Finite Group Databasebeta

← #27469875

Subgroups — #27469875

Order 13568.

772 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20277338C211yesmaximal, characteristic, complemented
2#20276133C211yesmaximal, characteristic, complemented
3#20276128C211yesmaximal, characteristic, complemented
4#20276123C211yesmaximal, characteristic, complemented
5#20276118C211yesmaximal, characteristic, complemented
6#20275532C211yesmaximal, characteristic, complemented
7#20275527C211yesmaximal, characteristic, complemented
8#20275038C211yesmaximal, characteristic
9#20275033C211yesmaximal, characteristic
10#20275028C211yesmaximal, characteristic, complemented
11#20275023C211yesmaximal, characteristic, complemented
12#20274850C211yesmaximal, characteristic, complemented
13#20274842C211yesmaximal, characteristic, complemented
14#20274481C211yesmaximal, characteristic, complemented
15C53×#2620C211yesmaximal, characteristic
16C2×#13133176C2211yescharacteristic
17C22×C4⋊Dic53C2211yescharacteristic
18C2×#13133141C2211yescharacteristic
19C2×#13133141C2211yescharacteristic
20C2×#13133141C2211yescharacteristic
21C2×#13133141C2211yescharacteristic
22C2×#13133138C2211yescharacteristic
23C2×#13133138C2211yescharacteristic
24C2×#13133138C2211yescharacteristic
25C2×#13133138C2211yescharacteristic
26C2×C22⋊#88322C2211yescharacteristic
27C2×C22⋊#88322C2211yescharacteristic
28C2×#13133128C2211yescharacteristic
29C2×C4×#88322C2211yescharacteristic
30#16810522—21no—
31#16810522—21no—
32#16810498—21no—
33#16810498—21no—
34#16810493C2211yescharacteristic
35#16810493C2211yescharacteristic
36#16810489C2211yescharacteristic
37#16810489C2211yescharacteristic
38#16810385C2211yescharacteristic
39#16810385C2211yescharacteristic
40#16810385C2211yescharacteristic
41#16810385C2211yescharacteristic
42#16810385C2211yescharacteristic
43#16810385C2211yescharacteristic
44#16810385C2211yescharacteristic
45#16810385C2211yescharacteristic
46#16810378C2211yescharacteristic
47#16810378C2211yescharacteristic
48C2×#124×C53C2211yescharacteristic
49C2×C22⋊Q8×C53C2211yescharacteristic
50C2×#121×C53C2211yescharacteristic
51C2×#117×C53C2211yescharacteristic
52C53×#396—21no—
53C53×#396—21no—
54C53×#394—21no—
55C53×#394—21no—
56C53×#391C2211yescharacteristic
57C53×#390C2211yescharacteristic
58C53×#384C2211yescharacteristic
59C23×Dic53—21no—
60C22×#88322C2311yescharacteristic
61C22×#88322C2311yescharacteristic
62C22×#88322C2311yescharacteristic
63C22×#88322C2311yescharacteristic
64#13133176—41no—
65#13133176—41no—
66C2×C4⋊Dic53—21no—
67C2×C4⋊Dic53—21no—
68C2×C4⋊Dic53—21no—
69C2×C4⋊Dic53—21no—
70C2×C4⋊Dic53—21no—
71C2×C4⋊Dic53—41no—
72C2×C4⋊Dic53—41no—
73C2×C4⋊Dic53C2311yescharacteristic
74C2×C4⋊Dic53C2311yescharacteristic
75C2×C4⋊Dic53C2311yescharacteristic
76C2×C4⋊Dic53C2311yescharacteristic
77C2×C22⋊Dic53—21no—
78C2×C4×Dic53—21no—
79#13133141—41no—
80#13133141—41no—
81#13133141—41no—
82#13133141—41no—
