Finite Group Databasebeta

← #27428441

Subgroups — #27428441

Order 13568.

754 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20279012C211yesmaximal, characteristic
2#20276282C211yesmaximal, characteristic, complemented
3#20276197C211yesmaximal, characteristic, complemented
4#20276151C211yesmaximal, characteristic, complemented
5#20276141C211yesmaximal, characteristic, complemented
6#20275307C211yesmaximal, characteristic, complemented
7#20275286C211yesmaximal, characteristic, complemented
8#20275008C211yesmaximal, characteristic
9#20274914C211yesmaximal, characteristic, complemented
10#20274903C211yesmaximal, characteristic, complemented
11#20274807C211yesmaximal, characteristic, complemented
12#20274805C211yesmaximal, characteristic, complemented
13#20274742C211yesmaximal, characteristic, complemented
14#20274431C211yesmaximal, characteristic, complemented
15C53×#2352C211yesmaximal, characteristic
16C23×#88322C2211yescharacteristic
17C2×#13133176C2211yescharacteristic
18C2×#13133150C2211yescharacteristic
19C2×#13133150C2211yescharacteristic
20C2×#13133150C2211yescharacteristic
21C2×#13133148C2211yescharacteristic
22C2×C22⋊#88322C2211yescharacteristic
23C2×C22⋊#88322C2211yescharacteristic
24C2×C22⋊#88322C2211yescharacteristic
25#16810551—21no—
26#16810537—21no—
27#16810521C2211yescharacteristic
28#16810521C2211yescharacteristic
29#16810511—21no—
30#16810504—21no—
31#16810483C2211yescharacteristic
32#16810483C2211yescharacteristic
33#16810483C2211yescharacteristic
34#16810385C2211yescharacteristic
35#16810385C2211yescharacteristic
36#16810385C2211yescharacteristic
37#16810385C2211yescharacteristic
38#16810385C2211yescharacteristic
39#16810385C2211yescharacteristic
40#16810385C2211yescharacteristic
41#16810385C2211yescharacteristic
42#16810385C2211yescharacteristic
43#16810385C2211yescharacteristic
44#16810385C2211yescharacteristic
45#16810385C2211yescharacteristic
46#16810378C2211yescharacteristic
47#16810378C2211yescharacteristic
48C2×C4×D4×C53C2211yescharacteristic
49C2×#117×C53C2211yescharacteristic
50C22×C4⋊C4×C53C2211yescharacteristic
51C53×#388—21no—
52C53×#387—21no—
53C53×#386—21no—
54C53×#382C2211yescharacteristic
55C53×#380C2211yescharacteristic
56C53×#380C2211yescharacteristic
57C4×C4⋊C4×C53C2211yescharacteristic
58C4×C22⋊C4×C53—21no—
59C22×#88322—22no—
60C22×#88322—22no—
61C22×#88322—41no—
62C22×#88322C2311yescharacteristic
63C22×#88322C2311yescharacteristic
64C22×#88322C2311yescharacteristic
65C22×#88322C2311yescharacteristic
66#13133176—22no—
67#13133176—22no—
68C2×C4⋊Dic53C2311yescharacteristic
69C2×C4⋊Dic53C2311yescharacteristic
70C2×C4⋊Dic53C2311yescharacteristic
71C2×C4⋊Dic53C2311yescharacteristic
72C2×#11719919—21no—
73C2×#11719919—21no—
74C2×C22⋊Dic53—21no—
75C2×C22⋊Dic53—21no—
76C2×C22⋊Dic53—21no—
77C2×C22⋊Dic53—21no—
78C2×C4×Dic53—21no—
79#13133150—22no—
80#13133150—22no—
81#13133150—41no—
82#13133148—41no—
83C22⋊#88322—22no—
84C22⋊#88322—22no—
85C22⋊#88322—41no—
86#13133124—21no—
87#13133124—21no—
88#13133124—21no—
89#13133124—21no—
90#13133124—21no—
91C22×D4×C53D411yescharacteristic
92C23×C4×C53—21no—
