Finite Group Databasebeta

← #27429101

Subgroups — #27429101

Order 13568.

756 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20277664C211yesmaximal, characteristic
2#20276152C211yesmaximal, characteristic, complemented
3#20276146C211yesmaximal, characteristic, complemented
4#20275915C211yesmaximal, characteristic, complemented
5#20275908C211yesmaximal, characteristic, complemented
6#20275514C211yesmaximal, characteristic
7#20275481C211yesmaximal, characteristic, complemented
8#20275028C211yesmaximal, characteristic, complemented
9#20275028C211yesmaximal, characteristic, complemented
10#20274848C211yesmaximal, characteristic, complemented
11#20274807C211yesmaximal, characteristic, complemented
12#20274807C211yesmaximal, characteristic, complemented
13#20274720C211yesmaximal, characteristic, complemented
14#20274545C211yesmaximal, characteristic, complemented
15C53×#2536C211yesmaximal, characteristic
16C2×#13133176C2211yescharacteristic
17C2×#13133176C2211yescharacteristic
18C2×#13133150C2211yescharacteristic
19C2×#13133148C2211yescharacteristic
20C2×C4⋊#88322C2211yescharacteristic
21C2×#13133141C2211yescharacteristic
22C2×#13133141C2211yescharacteristic
23C2×#13133141C2211yescharacteristic
24C2×#13133141C2211yescharacteristic
25C2×C22⋊#88322C2211yescharacteristic
26C2×C22⋊#88322C2211yescharacteristic
27C2×C22⋊#88322C2211yescharacteristic
28C2×C22⋊#88322C2211yescharacteristic
29C2×#13133128C2211yescharacteristic
30#16810526—21no—
31#16810516—21no—
32#16810516—21no—
33#16810493C2211yescharacteristic
34#16810483C2211yescharacteristic
35#16810483C2211yescharacteristic
36C4⋊C4×Dic53—21no—
37C4×C4⋊Dic53C2211yescharacteristic
38#16810385C2211yescharacteristic
39#16810385C2211yescharacteristic
40#16810385C2211yescharacteristic
41#16810385C2211yescharacteristic
42#16810385C2211yescharacteristic
43#16810385C2211yescharacteristic
44#16810385C2211yescharacteristic
45#16810385C2211yescharacteristic
46#16810385C2211yescharacteristic
47#16810385C2211yescharacteristic
48C2×C22⋊Q8×C53C2211yescharacteristic
49C2×C22⋊Q8×C53C2211yescharacteristic
50C2×C4×Q8×C53C2211yescharacteristic
51C2×C4×D4×C53C2211yescharacteristic
52C53×#398—21no—
53C53×#394—21no—
54C53×#388—21no—
55C53×#388—21no—
56C53×#382C2211yescharacteristic
57C53×#380C2211yescharacteristic
58C53×#380C2211yescharacteristic
59C22×#88322C2311yescharacteristic
60C22×#88322C2311yescharacteristic
61C22×#88322C2311yescharacteristic
62C22×#88322C2311yescharacteristic
63#13133176—22no—
64#13133176—22no—
65#13133176—41no—
66#13133176—41no—
67C2×C4⋊Dic53—21no—
68C2×C4⋊Dic53—21no—
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×C22⋊Dic53—21no—
77C2×C4×Dic53—21no—
78C2×C4×Dic53—21no—
79C2×C4×Dic53—21no—
80#13133150—41no—
81#13133148—41no—
82C4⋊#88322—22no—
83#13133141—22no—
84#13133141—22no—
