Finite Group Databasebeta

← #27428505

Subgroups — #27428505

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#20277644C211yesmaximal, characteristic
3#20276284C211yesmaximal, characteristic, complemented
4#20276198C211yesmaximal, characteristic, complemented
5#20276148C211yesmaximal, characteristic, complemented
6#20276143C211yesmaximal, characteristic, complemented
7#20275301C211yesmaximal, characteristic, complemented
8#20275294C211yesmaximal, characteristic, complemented
9#20274916C211yesmaximal, characteristic, complemented
10#20274910C211yesmaximal, characteristic, complemented
11#20274811C211yesmaximal, characteristic, complemented
12#20274802C211yesmaximal, characteristic, complemented
13#20274740C211yesmaximal, characteristic, complemented
14#20274432C211yesmaximal, characteristic, complemented
15C53×#1869C211yesmaximal, characteristic
16C2×#13133176C2211yescharacteristic
17C2×#13133176C2211yescharacteristic
18C22×C4⋊Dic53C2211yescharacteristic
19C2×#13133151C2211yescharacteristic
20C2×#13133149C2211yescharacteristic
21C2×#13133149C2211yescharacteristic
22C2×#13133149C2211yescharacteristic
23#16810551—21no—
24#16810537—21no—
25#16810521C2211yescharacteristic
26#16810521C2211yescharacteristic
27#16810516—21no—
28#16810511—21no—
29#16810509—21no—
30#16810504—21no—
31#16810503—21no—
32#16810489C2211yescharacteristic
33#16810483C2211yescharacteristic
34#16810483C2211yescharacteristic
35#16810483C2211yescharacteristic
36#16810483C2211yescharacteristic
37#16810483C2211yescharacteristic
38C4×C4⋊Dic53C2211yescharacteristic
39C22⋊C4×Dic53—21no—
40#16810386C2211yescharacteristic
41#16810386C2211yescharacteristic
42#16810386C2211yescharacteristic
43#16810386C2211yescharacteristic
44#16810386C2211yescharacteristic
45#16810386C2211yescharacteristic
46#16810386C2211yescharacteristic
47#16810386C2211yescharacteristic
48#16810386C2211yescharacteristic
49#16810386C2211yescharacteristic
50#16810386C2211yescharacteristic
51#16810386C2211yescharacteristic
52C23×C8×C53C2211yescharacteristic
53C2×C4×D4×C53C2211yescharacteristic
54C2×C22⋊C8×C53C2211yescharacteristic
55C2×C22⋊C8×C53C2211yescharacteristic
56C2×C22⋊C8×C53C2211yescharacteristic
57C53×#336C2211yescharacteristic
58C53×#336C2211yescharacteristic
59C23×Dic53—21no—
60#13133176—22no—
61#13133176—22no—
62#13133176—22no—
63#13133176—41no—
64C2×C4⋊Dic53—21no—
65C2×C4⋊Dic53—22no—
66C2×C4⋊Dic53—41no—
67C2×C4⋊Dic53C2311yescharacteristic
68C2×C4⋊Dic53C2311yescharacteristic
69C2×C4⋊Dic53C2311yescharacteristic
70C2×C4⋊Dic53C2311yescharacteristic
71C2×C4⋊Dic53C2311yescharacteristic
72C2×C4⋊Dic53C2311yescharacteristic
73C2×C4⋊Dic53C2311yescharacteristic
74C2×C4⋊Dic53C2311yescharacteristic
75C2×#11719919—21no—
76C2×#11719919—21no—
77C2×C22⋊Dic53—21no—
78C2×C22⋊Dic53—21no—
79C2×C22⋊Dic53—21no—
80C2×C22⋊Dic53—21no—
81C2×C22⋊Dic53—21no—
82C2×C22⋊Dic53—21no—
83C2×C22⋊Dic53—21no—
84C2×C4×Dic53—21no—
85C2×C4×Dic53—21no—
86C2×C4×Dic53—21no—
87#13133151—41no—
88#13133149—22no—
89#13133149—22no—
90#13133149—41no—
91#13133124—21no—
92#13133124—21no—
93#13133124—21no—
94#13133124—21no—
95#13133124—21no—
96#13133124—21no—
97#13133124—21no—
98#13133124—21no—
99#13133124—21no—
