Finite Group Databasebeta

← #27449755

Subgroups — #27449755

Order 13568.

794 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810961C211yesmaximal, characteristic
2C2×Q16×Dic53C211yesmaximal, characteristic, complemented
3#20278214C211yesmaximal, characteristic
4#20277746C211yesmaximal, characteristic
5#20276265C211yesmaximal, characteristic, complemented
6#20276035C211yesmaximal, characteristic, complemented
7#20275915C211yesmaximal, characteristic
8#20275486C211yesmaximal, characteristic, complemented
9#20275447C211yesmaximal, characteristic, complemented
10#20275359C211yesmaximal, characteristic, complemented
11#20275324C211yesmaximal, characteristic
12#20274985C211yesmaximal, characteristic, complemented
13#20274981C211yesmaximal, characteristic, complemented
14#20274916C211yesmaximal, characteristic
15C53×#2001C211yesmaximal, characteristic
16C2×Q8×Dic53C2211yescharacteristic
17C2×Q8×Dic53C2211yescharacteristic
18C2×#13133176C2211yescharacteristic
19#16810961—24no—
20Q16×Dic53—24no—
21C2×#13133151C2211yescharacteristic
22C2×#13133150C2211yescharacteristic
23C2×#13133150C2211yescharacteristic
24C2×#13133149C2211yescharacteristic
25C2×#13133148C2211yescharacteristic
26C2×#13133146C2211yescharacteristic
27C2×#13133141C2211yescharacteristic
28C2×#13133141C2211yescharacteristic
29C2×#13133141C2211yescharacteristic
30C2×#13133141C2211yescharacteristic
31C2×C22⋊#88322C2211yescharacteristic
32C2×C8×Dic53C2211yescharacteristic
33#16810562—21no—
34#16810548—21no—
35#16810530C2211yescharacteristic
36#16810530C2211yescharacteristic
37#16810522C2211yescharacteristic
38#16810519C2211yescharacteristic
39#16810516—21no—
40#16810516—21no—
41#16810511—21no—
42#16810509—21no—
43#16810504—21no—
44#16810488C2211yescharacteristic
45#16810483C2211yescharacteristic
46C4⋊C4×Dic53—21no—
47C4⋊C4×Dic53C2211yescharacteristic
48C4×C22⋊Dic53C2211yescharacteristic
49#16810386C2211yescharacteristic
50#16810386C2211yescharacteristic
51#16810385C2211yescharacteristic
52#16810377C2211yescharacteristic
53#16810377C2211yescharacteristic
54C22×Q16×C53C2211yescharacteristic
55C2×C22⋊Q8×C53C2211yescharacteristic
56C2×C22⋊Q8×C53C2211yescharacteristic
57C2×#103×C53C2211yescharacteristic
58C2×#103×C53C2211yescharacteristic
59C2×C22⋊C8×C53C2211yescharacteristic
60C53×#340C2211yescharacteristic
61C22×#88322C2311yescharacteristic
62C22×#88322C2311yescharacteristic
63Q8×Dic53—22no—
64Q8×Dic53—22no—
65Q8×Dic53—41no—
66Q8×Dic53—41no—
67#13133176—22no—
68#13133176—41no—
69C2×C4⋊Dic53—21no—
70C2×C4⋊Dic53—21no—
71C2×C4⋊Dic53—21no—
72C2×C4⋊Dic53C2311yescharacteristic
73C2×C4⋊Dic53C2311yescharacteristic
74C2×C4⋊Dic53C2311yescharacteristic
75C2×C4⋊Dic53C2311yescharacteristic
76C2×#11719919—21no—
77C2×C22⋊Dic53—21no—
78C2×C22⋊Dic53—21no—
79C2×C22⋊Dic53D411yescharacteristic
80C2×C22⋊Dic53D411yescharacteristic
81C2×C4×Dic53—21no—
82C2×C4×Dic53—21no—
