—
143C22×Dic53—21no—
144C22×Dic53—21no—
145C22×Dic53—21no—
146C22×Dic53C2×D412yes—
147C22×Dic53C2×D412yes—
148C2×D212—21no—
149C2×D212—21no—
150C2×D212—21no—
151C2×D212—21no—
152C2×D212—21no—
153C2×D212—21no—
154C2×Dic106—21no—
155C2×Dic106—21no—
156C2×Dic106—21no—
157C2×Dic106—21no—
158C2×Dic106—21no—
159C2×Dic106—21no—
160M16×C53—28no—
161C2×C8×C53—21no—
162C2×C8×C53—21no—
163C2×C8×C53C2×D412yes—
164C2×C8×C53C2×D412yes—
165C2×C8×C53#4112yes—
166C2×C8×C53#4112yes—
167C4⋊C4×C53—21no—
168C4⋊C4×C53—21no—
169C4⋊C4×C53—21no—
170C4⋊C4×C53—21no—
171C4⋊C4×C53—21no—
172C4⋊C4×C53—21no—
173C4⋊C4×C53—21no—
174C4⋊C4×C53—21no—
175C4⋊C4×C53—42no—
176C22⋊C4×C53—42no—
177C42×C53—42no—
178C22⋊Dic53—24no—
179C22⋊Dic53—42no—
180#11719921—44no—
181#11719921—42no—
182#11719921—44no—
183C4⋊Dic53—21no—
184C4⋊Dic53—21no—
185C4⋊Dic53—21no—
186C4⋊Dic53—21no—
187#11719919—44no—
188#11719919—42no—
189#11719919—44no—
190#11719919—44no—
191C4×Dic53—21no—
192C4×Dic53—21no—
193C4×Dic53—21no—
194C4×Dic53—21no—
195C2×#88322—21no—
196C2×#88322—21no—
197C2×#88322—21no—
198C2×#88322—21no—
199C2×#88322C2×D412yes—
200C2×#88322C2×D412yes—
201#11719913—44no—
202C23×C53—21no—
203C23×C53—21no—
204C23×C53—22no—
205C23×C53—22no—
206C23×C53#12712yes—
207C23×C53C22×D411yescharacteristic
208C22×D53—41no—
209C22×D53—41no—
210C22×D53—41no—
211C22×D53—42no—
212C22×D53—42no—
213C2×C4×C53—21no—
214C2×C4×C53—21no—
215C2×C4×C53—22no—
216C2×C4×C53—22no—
217C2×C4×C53—22no—
218C2×C4×C53—22no—
219C2×C4×C53—41no—
220C2×C4×C53—41no—
221C2×C4×C53—41no—
222C2×C4×C53—41no—
223C2×C4×C53—42no—
224C2×C4×C53—42no—
225C2×C4×C53—42no—
226C2×C4×C53—42no—
227C2×C4×C53—84no—
228C2×C4×C53#12712yes—
229C2×C4×C53C22×D411yescharacteristic
230C2×C4×C53C22×D411yescharacteristic
231C2×C4×C53C2×#4111yescharacteristic
232C2×C4×C53C2×#4111yescharacteristic
233C2×C4×C532−1+411yescharacteristic
234C2×C4×C532−1+411yescharacteristic
235C22⋊D53—84no—
236C2×Dic53—22no—
237C2×Dic53—22no—
238C2×Dic53—22no—
239C2×Dic53—22no—
240C2×Dic53—22no—
241C2×Dic53—41no—
242C2×Dic53—41no—
243C2×Dic53—41no—
244C2×Dic53—41no—
245C2×Dic53—41no—
246C2×Dic53—42no—
247C2×Dic53—42no—
248C2×Dic53AGL(1,ℤ/8)14yes—
249D212—41no—
250D212—41no—
251D212—41no—
252D212—41no—
253Dic106—41no—
254Dic106—41no—
255Dic106—41no—
256Dic106—41no—
257C8×C53—22no—
258C8×C53#12314yescomplemented
259#88322—22no—
260#88322—22no—
261#27944—531nomaximal, Sylow
262C22×C53—22no—
263C22×C53—22no—
264C22×C53—22no—
265C22×C53—42no—
266C22×C53—42no—
267C22×C53C2×#12711yescharacteristic
268C22×C53#54711yescharacteristic
269C22×C53#54711yescharacteristic
270C22×C53C2×AGL(1,ℤ/8)12yes—
271C22×C53#57812yes—
272C2×D53—82no—
273C2×D53—82no—
274C2×D53—81no—
275C2×D53—81no—
276C2×D53—81no—
277C4×C53—22no—
278C4×C53—81no—
279C4×C53—81no—
280C4×C53—81no—
281C4×C53—81no—
282C4×C53C2×#12311yescharacteristic
283C4×C53#53611yescharacteristic
284C4×C53#54711yescharacteristic
285C4×C53#54711yescharacteristic
286Dic53—24no—
287Dic53—42no—
288Dic53—81no—
289Dic53—81no—
290Dic53—81no—
291C2×#484—531no—
292C2×#482—531no—
293C2×#468—531no—
294C2×#428D5311yescharacteristic, complemented
295#2501—531no—
296#2499—531no—
297#2096—531no—
298#2095—531no—
299#2093—531no—
