Finite Group Databasebeta

← #27490017

Subgroups — #27490017

Order 13568.

1304 conjugacy classes of subgroups.

The subgroup lattice has 688 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810823C211yesmaximal, characteristic, complemented
2C2×#16810818C211yesmaximal, characteristic, complemented
3#20283060C211yesmaximal, characteristic, complemented
4#20275858C212yesmaximal, complemented
5#20275792C212yesmaximal, complemented
6#20275716C212yesmaximal, complemented
7#20275711C212yesmaximal, complemented
8#20274720C211yesmaximal, characteristic, complemented
9#20274715C211yesmaximal, characteristic, complemented
10#20274695C211yesmaximal, characteristic, complemented
11C2×C53×#529C211yesmaximal, characteristic, complemented
12C22×#11719924C2212yes—
13C22×#11719922C2212yes—
14C2×#13133214C2211yescharacteristic
15C2×#13133191—21no—
16C2×#13133191—21no—
17C2×#13133191C2212yes—
18#16810823—24no—
19#16810818—24no—
20C2×C4⋊#88322C2211yescharacteristic
21C2×C4⋊#88322C2211yescharacteristic
22C2×#13133141C2212yes—
23C2×#13133138C2212yes—
24C2×#13133137C2212yes—
25C2×#13133137C2212yes—
26C2×C22⋊#88322C2212yes—
27#16810525—21no—
28#16810525—21no—
29#16810525C2212yes—
30#16810493C2211yescharacteristic
31#16810385C2212yes—
32#16810385C2212yes—
33#16810378C2212yes—
34C53×#529—24no—
35C53×#529—24no—
36C2×#126×C53—21no—
37C2×#125×C53C2211yescharacteristic
38C2×#124×C53C2211yescharacteristic
39C2×#123×C53—21no—
40C2×C22⋊Q8×C53C2211yescharacteristic
41C2×#121×C53C2211yescharacteristic
42C2×C4×Q8×C53C2211yescharacteristic
43C2×C4×D4×C53—21no—
44C2×C4×D4×C53C2211yescharacteristic
45C2×#117×C53—21no—
46C22×C42×C53C2211yescharacteristic
47C22×D212—21no—
48C22×D212—21no—
49C22×D212C2312yes—
50C22×C22⋊D53—22no—
51C2×#11719924—24no—
52C2×#11719924—42no—
53C2×#11719924—42no—
54C2×#11719922—24no—
55C2×#11719922—42no—
56C2×#11719922—42no—
57C22×#88322C2312yes—
58C22×#88322C2312yes—
59#13133214—22no—
60#13133214—22no—
61#13133191—42no—
62#13133191—42no—
63#13133191—44no—
64#13133191—44no—
65C2×C4⋊Dic53—21no—
66C2×C4⋊Dic53—21no—
67C2×C4⋊Dic53C2312yes—
68C2×#11719921—22no—
69C2×#11719921—22no—
70C2×#11719921—22no—
71C4⋊#88322—22no—
72C4⋊#88322—22no—
73#13133141—42no—
74#13133138—42no—
75#13133137—24no—
76#13133137—24no—
77C22⋊#88322—42no—
78C22×Q8×C53C2311yescharacteristic
79C22×D4×C53—21no—
80C22×D4×C53C2311yescharacteristic
81C23×C4×C53—21no—
82C23×C4×C53D411yescharacteristic
83C23×C4×C53D411yescharacteristic
84C23×C4×C53C2311yescharacteristic
85#126×C53—28no—
86#125×C53—22no—
87#125×C53—22no—
88#124×C53—22no—
