Finite Group Databasebeta

← #27449687

Subgroups — #27449687

Order 13568.

850 conjugacy classes of subgroups.

The subgroup lattice has 676 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810964C211yesmaximal, characteristic, complemented
2C2×C4×#11719923C211yesmaximal, characteristic, complemented
3#20278218C211yesmaximal, characteristic, complemented
4#20276236C211yesmaximal, characteristic, complemented
5#20276032C211yesmaximal, characteristic, complemented
6#20275901C211yesmaximal, characteristic
7#20275718C211yesmaximal, characteristic
8#20275483C211yesmaximal, characteristic, complemented
9#20275440C211yesmaximal, characteristic, complemented
10#20275349C211yesmaximal, characteristic, complemented
11#20275323C211yesmaximal, characteristic, complemented
12#20274968C211yesmaximal, characteristic, complemented
13#20274962C211yesmaximal, characteristic, complemented
14#20274914C211yesmaximal, characteristic, complemented
15C53×#2363C211yesmaximal, characteristic
16C22×#11719923C2211yescharacteristic
17C2×#13133196C2211yescharacteristic
18C2×C4×Dic106C2211yescharacteristic
19#16810964—24no—
20C4×#11719923—24no—
21C2×#13133150C2211yescharacteristic
22C2×#13133150C2211yescharacteristic
23C2×#13133148C2211yescharacteristic
24C2×#13133148C2211yescharacteristic
25C2×#13133148C2211yescharacteristic
26C2×C4⋊#88322C2211yescharacteristic
27C2×#13133140C2211yescharacteristic
28C2×#13133140C2211yescharacteristic
29C2×#13133140C2211yescharacteristic
30C2×#13133138C2211yescharacteristic
31C2×#13133138C2211yescharacteristic
32C2×#13133138C2211yescharacteristic
33C2×C22⋊#88322C2211yescharacteristic
34C2×C22⋊#88322C2211yescharacteristic
35C2×C4×#88322C2211yescharacteristic
36#16810564—21no—
37#16810551—21no—
38#16810529C2211yescharacteristic
39#16810521C2211yescharacteristic
40#16810517—21no—
41#16810505—21no—
42#16810489C2211yescharacteristic
43#16810483C2211yescharacteristic
44#16810385C2211yescharacteristic
45#16810385C2211yescharacteristic
46#16810385C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810378C2211yescharacteristic
49#16810378C2211yescharacteristic
50C2×#121×C53C2211yescharacteristic
51C2×C4×D4×C53C2211yescharacteristic
52C2×#117×C53C2211yescharacteristic
53C53×#390—21no—
54C53×#390C2211yescharacteristic
55C53×#389C2211yescharacteristic
56C53×#388—21no—
57C53×#386—21no—
58C4×C4⋊C4×C53C2211yescharacteristic
59C4×C22⋊C4×C53—21no—
60C4×C22⋊C4×C53C2211yescharacteristic
61C22×Dic106C2311yescharacteristic
62C2×#11719923—22no—
63C2×#11719923—22no—
64C2×#11719923—22no—
65C22×#88322C2311yescharacteristic
66C22×#88322C2311yescharacteristic
67C22×#88322C2311yescharacteristic
68C22×#88322C2311yescharacteristic
69#13133196—22no—
70#13133196—41no—
71C4×Dic106—22no—
72C4×Dic106—41no—
73C2×C4⋊Dic53C2311yescharacteristic
74C2×C4⋊Dic53C2311yescharacteristic
75C2×C4⋊Dic53C2311yescharacteristic
76C2×#11719919—21no—
77C2×#11719919—21no—
78C2×#11719919—21no—
79C2×C22⋊Dic53—21no—
80C2×C22⋊Dic53—21no—
81C2×C4×Dic53—21no—
82#13133150—22no—
83#13133150—41no—
84#13133148—22no—
85#13133148—22no—
86#13133148—22no—
87C4⋊#88322—22no—
88#13133140—22no—
