Finite Group Databasebeta

← #27446111

Subgroups — #27446111

Order 13568.

732 conjugacy classes of subgroups.

The subgroup lattice has 634 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20277644C211yesmaximal, characteristic
2#20276152C211yesmaximal, characteristic, complemented
3#20276152C211yesmaximal, characteristic, complemented
4#20276151C211yesmaximal, characteristic, complemented
5#20276141C211yesmaximal, characteristic, complemented
6#20275681C211yesmaximal, characteristic, complemented
7#20275673C211yesmaximal, characteristic, complemented
8#20275510C211yesmaximal, characteristic
9#20275483C211yesmaximal, characteristic, complemented
10#20274912C211yesmaximal, characteristic, complemented
11#20274907C211yesmaximal, characteristic, complemented
12#20274807C211yesmaximal, characteristic, complemented
13#20274803C211yesmaximal, characteristic, complemented
14C4×#13133148C211yesmaximal, characteristic, complemented
15C53×#2562C211yesmaximal, characteristic
16C2×#13133176C2211yescharacteristic
17C22×C4⋊Dic53C2211yescharacteristic
18C2×#13133150C2211yescharacteristic
19C2×#13133148C2211yescharacteristic
20C2×#13133148C2211yescharacteristic
21C2×#13133148C2211yescharacteristic
22C2×C4⋊#88322C2211yescharacteristic
23C2×C22⋊#88322C2211yescharacteristic
24C2×C22⋊#88322C2211yescharacteristic
25C2×C22⋊#88322C2211yescharacteristic
26C2×C22⋊#88322C2211yescharacteristic
27C2×C4×#88322C2211yescharacteristic
28#16810516—21no—
29#16810509—21no—
30#16810503—21no—
31#16810489C2211yescharacteristic
32#16810483C2211yescharacteristic
33#16810483C2211yescharacteristic
34C4×C4⋊Dic53C2211yescharacteristic
35C22⋊C4×Dic53—21no—
36#16810385C2211yescharacteristic
37#16810385C2211yescharacteristic
38#16810385C2211yescharacteristic
39#16810385C2211yescharacteristic
40#16810385C2211yescharacteristic
41#16810385C2211yescharacteristic
42#16810385C2211yescharacteristic
43#16810385C2211yescharacteristic
44#16810385C2211yescharacteristic
45#16810385C2211yescharacteristic
46#16810385C2211yescharacteristic
47#16810385C2211yescharacteristic
48C2×#125×C53C2211yescharacteristic
49C2×C4×D4×C53C2211yescharacteristic
50C53×#400—21no—
51C53×#394—21no—
52C53×#388—21no—
53C53×#388—21no—
54C53×#380C2211yescharacteristic
55C53×#380C2211yescharacteristic
56C53×#380C2211yescharacteristic
57C53×#380C2211yescharacteristic
58C4×C4⋊C4×C53C2211yescharacteristic
59C23×Dic53—21no—
60C22×#88322C2311yescharacteristic
61C22×#88322C2311yescharacteristic
62C22×#88322C2311yescharacteristic
63C22×#88322C2311yescharacteristic
64#13133176—22no—
65#13133176—41no—
66C2×C4⋊Dic53—21no—
67C2×C4⋊Dic53—22no—
68C2×C4⋊Dic53—41no—
69C2×C4⋊Dic53C2311yescharacteristic
70C2×C4⋊Dic53C2311yescharacteristic
71C2×C4⋊Dic53C2311yescharacteristic
72C2×C4⋊Dic53C2311yescharacteristic
73C2×C22⋊Dic53—21no—