93C23×C4×C53C2311yescharacteristic
94C23×C4×C53C2311yescharacteristic
95C4×D4×C53—22no—
96C4×D4×C53—22no—
97#117×C53—22no—
98#117×C53—41no—
99C2×C4⋊C4×C53—21no—
100C2×C4⋊C4×C53—22no—
101C2×C4⋊C4×C53—41no—
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×C53D411yescharacteristic
111C2×C22⋊C4×C53D411yescharacteristic
112C2×C42×C53—21no—
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—42no—
140C22×C4×C53—42no—
141C22×C4×C53—81no—
142C22×C4×C53C2×D411yescharacteristic
143C22×C4×C53C2×D411yescharacteristic
144C22×C4×C53C2×D411yescharacteristic
145C22×C4×C53C2×D411yescharacteristic
146C22×C4×C53#4112yes—
147C22×C4×C53#4112yes—
148C22×C4×C53C2411yescharacteristic
149C22×Dic53—21no—
150C22×Dic53—21no—
151C22×Dic53—21no—
152C22×Dic53—21no—
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—42no—
162C4⋊C4×C53#4112yes—
163C22⋊C4×C53—22no—
164C22⋊C4×C53—22no—
165C22⋊C4×C53—42no—
166C22⋊C4×C53—42no—
167C22⋊C4×C53—81no—
168C42×C53—21no—
169C42×C53—21no—
170C42×C53—42no—
171C42×C53—81no—
172C22⋊Dic53—42no—
173C22⋊Dic53—42no—
174C22⋊Dic53—81no—
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—21no—
184#11719919—42no—
185#11719919—81no—
186C4×Dic53—42no—
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—44no—
199C2×#88322#4112yes—
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#12312yes—
213C23×C53#13512yes—
214C23×C53C22×D411yescharacteristic
215D4×C53—44no—
216C2×C4×C53—21no—
217C2×C4×C53—21no—
218C2×C4×C53—22no—
219C2×C4×C53—22no—
220C2×C4×C53—22no—
221C2×C4×C53—22no—
222C2×C4×C53—21no—
223C2×C4×C53—21no—
224C2×C4×C53—21no—
225C2×C4×C53—21no—
226C2×C4×C53—22no—
227C2×C4×C53—21no—
228C2×C4×C53—21no—
229C2×C4×C53—21no—
230C2×C4×C53—21no—
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×C53—41no—
246C2×C4×C53—41no—
247C2×C4×C53—84no—
248C2×C4×C53#12312yes—
249C2×C4×C53C2×#4111yescharacteristic
250C2×C4×C53C2×#4111yescharacteristic
251C2×C4×C53C2×#4111yescharacteristic
252C2×C4×C53C2×#4111yescharacteristic
253C2×C4×C53H(5,𝔽2)11yescharacteristic
254C2×C4×C532−1+411yescharacteristic
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—41no—
266C2×Dic53—41no—
267#88322—22no—
268#88322—41no—
269#88322—41no—
270#88322—41no—
271#24934—531nomaximal, Sylow
272C22×C53—22no—
273C22×C53—22no—
274C22×C53—22no—
275C22×C53—22no—
276C22×C53—22no—
277C22×C53—22no—
278C22×C53—41no—
279C22×C53—41no—
280C22×C53—41no—
281C22×C53—41no—
282C22×C53—41no—
283C22×C53—41no—
284C22×C53—41no—
285C22×C53—41no—
286C22×C53C2×#12311yescharacteristic
287C22×C53#54611yescharacteristic
288C22×C53#54711yescharacteristic
289C22×C53C2×#13511yescharacteristic
290C22×C53#57511yescharacteristic
291C22×C53#57611yescharacteristic
292C22×C53#57811yescharacteristic
293C4×C53—22no—
294C4×C53—41no—
295C4×C53—41no—
296C4×C53—41no—
297C4×C53—42no—
298C4×C53—81no—
299C4×C53—81no—
300C4×C53—81no—
301C4×C53#52911yescharacteristic
302C4×C53#53911yescharacteristic
303C4×C53#55111yescharacteristic
304C4×C53#55211yescharacteristic
305Dic53—81no—
306Dic53—81no—
307Dic53—81no—
308Dic53—81no—
309#2485—531no—
310#2352D5311yescharacteristic
311#2101—531no—
312#2085—531no—
313#2077—531no—
314#2075—531no—