89#124×C53—22no—
90#123×C53—28no—
91C22⋊Q8×C53—22no—
92C22⋊Q8×C53—22no—
93#121×C53—22no—
94#121×C53—22no—
95C4×Q8×C53—22no—
96C4×Q8×C53D414yescomplemented
97C4×D4×C53—28no—
98C4×D4×C53—22no—
99C4×D4×C53D414yescomplemented
100#117×C53—28no—
101C2×C4⋊C4×C53—21no—
102C2×C4⋊C4×C53—21no—
103C2×C4⋊C4×C53—21no—
104C2×C4⋊C4×C53—21no—
105C2×C4⋊C4×C53C2311yescharacteristic
106C2×C4⋊C4×C53C2311yescharacteristic
107C2×C22⋊C4×C53—21no—
108C2×C22⋊C4×C53—21no—
109C2×C22⋊C4×C53—21no—
110C2×C22⋊C4×C53—21no—
111C2×C22⋊C4×C53—21no—
112C2×C42×C53—22no—
113C2×C42×C53—22no—
114C2×C42×C53—21no—
115C2×C42×C53D411yescharacteristic
116C2×C42×C53D411yescharacteristic
117C2×C42×C53C2311yescharacteristic
118C2×C42×C53C2311yescharacteristic
119C24×C53—21no—
120C24×C53C2×D411yescharacteristic
121C23×D53—22no—
122C2×Q8×C53—21no—
123C2×Q8×C53—21no—
124C2×Q8×C53—21no—
125C2×Q8×C53—21no—
126C2×Q8×C53—21no—
127C2×Q8×C53C2×D412yes—
128C2×D4×C53—24no—
129C2×D4×C53—21no—
130C2×D4×C53—21no—
131C2×D4×C53—21no—
132C2×D4×C53—21no—
133C2×D4×C53—21no—
134C2×D4×C53—42no—
135C2×D4×C53—42no—
136C2×D4×C53C2×D412yes—
137C22×C4×C53—22no—
138C22×C4×C53—21no—
139C22×C4×C53—21no—
140C22×C4×C53—21no—
141C22×C4×C53—22no—
142C22×C4×C53—22no—
143C22×C4×C53—24no—
144C22×C4×C53—22no—
145C22×C4×C53—22no—
146C22×C4×C53—21no—
147C22×C4×C53—21no—
148C22×C4×C53—21no—
149C22×C4×C53—21no—
150C22×C4×C53—21no—
151C22×C4×C53—21no—
152C22×C4×C53—41no—
153C22×C4×C53—42no—
154C22×C4×C53—42no—
155C22×C4×C53C2×D411yescharacteristic
156C22×C4×C53C2×D411yescharacteristic
157C22×C4×C53C2×D411yescharacteristic
158C22×C4×C53C2×D411yescharacteristic
159C22×C4×C53C2×D411yescharacteristic
160C22×C4×C53C2411yescharacteristic
161C2×C22⋊D53—44no—
162C2×C22⋊D53—44no—
163C2×C22⋊D53—44no—
164C22×Dic53—22no—
165C2×D212—22no—
166C2×D212—22no—
167C2×D212—22no—
168C2×D212—22no—
169C2×D212—22no—
170C2×D212—22no—
171C2×D212—42no—
172C2×D212—42no—
173C2×D212—42no—
174C2×D212—42no—
175C2×D212—42no—
176C2×D212—42no—
177C4⋊C4×C53—24no—
178C4⋊C4×C53—24no—
179C4⋊C4×C53—24no—
180C4⋊C4×C53—24no—
181C4⋊C4×C53—21no—
182C4⋊C4×C53—21no—
183C4⋊C4×C53C2×D412yes—
184C4⋊C4×C53C2×D412yes—
185C22⋊C4×C53—24no—
186C22⋊C4×C53—24no—
187C22⋊C4×C53—24no—
188C22⋊C4×C53—24no—
189C22⋊C4×C53—24no—
190C42×C53—22no—
191C42×C53—22no—
192C42×C53—24no—
193C42×C53—21no—
194C42×C53—21no—
195C42×C53C2×D412yes—
196C42×C53C2×D412yes—
197#11719924—48no—
198#11719922—48no—
199#11719921—44no—
200#11719921—44no—
201#11719921—44no—
202C4⋊Dic53—22no—
203C4⋊Dic53—22no—
204C4⋊Dic53—42no—
205C4⋊Dic53—42no—
206C2×#88322—22no—
207C2×#88322—22no—