83C2×C4×Dic53—21no—
84C2×C4×Dic53—21no—
85C2×C4×Dic53C2311yescharacteristic
86C2×C4×Dic53C2311yescharacteristic
87#13133151—22no—
88#13133150—22no—
89#13133150—22no—
90#13133149—41no—
91#13133148—22no—
92#13133146—22no—
93#13133141—22no—
94#13133141—22no—
95#13133141—22no—
96#13133141D414yes—
97C22⋊#88322—22no—
98C8×Dic53—22no—
99#13133124—21no—
100#13133124—21no—
101#13133124—21no—
102#13133124—21no—
103#13133124—21no—
104#13133124—21no—
105#13133124—21no—
106#13133124—21no—
107#13133124—21no—
108#13133124—21no—
109#13133124D411yescharacteristic
110#13133124D411yescharacteristic
111C22×Q8×C53C2311yescharacteristic
112C22×Q8×C53C2311yescharacteristic
113C23×C4×C53C2311yescharacteristic
114C2×Q16×C53—22no—
115C2×Q16×C53—22no—
116C2×Q16×C53—22no—
117C22×C8×C53C2311yescharacteristic
118C22×C8×C53C2311yescharacteristic
119C22⋊Q8×C53—22no—
120C22⋊Q8×C53—22no—
121C22⋊Q8×C53—22no—
122C22⋊Q8×C53—41no—
123C2×C4⋊C4×C53—21no—
124C2×C4⋊C4×C53—21no—
125C2×C4⋊C4×C53C2311yescharacteristic
126C2×C4⋊C4×C53C2311yescharacteristic
127C2×C22⋊C4×C53—21no—
128C2×C22⋊C4×C53—21no—
129#103×C53—22no—
130#103×C53—41no—
131C22⋊C8×C53—22no—
132C24×C53C2×D411yescharacteristic
133C2×Q8×C53—21no—
134C2×Q8×C53—21no—
135C2×Q8×C53—21no—
136C2×Q8×C53—21no—
137C2×Q8×C53—21no—
138C2×Q8×C53—21no—
139C2×Q8×C53—21no—
140C2×Q8×C53—21no—
141C2×Q8×C53—21no—
142C2×Q8×C53—21no—
143C2×Q8×C53C2×D412yes—
144C2×Q8×C53#4112yes—
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—21no—
153C22×C4×C53—21no—
154C22×C4×C53—21no—
155C22×C4×C53C2×D411yescharacteristic
156C22×C4×C53C2411yescharacteristic
157C22×Dic53—21no—
158C22×Dic53—21no—
159C22×Dic53—21no—
160C22×Dic53—21no—
161C22×Dic53C2×D411yescharacteristic
162C22×Dic53C2×D411yescharacteristic
163C22×Dic53C2×D411yescharacteristic
164C22×Dic53C2×D411yescharacteristic
165Q16×C53—28no—
166C2×C8×C53—21no—
167C2×C8×C53—21no—
168C2×C8×C53—21no—
169C2×C8×C53—21no—
170C2×C8×C53—21no—
171C2×C8×C53#4112yes—
172C4⋊C4×C53—24no—
173C4⋊C4×C53—21no—
174C4⋊C4×C53—21no—
175C4⋊C4×C53—21no—
176C4⋊C4×C53—42no—
177C4⋊C4×C53#4112yes—
178C22⋊C4×C53—24no—
179C22⋊C4×C53—42no—
180C22⋊Dic53—22no—
181C22⋊Dic53—41no—
182C22⋊Dic53—42no—
183C22⋊Dic53—42no—
184C4⋊Dic53—21no—
185C4⋊Dic53—21no—
186C4⋊Dic53—21no—
187C4⋊Dic53—21no—
188C4⋊Dic53—21no—
189C4⋊Dic53—21no—
190C4⋊Dic53—21no—
191C4⋊Dic53—42no—
192C4⋊Dic53—42no—
193C4⋊Dic53—42no—
194C4⋊Dic53C2×D412yes—
195#11719919—42no—
196C4×Dic53—21no—
197C4×Dic53—21no—
198C4×Dic53—21no—
199C4×Dic53—21no—
200C4×Dic53—42no—
201C4×Dic53—42no—
202C4×Dic53—42no—
203C4×Dic53—42no—
204C2×#88322—21no—
205C2×#88322—21no—
206C2×#88322—21no—
207C2×#88322—21no—
208C2×#88322—21no—
209C2×#88322C2×D412yes—
210C23×C53—21no—
211C23×C53—21no—
212C23×C53—21no—
213C23×C53—21no—
214C23×C53—21no—
215C23×C53—21no—
216C23×C53—21no—
217C23×C53C22×D411yescharacteristic
218Q8×C53—22no—
219Q8×C53—22no—
220Q8×C53—22no—
221Q8×C53—22no—