315#1921—531no—
316#1918—531no—
317#1869—531no—
318#1854—531no—
319#1852—531no—
320#1825—531no—
321#1824—531no—
322#1808—531no—
323#1735—531no—
324C2×C53—42no—
325C2×C53—42no—
326C2×C53#241911yescharacteristic
327C2×C53#309111yescharacteristic
328C2×C53#309811yescharacteristic
329C2×C53#330011yescharacteristic
330C2×C53#331911yescharacteristic
331C2×C53#333911yescharacteristic
332C2×C53#334911yescharacteristic
333C23×C8—531no—
334C2×C4×D4C2×D5311yescharacteristic
335C2×#117—531no—
336C2×#117C2×D5311yescharacteristic
337C22×C4⋊C4C2×D5311yescharacteristic
338C2×#107—531no—
339C2×#107—531no—
340C2×#107—531no—
341C2×#106—531no—
342C2×C22⋊C8—531no—
343C2×C22⋊C8—531no—
344C2×C22⋊C8—531no—
345#397—1061no—
346#394—1061no—
347#389—531no—
348#389—531no—
349#388—21no—
350#388—1061no—
351#387—21no—
352#387—1061no—
353#386—21no—
354#382C2×D5311yescharacteristic
355#380—531no—
356#380—531no—
357#380—531no—
358#380C2×D5311yescharacteristic
359#380C2×D5311yescharacteristic
360C4×C4⋊C4C2×D5311yescharacteristic
361C4×C22⋊C4—21no—
362#340—531no—
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#336—531no—
375#336—531no—
376C53#2493411yescharacteristic, complemented, Sylow
377C22×D4C22⋊D5311yescharacteristic
378C23×C4—21no—
379C23×C4C22×D5311yescharacteristic
380C23×C4C22×D5311yescharacteristic
381C22×C8—531no—
382C22×C8—531no—
383C22×C8—531no—
384C22×C8—531no—
385C22×C8—1062no—
386C22×C8—1062no—
387C22×C8—2121no—
388C4×D4—22no—
389C4×D4—22no—
390#117—22no—
391#117—41no—
392#117—1062no—
393#117—1062no—
394C2×C4⋊C4—22no—
395C2×C4⋊C4—21no—
396C2×C4⋊C4—41no—
397C2×C4⋊C4—531no—
398C2×C4⋊C4—531no—
399C2×C4⋊C4—531no—
400C2×C4⋊C4—531no—
401C2×C4⋊C4—1061no—
402C2×C4⋊C4—1061no—
403C2×C4⋊C4C22⋊D5311yescharacteristic
404C2×C4⋊C4C22×D5311yescharacteristic
405C2×C4⋊C4C22×D5311yescharacteristic
406C2×C4⋊C4C22×D5311yescharacteristic
407C2×C4⋊C4C22×D5311yescharacteristic
408C2×C22⋊C4—21no—
409C2×C22⋊C4—21no—
410C2×C22⋊C4—21no—
411C2×C22⋊C4—1061no—
412C2×C22⋊C4—1061no—
413C2×C22⋊C4—1061no—
414C2×C22⋊C4—1061no—
415C2×C22⋊C4C22⋊D5311yescharacteristic
416C2×C22⋊C4C22⋊D5311yescharacteristic
417C2×C42—21no—
418C2×C42—21no—
419C2×C42—1061no—
420C2×C42C22×D5311yescharacteristic
421#107—1062no—
422#107—1062no—
423#107—2121no—
424#106—2121no—
425C22⋊C8—1062no—
426C22⋊C8—1062no—
427C22⋊C8—2121no—
428#95—21no—
429#95—21no—
430#95—21no—
431#95—21no—
432#95—21no—
433#95—1061no—
434#95—1061no—
435#95—1061no—
436#95—1061no—
437#95—1061no—
438C24C2×C22⋊D5311yescharacteristic
439C24C2×C22⋊D5311yescharacteristic
440C2×D4—22no—
441C2×D4—22no—
442C2×D4—41no—
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—42no—
458C22×C4—42no—
459C22×C4—81no—
460C22×C4—1061no—
461C22×C4—1061no—
462C22×C4—1061no—
463C22×C4—1061no—
464C22×C4#1171994512yes—
465C22×C4#1171994512yes—
466C22×C4C2×C22⋊D5311yescharacteristic
467C22×C4C2×C22⋊D5311yescharacteristic
468C22×C4C2×C22⋊D5311yescharacteristic
469C22×C4C2×C22⋊D5311yescharacteristic
470C22×C4C23×D5311yescharacteristic
471C2×C8—532no—
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—
482C2×C8—1061no—
483C2×C8—2124no—
484C4⋊C4—22no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—42no—