85#13133141—41no—
86#13133141—41no—
87C22⋊#88322—41no—
88C22⋊#88322—41no—
89C22⋊#88322—41no—
90C22⋊#88322—41no—
91#13133128—22no—
92#13133124—21no—
93#13133124—21no—
94#13133124—21no—
95C22×Q8×C53C2311yescharacteristic
96C22×D4×C53D411yescharacteristic
97C23×C4×C53C2311yescharacteristic
98C23×C4×C53C2311yescharacteristic
99C22⋊Q8×C53—22no—
100C22⋊Q8×C53—22no—
101C22⋊Q8×C53—41no—
102C22⋊Q8×C53—41no—
103C4×Q8×C53—22no—
104C4×Q8×C53—22no—
105C4×D4×C53—22no—
106C4×D4×C53—22no—
107C2×C4⋊C4×C53—21no—
108C2×C4⋊C4×C53—21no—
109C2×C4⋊C4×C53—21no—
110C2×C4⋊C4×C53D411yescharacteristic
111C2×C4⋊C4×C53C2311yescharacteristic
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×C53D411yescharacteristic
119C2×C22⋊C4×C53D411yescharacteristic
120C2×C42×C53—21no—
121C2×C42×C53C2311yescharacteristic
122#95×C53—21no—
123#95×C53—21no—
124#95×C53—21no—
125#95×C53—21no—
126C24×C53C2×D411yescharacteristic
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#4112yes—
134C2×D4×C53—22no—
135C2×D4×C53—41no—
136C2×D4×C53—41no—
137C22×C4×C53—21no—
138C22×C4×C53—21no—
139C22×C4×C53—21no—
140C22×C4×C53—21no—
141C22×C4×C53—21no—
142C22×C4×C53—21no—
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—21no—
153C22×C4×C53C2×D411yescharacteristic
154C22×C4×C53C2×D411yescharacteristic
155C22×C4×C53C2×D411yescharacteristic
156C22×C4×C53C2×D411yescharacteristic
157C22×C4×C53C2411yescharacteristic
158C22×Dic53—21no—
159C22×Dic53—21no—
160C22×Dic53—21no—
161C22×Dic53—21no—
162C4⋊C4×C53—22no—
163C4⋊C4×C53—21no—
164C4⋊C4×C53—21no—
165C4⋊C4×C53—21no—
166C4⋊C4×C53—21no—
167C4⋊C4×C53—21no—
168C4⋊C4×C53—21no—
169C4⋊C4×C53—42no—
170C4⋊C4×C53—42no—
171C4⋊C4×C53—42no—
172C22⋊C4×C53—22no—
173C22⋊C4×C53—22no—
174C22⋊C4×C53—42no—
175C22⋊C4×C53—42no—
176C22⋊C4×C53—42no—
177C22⋊C4×C53—42no—
178C42×C53—42no—
179C42×C53#4112yes—
180C42×C53#4112yes—
181C22⋊Dic53—42no—
182C22⋊Dic53—42no—
183C22⋊Dic53—81no—
184C22⋊Dic53—81no—
185C4⋊Dic53—21no—
186C4⋊Dic53—21no—
187C4⋊Dic53—21no—
188C4⋊Dic53—21no—
189C4⋊Dic53—21no—
190C4⋊Dic53—21no—
191C4⋊Dic53—21no—
192C4⋊Dic53—42no—
193C4⋊Dic53—81no—
194C4⋊Dic53#4112yes—
195C4×Dic53—42no—
196C4×Dic53—42no—
197C4×Dic53—42no—
198C2×#88322—21no—
199C2×#88322—21no—
200C2×#88322—21no—
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—
210C23×C53—21no—
211C23×C53—21no—
212C23×C53—21no—
213C23×C53—21no—
214C23×C53—21no—
215C23×C53—21no—
216C23×C53—21no—
217C23×C53—21no—
218C23×C53—21no—
219C23×C53—21no—
220C23×C53—21no—
221C23×C53—21no—
222C23×C53—21no—
223C23×C53—21no—
224C23×C53C22×D411yescharacteristic
225Q8×C53—22no—
226Q8×C53—22no—
227Q8×C53—41no—
228Q8×C53—41no—
229D4×C53—44no—
230C2×C4×C53—21no—
231C2×C4×C53—21no—
232C2×C4×C53—21no—
233C2×C4×C53—21no—
234C2×C4×C53—21no—
235C2×C4×C53—21no—
236C2×C4×C53—21no—
237C2×C4×C53—21no—
238C2×C4×C53—21no—