222Q8×C53—41no—
223Q8×C53—41no—
224Q8×C53—41no—
225Q8×C53—41no—
226C2×C4×C53—21no—
227C2×C4×C53—22no—
228C2×C4×C53—22no—
229C2×C4×C53—21no—
230C2×C4×C53—22no—
231C2×C4×C53—21no—
232C2×C4×C53—41no—
233C2×C4×C53—41no—
234C2×C4×C53—41no—
235C2×C4×C53—41no—
236C2×C4×C53—41no—
237C2×C4×C53—41no—
238C2×C4×C53—41no—
239C2×C4×C53—41no—
240C2×C4×C53—41no—
241C2×C4×C53—41no—
242C2×C4×C53—41no—
243C2×C4×C53—41no—
244C2×C4×C53—41no—
245C2×C4×C53C22×D411yescharacteristic
246C2×C4×C53C2×#4111yescharacteristic
247C2×C4×C53C2×#4111yescharacteristic
248C2×C4×C53C2×#4111yescharacteristic
249C2×C4×C53H(5,𝔽2)11yescharacteristic
250C2×C4×C532−1+411yescharacteristic
251C2×Dic53—21no—
252C2×Dic53—21no—
253C2×Dic53—22no—
254C2×Dic53—21no—
255C2×Dic53—21no—
256C2×Dic53—21no—
257C2×Dic53—21no—
258C2×Dic53—21no—
259C2×Dic53—21no—
260C2×Dic53—21no—
261C2×Dic53—21no—
262C2×Dic53—21no—
263C2×Dic53—41no—
264C2×Dic53—41no—
265C2×Dic53—41no—
266C2×Dic53—41no—
267C2×Dic53—41no—
268C2×Dic53—41no—
269C2×Dic53—41no—
270C2×Dic53—41no—
271C2×Dic53—41no—
272C2×Dic53—41no—
273C2×Dic53—41no—
274C2×Dic53#13512yes—
275C8×C53—22no—
276C8×C53—41no—
277#88322—22no—
278#88322—41no—
279#26561—531nomaximal, Sylow
280C22×C53—41no—
281C22×C53—41no—
282C22×C53—41no—
283C22×C53—41no—
284C22×C53—41no—
285C22×C53—41no—
286C22×C53—41no—
287C22×C53#52511yescharacteristic
288C22×C53D4211yescharacteristic
289C22×C53#54711yescharacteristic
290C22×C53C2×#13511yescharacteristic
291C22×C53#57511yescharacteristic
292C22×C53#57811yescharacteristic
293C22×C53#57811yescharacteristic
294C4×C53—42no—
295C4×C53—41no—
296C4×C53—42no—
297C4×C53—42no—
298C4×C53—81no—
299C4×C53#52511yescharacteristic
300C4×C53#53911yescharacteristic
301C4×C53#54711yescharacteristic
302C4×C53#55911yescharacteristic
303Dic53—22no—
304Dic53—41no—
305Dic53—41no—
306Dic53—41no—
307Dic53—42no—
308Dic53—81no—
309Dic53—81no—
310Dic53—81no—
311C2×#485—531no—
312C2×C4×Q16—531no—
313#2421—531no—
314#2362—531no—
315#2097—531no—
316#2056—531no—
317#2036—531no—
318#2001D5311yescharacteristic
319#1949—531no—
320#1942—531no—
321#1928—531no—
322#1923—531no—
323#1865—531no—
324#1865—531no—
325#1854—531no—
326C2×C53—81no—
327C2×C53#245511yescharacteristic
328C2×C53#310211yescharacteristic
329C2×C53#312711yescharacteristic
330C2×C53#326911yescharacteristic
331C2×C53#328511yescharacteristic
332C2×C53#334211yescharacteristic
333C2×C53#335211yescharacteristic
334C22×Q16C2×D5311yescharacteristic
335C2×C22⋊Q8C2×D5311yescharacteristic
336C2×C22⋊Q8C2×D5311yescharacteristic
337C2×C4×Q8—531no—
338C2×C4×Q8—531no—
339C2×#117—531no—
340#485—1064no—
341C4×Q16—1064no—
342C2×#107—531no—
343C2×#107—531no—
344C2×#107—531no—
345C2×#106—531no—
346C2×#106—531no—
347C2×C4⋊C8—531no—
348C2×#103—531no—
349C2×#103—531no—
350C2×#103—531no—
351C2×#103—531no—
352C2×#103C2×D5311yescharacteristic
353C2×#103C2×D5311yescharacteristic
354C2×C22⋊C8—531no—