300#2092—531no—
301#1973—531no—
302#1968—531no—
303#1847—531no—
304#1845—531no—
305#1753—531no—
306C2×C53—42no—
307C2×C53#250111yescharacteristic
308C2×C53#315111yescharacteristic
309C2×C53#315611yescharacteristic
310C2×C53#334312yes—
311C2×C53#335312yes—
312D53—161no—
313C22×SD16—531no—
314C22×M16C2×D5311yescharacteristic
315C2×#123—1062no—
316C2×C22⋊Q8—531no—
317C2×#121—531no—
318C2×#117C2×D5311yescharacteristic
319C22×C4⋊C4C2×D5311yescharacteristic
320#484—1064no—
321#482—1064no—
322#468—1064no—
323#428—24no—
324C2×#107—531no—
325C2×#107—531no—
326C2×#107—531no—
327C2×#107—531no—
328C2×#107C2×D5311yescharacteristic
329C2×#107C2×D5311yescharacteristic
330C2×#106—531no—
331C2×#106C2×D5311yescharacteristic
332C2×#106C2×D5311yescharacteristic
333C2×#103—531no—
334C2×#103—531no—
335C2×#103—531no—
336C2×#102—531no—
337C2×#102—531no—
338C2×#102—531no—
339C2×C22⋊C8—531no—
340C2×#97—531no—
341C2×C4×C8—531no—
342#399—1062no—
343#397—1062no—
344#391—531no—
345#391—531no—
346#390—531no—
347#390—531no—
348#387—1062no—
349#384—531no—
350#384—531no—
351#384—531no—
352#382—531no—
353C4×C22⋊C4—531no—
354#336—531no—
355#336—531no—
356C53#2794411yescharacteristic, complemented, Sylow
357C22×Q8—531no—
358C22×D4—531no—
359C22×D4—1061no—
360C23×C4—21no—
361C23×C4C22×D5311yescharacteristic
362C2×SD16—1062no—
363C2×SD16—1062no—
364C2×SD16—1062no—
365C2×M16—22no—
366C2×M16—22no—
367C2×M16C22⋊D5314yes—
368C22×C8—531no—
369C22×C8—531no—
370C22×C8C22×D5311yescharacteristic
371C22×C8C22×D5311yescharacteristic
372#123—2128no—
373C22⋊Q8—1062no—
374C22⋊Q8—1062no—
375#121—1062no—
376#121—1062no—
377#117—22no—
378#117—22no—
379C2×C4⋊C4—22no—
380C2×C4⋊C4—22no—
381C2×C4⋊C4—21no—
382C2×C4⋊C4—531no—
383C2×C4⋊C4—531no—
384C2×C4⋊C4—1062no—
385C2×C4⋊C4—1062no—
386C2×C4⋊C4—1062no—
387C2×C4⋊C4—1061no—
388C2×C4⋊C4C22×D5311yescharacteristic
389C2×C4⋊C4C22×D5311yescharacteristic
390C2×C4⋊C4C22×D5311yescharacteristic
391C2×C4⋊C4C22×D5311yescharacteristic
392C2×C22⋊C4—21no—
393C2×C22⋊C4—532no—
394C2×C22⋊C4—1062no—
395C2×C22⋊C4—1062no—
396C2×C22⋊C4—1061no—
397C2×C22⋊C4—1061no—
398C2×C42—21no—
399C2×C42—531no—
400C2×C42—531no—
401#107—22no—
402#107—22no—
403#107—1062no—
404#107—1062no—
405#107—1062no—
406#107—1062no—
407#106—22no—
408#106—22no—
409#106—1062no—
410#103—1062no—
411#103—1062no—
412#103—1062no—
413#102—1062no—
414#102—1062no—
415#102—1062no—
416C22⋊C8—534no—
417#97—1062no—
418C4×C8—1062no—
419#95—532no—
420#95—1062no—
421#95—1062no—
422#95—1062no—
423C24—1061no—
424C24D4×D5311yescharacteristic
425C2×Q8—1061no—
426C2×Q8—1061no—
427C2×Q8—1061no—
428C2×Q8—1061no—
429C2×Q8—1061no—
430C2×Q8—1061no—
431C2×D4—1061no—
432C2×D4—1061no—
433C2×D4—1061no—
434C2×D4—1061no—
435C2×D4—1061no—
436C2×D4—1061no—
437C2×D4—2124no—
438C2×D4—2122no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—44no—
448C22×C4—42no—
449C22×C4—532no—
450C22×C4—532no—
451C22×C4—1061no—
452C22×C4—1061no—
453C22×C4—1061no—
454C22×C4D4×D5311yescharacteristic
455C22×C4D4×D5312yes—
456C22×C4C2×C22⋊D5312yes—
457C22×C4C23×D5311yescharacteristic
458SD16—2124no—
459M16—28no—
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—532no—
463C2×C8—532no—
464C2×C8—1061no—
465C2×C8—1061no—
