Finite Group Databasebeta

← #27429786

Subgroups — #27429786

Order 13568.

850 conjugacy classes of subgroups.

The subgroup lattice has 674 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×SD16×Dic53C211yesmaximal, characteristic, complemented
2#20278214C211yesmaximal, characteristic, complemented
3#20278203C211yesmaximal, characteristic
4#20276136C211yesmaximal, characteristic, complemented
5#20276131C211yesmaximal, characteristic, complemented
6#20275915C211yesmaximal, characteristic, complemented
7#20275894C211yesmaximal, characteristic
8#20275571C211yesmaximal, characteristic, complemented
9#20275441C211yesmaximal, characteristic, complemented
10#20274967C211yesmaximal, characteristic, complemented
11#20274960C211yesmaximal, characteristic, complemented
12#20274876C211yesmaximal, characteristic, complemented
13#20274740C211yesmaximal, characteristic, complemented
14C8×C22⋊Dic53C211yesmaximal, characteristic, complemented
15C2×C53×#465C211yesmaximal, characteristic
16C2×Q8×Dic53C2211yescharacteristic
17C2×D4×Dic53C2211yescharacteristic
18SD16×Dic53—24no—
19C2×#13133150C2211yescharacteristic
20C2×#13133150C2211yescharacteristic
21C2×#13133149C2211yescharacteristic
22C2×#13133149C2211yescharacteristic
23C2×#13133141C2211yescharacteristic
24C2×#13133141C2211yescharacteristic
25C2×#13133138C2211yescharacteristic
26C2×#13133138C2211yescharacteristic
27C2×C22⋊#88322C2211yescharacteristic
28C2×C8×Dic53C2211yescharacteristic
29C2×C8×Dic53C2211yescharacteristic
30#16810562—21no—
31#16810551—21no—
32#16810548—21no—
33#16810537—21no—
34#16810530C2211yescharacteristic
35#16810526C2211yescharacteristic
36#16810519C2211yescharacteristic
37#16810519C2211yescharacteristic
38#16810511—21no—
39#16810504—21no—
40#16810503—21no—
41#16810488C2211yescharacteristic
42#16810488C2211yescharacteristic
43#16810484—21no—
44#16810483C2211yescharacteristic
45#16810483C2211yescharacteristic
46C4×C22⋊Dic53C2211yescharacteristic
47#16810386C2211yescharacteristic
48#16810386C2211yescharacteristic
49#16810385C2211yescharacteristic
50#16810385C2211yescharacteristic
51#16810377C2211yescharacteristic
52#16810377C2211yescharacteristic
53C22×SD16×C53C2211yescharacteristic
54C23×C8×C53C2211yescharacteristic
55C2×C22⋊Q8×C53C2211yescharacteristic
56C2×#121×C53C2211yescharacteristic
57C53×#465—24no—
58C2×#106×C53C2211yescharacteristic
59C2×#103×C53C2211yescharacteristic
60C2×#102×C53C2211yescharacteristic
61C23×Dic53—21no—
62C22×#88322C2311yescharacteristic
63C22×#88322C2311yescharacteristic
64Q8×Dic53—22no—
65Q8×Dic53—41no—
66D4×Dic53—22no—
67D4×Dic53—41no—
68C2×C4⋊Dic53—21no—
69C2×C4⋊Dic53C2311yescharacteristic
70C2×C4⋊Dic53C2311yescharacteristic
71C2×C4⋊Dic53C2311yescharacteristic
72C2×C4⋊Dic53C2311yescharacteristic
73C2×#11719919—21no—
74C2×#11719919—21no—
75C2×C22⋊Dic53—21no—
76C2×C22⋊Dic53—21no—
77C2×C22⋊Dic53—21no—
78C2×C22⋊Dic53—21no—
79C2×C22⋊Dic53—21no—
80C2×C22⋊Dic53D411yescharacteristic
81C2×C22⋊Dic53D411yescharacteristic
82C2×C4×Dic53—21no—
83C2×C4×Dic53C2311yescharacteristic