83#13133138—41no—
84#13133138—41no—
85#13133138—41no—
86#13133138—41no—
87C22⋊#88322—22no—
88C22⋊#88322—22no—
89#13133128—22no—
90C4×#88322—22no—
91#13133124—21no—
92#13133124—21no—
93#13133124—21no—
94#13133124—21no—
95C22×Q8×C53C2311yescharacteristic
96C22×D4×C53—21no—
97C22×D4×C53C2311yescharacteristic
98C23×C4×C53C2311yescharacteristic
99#124×C53—22no—
100#124×C53—22no—
101C22⋊Q8×C53—22no—
102C22⋊Q8×C53—22no—
103#121×C53—22no—
104#121×C53—22no—
105#117×C53—22no—
106#117×C53—22no—
107C2×C4⋊C4×C53—21no—
108C2×C4⋊C4×C53—21no—
109C2×C4⋊C4×C53—21no—
110C2×C4⋊C4×C53D411yescharacteristic
111C2×C4⋊C4×C53D411yescharacteristic
112C2×C4⋊C4×C53C2311yescharacteristic
113C2×C4⋊C4×C53C2311yescharacteristic
114C2×C22⋊C4×C53—21no—
115C2×C22⋊C4×C53—21no—
116C2×C22⋊C4×C53—21no—
117C2×C22⋊C4×C53—21no—
118C2×C22⋊C4×C53—21no—
119C2×C22⋊C4×C53—21no—
120C2×C22⋊C4×C53D411yescharacteristic
121C2×C22⋊C4×C53D411yescharacteristic
122C2×C42×C53C2311yescharacteristic
123C2×C42×C53C2311yescharacteristic
124#95×C53—21no—
125#95×C53—21no—
126C24×C53—21no—
127C24×C53C2×D411yescharacteristic
128C2×Q8×C53—21no—
129C2×Q8×C53—21no—
130C2×Q8×C53—21no—
131C2×Q8×C53—21no—
132C2×Q8×C53—21no—
133C2×Q8×C53—21no—
134C2×D4×C53—21no—
135C2×D4×C53—21no—
136C2×D4×C53—21no—
137C2×D4×C53—21no—
138C2×D4×C53—21no—
139C2×D4×C53—21no—
140C2×D4×C53—42no—
141C2×D4×C53—81no—
142C2×D4×C53—81no—
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×C53C2×D411yescharacteristic
151C22×C4×C53C2×D411yescharacteristic
152C22×C4×C53C2×D411yescharacteristic
153C22×C4×C53C2×D411yescharacteristic
154C22×C4×C53C2×D411yescharacteristic
155C22×C4×C53#4112yes—
156C22×C4×C53#4112yes—
157C22×C4×C53C2411yescharacteristic
158C22×Dic53—21no—
159C22×Dic53—21no—
160C22×Dic53—21no—
161C22×Dic53—21no—
162C22×Dic53—42no—
163C22×Dic53—81no—
164C22×Dic53—81no—
165C4⋊C4×C53—22no—
166C4⋊C4×C53—22no—
167C4⋊C4×C53—21no—
168C4⋊C4×C53—21no—
169C4⋊C4×C53—21no—
170C4⋊C4×C53—21no—
171C4⋊C4×C53—42no—
172C4⋊C4×C53—81no—
173C4⋊C4×C53—81no—
174C22⋊C4×C53—22no—
175C22⋊C4×C53—22no—
176C22⋊C4×C53—42no—
177C22⋊C4×C53—42no—
178C22⋊C4×C53—42no—
179C22⋊C4×C53—42no—
180C22⋊C4×C53—81no—
181C22⋊C4×C53—81no—
182C42×C53—21no—
183C42×C53—21no—
184C42×C53#4112yes—
185C42×C53#4112yes—
186C22⋊Dic53—42no—
187C4⋊Dic53—21no—
188C4⋊Dic53—21no—
189C4⋊Dic53—21no—
190C4⋊Dic53—21no—
191C4⋊Dic53—21no—
192C4⋊Dic53—21no—
193C4⋊Dic53—21no—
194C4⋊Dic53—21no—
195C4⋊Dic53—42no—
196C4⋊Dic53—81no—
197C4⋊Dic53—81no—
198C4⋊Dic53—81no—
199C4⋊Dic53—81no—
200C4×Dic53—42no—
201C2×#88322—21no—
202C2×#88322—21no—
203C2×#88322—21no—
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—
213C23×C53—21no—
214C23×C53—21no—
215C23×C53—21no—
216C23×C53—21no—
217C23×C53—21no—
218C23×C53—21no—
219C23×C53—41no—
220C23×C53—41no—
221C23×C53—41no—
222C23×C53—41no—
223C23×C53—41no—
224C23×C53—41no—
225C23×C53—41no—
226C23×C53#12412yes—
227C23×C53C22×D411yescharacteristic
228Q8×C53—41no—