89#13133140—22no—
90#13133140D414yescomplemented
91#13133138—22no—
92#13133138—22no—
93#13133138—41no—
94C22⋊#88322—22no—
95C22⋊#88322—22no—
96C4×#88322—22no—
97#13133124—21no—
98#13133124—21no—
99#13133124—21no—
100#13133124—21no—
101#13133124—21no—
102#13133124—21no—
103C22×D4×C53—21no—
104C22×D4×C53C2311yescharacteristic
105C23×C4×C53—21no—
106C23×C4×C53C2311yescharacteristic
107#121×C53—22no—
108#121×C53—22no—
109C4×D4×C53—22no—
110C4×D4×C53—41no—
111#117×C53—22no—
112#117×C53—41no—
113C2×C4⋊C4×C53—21no—
114C2×C4⋊C4×C53C2311yescharacteristic
115C2×C4⋊C4×C53C2311yescharacteristic
116C2×C4⋊C4×C53C2311yescharacteristic
117C2×C22⋊C4×C53—21no—
118C2×C22⋊C4×C53—21no—
119C2×C22⋊C4×C53—21no—
120C2×C22⋊C4×C53—21no—
121C2×C22⋊C4×C53—21no—
122C2×C22⋊C4×C53D411yescharacteristic
123C2×C22⋊C4×C53D411yescharacteristic
124C2×C42×C53—21no—
125C2×C42×C53—21no—
126C2×C42×C53C2311yescharacteristic
127C2×C42×C53C2311yescharacteristic
128#95×C53—21no—
129#95×C53—21no—
130#95×C53D411yescharacteristic
131#95×C53D411yescharacteristic
132C24×C53—21no—
133C24×C53C2×D411yescharacteristic
134C2×D4×C53—24no—
135C2×D4×C53—21no—
136C2×D4×C53—21no—
137C2×D4×C53—21no—
138C2×D4×C53—21no—
139C2×D4×C53—21no—
140C2×D4×C53—81no—
141C2×D4×C53—81no—
142C2×D4×C53#4112yes—
143C22×C4×C53—21no—
144C22×C4×C53—21no—
145C22×C4×C53—21no—
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—42no—
153C22×C4×C53—42no—
154C22×C4×C53—81no—
155C22×C4×C53C2×D411yescharacteristic
156C22×C4×C53C2×D411yescharacteristic
157C22×C4×C53C2×D411yescharacteristic
158C22×C4×C53C2×D411yescharacteristic
159C22×C4×C53C2×D411yescharacteristic
160C22×C4×C53C2411yescharacteristic
161C22×Dic53—21no—
162C22×Dic53—21no—
163C22×Dic53—21no—
164C22×Dic53—21no—
165C2×Dic106—21no—
166C2×Dic106—21no—
167C2×Dic106—21no—
168C2×Dic106—21no—
169C2×Dic106—21no—
170C2×Dic106C2×D412yes—
171C4⋊C4×C53—21no—
172C4⋊C4×C53—21no—
173C4⋊C4×C53—21no—
174C4⋊C4×C53—21no—
175C4⋊C4×C53—81no—
176C4⋊C4×C53C2×D412yes—
177C4⋊C4×C53#4112yes—
178C22⋊C4×C53—24no—
179C22⋊C4×C53—22no—
180C22⋊C4×C53—41no—
181C22⋊C4×C53—42no—
182C22⋊C4×C53—42no—
183C22⋊C4×C53—42no—
184C22⋊C4×C53—81no—
185C42×C53—21no—
186C42×C53—21no—
187C42×C53—21no—
188C42×C53—21no—
189C42×C53—42no—
190C42×C53—81no—
191C22⋊Dic53—42no—
192C22⋊Dic53—42no—
193#11719923—28no—
194C4⋊Dic53—21no—
195C4⋊Dic53—21no—
196C4⋊Dic53—21no—
197C4⋊Dic53—21no—
198C4⋊Dic53—21no—
199C4⋊Dic53—21no—
200#11719919—42no—
201#11719919—42no—
202#11719919—42no—
203C4×Dic53—42no—
204C2×#88322—21no—
205C2×#88322—21no—
206C2×#88322—21no—
207C2×#88322—21no—
208C2×#88322—21no—
209C2×#88322—21no—
210C2×#88322—21no—
211C2×#88322—21no—
212C2×#88322—21no—
213C2×#88322—21no—
214C2×#88322C2×D412yes—
215C2×#88322#4112yes—
216C23×C53—21no—
217C23×C53—22no—
218C23×C53—21no—
219C23×C53—21no—
220C23×C53—21no—
221C23×C53—21no—
222C23×C53—21no—
223C23×C53—21no—
224C23×C53—41no—
225C23×C53—41no—
226C23×C53—41no—
227C23×C53—41no—
228C23×C53—41no—
229C23×C53—41no—
230C23×C53C22×D411yescharacteristic
231D4×C53—22no—
232D4×C53—22no—
233D4×C53—41no—
234D4×C53—41no—
235D4×C53—84no—
236C2×C4×C53—21no—
237C2×C4×C53—21no—