84C2×C4×Dic53C2311yescharacteristic
85#13133150—41no—
86#13133150—41no—
87#13133149—22no—
88#13133149—22no—
89#13133141—22no—
90#13133141—22no—
91#13133138—22no—
92#13133138—22no—
93C22⋊#88322—22no—
94C8×Dic53—22no—
95C8×Dic53—22no—
96#13133124—21no—
97#13133124—21no—
98#13133124—21no—
99#13133124—21no—
100#13133124—21no—
101#13133124—21no—
102#13133124—21no—
103#13133124—21no—
104#13133124—21no—
105#13133124—21no—
106#13133124D411yescharacteristic
107#13133124D411yescharacteristic
108C22×Q8×C53C2311yescharacteristic
109C22×D4×C53—21no—
110C22×D4×C53C2311yescharacteristic
111C23×C4×C53C2311yescharacteristic
112C2×SD16×C53—22no—
113C2×SD16×C53—22no—
114C2×SD16×C53D414yes—
115C22×C8×C53—22no—
116C22×C8×C53—22no—
117C22×C8×C53—22no—
118C22×C8×C53C2311yescharacteristic
119C22×C8×C53C2311yescharacteristic
120C22⋊Q8×C53—22no—
121C22⋊Q8×C53—41no—
122#121×C53—22no—
123#121×C53—41no—
124C2×C4⋊C4×C53—21no—
125C2×C4⋊C4×C53C2311yescharacteristic
126C2×C4⋊C4×C53C2311yescharacteristic
127C2×C22⋊C4×C53—21no—
128C2×C22⋊C4×C53—21no—
129#106×C53—22no—
130#103×C53—22no—
131#102×C53—22no—
132C24×C53—21no—
133C24×C53C2×D411yescharacteristic
134C2×Q8×C53—21no—
135C2×Q8×C53—21no—
136C2×Q8×C53—21no—
137C2×Q8×C53—21no—
138C2×Q8×C53—21no—
139C2×Q8×C53C2×D412yes—
140C2×D4×C53—21no—
141C2×D4×C53—21no—
142C2×D4×C53—21no—
143C2×D4×C53—21no—
144C2×D4×C53—21no—
145C2×D4×C53—42no—
146C2×D4×C53—42no—
147C2×D4×C53—81no—
148C2×D4×C53C2×D412yes—
149C22×C4×C53—21no—
150C22×C4×C53—21no—
151C22×C4×C53—21no—
152C22×C4×C53—21no—
153C22×C4×C53—21no—
154C22×C4×C53—21no—
155C22×C4×C53—21no—
156C22×C4×C53—21no—
157C22×C4×C53—21no—
158C22×C4×C53C2×D411yescharacteristic
159C22×C4×C53C2411yescharacteristic
160C22×Dic53—21no—
161C22×Dic53—21no—
162C22×Dic53—21no—
163C22×Dic53—21no—
164C22×Dic53—42no—
165C22×Dic53—42no—
166C22×Dic53—81no—
167C22×Dic53C2×D411yescharacteristic
168C22×Dic53C2×D411yescharacteristic
169C22×Dic53C2×D411yescharacteristic
170C22×Dic53C2×D411yescharacteristic
171SD16×C53—28no—
172C2×C8×C53—21no—
173C2×C8×C53—21no—
174C2×C8×C53—21no—
175C2×C8×C53—44no—
176C2×C8×C53C2×D412yes—
177C2×C8×C53#4112yes—
178C2×C8×C53#4112yes—
179C4⋊C4×C53—21no—
180C4⋊C4×C53—21no—
181C4⋊C4×C53—21no—
182C4⋊C4×C53—21no—
183C4⋊C4×C53—42no—
184C22⋊C4×C53—42no—
185C22⋊C4×C53—42no—
186C22⋊Dic53—22no—
187C22⋊Dic53—22no—
188C22⋊Dic53—42no—
189C22⋊Dic53—42no—
190C22⋊Dic53—81no—
191C22⋊Dic53—81no—
192C22⋊Dic53—81no—
193C4⋊Dic53—21no—
194C4⋊Dic53—21no—
195C4⋊Dic53—21no—
196C4⋊Dic53—21no—
197C4⋊Dic53—21no—
198C4⋊Dic53—21no—
199C4⋊Dic53—21no—
200C4⋊Dic53—21no—
201C4⋊Dic53—42no—
202#11719919—42no—
203#11719919—81no—
204C4×Dic53—21no—
205C4×Dic53—21no—
206C4×Dic53—21no—
207C4×Dic53—21no—
208C4×Dic53—42no—
209C2×#88322—21no—
210C2×#88322—21no—
211C2×#88322—21no—
212C2×#88322—21no—
213C2×#88322—21no—
214C2×#88322#4112yes—
215C23×C53—21no—
216C23×C53—21no—
217C23×C53—21no—
218C23×C53—21no—
219C23×C53—21no—
220C23×C53—21no—
221C23×C53—22no—
222C23×C53—41no—
223C23×C53—41no—
224C23×C53—41no—
225C23×C53—41no—
226C23×C53—41no—
227C23×C53—41no—