74C2×C22⋊Dic53—21no—
75C2×C22⋊Dic53—21no—
76C2×C4×Dic53—21no—
77C2×C4×Dic53—21no—
78#13133150—41no—
79#13133148—22no—
80#13133148—22no—
81#13133148—41no—
82C4⋊#88322—22no—
83C22⋊#88322—22no—
84C22⋊#88322—22no—
85C22⋊#88322—41no—
86C22⋊#88322—41no—
87C4×#88322—22no—
88#13133124—21no—
89#13133124—21no—
90#13133124—21no—
91#13133124—21no—
92#13133124—21no—
93C22×D4×C53D411yescharacteristic
94C23×C4×C53C2311yescharacteristic
95C23×C4×C53C2311yescharacteristic
96#125×C53—22no—
97#125×C53—22no—
98C4×D4×C53—22no—
99C4×D4×C53—22no—
100C2×C4⋊C4×C53—21no—
101C2×C4⋊C4×C53—21no—
102C2×C4⋊C4×C53D411yescharacteristic
103C2×C4⋊C4×C53C2311yescharacteristic
104C2×C4⋊C4×C53C2311yescharacteristic
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×C53D411yescharacteristic
112C2×C22⋊C4×C53D411yescharacteristic
113C2×C42×C53—21no—
114C2×C42×C53C2311yescharacteristic
115#95×C53—21no—
116#95×C53—21no—
117#95×C53—21no—
118#95×C53—21no—
119#95×C53—21no—
120C24×C53C2×D411yescharacteristic
121C24×C53C2×D411yescharacteristic
122C2×D4×C53—22no—
123C2×D4×C53—22no—
124C2×D4×C53—41no—
125C22×C4×C53—21no—
126C22×C4×C53—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—21no—
136C22×C4×C53—21no—
137C22×C4×C53—21no—
138C22×C4×C53—21no—
139C22×C4×C53—21no—
140C22×C4×C53—21no—
141C22×C4×C53C2×D411yescharacteristic
142C22×C4×C53C2×D411yescharacteristic
143C22×C4×C53C2×D411yescharacteristic
144C22×C4×C53C2×D411yescharacteristic
145C22×C4×C53C2411yescharacteristic
146C22×Dic53—21no—
147C22×Dic53—21no—
148C22×Dic53—21no—
149C22×Dic53—21no—
150C22×Dic53—42no—
151C22×Dic53—42no—
152C22×Dic53—81no—
153C4⋊C4×C53—22no—
154C4⋊C4×C53—21no—
155C4⋊C4×C53—21no—
156C4⋊C4×C53—21no—
157C4⋊C4×C53—21no—
158C4⋊C4×C53—21no—
159C4⋊C4×C53—21no—
160C4⋊C4×C53—21no—
161C4⋊C4×C53—21no—
162C4⋊C4×C53—42no—
163C4⋊C4×C53—42no—
164C22⋊C4×C53—22no—
165C22⋊C4×C53—22no—
166C22⋊C4×C53—42no—
167C22⋊C4×C53—42no—
168C22⋊C4×C53—81no—
169C22⋊C4×C53—81no—
170C42×C53—81no—
171C42×C53#4112yes—
172C42×C53#4112yes—
173C22⋊Dic53—42no—
174C22⋊Dic53—42no—
175C22⋊Dic53—81no—
176C4⋊Dic53—21no—
177C4⋊Dic53—21no—
178C4⋊Dic53—21no—
179C4⋊Dic53—21no—
180C4⋊Dic53—21no—
181C4⋊Dic53—21no—
182C4⋊Dic53—21no—
183C4⋊Dic53—42no—
184C4⋊Dic53#4112yes—
185C4×Dic53—42no—
186C4×Dic53—81no—
187C2×#88322—21no—
188C2×#88322—21no—
189C2×#88322—21no—
190C2×#88322—21no—
191C2×#88322—21no—
192C2×#88322—21no—
193C2×#88322—21no—
194C2×#88322—21no—
195C2×#88322—21no—
196C2×#88322—21no—
197C2×#88322—21no—
198C2×#88322#4112yes—
199C23×C53—21no—
200C23×C53—21no—
201C23×C53—21no—
202C23×C53—21no—
203C23×C53—21no—
204C23×C53—21no—
205C23×C53—21no—
206C23×C53—21no—
207C23×C53—21no—
208C23×C53—21no—
209C23×C53—21no—
210C23×C53—21no—
211C23×C53—21no—
212C23×C53#13512yes—
213C23×C53C22×D411yescharacteristic
214D4×C53—44no—
215C2×C4×C53—21no—