208C2×#88322—22no—
209C2×#88322—22no—
210C2×#88322—22no—
211C2×#88322—22no—
212C23×C53—21no—
213C23×C53—21no—
214C23×C53—21no—
215C23×C53—21no—
216C23×C53—21no—
217C23×C53—21no—
218C23×C53—22no—
219C23×C53—41no—
220C23×C53—41no—
221C23×C53—41no—
222C23×C53—41no—
223C23×C53—41no—
224C23×C53—41no—
225C23×C53AGL(1,ℤ/8)12yes—
226C23×C53C22×D411yescharacteristic
227C22×D53—42no—
228C22×D53—42no—
229C22×D53—42no—
230C22×D53—42no—
231C22×D53—42no—
232C22×D53—42no—
233C22×D53—42no—
234Q8×C53—22no—
235Q8×C53—22no—
236Q8×C53—41no—
237Q8×C53—41no—
238D4×C53—22no—
239D4×C53—22no—
240D4×C53—48no—
241D4×C53—41no—
242D4×C53—41no—
243C2×C4×C53—21no—
244C2×C4×C53—21no—
245C2×C4×C53—22no—
246C2×C4×C53—21no—
247C2×C4×C53—21no—
248C2×C4×C53—21no—
249C2×C4×C53—22no—
250C2×C4×C53—22no—
251C2×C4×C53—21no—
252C2×C4×C53—21no—
253C2×C4×C53—21no—
254C2×C4×C53—21no—
255C2×C4×C53—21no—
256C2×C4×C53—21no—
257C2×C4×C53—44no—
258C2×C4×C53—41no—
259C2×C4×C53—41no—
260C2×C4×C53—41no—
261C2×C4×C53—41no—
262C2×C4×C53—41no—
263C2×C4×C53—41no—
264C2×C4×C53—41no—
265C2×C4×C53—41no—
266C2×C4×C53—41no—
267C2×C4×C53—41no—
268C2×C4×C53—44no—
269C2×C4×C53—48no—
270C2×C4×C53#12012yes—
271C2×C4×C53#12012yes—
272C2×C4×C53#13512yes—
273C2×C4×C53AGL(1,ℤ/8)12yes—
274C2×C4×C53C22×D411yescharacteristic
275C2×C4×C53C22×D411yescharacteristic
276C2×C4×C53H(5,𝔽2)11yescharacteristic
277C2×C4×C53H(5,𝔽2)11yescharacteristic
278C2×C4×C53H(5,𝔽2)11yescharacteristic
279C2×C4×C532−1+411yescharacteristic
280C22⋊D53—88no—
281C2×Dic53—42no—
282C2×Dic53—42no—
283C2×Dic53—42no—
284D212—42no—
285D212—42no—
286D212—42no—
287D212—42no—
288D212—84no—
289D212—84no—
290#88322—42no—
291#88322—42no—
292#29662—531nomaximal, Sylow
293C22×C53—22no—
294C22×C53—22no—
295C22×C53—22no—
296C22×C53—41no—
297C22×C53—41no—
298C22×C53—41no—
299C22×C53—41no—
300C22×C53—42no—
301C22×C53—42no—
302C22×C53—42no—
303C22×C53—81no—
304C22×C53—81no—
305C22×C53—81no—
306C22×C53—81no—
307C22×C53C2×#12011yescharacteristic
308C22×C53#53511yescharacteristic
309C22×C53#53711yescharacteristic
310C22×C53C2×#13511yescharacteristic
311C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
312C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
313C22×C53#57511yescharacteristic
314C2×D53—82no—
315C2×D53—82no—
316C2×D53—82no—
317C2×D53—82no—
318C2×D53—82no—
319C2×D53—82no—
320C2×D53—82no—
321C4×C53—22no—
322C4×C53—41no—
323C4×C53—42no—
324C4×C53—41no—
325C4×C53—42no—
326C4×C53—41no—
327C4×C53—42no—
328C4×C53#44914yes—
329C4×C53#53511yescharacteristic
330C4×C53#53511yescharacteristic