229Q8×C53—41no—
230Q8×C53—41no—
231Q8×C53—41no—
232D4×C53—41no—
233D4×C53—41no—
234D4×C53—41no—
235D4×C53—41no—
236D4×C53—84no—
237C2×C4×C53—21no—
238C2×C4×C53—21no—
239C2×C4×C53—21no—
240C2×C4×C53—21no—
241C2×C4×C53—21no—
242C2×C4×C53—21no—
243C2×C4×C53—22no—
244C2×C4×C53—22no—
245C2×C4×C53—22no—
246C2×C4×C53—22no—
247C2×C4×C53—21no—
248C2×C4×C53—21no—
249C2×C4×C53—21no—
250C2×C4×C53—21no—
251C2×C4×C53—21no—
252C2×C4×C53—21no—
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#12312yes—
263C2×C4×C53#12312yes—
264C2×C4×C53#12412yes—
265C2×C4×C53C2×#4111yescharacteristic
266C2×C4×C53C2×#4111yescharacteristic
267C2×C4×C53C2×#4111yescharacteristic
268C2×C4×C53C2×#4111yescharacteristic
269C2×C4×C53H(5,𝔽2)11yescharacteristic
270C2×C4×C53H(5,𝔽2)11yescharacteristic
271C2×Dic53—41no—
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—84no—
284#88322—41no—
285#88322—41no—
286#88322—41no—
287#88322—41no—
288#27984—531nomaximal, Sylow
289C22×C53—22no—
290C22×C53—22no—
291C22×C53—22no—
292C22×C53—41no—
293C22×C53—41no—
294C22×C53—41no—
295C22×C53—41no—
296C22×C53—81no—
297C22×C53—81no—
298C22×C53—81no—
299C22×C53—81no—
300C22×C53—81no—
301C22×C53—81no—
302C22×C53—81no—
303C22×C53C2×#12311yescharacteristic
304C22×C53C2×#12411yescharacteristic
305C22×C53#53511yescharacteristic
306C22×C53#57611yescharacteristic
307C22×C53#57711yescharacteristic
308C22×C53#57711yescharacteristic
309C22×C53#57811yescharacteristic
310C4×C53—22no—
311C4×C53—22no—
312C4×C53—22no—
313C4×C53—41no—
314C4×C53—41no—
315C4×C53—81no—
316C4×C53—81no—
317C4×C53—81no—
318C4×C53#55311yescharacteristic
319C4×C53#55311yescharacteristic
320C4×C53#55311yescharacteristic
321C4×C53#55311yescharacteristic
322Dic53—81no—
323Dic53—81no—
324Dic53—81no—
325Dic53—81no—
326#2620D5311yescharacteristic
327#2317—531no—
328#2074—531no—
329#2073—531no—
330#2072—531no—
331#2071—531no—
332#1959—531no—
333#1958—531no—
334#1875—531no—
335#1874—531no—
336#1873—531no—
337#1872—531no—
338#1836—531no—
339#1834—531no—
340#1750—531no—
341C2×C53—42no—
342C2×C53—161no—
343C2×C53#250411yescharacteristic
344C2×C53#309811yescharacteristic
345C2×C53#309911yescharacteristic
346C2×C53#313611yescharacteristic
347C2×C53#313711yescharacteristic
348C2×C53#322411yescharacteristic
349C2×C53#322511yescharacteristic
350C2×#124C2×D5311yescharacteristic
351C2×C22⋊Q8C2×D5311yescharacteristic
352C2×#121C2×D5311yescharacteristic
353C2×#117—531no—
354C2×#117C2×D5311yescharacteristic
355C22×C4⋊C4—531no—
356C2×#103—531no—
357C2×#103—531no—
358C2×#103—531no—
359C2×#103—531no—
360C2×#102—531no—
361C2×#102—531no—
362C2×#102—531no—
363C2×#102—531no—
364C2×C22⋊C8—531no—
365C2×C22⋊C8—531no—
366C2×#97—531no—
367C2×C4×C8—531no—
368#396—21no—
369#396—21no—
370#394—21no—
371#394—21no—
372#391C2×D5311yescharacteristic
373#390C2×D5311yescharacteristic
374#389—1061no—
375#389—1061no—
376#385—1061no—
377#385—1061no—
378#384—531no—
379#384—531no—