228C23×C53#13512yes—
229C23×C53C22×D411yescharacteristic
230Q8×C53—22no—
231Q8×C53—22no—
232Q8×C53—41no—
233Q8×C53—41no—
234D4×C53—22no—
235D4×C53—22no—
236D4×C53—41no—
237D4×C53—41no—
238D4×C53—84no—
239C2×C4×C53—21no—
240C2×C4×C53—21no—
241C2×C4×C53—22no—
242C2×C4×C53—21no—
243C2×C4×C53—41no—
244C2×C4×C53—41no—
245C2×C4×C53—41no—
246C2×C4×C53—41no—
247C2×C4×C53—41no—
248C2×C4×C53—41no—
249C2×C4×C53—41no—
250C2×C4×C53—41no—
251C2×C4×C53—41no—
252C2×C4×C53—41no—
253C2×C4×C53—41no—
254C2×C4×C53—41no—
255C2×C4×C53C22×D411yescharacteristic
256C2×C4×C53C2×#4111yescharacteristic
257C2×C4×C53C2×#4111yescharacteristic
258C2×C4×C53C2×#4111yescharacteristic
259C2×C4×C53H(5,𝔽2)11yescharacteristic
260C2×C4×C532−1+411yescharacteristic
261C2×Dic53—21no—
262C2×Dic53—21no—
263C2×Dic53—21no—
264C2×Dic53—21no—
265C2×Dic53—21no—
266C2×Dic53—21no—
267C2×Dic53—21no—
268C2×Dic53—21no—
269C2×Dic53—21no—
270C2×Dic53—21no—
271C2×Dic53—21no—
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—
283C2×Dic53—41no—
284C2×Dic53—84no—
285C2×Dic53#13512yes—
286C8×C53—41no—
287C8×C53#12314yes—
288#88322—22no—
289#88322—41no—
290#25031—531nomaximal, Sylow
291C22×C53—22no—
292C22×C53—22no—
293C22×C53—22no—
294C22×C53—41no—
295C22×C53—41no—
296C22×C53—41no—
297C22×C53—41no—
298C22×C53—42no—
299C22×C53—42no—
300C22×C53—42no—
301C22×C53—81no—
302C22×C53—81no—
303C22×C53—81no—
304C22×C53—81no—
305C22×C53#53211yescharacteristic
306C22×C53D4211yescharacteristic
307C22×C53#54711yescharacteristic
308C22×C53C2×#13511yescharacteristic
309C22×C53C2×#13511yescharacteristic
310C22×C53#57711yescharacteristic
311C22×C53#57711yescharacteristic
312C4×C53—41no—
313C4×C53—42no—
314C4×C53—81no—
315C4×C53—81no—
316C4×C53C2×#12311yescharacteristic
317C4×C53#54211yescharacteristic
318C4×C53#54711yescharacteristic
319C4×C53#55311yescharacteristic
320Dic53—22no—
321Dic53—41no—
322Dic53—41no—
323Dic53—41no—
324Dic53—81no—
325Dic53—81no—
326Dic53—81no—
327Dic53—81no—
328C2×#465D5311yescharacteristic
329C2×C4×SD16—531no—
330#2421—531no—
331#2420—531no—
332#2074—531no—
333#2073—531no—
334#2036—531no—
335#2033—531no—
336#1968—531no—
337#1941—531no—
338#1863—531no—
339#1862—531no—
340#1846—531no—
341#1807—531no—
342C8×C22⋊C4—531no—
343C2×C53—42no—
344C2×C53—82no—
345C2×C53#242211yescharacteristic
346C2×C53#316811yescharacteristic
347C2×C53#317411yescharacteristic
348C2×C53#328211yescharacteristic
349C2×C53#328211yescharacteristic
350C2×C53#333711yescharacteristic
351C2×C53#334011yescharacteristic
352C22×SD16C2×D5311yescharacteristic
353C23×C8C2×D5311yescharacteristic
354C2×C22⋊Q8C2×D5311yescharacteristic
355C2×#121C2×D5311yescharacteristic
356C2×C4×Q8—531no—
357C2×C4×D4—531no—
358#465—24no—
359C4×SD16—1064no—
360C2×#107—531no—
361C2×#107—531no—
362C2×#106—531no—
363C2×#106—531no—
364C2×#106C2×D5311yescharacteristic
365C2×#103—531no—
366C2×#103—531no—
367C2×#103C2×D5311yescharacteristic
368C2×#102—531no—
369C2×#102—531no—
370C2×#102C2×D5311yescharacteristic
371C2×C22⋊C8—531no—
372C2×C4×C8—531no—
373C2×C4×C8—531no—