355C2×C22⋊C8C2×D5311yescharacteristic
356C2×C4×C8—531no—
357#400—1061no—
358#397—1061no—
359#391—531no—
360#391—531no—
361#389—531no—
362#389—531no—
363#388—1061no—
364#388—1061no—
365#388—1061no—
366#387—1061no—
367#387—1061no—
368#382—531no—
369#380—531no—
370C4×C4⋊C4—531no—
371C4×C4⋊C4—1061no—
372C4×C22⋊C4—531no—
373#340—531no—
374#340—531no—
375#340—531no—
376#340C2×D5311yescharacteristic
377#336—531no—
378#336—531no—
379C53#2656111yescharacteristic, complemented, Sylow
380C22×Q8C22×D5311yescharacteristic
381C22×Q8C22×D5311yescharacteristic
382C23×C4C22×D5311yescharacteristic
383C2×Q16—22no—
384C2×Q16—22no—
385C2×Q16—22no—
386C22×C8—531no—
387C22×C8—531no—
388C22×C8C22×D5311yescharacteristic
389C22×C8C22×D5311yescharacteristic
390C22⋊Q8—22no—
391C22⋊Q8—22no—
392C22⋊Q8—22no—
393C22⋊Q8—41no—
394C4×Q8—1062no—
395C4×Q8—1062no—
396C4×Q8—2121no—
397C4×Q8—2121no—
398#117—1062no—
399#117—2121no—
400C2×C4⋊C4—21no—
401C2×C4⋊C4—21no—
402C2×C4⋊C4—531no—
403C2×C4⋊C4—531no—
404C2×C4⋊C4—531no—
405C2×C4⋊C4—531no—
406C2×C4⋊C4—1061no—
407C2×C4⋊C4—1061no—
408C2×C4⋊C4—1061no—
409C2×C4⋊C4—1061no—
410C2×C4⋊C4C22×D5311yescharacteristic
411C2×C4⋊C4C22×D5311yescharacteristic
412C2×C22⋊C4—21no—
413C2×C22⋊C4—21no—
414C2×C22⋊C4—531no—
415C2×C22⋊C4—531no—
416C2×C22⋊C4—1061no—
417C2×C22⋊C4—1061no—
418C2×C42—531no—
419C2×C42—531no—
420C2×C42—1061no—
421C2×C42—1061no—
422C2×C42—1061no—
423C2×C42—1061no—
424#107—1062no—
425#107—1062no—
426#107—1062no—
427#106—1062no—
428#106—2121no—
429C4⋊C8—1062no—
430#103—22no—
431#103—41no—
432#103—534no—
433#103—1062no—
434#103—1062no—
435#103—1062no—
436C22⋊C8—22no—
437C22⋊C8—1062no—
438C4×C8—1062no—
439#95—531no—
440#95—531no—
441#95—1061no—
442#95—1061no—
443#95—1061no—
444#95—1061no—
445#95—1061no—
446#95—1061no—
447#95—1061no—
448#95—1061no—
449#95—1061no—
450#95—1061no—
451C24D4×D5311yescharacteristic
452C2×Q8—21no—
453C2×Q8—21no—
454C2×Q8—21no—
455C2×Q8—21no—
456C2×Q8—21no—
457C2×Q8—21no—
458C2×Q8—21no—
459C2×Q8—21no—
460C2×Q8—21no—
461C2×Q8—21no—
462C2×Q8D4×D5312yes—
463C2×Q8#1171994712yes—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—531no—
475C22×C4—531no—
476C22×C4—531no—
477C22×C4—531no—
478C22×C4—1061no—
479C22×C4—1061no—
480C22×C4—1061no—
481C22×C4—1061no—
482C22×C4D4×D5311yescharacteristic
483C22×C4C23×D5311yescharacteristic
484Q16—28no—
485C2×C8—21no—
486C2×C8—21no—
487C2×C8—21no—
488C2×C8—21no—
489C2×C8—21no—
490C2×C8—532no—
491C2×C8—1061no—
492C2×C8—1061no—
493C2×C8—1061no—
494C2×C8—1061no—
495C2×C8—1061no—
496C2×C8#1171994712yes—
497C4⋊C4—24no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—42no—
502C4⋊C4—532no—
503C4⋊C4—1061no—
504C4⋊C4—1061no—
505C4⋊C4—1061no—
506C4⋊C4—1061no—
507C4⋊C4—1061no—
508C4⋊C4—1061no—
509C4⋊C4—1061no—
510C4⋊C4—2122no—
511C4⋊C4—2122no—
512C4⋊C4—2122no—
513C4⋊C4—2122no—
514C4⋊C4#1171994712yes—
515C22⋊C4—24no—
516C22⋊C4—42no—
517C22⋊C4—1062no—