216C2×C4×C53—21no—
217C2×C4×C53—21no—
218C2×C4×C53—21no—
219C2×C4×C53—21no—
220C2×C4×C53—21no—
221C2×C4×C53—21no—
222C2×C4×C53—21no—
223C2×C4×C53—21no—
224C2×C4×C53—21no—
225C2×C4×C53—41no—
226C2×C4×C53—41no—
227C2×C4×C53—41no—
228C2×C4×C53—41no—
229C2×C4×C53—41no—
230C2×C4×C53—41no—
231C2×C4×C53—41no—
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×C53#12312yes—
246C2×C4×C53#12312yes—
247C2×C4×C53C2×#4111yescharacteristic
248C2×C4×C53C2×#4111yescharacteristic
249C2×C4×C53C2×#4111yescharacteristic
250C2×C4×C53C2×#4111yescharacteristic
251C2×C4×C53H(5,𝔽2)11yescharacteristic
252C2×C4×C532−1+411yescharacteristic
253C2×Dic53—22no—
254C2×Dic53—41no—
255C2×Dic53—41no—
256C2×Dic53—41no—
257C2×Dic53—41no—
258C2×Dic53—41no—
259C2×Dic53—41no—
260C2×Dic53—41no—
261C2×Dic53—41no—
262C2×Dic53—41no—
263C2×Dic53—41no—
264C2×Dic53—41no—
265C2×Dic53—84no—
266#88322—22no—
267#88322—41no—
268#88322—41no—
269#88322—41no—
270#26304—531nomaximal, Sylow
271C22×C53—22no—
272C22×C53—22no—
273C22×C53—22no—
274C22×C53—41no—
275C22×C53—41no—
276C22×C53—41no—
277C22×C53—41no—
278C22×C53—41no—
279C22×C53—41no—
280C22×C53—41no—
281C22×C53—41no—
282C22×C53—41no—
283C22×C53—41no—
284C22×C53—41no—
285C22×C53C2×#12311yescharacteristic
286C22×C53#54611yescharacteristic
287C22×C53#54711yescharacteristic
288C22×C53C2×#13511yescharacteristic
289C22×C53#57511yescharacteristic
290C22×C53#57711yescharacteristic
291C22×C53#57711yescharacteristic
292C4×C53—22no—
293C4×C53—22no—
294C4×C53—41no—
295C4×C53—41no—
296C4×C53—81no—
297C4×C53—81no—
298C4×C53—81no—
299C4×C53—81no—
300C4×C53#52911yescharacteristic
301C4×C53#54211yescharacteristic
302C4×C53#55611yescharacteristic
303C4×C53#55911yescharacteristic
304Dic53—42no—
305Dic53—81no—
306Dic53—81no—
307Dic53—81no—
308#2562D5311yescharacteristic
309#2352—531no—
310#2077—531no—
311#2077—531no—
312#2077—531no—
313#2075—531no—
314#1991—531no—
315#1989—531no—
316#1954—531no—
317#1949—531no—
318#1854—531no—
319#1853—531no—
320#1825—531no—
321#1824—531no—
322C4×#106—531no—
323C2×C53—42no—
324C2×C53—81no—
325C2×C53#244711yescharacteristic
326C2×C53#309111yescharacteristic
327C2×C53#309911yescharacteristic
328C2×C53#330111yescharacteristic
329C2×C53#332311yescharacteristic
330C2×C53#333711yescharacteristic
331C2×C53#335111yescharacteristic
332C2×#125C2×D5311yescharacteristic
333C2×C4×D4C2×D5311yescharacteristic
334C2×#117—531no—
335C22×C4⋊C4—531no—
336C2×#107—531no—
337C2×#106—531no—
338C2×#106—531no—
339C2×#106—531no—
340C2×C4⋊C8—531no—
341C2×C22⋊C8—531no—
342C2×C22⋊C8—531no—
343C2×C22⋊C8—531no—
344C2×C22⋊C8—531no—
345C2×C4×C8—531no—
346#400—21no—
347#394—21no—
348#388—21no—
349#388—21no—
350#388—1061no—
351#387—1061no—
352#386—1061no—
353#382—531no—
354#380—531no—
355#380—531no—
356#380C2×D5311yescharacteristic