100#13133124—21no—
101C22×D4×C53D411yescharacteristic
102C23×C4×C53C2311yescharacteristic
103C23×C4×C53C2311yescharacteristic
104C22×C8×C53—22no—
105C22×C8×C53—22no—
106C22×C8×C53—41no—
107C22×C8×C53C2311yescharacteristic
108C22×C8×C53C2311yescharacteristic
109C22×C8×C53C2311yescharacteristic
110C22×C8×C53C2311yescharacteristic
111C4×D4×C53—22no—
112C4×D4×C53—22no—
113C2×C4⋊C4×C53D411yescharacteristic
114C2×C22⋊C4×C53D411yescharacteristic
115C2×C22⋊C4×C53D411yescharacteristic
116C2×C42×C53C2311yescharacteristic
117C22⋊C8×C53—22no—
118C22⋊C8×C53—22no—
119C22⋊C8×C53—41no—
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×C53C2×D411yescharacteristic
136C22×C4×C53C2×D411yescharacteristic
137C22×C4×C53C2×D411yescharacteristic
138C22×C4×C53C2×D411yescharacteristic
139C22×C4×C53#4112yes—
140C22×C4×C53#4112yes—
141C22×C4×C53C2411yescharacteristic
142C22×Dic53—21no—
143C22×Dic53—21no—
144C22×Dic53—21no—
145C22×Dic53—21no—
146C22×Dic53—21no—
147C22×Dic53—21no—
148C22×Dic53—21no—
149C22×Dic53—21no—
150C22×Dic53—42no—
151C22×Dic53—42no—
152C22×Dic53—81no—
153C2×C8×C53—21no—
154C2×C8×C53—21no—
155C2×C8×C53—21no—
156C2×C8×C53—21no—
157C2×C8×C53—21no—
158C2×C8×C53—21no—
159C2×C8×C53—21no—
160C2×C8×C53—21no—
161C2×C8×C53—21no—
162C2×C8×C53—21no—
163C2×C8×C53—21no—
164C2×C8×C53—44no—
165C2×C8×C53#4112yes—
166C4⋊C4×C53—22no—
167C22⋊C4×C53—22no—
168C22⋊C4×C53—22no—
169C42×C53—21no—
170C42×C53—21no—
171C22⋊Dic53—42no—
172C22⋊Dic53—42no—
173C22⋊Dic53—42no—
174C22⋊Dic53—42no—
175C22⋊Dic53—81no—
176C22⋊Dic53—81no—
177C22⋊Dic53—81no—
178C4⋊Dic53—21no—
179C4⋊Dic53—21no—
180C4⋊Dic53—21no—
181C4⋊Dic53—21no—
182C4⋊Dic53—21no—
183C4⋊Dic53—21no—
184C4⋊Dic53—21no—
185C4⋊Dic53—21no—
186C4⋊Dic53—21no—
187C4⋊Dic53—21no—
188C4⋊Dic53—21no—
189C4⋊Dic53—21no—
190C4⋊Dic53—21no—
191C4⋊Dic53—21no—
192C4⋊Dic53—21no—
193C4⋊Dic53—42no—
194C4⋊Dic53#4112yes—
195#11719919—42no—
196#11719919—81no—
197C4×Dic53—42no—
198C4×Dic53—42no—
199C4×Dic53—81no—
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—21no—
227C2×C4×C53—21no—
228C2×C4×C53—21no—
229C2×C4×C53—21no—
230C2×C4×C53—21no—
231C2×C4×C53—41no—
232C2×C4×C53—41no—
233C2×C4×C53—41no—
234C2×C4×C53—41no—
235C2×C4×C53#12312yes—
236C2×C4×C53C2×#4111yescharacteristic
237C2×C4×C53C2×#4111yescharacteristic
238C2×C4×C53C2×#4111yescharacteristic
239C2×C4×C53C2×#4111yescharacteristic
240C2×C4×C53H(5,𝔽2)11yescharacteristic
241C2×C4×C532−1+411yescharacteristic
242C2×Dic53—22no—
243C2×Dic53—41no—
244C2×Dic53—41no—
245C2×Dic53—41no—
246C2×Dic53—41no—
247C2×Dic53—41no—
248C2×Dic53—41no—
249C2×Dic53—41no—
250C2×Dic53—41no—
251C2×Dic53—41no—
252C2×Dic53—41no—
253C2×Dic53—41no—
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—41no—
266C2×Dic53—84no—
267C8×C53—22no—
268C8×C53—41no—
269C8×C53—41no—
270C8×C53—41no—
271#24938—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#57711yescharacteristic
292C22×C53#57711yescharacteristic
293C4×C53—22no—
294C4×C53—41no—
295C4×C53—41no—
296C4×C53—41no—
297C4×C53#52911yescharacteristic
298C4×C53#53911yescharacteristic
299C4×C53#55111yescharacteristic
300C4×C53#55211yescharacteristic
301Dic53—42no—
302Dic53—81no—
303Dic53—81no—
304Dic53—81no—
305Dic53—81no—
306Dic53—81no—