518C22⋊C4—2121no—
519C22⋊C4—2122no—
520C22⋊C4—2122no—
521C42—1061no—
522C42—1061no—
523C42—1061no—
524C42—1061no—
525C42—2122no—
526C42—2122no—
527C42—2122no—
528C42—2122no—
529C23—21no—
530C23—21no—
531C23—21no—
532C23—21no—
533C23—21no—
534C23—21no—
535C23—21no—
536C23C2×D4×D5311yescharacteristic
537Q8—22no—
538Q8—22no—
539Q8—22no—
540Q8—22no—
541Q8—41no—
542Q8—41no—
543Q8—41no—
544Q8—41no—
545C2×C4—21no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—22no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—532no—
565C2×C4—1061no—
566C2×C4—1061no—
567C2×C4—1061no—
568C2×C4—1061no—
569C2×C4—1061no—
570C2×C4—1061no—
571C2×C4—1061no—
572C2×C4—1061no—
573C2×C4—1061no—
574C2×C4—1061no—
575C2×C4—1061no—
576C2×C4—1062no—
577C2×C4—2121no—
578C2×C4—2121no—
579C2×C4—2121no—
580C2×C4—2121no—
581C2×C4—2121no—
582C2×C4—2121no—
583C2×C4—2121no—
584C2×C4—2121no—
585C2×C4—2121no—
586C2×C4—2121no—
587C2×C4—2121no—
588C2×C4C2×D4×D5311yescharacteristic
589C2×C4C2×#1171994711yescharacteristic
590C2×C4C2×#1171994711yescharacteristic
591C2×C4C2×#1171994711yescharacteristic
592C2×C4#1313326211yescharacteristic
593C2×C4#1313326411yescharacteristic
594C8—22no—
595C8—41no—
596C8—1062no—
597C8—2121no—
598C22—41no—
599C22—41no—
600C22—41no—
601C22—41no—
602C22—41no—
603C22—41no—
604C22—41no—
605C22#1681112911yescharacteristic
606C22#1681125511yescharacteristic
607C22#1681127311yescharacteristic
608C22#135×D5311yescharacteristic
609C22#1681142211yescharacteristic
610C22#1681143611yescharacteristic
611C22#1681143911yescharacteristic
612C4—42no—
613C4—41no—
614C4—42no—
615C4—42no—
616C4—81no—
617C4—1062no—
618C4—4241no—
619C4—4241no—
620C4—4241no—
621C4—2121no—
622C4—2121no—
623C4—2122no—
624C4—2121no—
625C4#1681112911yescharacteristic
626C4#1681121811yescharacteristic
627C4#1681127311yescharacteristic
628C4#1681135911yescharacteristic
629C2—81no—
630C2#2027859111yescharacteristic
631C2#2028473411yescharacteristic
632C2#2028500911yescharacteristic
633C2#2028627311yescharacteristic
634C2#2028645411yescharacteristic
635C2#2028714211yescharacteristic
636C2#2028728511yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2001C2
socleC23×C53C22×D4
prime radicalC24×C53C2×D4
hypercenter#2001D53
hypocenterC53#26561
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2001

Complements

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

NH ≅ G/Ntype
2. C2×Q16×Dic53C2semidirect
5. #20276265C2semidirect
6. #20276035C2semidirect
8. #20275486C2semidirect
9. #20275447C2semidirect
10. #20275359C2semidirect
12. #20274985C2semidirect
13. #20274981C2semidirect
379. C53#26561semidirect

Maximal subgroups

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

subgroupindex
C2×#168109612
C2×Q16×Dic532
#202782142
#202777462
#202762652
#202760352
#202759152
#202754862
#202754472
#202753592
#202753242
#202749852
#202749812
#202749162
C53×#20012
#2656153