357#380C2×D5311yescharacteristic
358#380C2×D5311yescharacteristic
359#380C2×D5311yescharacteristic
360C4×C4⋊C4—531no—
361C4×C4⋊C4C2×D5311yescharacteristic
362C4×C22⋊C4—1061no—
363#340—531no—
364#340—531no—
365#340—531no—
366#340—531no—
367#340—531no—
368#340—531no—
369#340—531no—
370#340—531no—
371#340—531no—
372#340—531no—
373#340—531no—
374#340—531no—
375C53#2630411yescharacteristic, complemented, Sylow
376C22×D4C22⋊D5311yescharacteristic
377C23×C4—1061no—
378C23×C4C22×D5311yescharacteristic
379C23×C4C22×D5311yescharacteristic
380C22×C8—531no—
381C22×C8—531no—
382C22×C8—531no—
383C22×C8—531no—
384#125—22no—
385#125—22no—
386C4×D4—22no—
387C4×D4—22no—
388#117—1062no—
389#117—2121no—
390C2×C4⋊C4—21no—
391C2×C4⋊C4—21no—
392C2×C4⋊C4—531no—
393C2×C4⋊C4—531no—
394C2×C4⋊C4—531no—
395C2×C4⋊C4—531no—
396C2×C4⋊C4—1062no—
397C2×C4⋊C4—1061no—
398C2×C4⋊C4—2121no—
399C2×C4⋊C4C22⋊D5311yescharacteristic
400C2×C4⋊C4C22×D5311yescharacteristic
401C2×C4⋊C4C22×D5311yescharacteristic
402C2×C4⋊C4C22×D5311yescharacteristic
403C2×C4⋊C4C22×D5311yescharacteristic
404C2×C22⋊C4—21no—
405C2×C22⋊C4—21no—
406C2×C22⋊C4—21no—
407C2×C22⋊C4—21no—
408C2×C22⋊C4—1061no—
409C2×C22⋊C4—1061no—
410C2×C22⋊C4—1061no—
411C2×C22⋊C4C22⋊D5311yescharacteristic
412C2×C22⋊C4C22⋊D5311yescharacteristic
413C2×C42—21no—
414C2×C42—1061no—
415C2×C42—1061no—
416C2×C42C22×D5311yescharacteristic
417#107—2121no—
418#106—1062no—
419#106—1062no—
420#106—2121no—
421C4⋊C8—1062no—
422C22⋊C8—1062no—
423C22⋊C8—1062no—
424C22⋊C8—2121no—
425C22⋊C8—2121no—
426C4×C8—1062no—
427#95—21no—
428#95—21no—
429#95—21no—
430#95—21no—
431#95—21no—
432#95—1061no—
433#95—1061no—
434#95—1061no—
435#95—1061no—
436#95—1061no—
437C24C2×C22⋊D5311yescharacteristic
438C24C2×C22⋊D5311yescharacteristic
439C2×D4—22no—
440C2×D4—22no—
441C2×D4—41no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—1061no—
459C22×C4—1061no—
460C22×C4—1061no—
461C22×C4—1061no—
462C22×C4—4241no—
463C22×C4—2122no—
464C22×C4—2122no—
465C22×C4C2×C22⋊D5311yescharacteristic
466C22×C4C2×C22⋊D5311yescharacteristic
467C22×C4C2×C22⋊D5311yescharacteristic
468C22×C4C2×C22⋊D5311yescharacteristic
469C22×C4C23×D5311yescharacteristic
470C2×C8—532no—
471C2×C8—1061no—
472C2×C8—1061no—
473C2×C8—1061no—
474C2×C8—1061no—
475C2×C8—1061no—
476C2×C8—1061no—
477C2×C8—1061no—
478C2×C8—1061no—
479C2×C8—1061no—
480C2×C8—1061no—
481C2×C8—1061no—
482C4⋊C4—22no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—42no—
492C4⋊C4—42no—
493C4⋊C4—532no—
494C4⋊C4—1061no—
495C4⋊C4—1061no—
496C4⋊C4—1061no—
497C4⋊C4—1061no—
498C4⋊C4—1061no—
499C4⋊C4—1061no—
500C4⋊C4—1061no—
501C4⋊C4—2122no—
502C22⋊C4—22no—
503C22⋊C4—22no—
504C22⋊C4—42no—
505C22⋊C4—42no—
506C22⋊C4—81no—
507C22⋊C4—81no—
508C22⋊C4—4241no—
509C22⋊C4—2122no—
510C22⋊C4—2122no—
511C42—81no—
512C42—4241no—
513C42—2122no—