374#400—1061no—
375#397—1061no—
376#397—1061no—
377#394—1061no—
378#391—531no—
379#390—531no—
380#389—531no—
381#389—531no—
382#388—1061no—
383#387—1061no—
384#386—1061no—
385#382—531no—
386#382—531no—
387#381—1061no—
388#380—531no—
389#380—531no—
390C4×C22⋊C4—531no—
391#340—531no—
392#340—531no—
393#340—531no—
394#340—531no—
395#336—531no—
396#336—531no—
397C53#2503111yescharacteristic, complemented, Sylow
398C22×Q8C22×D5311yescharacteristic
399C22×D4—21no—
400C22×D4C22×D5311yescharacteristic
401C23×C4—1061no—
402C23×C4C22×D5311yescharacteristic
403C2×SD16—22no—
404C2×SD16—22no—
405C2×SD16C22⋊D5314yes—
406C22×C8—22no—
407C22×C8—22no—
408C22×C8—22no—
409C22×C8—531no—
410C22×C8—531no—
411C22×C8C22×D5311yescharacteristic
412C22×C8C22×D5311yescharacteristic
413C22⋊Q8—22no—
414C22⋊Q8—41no—
415#121—22no—
416#121—41no—
417C4×Q8—1062no—
418C4×Q8—2121no—
419C4×D4—1062no—
420C4×D4—2121no—
421C2×C4⋊C4—21no—
422C2×C4⋊C4—531no—
423C2×C4⋊C4—531no—
424C2×C4⋊C4—531no—
425C2×C4⋊C4—531no—
426C2×C4⋊C4—1061no—
427C2×C4⋊C4—1061no—
428C2×C4⋊C4—1061no—
429C2×C4⋊C4C22×D5311yescharacteristic
430C2×C4⋊C4C22×D5311yescharacteristic
431C2×C22⋊C4—21no—
432C2×C22⋊C4—21no—
433C2×C22⋊C4—531no—
434C2×C22⋊C4—531no—
435C2×C22⋊C4—1061no—
436C2×C22⋊C4—1061no—
437C2×C22⋊C4—1061no—
438C2×C22⋊C4—1061no—
439C2×C22⋊C4—1061no—
440C2×C42—531no—
441C2×C42—531no—
442C2×C42—1061no—
443#107—2121no—
444#107—2121no—
445#106—22no—
446#106—1062no—
447#106—1062no—
448#103—22no—
449#103—1062no—
450#103—1062no—
451#102—22no—
452#102—1062no—
453#102—1062no—
454C22⋊C8—1062no—
455C4×C8—1062no—
456C4×C8—1062no—
457#95—531no—
458#95—531no—
459#95—1061no—
460#95—1061no—
461#95—1061no—
462#95—1061no—
463#95—1061no—
464#95—1061no—
465#95—1061no—
466#95—1061no—
467#95—1061no—
468#95—1061no—
469C24—21no—
470C24D4×D5311yescharacteristic
471C2×Q8—21no—
472C2×Q8—21no—
473C2×Q8—21no—
474C2×Q8—21no—
475C2×Q8—21no—
476C2×Q8C2×C22⋊D5312yes—
477C2×D4—21no—
478C2×D4—21no—
479C2×D4—21no—
480C2×D4—21no—
481C2×D4—21no—
482C2×D4—42no—
483C2×D4—42no—
484C2×D4—81no—
485C2×D4C2×C22⋊D5312yes—
486C22×C4—21no—
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—531no—
496C22×C4—531no—
497C22×C4—531no—
498C22×C4—531no—
499C22×C4—1061no—
500C22×C4—1061no—
501C22×C4—1061no—
502C22×C4—1061no—
503C22×C4—4241no—
504C22×C4—2122no—
505C22×C4—2122no—
506C22×C4D4×D5311yescharacteristic
507C22×C4C23×D5311yescharacteristic
508SD16—28no—
509C2×C8—21no—
510C2×C8—21no—
511C2×C8—21no—
512C2×C8—44no—
513C2×C8—532no—
514C2×C8—1061no—
515C2×C8—1061no—
516C2×C8—1061no—
517C2×C8—1061no—
518C2×C8—1061no—
519C2×C8#1171994712yes—
520C2×C8#1171994712yes—
521C2×C8C2×C22⋊D5312yes—
522C4⋊C4—21no—
523C4⋊C4—21no—
524C4⋊C4—21no—
525C4⋊C4—21no—
526C4⋊C4—42no—
527C4⋊C4—1061no—
528C4⋊C4—1061no—
529C4⋊C4—1061no—
530C4⋊C4—1061no—
531C4⋊C4—1061no—
532C4⋊C4—1061no—
533C4⋊C4—1061no—
534C4⋊C4—1061no—
535C4⋊C4—4241no—
536C4⋊C4—2122no—
537C4⋊C4—2122no—
538C22⋊C4—42no—
539C22⋊C4—42no—
540C22⋊C4—1062no—
541C22⋊C4—1062no—
542C22⋊C4—4241no—
543C22⋊C4—4241no—
544C22⋊C4—4241no—
545C22⋊C4—2122no—