331C4×C53#53711yescharacteristic
332C4×C53#53711yescharacteristic
333Dic53—82no—
334C2×#529D5311yescharacteristic, complemented
335C2×#460—531no—
336C2×#459—531no—
337#2856—531no—
338#2025—532no—
339#2012—532no—
340#2000—532no—
341#1999—532no—
342#1803—531no—
343#1802—531no—
344#1797—531no—
345C2×C53—42no—
346C2×C53—82no—
347C2×C53#261511yescharacteristic
348C2×C53C2×#44911yescharacteristic
349C2×C53#300611yescharacteristic
350C2×C53#321211yescharacteristic
351C2×C53#321711yescharacteristic
352C2×C53#324711yescharacteristic
353C2×C53#325111yescharacteristic
354D53—162no—
355C22×SD16—532no—
356C22×D8—532no—
357C2×#127—531no—
358#529—24no—
359#529—24no—
360C2×#126—21no—
361C2×#125C2×D5311yescharacteristic
362C2×#124C2×D5311yescharacteristic
363C2×#123—21no—
364C2×C22⋊Q8C2×D5311yescharacteristic
365C2×#121—532no—
366C2×#121—1061no—
367C2×#121—1061no—
368C2×#121C2×D5311yescharacteristic
369C2×C4×Q8C2×D5311yescharacteristic
370C2×C4×D4—21no—
371C2×C4×D4C2×D5311yescharacteristic
372C2×#117—21no—
373C22×C42C2×D5311yescharacteristic
374#460—1064no—
375#459—1064no—
376C2×C4⋊C8—531no—
377C2×C4⋊C8—531no—
378C2×#103—532no—
379C2×#102—532no—
380C2×#102—532no—
381C2×#102—532no—
382C2×C22⋊C8—532no—
383#390—532no—
384#390—1061no—
385#390—1061no—
386#384—531no—
387#340—532no—
388#340—532no—
389#336—532no—
390C53#2966211yescharacteristic, complemented, Sylow
391C22×Q8C22×D5311yescharacteristic
392C22×D4—21no—
393C22×D4—532no—
394C22×D4—1062no—
395C22×D4—1061no—
396C22×D4—1061no—
397C22×D4C22×D5311yescharacteristic
398C23×C4—21no—
399C23×C4C22⋊D5311yescharacteristic
400C23×C4C22⋊D5311yescharacteristic
401C23×C4C22×D5311yescharacteristic
402C2×SD16—1064no—
403C2×SD16—2122no—
404C2×SD16—2122no—
405C2×D8—1064no—
406C2×D8—2122no—
407C2×D8—2122no—
408C22×C8—532no—
409C22×C8—532no—
410#127—1062no—
411#127—1062no—
412#126—28no—
413#125—22no—
414#125—22no—
415#124—22no—
416#124—22no—
417#123—28no—
418C22⋊Q8—22no—
419C22⋊Q8—22no—
420#121—22no—
421#121—22no—
422#121—2122no—
423#121—2122no—
424#121—2124no—
425#121—2124no—
426C4×Q8—22no—
427C4×Q8C22⋊D5314yescomplemented
428C4×D4—28no—
429C4×D4—22no—
430C4×D4C22⋊D5314yescomplemented
431#117—28no—
432C2×C4⋊C4—21no—
433C2×C4⋊C4—21no—
434C2×C4⋊C4—21no—
435C2×C4⋊C4—21no—
436C2×C4⋊C4—532no—
437C2×C4⋊C4—1061no—
438C2×C4⋊C4—1061no—
439C2×C4⋊C4C22×D5311yescharacteristic
440C2×C4⋊C4C22×D5311yescharacteristic
441C2×C22⋊C4—21no—
442C2×C22⋊C4—21no—
443C2×C22⋊C4—21no—
444C2×C22⋊C4—21no—
445C2×C22⋊C4—21no—
446C2×C22⋊C4—1062no—
447C2×C22⋊C4—1062no—
448C2×C22⋊C4—1062no—
449C2×C42—21no—
450C2×C42—22no—