239C2×C4×C53—21no—
240C2×C4×C53—22no—
241C2×C4×C53—41no—
242C2×C4×C53—41no—
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×C53—41no—
256C2×C4×C53—41no—
257C2×C4×C53—41no—
258C2×C4×C53—41no—
259C2×C4×C53—41no—
260C2×C4×C53#12312yes—
261C2×C4×C53#12312yes—
262C2×C4×C53C2×#4111yescharacteristic
263C2×C4×C53C2×#4111yescharacteristic
264C2×C4×C53C2×#4111yescharacteristic
265C2×C4×C53C2×#4111yescharacteristic
266C2×C4×C53H(5,𝔽2)11yescharacteristic
267C2×C4×C53H(5,𝔽2)11yescharacteristic
268C2×Dic53—22no—
269C2×Dic53—41no—
270C2×Dic53—41no—
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—
280#88322—41no—
281#88322—41no—
282#88322—41no—
283#88322—41no—
284#24980—531nomaximal, Sylow
285C22×C53—41no—
286C22×C53—41no—
287C22×C53—41no—
288C22×C53—41no—
289C22×C53—41no—
290C22×C53—41no—
291C22×C53—41no—
292C22×C53—41no—
293C22×C53—41no—
294C22×C53—41no—
295C22×C53—41no—
296C22×C53—41no—
297C22×C53—41no—
298C22×C53—41no—
299C22×C53C2×#12311yescharacteristic
300C22×C53#54611yescharacteristic
301C22×C53#54611yescharacteristic
302C22×C53#57511yescharacteristic
303C22×C53#57511yescharacteristic
304C22×C53#57711yescharacteristic
305C22×C53#57811yescharacteristic
306C4×C53—22no—
307C4×C53—22no—
308C4×C53—41no—
309C4×C53—41no—
310C4×C53—42no—
311C4×C53—81no—
312C4×C53—81no—
313C4×C53—81no—
314C4×C53#52911yescharacteristic
315C4×C53#54011yescharacteristic
316C4×C53#55311yescharacteristic
317C4×C53#55511yescharacteristic
318Dic53—42no—
319Dic53—81no—
320Dic53—81no—
321Dic53—81no—
322#2536D5311yescharacteristic
323#2354—531no—
324#2077—531no—
325#2076—531no—
326#2036—531no—
327#2035—531no—
328#1955—531no—
329#1949—531no—
330#1873—531no—
331#1873—531no—
332#1836—531no—
333#1825—531no—
334#1825—531no—
335#1803—531no—
336#1763—531no—
337C2×C53—81no—
338C2×C53—81no—
339C2×C53#242011yescharacteristic
340C2×C53#309311yescharacteristic
341C2×C53#309711yescharacteristic
342C2×C53#331311yescharacteristic
343C2×C53#331511yescharacteristic
344C2×C53#331711yescharacteristic
345C2×C53#332111yescharacteristic
346C2×C22⋊Q8C2×D5311yescharacteristic
347C2×C22⋊Q8C2×D5311yescharacteristic
348C2×C4×Q8C2×D5311yescharacteristic
349C2×C4×D4C2×D5311yescharacteristic
350C2×#117—531no—
351C2×#117—531no—
352C2×#107—531no—
353C2×#106—531no—
354C2×C4⋊C8—531no—
355C2×#103—531no—
356C2×#103—531no—
357C2×#103—531no—
358C2×#103—531no—
359C2×C22⋊C8—531no—
360C2×C22⋊C8—531no—
361C2×C22⋊C8—531no—
362C2×C22⋊C8—531no—
363C2×#97—531no—
364#398—21no—
365#394—21no—
366#390—1061no—
367#388—21no—
368#388—21no—
369#388—1061no—
370#388—1061no—
371#384—531no—
372#382C2×D5311yescharacteristic
373#380—531no—
374#380—531no—
375#380C2×D5311yescharacteristic
376#380C2×D5311yescharacteristic
377C4×C4⋊C4—531no—
378C4×C4⋊C4—1061no—
379#340—531no—
380#340—531no—
381#340—531no—
382#340—531no—
383#340—531no—