238C2×C4×C53—21no—
239C2×C4×C53—22no—
240C2×C4×C53—21no—
241C2×C4×C53—22no—
242C2×C4×C53—21no—
243C2×C4×C53—21no—
244C2×C4×C53—21no—
245C2×C4×C53—21no—
246C2×C4×C53—21no—
247C2×C4×C53—21no—
248C2×C4×C53—21no—
249C2×C4×C53—21no—
250C2×C4×C53—21no—
251C2×C4×C53—21no—
252C2×C4×C53—41no—
253C2×C4×C53—41no—
254C2×C4×C53—41no—
255C2×C4×C53—41no—
256C2×C4×C53—41no—
257C2×C4×C53—41no—
258C2×C4×C53—41no—
259C2×C4×C53—41no—
260C2×C4×C53—41no—
261C2×C4×C53—41no—
262C2×C4×C53—41no—
263C2×C4×C53—84no—
264C2×C4×C53#13512yes—
265C2×C4×C53C22×D411yescharacteristic
266C2×C4×C53C2×#4111yescharacteristic
267C2×C4×C53C2×#4111yescharacteristic
268C2×C4×C53C2×#4111yescharacteristic
269C2×C4×C53H(5,𝔽2)11yescharacteristic
270C2×C4×C532−1+411yescharacteristic
271C2×Dic53—22no—
272C2×Dic53—41no—
273C2×Dic53—41no—
274C2×Dic53—41no—
275C2×Dic53—41no—
276C2×Dic53—41no—
277C2×Dic53—41no—
278C2×Dic53—41no—
279C2×Dic53—41no—
280C2×Dic53—41no—
281C2×Dic53—41no—
282C2×Dic53—41no—
283Dic106—22no—
284Dic106—22no—
285Dic106—41no—
286Dic106—41no—
287#88322—22no—
288#88322—22no—
289#88322—41no—
290#88322—41no—
291#26556—531nomaximal, Sylow
292C22×C53—41no—
293C22×C53—41no—
294C22×C53—41no—
295C22×C53—42no—
296C22×C53—41no—
297C22×C53—42no—
298C22×C53—41no—
299C22×C53—42no—
300C22×C53—41no—
301C22×C53—41no—
302C22×C53—81no—
303C22×C53—81no—
304C22×C53—81no—
305C22×C53—81no—
306C22×C53#52511yescharacteristic
307C22×C53D4211yescharacteristic
308C22×C53#54711yescharacteristic
309C22×C53C2×#13511yescharacteristic
310C22×C53#57511yescharacteristic
311C22×C53#57711yescharacteristic
312C22×C53#57711yescharacteristic
313C4×C53—22no—
314C4×C53—41no—
315C4×C53—41no—
316C4×C53—42no—
317C4×C53—41no—
318C4×C53—41no—
319C4×C53—42no—
320C4×C53—81no—
321C4×C53#52511yescharacteristic
322C4×C53#53911yescharacteristic
323C4×C53#54711yescharacteristic
324C4×C53#55911yescharacteristic
325Dic53—42no—
326Dic53—81no—
327Dic53—81no—
328Dic53—81no—
329C2×#485—531no—
330C2×C4×SD16—531no—
331#2421—531no—
332#2363D5311yescharacteristic
333#2092—531no—
334#2055—531no—
335#2034—531no—
336#2000—531no—
337#1949—531no—
338#1941—531no—
339#1927—531no—
340#1923—531no—
341#1863—531no—
342#1862—531no—
343#1854—531no—
344C2×C53—81no—
345C2×C53—82no—
346C2×C53#245511yescharacteristic
347C2×C53#310211yescharacteristic
348C2×C53#312311yescharacteristic
349C2×C53#327111yescharacteristic
350C2×C53#328211yescharacteristic
351C2×C53#334011yescharacteristic
352C2×C53#335111yescharacteristic
353C22×SD16—531no—
354C2×C22⋊Q8—531no—
355C2×#121C2×D5311yescharacteristic
356C2×C4×Q8—531no—
357C2×C4×D4C2×D5311yescharacteristic
358C2×#117C2×D5311yescharacteristic
359#485—1064no—
360C4×SD16—1064no—
361C2×#107—531no—
362C2×#107—531no—
363C2×#106—531no—
364C2×#106—531no—
365C2×#106—531no—
366C2×C4⋊C8—531no—
367C2×#103—531no—
368C2×#103—531no—
369C2×#103—531no—
370C2×#102—531no—
371C2×#102—531no—
372C2×#102—531no—
373C2×C22⋊C8—531no—
374C2×C22⋊C8—531no—
375C2×C4×C8—531no—
376#400—1061no—
377#397—1061no—
378#391—531no—
379#390—21no—
380#390C2×D5311yescharacteristic
381#389—531no—
382#389C2×D5311yescharacteristic
383#388—21no—