451C2×C42—22no—
452C2×C42C22⋊D5311yescharacteristic
453C2×C42C22⋊D5311yescharacteristic
454C2×C42C22×D5311yescharacteristic
455C2×C42C22×D5311yescharacteristic
456C4⋊C8—1062no—
457C4⋊C8—1062no—
458#103—2122no—
459#102—1064no—
460#102—1064no—
461#102—2122no—
462C22⋊C8—2122no—
463C24—21no—
464C24—1062no—
465C24C2×C22⋊D5311yescharacteristic
466C2×Q8—21no—
467C2×Q8—21no—
468C2×Q8—21no—
469C2×Q8—21no—
470C2×Q8—21no—
471C2×Q8C2×C22⋊D5312yes—
472C2×D4—24no—
473C2×D4—21no—
474C2×D4—21no—
475C2×D4—21no—
476C2×D4—21no—
477C2×D4—21no—
478C2×D4—42no—
479C2×D4—42no—
480C2×D4—1062no—
481C2×D4—1062no—
482C2×D4—1062no—
483C2×D4—1062no—
484C2×D4—1062no—
485C2×D4—1062no—
486C2×D4—2122no—
487C2×D4—2122no—
488C2×D4—2124no—
489C2×D4—2124no—
490C2×D4—2124no—
491C2×D4—2122no—
492C2×D4—2122no—
493C2×D4—2122no—
494C2×D4—2122no—
495C2×D4C2×C22⋊D5312yes—
496C22×C4—22no—
497C22×C4—21no—
498C22×C4—21no—
499C22×C4—21no—
500C22×C4—22no—
501C22×C4—22no—
502C22×C4—24no—
503C22×C4—22no—
504C22×C4—22no—
505C22×C4—21no—
506C22×C4—21no—
507C22×C4—21no—
508C22×C4—21no—
509C22×C4—21no—
510C22×C4—21no—
511C22×C4—41no—
512C22×C4—42no—
513C22×C4—42no—
514C22×C4—1062no—
515C22×C4C2×C22⋊D5311yescharacteristic
516C22×C4C2×C22⋊D5311yescharacteristic
517C22×C4C2×C22⋊D5311yescharacteristic
518C22×C4C2×C22⋊D5311yescharacteristic
519C22×C4C2×C22⋊D5311yescharacteristic
520C22×C4C23×D5311yescharacteristic
521SD16—2128no—
522D8—2128no—
523C2×C8—1062no—
524C2×C8—1062no—
525C2×C8—1062no—
526C2×C8—1062no—
527C2×C8—1062no—
528C2×C8—1062no—
529C4⋊C4—24no—
530C4⋊C4—24no—
531C4⋊C4—24no—
532C4⋊C4—24no—
533C4⋊C4—21no—
534C4⋊C4—21no—
535C4⋊C4—1062no—
536C4⋊C4—1062no—
537C4⋊C4—2122no—
538C4⋊C4—2122no—
539C4⋊C4C2×C22⋊D5312yes—
540C4⋊C4C2×C22⋊D5312yes—
541C22⋊C4—24no—
542C22⋊C4—24no—
543C22⋊C4—24no—
544C22⋊C4—24no—
545C22⋊C4—24no—
546C22⋊C4—2124no—
547C22⋊C4—2124no—
548C22⋊C4—2124no—
549C42—22no—
550C42—22no—
551C42—24no—
552C42—21no—
553C42—21no—
554C42C2×C22⋊D5312yes—
555C42C2×C22⋊D5312yes—
556C23—21no—
557C23—21no—
558C23—21no—
559C23—21no—
560C23—21no—
561C23—21no—
562C23—22no—
563C23—41no—
564C23—41no—
565C23—41no—
566C23—41no—
567C23—41no—
568C23—41no—
569C23—2122no—
570C23—2122no—
571C23—2122no—
572C23—2122no—
573C23—2122no—
574C23—2122no—
575C23—2122no—
576C23#1313324212yes—
577C23C22×C22⋊D5311yescharacteristic
578Q8—22no—
579Q8—22no—
580Q8—41no—
581Q8—41no—
582D4—22no—
583D4—22no—
584D4—48no—
585D4—41no—
586D4—41no—
587D4—4244no—
588D4—4248no—
589D4—4244no—
590D4—2122no—
591D4—2122no—
592D4—2122no—