384#340—531no—
385#340—531no—
386#340—531no—
387#340—531no—
388#340—531no—
389C53#2498011yescharacteristic, complemented, Sylow
390C22×Q8C22×D5311yescharacteristic
391C22×D4C22⋊D5311yescharacteristic
392C23×C4C22×D5311yescharacteristic
393C23×C4C22×D5311yescharacteristic
394C22×C8—531no—
395C22×C8—531no—
396C22×C8—531no—
397C22×C8—531no—
398C22⋊Q8—22no—
399C22⋊Q8—22no—
400C22⋊Q8—41no—
401C22⋊Q8—41no—
402C4×Q8—22no—
403C4×Q8—22no—
404C4×D4—22no—
405C4×D4—22no—
406#117—1062no—
407#117—1062no—
408#117—2121no—
409#117—2121no—
410C2×C4⋊C4—21no—
411C2×C4⋊C4—21no—
412C2×C4⋊C4—21no—
413C2×C4⋊C4—531no—
414C2×C4⋊C4—531no—
415C2×C4⋊C4—531no—
416C2×C4⋊C4—531no—
417C2×C4⋊C4—1061no—
418C2×C4⋊C4—1061no—
419C2×C4⋊C4C22⋊D5311yescharacteristic
420C2×C4⋊C4C22×D5311yescharacteristic
421C2×C4⋊C4C22×D5311yescharacteristic
422C2×C4⋊C4C22×D5311yescharacteristic
423C2×C22⋊C4—21no—
424C2×C22⋊C4—21no—
425C2×C22⋊C4—21no—
426C2×C22⋊C4—21no—
427C2×C22⋊C4—1061no—
428C2×C22⋊C4—1061no—
429C2×C22⋊C4—1061no—
430C2×C22⋊C4—1061no—
431C2×C22⋊C4C22⋊D5311yescharacteristic
432C2×C22⋊C4C22⋊D5311yescharacteristic
433C2×C42—21no—
434C2×C42—1061no—
435C2×C42—1061no—
436C2×C42—1061no—
437C2×C42C22×D5311yescharacteristic
438#107—2121no—
439#106—2121no—
440C4⋊C8—1062no—
441#103—1062no—
442#103—1062no—
443#103—2121no—
444#103—2121no—
445C22⋊C8—2121no—
446C22⋊C8—2121no—
447C22⋊C8—2121no—
448C22⋊C8—2121no—
449#97—1062no—
450#95—21no—
451#95—21no—
452#95—21no—
453#95—21no—
454#95—1061no—
455#95—1061no—
456#95—1061no—
457C24C2×C22⋊D5311yescharacteristic
458C24C2×C22⋊D5311yescharacteristic
459C2×Q8—21no—
460C2×Q8—21no—
461C2×Q8—21no—
462C2×Q8—21no—
463C2×Q8—21no—
464C2×Q8#1171994712yes—
465C2×D4—22no—
466C2×D4—41no—
467C2×D4—41no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—1061no—
485C22×C4—1061no—
486C22×C4—1061no—
487C22×C4—1061no—
488C22×C4C2×C22⋊D5311yescharacteristic
489C22×C4C2×C22⋊D5311yescharacteristic
490C22×C4C2×C22⋊D5311yescharacteristic
491C22×C4C2×C22⋊D5311yescharacteristic
492C22×C4C23×D5311yescharacteristic
493C2×C8—1061no—
494C2×C8—1061no—
495C2×C8—1061no—
496C2×C8—1061no—
497C2×C8—1061no—
498C2×C8—1061no—
499C2×C8—1061no—
500C2×C8—1061no—
501C2×C8—1061no—
502C2×C8—1061no—
503C2×C8—1061no—
504C2×C8—1061no—
505C4⋊C4—22no—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—21no—
512C4⋊C4—42no—
513C4⋊C4—42no—
514C4⋊C4—42no—
515C4⋊C4—532no—
516C4⋊C4—1061no—
517C4⋊C4—1061no—
518C4⋊C4—1061no—
519C4⋊C4—1061no—
520C4⋊C4—1061no—
521C4⋊C4—1061no—
522C4⋊C4—1061no—
523C4⋊C4—4241no—
524C4⋊C4—2122no—
525C22⋊C4—22no—
526C22⋊C4—22no—
527C22⋊C4—42no—
528C22⋊C4—42no—
529C22⋊C4—42no—
530C22⋊C4—42no—
531C22⋊C4—4241no—
532C22⋊C4—4241no—
533C22⋊C4—2122no—
534C22⋊C4—2122no—
535C42—42no—
536C42—2122no—
537C42—2122no—
538C42—2122no—
539C42#1171994712yes—
540C42#1171994712yes—