307Dic53—81no—
308Dic53—81no—
309#2485—531no—
310#2352—531no—
311#2101—531no—
312#2085—531no—
313#2077—531no—
314#2076—531no—
315#1920—531no—
316#1919—531no—
317#1869D5311yescharacteristic
318#1854—531no—
319#1853—531no—
320#1825—531no—
321#1823—531no—
322#1807—531no—
323#1735—531no—
324C2×C53—42no—
325C2×C53—42no—
326C2×C53#241911yescharacteristic
327C2×C53#309111yescharacteristic
328C2×C53#309911yescharacteristic
329C2×C53#329611yescharacteristic
330C2×C53#332111yescharacteristic
331C2×C53#333711yescharacteristic
332C2×C53#335011yescharacteristic
333C23×C8C2×D5311yescharacteristic
334C2×C4×D4C2×D5311yescharacteristic
335C2×#117—531no—
336C2×#117—531no—
337C22×C4⋊C4—531no—
338C2×#107—531no—
339C2×#106—531no—
340C2×#106—531no—
341C2×#106—531no—
342C2×C22⋊C8C2×D5311yescharacteristic
343C2×C22⋊C8C2×D5311yescharacteristic
344C2×C22⋊C8C2×D5311yescharacteristic
345#397—1061no—
346#394—1061no—
347#389—531no—
348#389—531no—
349#388—1061no—
350#388—1061no—
351#387—1061no—
352#387—1061no—
353#386—1061no—
354#382—531no—
355#380—531no—
356#380—531no—
357#380—531no—
358#380—531no—
359#380—531no—
360C4×C4⋊C4—531no—
361C4×C22⋊C4—1061no—
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#336C2×D5311yescharacteristic
375#336C2×D5311yescharacteristic
376C53#2493811yescharacteristic, complemented, Sylow
377C22×D4D21211yescharacteristic
378C23×C4—1061no—
379C23×C4C22×D5311yescharacteristic
380C23×C4C22×D5311yescharacteristic
381C22×C8—22no—
382C22×C8—22no—
383C22×C8—41no—
384C22×C8C22×D5311yescharacteristic
385C22×C8C22×D5311yescharacteristic
386C22×C8C22×D5311yescharacteristic
387C22×C8C22×D5311yescharacteristic
388C4×D4—22no—
389C4×D4—22no—
390#117—1062no—
391#117—1062no—
392#117—1062no—
393#117—2121no—
394C2×C4⋊C4—531no—
395C2×C4⋊C4—531no—
396C2×C4⋊C4—531no—
397C2×C4⋊C4—531no—
398C2×C4⋊C4—531no—
399C2×C4⋊C4—531no—
400C2×C4⋊C4—531no—
401C2×C4⋊C4—531no—
402C2×C4⋊C4—1062no—
403C2×C4⋊C4—1061no—
404C2×C4⋊C4—1061no—
405C2×C4⋊C4—1061no—
406C2×C4⋊C4—2121no—
407C2×C4⋊C4D21211yescharacteristic
408C2×C22⋊C4—1061no—
409C2×C22⋊C4—1061no—
410C2×C22⋊C4—1061no—
411C2×C22⋊C4—1061no—
412C2×C22⋊C4—1061no—
413C2×C22⋊C4—1061no—
414C2×C22⋊C4—1061no—
415C2×C22⋊C4D21211yescharacteristic
416C2×C22⋊C4D21211yescharacteristic
417C2×C42—1061no—
418C2×C42—1061no—
419C2×C42—1061no—
420C2×C42C22×D5311yescharacteristic
421#107—2121no—
422#106—1062no—
423#106—1062no—
424#106—2121no—
425C22⋊C8—22no—
426C22⋊C8—22no—
427C22⋊C8—41no—
428#95—1061no—
429#95—1061no—
430#95—1061no—
431#95—1061no—
432#95—1061no—
433#95—1061no—
434#95—1061no—
435#95—1061no—
436#95—1061no—
437#95—1061no—
438C24C2×D21211yescharacteristic
439C24C2×D21211yescharacteristic
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—1061no—
454C22×C4—1061no—
455C22×C4—1061no—
456C22×C4—1061no—
457C22×C4—1061no—
458C22×C4—1061no—
459C22×C4—1061no—
460C22×C4—1061no—
461C22×C4—4241no—
462C22×C4—2122no—
463C22×C4—2122no—
464C22×C4C2×D21211yescharacteristic
465C22×C4C2×D21211yescharacteristic
466C22×C4C2×D21211yescharacteristic
467C22×C4C2×D21211yescharacteristic
468C22×C4#1171994712yes—
469C22×C4#1171994712yes—
470C22×C4C23×D5311yescharacteristic
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—21no—