546C22⋊C4—2122no—
547C42—1061no—
548C42—1061no—
549C42—1061no—
550C42—1061no—
551C42—2122no—
552C23—21no—
553C23—21no—
554C23—21no—
555C23—21no—
556C23—21no—
557C23—21no—
558C23—22no—
559C23—41no—
560C23—41no—
561C23—41no—
562C23—41no—
563C23—41no—
564C23—41no—
565C23#1313323812yes—
566C23C2×D4×D5311yescharacteristic
567Q8—22no—
568Q8—22no—
569Q8—41no—
570Q8—41no—
571D4—22no—
572D4—22no—
573D4—41no—
574D4—41no—
575D4—84no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—22no—
579C2×C4—21no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—532no—
593C2×C4—1061no—
594C2×C4—1061no—
595C2×C4—1061no—
596C2×C4—1061no—
597C2×C4—1061no—
598C2×C4—1061no—
599C2×C4—1061no—
600C2×C4—1061no—
601C2×C4—1061no—
602C2×C4—1061no—
603C2×C4—1061no—
604C2×C4—4244no—
605C2×C4—2121no—
606C2×C4—2121no—
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×C4C22×C22⋊D5311yescharacteristic
618C2×C4#41×D5311yescharacteristic
619C2×C4C2×#1171994711yescharacteristic
620C2×C4C2×#1171994711yescharacteristic
621C2×C4#1313326211yescharacteristic
622C2×C4#1313326411yescharacteristic
623C8—41no—
624C8—1062no—
625C8—2121no—
626C8#1313319814yes—
627C22—22no—
628C22—22no—
629C22—22no—
630C22—41no—
631C22—41no—
632C22—41no—
633C22—41no—
634C22—42no—
635C22—42no—
636C22—42no—
637C22—81no—
638C22—81no—
639C22—81no—
640C22—81no—
641C22#1681116811yescharacteristic
642C22D4×C22⋊D5311yescharacteristic
643C22#1681127311yescharacteristic
644C22#135×D5311yescharacteristic
645C22C2×#1313323811yescharacteristic
646C22#1681142911yescharacteristic
647C22#1681143011yescharacteristic
648C4—41no—
649C4—42no—
650C4—81no—
651C4—81no—
652C4—1062no—
653C4—4241no—
654C4—4241no—
655C4—4241no—
656C4—4241no—
657C4—2121no—
658C4—2121no—
659C4—2121no—
660C4C2×#1313319811yescharacteristic
661C4#1681124011yescharacteristic
662C4#1681127211yescharacteristic
663C4#1681132011yescharacteristic
664C2—42no—
665C2—82no—
666C2#2027822511yescharacteristic
667C2#2028527811yescharacteristic
668C2#2028533611yescharacteristic
669C2#2028641611yescharacteristic
670C2#2028642211yescharacteristic
671C2#2028710011yescharacteristic
672C2#2028712311yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC2×C53×#465C2
socleC23×C53C22×D4
prime radicalC2×#121×C53C22
hypercenterC2×#465D53
hypocenterC53#25031
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#465

Complements

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

NH ≅ G/Ntype
1. C2×SD16×Dic53C2semidirect
2. #20278214C2semidirect
4. #20276136C2semidirect
5. #20276131C2semidirect
6. #20275915C2semidirect
8. #20275571C2semidirect
9. #20275441C2semidirect
10. #20274967C2semidirect
11. #20274960C2semidirect
12. #20274876C2semidirect
13. #20274740C2semidirect
14. C8×C22⋊Dic53C2semidirect
397. C53#25031semidirect

Maximal subgroups

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

subgroupindex
C2×SD16×Dic532
#202782142
#202782032
#202761362
#202761312
#202759152
#202758942
#202755712
#202754412
#202749672
#202749602
#202748762
#202747402
C8×C22⋊Dic532
C2×C53×#4652
#2503153