514C42#1171994712yes—
515C42#1171994712yes—
516C23—21no—
517C23—21no—
518C23—21no—
519C23—21no—
520C23—21no—
521C23—21no—
522C23—21no—
523C23—21no—
524C23—21no—
525C23—21no—
526C23—21no—
527C23—21no—
528C23—21no—
529C23#1313323712yes—
530C23C22×C22⋊D5311yescharacteristic
531D4—44no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
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—1062no—
563C2×C4—4244no—
564C2×C4—2121no—
565C2×C4—2121no—
566C2×C4—2121no—
567C2×C4—2121no—
568C2×C4—2121no—
569C2×C4—2121no—
570C2×C4—2121no—
571C2×C4—2121no—
572C2×C4—2121no—
573C2×C4—2121no—
574C2×C4—2121no—
575C2×C4#1313319812yes—
576C2×C4#1313319812yes—
577C2×C4#41×D5311yescharacteristic
578C2×C4#41×D5311yescharacteristic
579C2×C4C2×#1171994711yescharacteristic
580C2×C4C2×#1171994711yescharacteristic
581C2×C4#1313326311yescharacteristic
582C2×C4#1313326411yescharacteristic
583C8—1062no—
584C8—2121no—
585C8—2121no—
586C8—2121no—
587C22—22no—
588C22—22no—
589C22—22no—
590C22—41no—
591C22—41no—
592C22—41no—
593C22—41no—
594C22—41no—
595C22—41no—
596C22—41no—
597C22—41no—
598C22—41no—
599C22—41no—
600C22—41no—
601C22C2×#1313319811yescharacteristic
602C22#1681126111yescharacteristic
603C22#1681127211yescharacteristic
604C22C2×#1313323711yescharacteristic
605C22#1681142111yescharacteristic
606C22#1681143111yescharacteristic
607C22#1681143411yescharacteristic
608C4—22no—
609C4—22no—
610C4—41no—
611C4—41no—
612C4—81no—
613C4—81no—
614C4—81no—
615C4—81no—
616C4—4241no—
617C4—4241no—
618C4—4241no—
619C4—2122no—
620C4#1681115211yescharacteristic
621C4#1681124211yescharacteristic
622C4#1681134211yescharacteristic
623C4#1681135511yescharacteristic
624C2—42no—
625C2—81no—
626C2#2027849811yescharacteristic
627C2#2028462411yescharacteristic
628C2#2028470011yescharacteristic
629C2#2028661111yescharacteristic
630C2#2028692511yescharacteristic
631C2#2028709411yescharacteristic
632C2#2028726911yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2562C2
socleC23×C53C22×D4
prime radicalC22×D4×C53D4
hypercenter#2562D53
hypocenterC53#26304
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #2562

Complements

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

NH ≅ G/Ntype
2. #20276152C2semidirect
3. #20276152C2semidirect
4. #20276151C2semidirect
5. #20276141C2semidirect
6. #20275681C2semidirect
7. #20275673C2semidirect
9. #20275483C2semidirect
10. #20274912C2semidirect
11. #20274907C2semidirect
12. #20274807C2semidirect
13. #20274803C2semidirect
14. C4×#13133148C2semidirect
375. C53#26304semidirect

Maximal subgroups

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

subgroupindex
#202776442
#202761522
#202761512
#202761412
#202756812
#202756732
#202755102
#202754832
#202749122
#202749072
#202748072
#202748032
C4×#131331482
C53×#25622
#2630453