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—44no—
483C2×C8#1171994712yes—
484C4⋊C4—22no—
485C4⋊C4—532no—
486C4⋊C4—1061no—
487C4⋊C4—1061no—
488C4⋊C4—1061no—
489C4⋊C4—1061no—
490C4⋊C4—1061no—
491C4⋊C4—1061no—
492C4⋊C4—1061no—
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—2122no—
504C22⋊C4—22no—
505C22⋊C4—22no—
506C22⋊C4—4241no—
507C22⋊C4—4241no—
508C22⋊C4—4241no—
509C22⋊C4—2122no—
510C22⋊C4—2122no—
511C22⋊C4—2122no—
512C22⋊C4—2122no—
513C42—21no—
514C42—21no—
515C42—4241no—
516C42—2122no—
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#1313320112yes—
531C23#1313323912yes—
532C23C22×D21211yescharacteristic
533D4—44no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—21no—
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—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—1062no—
554C2×C4—4244no—
555C2×C4—2121no—
556C2×C4—2121no—
557C2×C4—2121no—
558C2×C4—2121no—
559C2×C4—2121no—
560C2×C4—2121no—
561C2×C4—2121no—
562C2×C4—2121no—
563C2×C4—2121no—
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—2121no—
576C2×C4—2121no—
577C2×C4—2121no—
578C2×C4#1313320112yes—
579C2×C4#41×D5311yescharacteristic
580C2×C4C2×#1171994711yescharacteristic
581C2×C4C2×#1171994711yescharacteristic
582C2×C4C2×#1171994711yescharacteristic
583C2×C4#1313326211yescharacteristic
584C2×C4#1313326411yescharacteristic
585C8—22no—
586C8—41no—
587C8—41no—
588C8—41no—
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×#1313320111yescharacteristic
604C22#1681126811yescharacteristic
605C22#1681127711yescharacteristic
606C22C2×#1313323911yescharacteristic
607C22#1681142311yescharacteristic
608C22#1681143511yescharacteristic
609C22#1681143511yescharacteristic
610C4—22no—
611C4—41no—
612C4—41no—
613C4—41no—
614C4—4241no—
615C4—4241no—
616C4—4241no—
617C4—4241no—
618C4—4241no—
619C4—4241no—
620C4—4241no—
621C4—2122no—
622C4#1681115411yescharacteristic
623C4#1681121811yescharacteristic
624C4#1681130411yescharacteristic
625C4#1681131011yescharacteristic
626C2—42no—
627C2—42no—
628C2#2027819011yescharacteristic
629C2#2028463011yescharacteristic
630C2#2028470611yescharacteristic
631C2#2028655211yescharacteristic
632C2#2028690311yescharacteristic
633C2#2028709611yescharacteristic
634C2#2028726211yescharacteristic
subgroupG/H
centerC23C22×D212
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#1869C2
socleC23×C53C22×D4
prime radicalC22×D4×C53D4
hypercenter#1869D53
hypocenterC53#24938
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #1869

Complements

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

NH ≅ G/Ntype
3. #20276284C2semidirect
4. #20276198C2semidirect
5. #20276148C2semidirect
6. #20276143C2semidirect
7. #20275301C2semidirect
8. #20275294C2semidirect
9. #20274916C2semidirect
10. #20274910C2semidirect
11. #20274811C2semidirect
12. #20274802C2semidirect
13. #20274740C2semidirect
14. #20274432C2semidirect
376. C53#24938semidirect

Maximal subgroups

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

subgroupindex
#202790122
#202776442
#202762842
#202761982
#202761482
#202761432
#202753012
#202752942
#202749162
#202749102
#202748112
#202748022
#202747402
#202744322
C53×#18692
#2493853