384#388—1061no—
385#387—1061no—
386#386—21no—
387#382—531no—
388#380—531no—
389C4×C4⋊C4C2×D5311yescharacteristic
390C4×C22⋊C4—21no—
391C4×C22⋊C4C2×D5311yescharacteristic
392#340—531no—
393#340—531no—
394#340—531no—
395#340—531no—
396#336—531no—
397#336—531no—
398C53#2655611yescharacteristic, complemented, Sylow
399C22×Q8—531no—
400C22×D4—21no—
401C22×D4C22×D5311yescharacteristic
402C23×C4—21no—
403C23×C4C22×D5311yescharacteristic
404C2×SD16—1062no—
405C2×SD16—1062no—
406C2×SD16—1062no—
407C22×C8—531no—
408C22×C8—531no—
409C22×C8—531no—
410C22×C8—531no—
411C22⋊Q8—1062no—
412C22⋊Q8—2121no—
413#121—22no—
414#121—22no—
415C4×Q8—1062no—
416C4×Q8—2121no—
417C4×D4—22no—
418C4×D4—41no—
419#117—22no—
420#117—41no—
421C2×C4⋊C4—21no—
422C2×C4⋊C4—531no—
423C2×C4⋊C4—531no—
424C2×C4⋊C4—531no—
425C2×C4⋊C4—1061no—
426C2×C4⋊C4—1061no—
427C2×C4⋊C4—1061no—
428C2×C4⋊C4C22×D5311yescharacteristic
429C2×C4⋊C4C22×D5311yescharacteristic
430C2×C4⋊C4C22×D5311yescharacteristic
431C2×C22⋊C4—21no—
432C2×C22⋊C4—21no—
433C2×C22⋊C4—21no—
434C2×C22⋊C4—21no—
435C2×C22⋊C4—21no—
436C2×C22⋊C4—1061no—
437C2×C22⋊C4—1061no—
438C2×C22⋊C4C22⋊D5311yescharacteristic
439C2×C22⋊C4C22⋊D5311yescharacteristic
440C2×C42—21no—
441C2×C42—21no—
442C2×C42—1061no—
443C2×C42C22×D5311yescharacteristic
444C2×C42C22×D5311yescharacteristic
445#107—1062no—
446#107—2121no—
447#106—1062no—
448#106—1062no—
449#106—1062no—
450C4⋊C8—1062no—
451#103—534no—
452#103—1062no—
453#103—1062no—
454#102—1062no—
455#102—1062no—
456#102—2121no—
457C22⋊C8—1062no—
458C22⋊C8—1062no—
459C4×C8—1062no—
460#95—21no—
461#95—21no—
462#95—1061no—
463#95—1061no—
464#95—1061no—
465#95—1061no—
466#95—1061no—
467#95—1061no—
468#95C22⋊D5311yescharacteristic
469#95C22⋊D5311yescharacteristic
470C24—21no—
471C24C2×C22⋊D5311yescharacteristic
472C2×Q8—532no—
473C2×Q8—1061no—
474C2×Q8—1061no—
475C2×Q8—1061no—
476C2×Q8—1061no—
477C2×Q8—1061no—
478C2×D4—24no—
479C2×D4—21no—
480C2×D4—21no—
481C2×D4—21no—
482C2×D4—21no—
483C2×D4—21no—
484C2×D4—81no—
485C2×D4—81no—
486C2×D4#1171994512yes—
487C22×C4—21no—
488C22×C4—21no—
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—21no—
492C22×C4—21no—
493C22×C4—21no—
494C22×C4—21no—
495C22×C4—21no—
496C22×C4—42no—
497C22×C4—42no—
498C22×C4—81no—
499C22×C4—1061no—
500C22×C4—1061no—
501C22×C4—1061no—
502C22×C4—1061no—
503C22×C4C2×C22⋊D5311yescharacteristic
504C22×C4C2×C22⋊D5311yescharacteristic
505C22×C4C2×C22⋊D5311yescharacteristic
506C22×C4C2×C22⋊D5311yescharacteristic
507C22×C4C2×C22⋊D5311yescharacteristic
508C22×C4C23×D5311yescharacteristic
509SD16—1068no—
510C2×C8—532no—
511C2×C8—532no—
512C2×C8—1061no—
513C2×C8—1061no—
514C2×C8—1061no—
515C2×C8—1061no—
516C2×C8—1061no—
517C2×C8—1061no—
518C2×C8—1061no—
519C2×C8—1061no—
520C2×C8—1061no—
521C2×C8—1061no—
522C4⋊C4—21no—
523C4⋊C4—21no—
524C4⋊C4—21no—
525C4⋊C4—21no—
526C4⋊C4—81no—
527C4⋊C4—1061no—
528C4⋊C4—1061no—
529C4⋊C4—1061no—
530C4⋊C4—1061no—
531C4⋊C4—1061no—
532C4⋊C4—1061no—
533C4⋊C4—2122no—
534C4⋊C4—2122no—
535C4⋊C4—2122no—
536C4⋊C4#1171994512yes—
537C4⋊C4D4×D5312yes—
538C22⋊C4—24no—
539C22⋊C4—22no—
540C22⋊C4—41no—
541C22⋊C4—42no—