380#384C2×D5311yescharacteristic
381#382—531no—
382#382—531no—
383#340—531no—
384#340—531no—
385#340—531no—
386#340—531no—
387#340—531no—
388#340—531no—
389#340—531no—
390#340—531no—
391#336—531no—
392#336—531no—
393C53#2798411yescharacteristic, complemented, Sylow
394C22×Q8C22×D5311yescharacteristic
395C22×D4—21no—
396C22×D4C22×D5311yescharacteristic
397C23×C4—1061no—
398C23×C4C22×D5311yescharacteristic
399C22×C8—531no—
400C22×C8—531no—
401C22×C8—531no—
402C22×C8—531no—
403#124—22no—
404#124—22no—
405C22⋊Q8—22no—
406C22⋊Q8—22no—
407#121—22no—
408#121—22no—
409#117—22no—
410#117—22no—
411#117—2121no—
412#117—2121no—
413C2×C4⋊C4—21no—
414C2×C4⋊C4—21no—
415C2×C4⋊C4—21no—
416C2×C4⋊C4—531no—
417C2×C4⋊C4—531no—
418C2×C4⋊C4—531no—
419C2×C4⋊C4—531no—
420C2×C4⋊C4—1061no—
421C2×C4⋊C4—1061no—
422C2×C4⋊C4—1061no—
423C2×C4⋊C4—1061no—
424C2×C4⋊C4—1061no—
425C2×C4⋊C4—2121no—
426C2×C4⋊C4—2121no—
427C2×C4⋊C4C22⋊D5311yescharacteristic
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—21no—
437C2×C22⋊C4—1061no—
438C2×C22⋊C4C22⋊D5311yescharacteristic
439C2×C22⋊C4C22⋊D5311yescharacteristic
440C2×C42—1061no—
441C2×C42C22×D5311yescharacteristic
442C2×C42C22×D5311yescharacteristic
443#103—2121no—
444#103—2121no—
445#103—2121no—
446#103—2121no—
447#102—2121no—
448#102—2121no—
449#102—2121no—
450#102—2121no—
451C22⋊C8—1062no—
452C22⋊C8—1062no—
453#97—1062no—
454C4×C8—1062no—
455#95—21no—
456#95—21no—
457#95—1061no—
458#95—1061no—
459#95—1061no—
460#95—1061no—
461C24—21no—
462C24C2×C22⋊D5311yescharacteristic
463C2×Q8—21no—
464C2×Q8—21no—
465C2×Q8—21no—
466C2×Q8—21no—
467C2×Q8—21no—
468C2×Q8—21no—
469C2×D4—21no—
470C2×D4—21no—
471C2×D4—21no—
472C2×D4—21no—
473C2×D4—21no—
474C2×D4—21no—
475C2×D4—42no—
476C2×D4—81no—
477C2×D4—81no—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—1061no—
486C22×C4—1061no—
487C22×C4—1061no—
488C22×C4—1061no—
489C22×C4—4241no—
490C22×C4—4241no—
491C22×C4—2122no—
492C22×C4#1171994712yes—
493C22×C4#1171994712yes—
494C22×C4C2×C22⋊D5311yescharacteristic
495C22×C4C2×C22⋊D5311yescharacteristic
496C22×C4C2×C22⋊D5311yescharacteristic
497C22×C4C2×C22⋊D5311yescharacteristic
498C22×C4C2×C22⋊D5311yescharacteristic
499C22×C4C23×D5311yescharacteristic
500C2×C8—1061no—
501C2×C8—1061no—
502C2×C8—1061no—
503C2×C8—1061no—
504C2×C8—1061no—
505C2×C8—1061no—
506C2×C8—1061no—
507C2×C8—1061no—
508C2×C8—1061no—
509C2×C8—1061no—
510C2×C8—1061no—
511C2×C8—1061no—
512C4⋊C4—22no—
513C4⋊C4—22no—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—21no—
518C4⋊C4—42no—
519C4⋊C4—81no—
520C4⋊C4—81no—
521C4⋊C4—1061no—
522C4⋊C4—1061no—
523C4⋊C4—1061no—
524C4⋊C4—1061no—
525C4⋊C4—1061no—
526C4⋊C4—1061no—
527C4⋊C4—1061no—
528C4⋊C4—1061no—
529C4⋊C4—4241no—
530C4⋊C4—4241no—
531C4⋊C4—4241no—
532C4⋊C4—4241no—
533C4⋊C4—2122no—
534C22⋊C4—22no—
535C22⋊C4—22no—
536C22⋊C4—42no—
537C22⋊C4—42no—
538C22⋊C4—42no—
539C22⋊C4—42no—
540C22⋊C4—81no—
541C22⋊C4—81no—
542C22⋊C4—2122no—