541C23—21no—
542C23—21no—
543C23—21no—
544C23—21no—
545C23—21no—
546C23—21no—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23—21no—
552C23—21no—
553C23—21no—
554C23—21no—
555C23C22×C22⋊D5311yescharacteristic
556Q8—22no—
557Q8—22no—
558Q8—41no—
559Q8—41no—
560D4—44no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—22no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
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—1062no—
592C2×C4—2121no—
593C2×C4—2121no—
594C2×C4—2121no—
595C2×C4—2121no—
596C2×C4—2121no—
597C2×C4—2121no—
598C2×C4—2121no—
599C2×C4—2121no—
600C2×C4—2121no—
601C2×C4—2121no—
602C2×C4—2121no—
603C2×C4#1313319812yes—
604C2×C4#1313319812yes—
605C2×C4#41×D5311yescharacteristic
606C2×C4C2×#1171994711yescharacteristic
607C2×C4C2×#1171994711yescharacteristic
608C2×C4C2×#1171994711yescharacteristic
609C2×C4#1313326211yescharacteristic
610C2×C4#1313326311yescharacteristic
611C8—2121no—
612C8—2121no—
613C8—2121no—
614C8—2121no—
615C22—41no—
616C22—41no—
617C22—41no—
618C22—41no—
619C22—41no—
620C22—41no—
621C22—41no—
622C22—41no—
623C22—41no—
624C22—41no—
625C22—41no—
626C22—41no—
627C22—41no—
628C22—41no—
629C22C2×#1313319811yescharacteristic
630C22#1681126111yescharacteristic
631C22#1681126711yescharacteristic
632C22#1681142111yescharacteristic
633C22#1681142111yescharacteristic
634C22#1681143111yescharacteristic
635C22#1681143811yescharacteristic
636C4—22no—
637C4—22no—
638C4—41no—
639C4—41no—
640C4—42no—
641C4—81no—
642C4—81no—
643C4—81no—
644C4—4241no—
645C4—4241no—
646C4—4241no—
647C4—2122no—
648C4#1681115411yescharacteristic
649C4#1681122611yescharacteristic
650C4#1681132511yescharacteristic
651C4#1681133711yescharacteristic
652C2—81no—
653C2—81no—
654C2#2027820311yescharacteristic
655C2#2028464411yescharacteristic
656C2#2028467811yescharacteristic
657C2#2028678111yescharacteristic
658C2#2028680511yescharacteristic
659C2#2028683511yescharacteristic
660C2#2028690111yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2536C2
socleC23×C53C22×D4
prime radicalC22×D4×C53D4
hypercenter#2536D53
hypocenterC53#24980
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #2536

Complements

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

NH ≅ G/Ntype
2. #20276152C2semidirect
3. #20276146C2semidirect
4. #20275915C2semidirect
5. #20275908C2semidirect
7. #20275481C2semidirect
8. #20275028C2semidirect
9. #20275028C2semidirect
10. #20274848C2semidirect
11. #20274807C2semidirect
12. #20274807C2semidirect
13. #20274720C2semidirect
14. #20274545C2semidirect
389. C53#24980semidirect

Maximal subgroups

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

subgroupindex
#202776642
#202761522
#202761462
#202759152
#202759082
#202755142
#202754812
#202750282
#202748482
#202748072
#202747202
#202745452
C53×#25362
#2498053