542C22⋊C4—42no—
543C22⋊C4—42no—
544C22⋊C4—81no—
545C22⋊C4—2122no—
546C22⋊C4—2122no—
547C42—21no—
548C42—21no—
549C42—21no—
550C42—21no—
551C42—42no—
552C42—81no—
553C42—2122no—
554C23—21no—
555C23—22no—
556C23—21no—
557C23—21no—
558C23—21no—
559C23—21no—
560C23—21no—
561C23—21no—
562C23—41no—
563C23—41no—
564C23—41no—
565C23—41no—
566C23—41no—
567C23—41no—
568C23C22×C22⋊D5311yescharacteristic
569Q8—1062no—
570Q8—1062no—
571Q8—2121no—
572Q8—2121no—
573D4—22no—
574D4—22no—
575D4—41no—
576D4—41no—
577D4—84no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—22no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—22no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—84no—
606C2×C4—1062no—
607C2×C4—2121no—
608C2×C4—2121no—
609C2×C4—2121no—
610C2×C4—2121no—
611C2×C4—2121no—
612C2×C4—2121no—
613C2×C4—2121no—
614C2×C4—2121no—
615C2×C4—2121no—
616C2×C4—2121no—
617C2×C4—2121no—
618C2×C4#1313323712yes—
619C2×C4C2×D4×D5311yescharacteristic
620C2×C4#41×D5311yescharacteristic
621C2×C4C2×#1171994511yescharacteristic
622C2×C4C2×#1171994511yescharacteristic
623C2×C4#1313326211yescharacteristic
624C2×C4#1313326411yescharacteristic
625C8—1062no—
626C8—1062no—
627C8—2121no—
628C8—2121no—
629C22—41no—
630C22—41no—
631C22—42no—
632C22—41no—
633C22—42no—
634C22—41no—
635C22—42no—
636C22—41no—
637C22—41no—
638C22—41no—
639C22—81no—
640C22—81no—
641C22—81no—
642C22—81no—
643C22#1681113411yescharacteristic
644C22D4×C22⋊D5311yescharacteristic
645C22#1681127211yescharacteristic
646C22C2×#1313323711yescharacteristic
647C22#1681142111yescharacteristic
648C22#1681143411yescharacteristic
649C22#1681143411yescharacteristic
650C4—22no—
651C4—41no—
652C4—42no—
653C4—41no—
654C4—41no—
655C4—41no—
656C4—42no—
657C4—81no—
658C4—4241no—
659C4—4241no—
660C4—4241no—
661C4—2122no—
662C4#1681113211yescharacteristic
663C4#1681121511yescharacteristic
664C4#1681127811yescharacteristic
665C4#1681135611yescharacteristic
666C2—81no—
667C2—82no—
668C2#2027859211yescharacteristic
669C2#2028473311yescharacteristic
670C2#2028496711yescharacteristic
671C2#2028629611yescharacteristic
672C2#2028641811yescharacteristic
673C2#2028712811yescharacteristic
674C2#2028726911yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2363C2
socleC23×C53C22×D4
prime radicalC2×#121×C53C22
hypercenter#2363D53
hypocenterC53#26556
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2363

Complements

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

NH ≅ G/Ntype
1. C2×#16810964C2semidirect
2. C2×C4×#11719923C2semidirect
3. #20278218C2semidirect
4. #20276236C2semidirect
5. #20276032C2semidirect
8. #20275483C2semidirect
9. #20275440C2semidirect
10. #20275349C2semidirect
11. #20275323C2semidirect
12. #20274968C2semidirect
13. #20274962C2semidirect
14. #20274914C2semidirect
90. #13133140D4semidirect
398. C53#26556semidirect

Maximal subgroups

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

subgroupindex
C2×#168109642
C2×C4×#117199232
#202782182
#202762362
#202760322
#202759012
#202757182
#202754832
#202754402
#202753492
#202753232
#202749682
#202749622
#202749142
C53×#23632
#2655653