543C42—21no—
544C42—21no—
545C42—2122no—
546C42#1171994712yes—
547C42#1171994712yes—
548C23—21no—
549C23—21no—
550C23—21no—
551C23—21no—
552C23—21no—
553C23—21no—
554C23—41no—
555C23—41no—
556C23—41no—
557C23—41no—
558C23—41no—
559C23—41no—
560C23—41no—
561C23#1313320412yes—
562C23C22×C22⋊D5311yescharacteristic
563Q8—41no—
564Q8—41no—
565Q8—41no—
566Q8—41no—
567D4—41no—
568D4—41no—
569D4—41no—
570D4—41no—
571D4—84no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—4244no—
598C2×C4—2121no—
599C2×C4—2121no—
600C2×C4—2121no—
601C2×C4—2121no—
602C2×C4—2121no—
603C2×C4—2121no—
604C2×C4—2121no—
605C2×C4—2121no—
606C2×C4—2121no—
607C2×C4—2121no—
608C2×C4—2121no—
609C2×C4—2121no—
610C2×C4#1313319812yes—
611C2×C4#1313319812yes—
612C2×C4#1313320412yes—
613C2×C4C2×#1171994711yescharacteristic
614C2×C4C2×#1171994711yescharacteristic
615C2×C4C2×#1171994711yescharacteristic
616C2×C4C2×#1171994711yescharacteristic
617C2×C4#1313326311yescharacteristic
618C2×C4#1313326311yescharacteristic
619C8—2121no—
620C8—2121no—
621C8—2121no—
622C8—2121no—
623C22—22no—
624C22—22no—
625C22—22no—
626C22—41no—
627C22—41no—
628C22—41no—
629C22—41no—
630C22—81no—
631C22—81no—
632C22—81no—
633C22—81no—
634C22—81no—
635C22—81no—
636C22—81no—
637C22C2×#1313319811yescharacteristic
638C22C2×#1313320411yescharacteristic
639C22#1681118711yescharacteristic
640C22#1681142611yescharacteristic
641C22#1681143111yescharacteristic
642C22#1681143411yescharacteristic
643C22#1681143811yescharacteristic
644C4—22no—
645C4—22no—
646C4—22no—
647C4—41no—
648C4—41no—
649C4—81no—
650C4—81no—
651C4—81no—
652C4—4241no—
653C4—4241no—
654C4—4241no—
655C4—4241no—
656C4#1681132511yescharacteristic
657C4#1681132511yescharacteristic
658C4#1681132511yescharacteristic
659C4#1681132511yescharacteristic
660C2—42no—
661C2—161no—
662C2#2027920811yescharacteristic
663C2#2028468911yescharacteristic
664C2#2028470011yescharacteristic
665C2#2028506911yescharacteristic
666C2#2028508011yescharacteristic
667C2#2028580311yescharacteristic
668C2#2028581811yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2620C2
socleC23×C53C22×D4
prime radicalC2×#121×C53C22
hypercenter#2620D53
hypocenterC53#27984
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2620

Complements

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

NH ≅ G/Ntype
1. #20277338C2semidirect
2. #20276133C2semidirect
3. #20276128C2semidirect
4. #20276123C2semidirect
5. #20276118C2semidirect
6. #20275532C2semidirect
7. #20275527C2semidirect
10. #20275028C2semidirect
11. #20275023C2semidirect
12. #20274850C2semidirect
13. #20274842C2semidirect
14. #20274481C2semidirect
393. C53#27984semidirect

Maximal subgroups

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

subgroupindex
#202773382
#202761332
#202761282
#202761232
#202761182
#202755322
#202755272
#202750382
#202750332
#202750282
#202750232
#202748502
#202748422
#202744812
C53×#26202
#2798453