593D4—2122no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—44no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—44no—
616C2×C4—48no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—2122no—
622C2×C4—2122no—
623C2×C4—2122no—
624C2×C4#1313318612yes—
625C2×C4#1313318612yes—
626C2×C4#1313323712yes—
627C2×C4#1313324212yes—
628C2×C4C22×C22⋊D5311yescharacteristic
629C2×C4C22×C22⋊D5311yescharacteristic
630C2×C4#1313326311yescharacteristic
631C2×C4#1313326311yescharacteristic
632C2×C4#1313326311yescharacteristic
633C2×C4#1313326411yescharacteristic
634C8—2122no—
635C8—2122no—
636C22—22no—
637C22—22no—
638C22—22no—
639C22—41no—
640C22—41no—
641C22—41no—
642C22—41no—
643C22—42no—
644C22—42no—
645C22—42no—
646C22—81no—
647C22—81no—
648C22—81no—
649C22—81no—
650C22—4242no—
651C22—4242no—
652C22—4242no—
653C22—4242no—
654C22—4242no—
655C22—4242no—
656C22—4242no—
657C22C2×#1313318611yescharacteristic
658C22#1681118711yescharacteristic
659C22#1681120111yescharacteristic
660C22C2×#1313323711yescharacteristic
661C22C2×#1313324211yescharacteristic
662C22C2×#1313324211yescharacteristic
663C22#1681142111yescharacteristic
664C4—22no—
665C4—41no—
666C4—42no—
667C4—41no—
668C4—42no—
669C4—41no—
670C4—42no—
671C4—4242no—
672C4#1681076814yes—
673C4#1681118711yescharacteristic
674C4#1681118711yescharacteristic
675C4#1681120111yescharacteristic
676C4#1681120111yescharacteristic
677C2—42no—
678C2—82no—
679C2—8482no—
680C2#2028043911yescharacteristic
681C2C2×#1681076811yescharacteristic
682C2#2028396711yescharacteristic
683C2#2028568711yescharacteristic
684C2#2028573011yescharacteristic
685C2#2028604111yescharacteristic
686C2#2028607711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC2×C53×#529C2
socleC23×C53C22×D4
hypercenterC2×#529D53
hypocenterC53#29662
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#529

Complements

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

NH ≅ G/Ntype
1. C2×#16810823C2semidirect
2. C2×#16810818C2semidirect
3. #20283060C2semidirect
4. #20275858C2semidirect
5. #20275792C2semidirect
6. #20275716C2semidirect
7. #20275711C2semidirect
8. #20274720C2semidirect
9. #20274715C2semidirect
10. #20274695C2semidirect
11. C2×C53×#529C2semidirect
96. C4×Q8×C53D4semidirect
99. C4×D4×C53D4semidirect
334. C2×#529D53semidirect
390. C53#29662semidirect
427. C4×Q8C22⋊D53semidirect
430. C4×D4C22⋊D53semidirect

Maximal subgroups

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

subgroupindex
C2×#168108232
C2×#168108182
#202830602
#202758582
#202757922
#202757162
#202757112
#202747202
#202747152
#202746952
C2×C53×#5292
#2966253