493C4⋊C4—1061no—
494C4⋊C4—1061no—
495C4⋊C4—1061no—
496C4⋊C4—1061no—
497C4⋊C4—1061no—
498C4⋊C4—1061no—
499C4⋊C4—1061no—
500C4⋊C4—1061no—
501C4⋊C4—4241no—
502C4⋊C4—2122no—
503C4⋊C4#1171994712yes—
504C22⋊C4—22no—
505C22⋊C4—22no—
506C22⋊C4—42no—
507C22⋊C4—42no—
508C22⋊C4—81no—
509C22⋊C4—4241no—
510C22⋊C4—4241no—
511C22⋊C4—2122no—
512C22⋊C4—2122no—
513C42—21no—
514C42—21no—
515C42—42no—
516C42—81no—
517C42—2122no—
518C23—21no—
519C23—21no—
520C23—21no—
521C23—21no—
522C23—21no—
523C23—21no—
524C23—21no—
525C23—21no—
526C23—21no—
527C23—21no—
528C23—21no—
529C23—21no—
530C23#1313320012yes—
531C23#1313323712yes—
532C23C22×C22⋊D5311yescharacteristic
533D4—44no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
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—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—84no—
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—2121no—
576C2×C4—2121no—
577C2×C4—2121no—
578C2×C4#1313320012yes—
579C2×C4#41×D5311yescharacteristic
580C2×C4C2×#1171994711yescharacteristic
581C2×C4C2×#1171994511yescharacteristic
582C2×C4C2×#1171994511yescharacteristic
583C2×C4#1313326211yescharacteristic
584C2×C4#1313326411yescharacteristic
585C8—1062no—
586C8—2121no—
587C8—2121no—
588C8—2121no—
589C22—22no—
590C22—22no—
591C22—22no—
592C22—22no—
593C22—22no—
594C22—22no—
595C22—41no—
596C22—41no—
597C22—41no—
598C22—41no—
599C22—41no—
600C22—41no—
601C22—41no—
602C22—41no—
603C22C2×#1313320011yescharacteristic
604C22#1681126711yescharacteristic
605C22#1681127211yescharacteristic
606C22C2×#1313323711yescharacteristic
607C22#1681142111yescharacteristic
608C22#1681142611yescharacteristic
609C22#1681143811yescharacteristic
610C4—22no—
611C4—41no—
612C4—42no—
613C4—41no—
614C4—41no—
615C4—81no—
616C4—81no—
617C4—81no—
618C4—4241no—
619C4—4241no—
620C4—4241no—
621C4—4241no—
622C4#1681115311yescharacteristic
623C4#1681121511yescharacteristic
624C4#1681130211yescharacteristic
625C4#1681131311yescharacteristic
626C2—42no—
627C2—42no—
628C2#2027818611yescharacteristic
629C2#2028462811yescharacteristic
630C2#2028469311yescharacteristic
631C2#2028660211yescharacteristic
632C2#2028687111yescharacteristic
633C2#2028711111yescharacteristic
634C2#2028724511yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2352C2
socleC23×C53C22×D4
prime radicalC22×D4×C53D4
hypercenter#2352D53
hypocenterC53#24934
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #2352

Complements

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

NH ≅ G/Ntype
2. #20276282C2semidirect
3. #20276197C2semidirect
4. #20276151C2semidirect
5. #20276141C2semidirect
6. #20275307C2semidirect
7. #20275286C2semidirect
9. #20274914C2semidirect
10. #20274903C2semidirect
11. #20274807C2semidirect
12. #20274805C2semidirect
13. #20274742C2semidirect
14. #20274431C2semidirect
376. C53#24934semidirect

Maximal subgroups

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

subgroupindex
#202790122
#202762822
#202761972
#202761512
#202761412
#202753072
#202752862
#202750082
#202749142
#202749032
#202748072
#202748052
#202747422
#202744312
C53×#23522
#2493453