466C2×C8—1061no—
467C2×C8—1061no—
468C2×C8#1171994512yes—
469C2×C8#1171994512yes—
470C2×C8C2×C22⋊D5312yes—
471C2×C8C2×C22⋊D5312yes—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—42no—
481C4⋊C4—1061no—
482C4⋊C4—1061no—
483C4⋊C4—1061no—
484C4⋊C4—1061no—
485C4⋊C4—2124no—
486C4⋊C4—2124no—
487C4⋊C4—2124no—
488C4⋊C4—2122no—
489C22⋊C4—42no—
490C22⋊C4—1064no—
491C22⋊C4—2124no—
492C22⋊C4—2124no—
493C22⋊C4—2122no—
494C22⋊C4—2122no—
495C42—42no—
496C42—1061no—
497C42—1061no—
498C42—1061no—
499C42—1061no—
500C23—21no—
501C23—21no—
502C23—22no—
503C23—22no—
504C23—2121no—
505C23—2121no—
506C23—2122no—
507C23—2121no—
508C23—2122no—
509C23#1313321312yes—
510C23C2×D4×D5311yescharacteristic
511Q8—2121no—
512Q8—2121no—
513Q8—2121no—
514Q8—2121no—
515D4—4244no—
516D4—2121no—
517D4—2121no—
518D4—2121no—
519D4—2121no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—42no—
534C2×C4—84no—
535C2×C4—534no—
536C2×C4—1062no—
537C2×C4—1062no—
538C2×C4—1062no—
539C2×C4—1062no—
540C2×C4—1062no—
541C2×C4—2121no—
542C2×C4—2121no—
543C2×C4—2122no—
544C2×C4—2122no—
545C2×C4—2121no—
546C2×C4—2121no—
547C2×C4—2121no—
548C2×C4#1313321312yes—
549C2×C4C2×D4×D5311yescharacteristic
550C2×C4C22×C22⋊D5311yescharacteristic
551C2×C4C2×#1171994511yescharacteristic
552C2×C4C2×#1171994511yescharacteristic
553C2×C4#1313326511yescharacteristic
554C2×C4#1313326511yescharacteristic
555C8—22no—
556C8—1062no—
557C8—1062no—
558C8#1313320014yescomplemented
559C22—22no—
560C22—22no—
561C22—22no—
562C22—42no—
563C22—42no—
564C22—4241no—
565C22—4242no—
566C22—4241no—
567C22—4242no—
568C22—4241no—
569C22C2×#1313321311yescharacteristic
570C22#1681127411yescharacteristic
571C22#1681127411yescharacteristic
572C22AGL(1,ℤ/8)×D5312yes—
573C22#1681143912yes—
574C4—22no—
575C4—81no—
576C4—81no—
577C4—81no—
578C4—81no—
579C4—1064no—
580C4—4241no—
581C4—4241no—
582C4—4241no—
583C4—2122no—
584C4C2×#1313320011yescharacteristic
585C4#1681119711yescharacteristic
586C4#1681127411yescharacteristic
587C4#1681127411yescharacteristic
588C2—42no—
589C2—8481no—
590C2#2027918211yescharacteristic
591C2#2028516311yescharacteristic
592C2#2028520811yescharacteristic
593C2#2028715612yes—
594C2#2028729512yes—
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC2×C53×#428C2
socleC23×C53C22×D4
prime radicalC2×#13133191C22
hypercenterC2×#428D53
hypocenterC53#27944
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#428

Complements

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

NH ≅ G/Ntype
1. C2×#16810958C2semidirect
5. #20279164C2semidirect
6. #20276255C2semidirect
7. #20276250C2semidirect
8. #20276240C2semidirect
9. #20276237C2semidirect
10. #20275594C2semidirect
11. #20275567C2semidirect
12. #20274880C2semidirect
13. #20274873C2semidirect
14. #20274496C2semidirect
15. C2×C53×#428C2semidirect
258. C8×C53#123semidirect
294. C2×#428D53semidirect
356. C53#27944semidirect
558. C8#13133200semidirect

Maximal subgroups

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

subgroupindex
C2×#168109582
C2×#168109472
C2×#168108692
#202791822
#202791642
#202762552
#202762502
#202762402
#202762372
#202755942
#202755672
#202748802
#202748732
#202744962
C2×C53×#4282
#2794453