Finite Group Databasebeta

← #27446113

Subgroups — #27446113

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#20279881C211yesmaximal, characteristic
2#20277644C211yesmaximal, characteristic
3#20276148C211yesmaximal, characteristic, complemented
4#20276148C211yesmaximal, characteristic, complemented
5#20276148C211yesmaximal, characteristic, complemented
6#20276138C211yesmaximal, characteristic, complemented
7#20275680C211yesmaximal, characteristic, complemented
8#20275670C211yesmaximal, characteristic, complemented
9#20275485C211yesmaximal, characteristic, complemented
10#20274916C211yesmaximal, characteristic, complemented
11#20274910C211yesmaximal, characteristic, complemented
12#20274811C211yesmaximal, characteristic, complemented
13#20274806C211yesmaximal, characteristic, complemented
14C4×#13133149C211yesmaximal, characteristic, complemented
15C53×#1954C211yesmaximal, characteristic
16C2×#13133209C2211yescharacteristic
17C2×#13133176C2211yescharacteristic
18C22×C4⋊Dic53C2211yescharacteristic
19C2×#13133151C2211yescharacteristic
20C2×#13133149C2211yescharacteristic
21C2×#13133149C2211yescharacteristic
22C2×#13133149C2211yescharacteristic
23#16810561—21no—
24#16810537—21no—
25#16810517—21no—
26#16810517—21no—
27#16810516—21no—
28#16810509—21no—
29#16810503—21no—
30#16810489C2211yescharacteristic
31#16810483C2211yescharacteristic
32#16810483C2211yescharacteristic
33#16810483C2211yescharacteristic
34#16810483C2211yescharacteristic
35#16810483C2211yescharacteristic
36#16810483C2211yescharacteristic
37C4×C4⋊Dic53C2211yescharacteristic
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
52C2×C4×D4×C53C2211yescharacteristic
53C2×C4⋊C8×C53C2211yescharacteristic
54C2×C22⋊C8×C53C2211yescharacteristic
55C2×C22⋊C8×C53C2211yescharacteristic
56C2×C22⋊C8×C53C2211yescharacteristic
57C2×C22⋊C8×C53C2211yescharacteristic
58C2×C4×C8×C53C2211yescharacteristic
59C23×Dic53—21no—
60#13133209—22no—
61#13133209—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×C53C2311yescharacteristic
105C22×C8×C53C2311yescharacteristic
106C22×C8×C53C2311yescharacteristic
107C22×C8×C53C2311yescharacteristic
108C4×D4×C53—22no—
109C4×D4×C53—22no—
110C2×C4⋊C4×C53D411yescharacteristic
111C2×C22⋊C4×C53D411yescharacteristic
112C2×C22⋊C4×C53D411yescharacteristic
113C2×C42×C53C2311yescharacteristic
114C4⋊C8×C53—22no—
115C22⋊C8×C53—22no—
116C22⋊C8×C53—22no—
117C22⋊C8×C53—41no—
118C22⋊C8×C53—41no—
119C4×C8×C53—22no—
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×C53C2×D411yescharacteristic
138C22×C4×C53C2×D411yescharacteristic
139C22×C4×C53C2×D411yescharacteristic
140C22×C4×C53C2×D411yescharacteristic
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#4112yes—
165C4⋊C4×C53—22no—
166C22⋊C4×C53—22no—
167C22⋊C4×C53—22no—
168C42×C53#4112yes—
169C42×C53#4112yes—
170C22⋊Dic53—42no—
171C22⋊Dic53—42no—
172C22⋊Dic53—42no—
173C22⋊Dic53—42no—
174C22⋊Dic53—81no—
175C22⋊Dic53—81no—
176C22⋊Dic53—81no—
177C4⋊Dic53—21no—
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—42no—
193C4⋊Dic53#4112yes—
194#11719919—42no—
195#11719919—42no—
196C4×Dic53—42no—
197C4×Dic53—81no—
198C4×Dic53—81no—
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#12312yes—
234C2×C4×C53#12312yes—
235C2×C4×C53C2×#4111yescharacteristic
236C2×C4×C53C2×#4111yescharacteristic
237C2×C4×C53C2×#4111yescharacteristic
238C2×C4×C53C2×#4111yescharacteristic
239C2×C4×C53H(5,𝔽2)11yescharacteristic
240C2×C4×C532−1+411yescharacteristic
241C2×Dic53—22no—
242C2×Dic53—41no—
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—84no—
266C8×C53—22no—
267C8×C53—41no—
268C8×C53—41no—
269C8×C53—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#52911yescharacteristic
297C4×C53#54211yescharacteristic
298C4×C53#55611yescharacteristic
299C4×C53#55911yescharacteristic
300Dic53—42no—
301Dic53—81no—
302Dic53—81no—
303Dic53—81no—
304Dic53—81no—
305Dic53—81no—
306Dic53—81no—
307Dic53—81no—
308#2562—531no—
309#2352—531no—
310#2077—531no—
311#2077—531no—
312#2077—531no—
313#2075—531no—
314#1991—531no—
315#1989—531no—
316#1954D5311yescharacteristic
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×#125—531no—
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⋊C8C2×D5311yescharacteristic
341C2×C22⋊C8C2×D5311yescharacteristic
342C2×C22⋊C8C2×D5311yescharacteristic
343C2×C22⋊C8C2×D5311yescharacteristic
344C2×C22⋊C8C2×D5311yescharacteristic
345C2×C4×C8C2×D5311yescharacteristic
346#400—1061no—
347#394—1061no—
348#388—1061no—
349#388—1061no—
350#388—1061no—
351#387—1061no—
352#386—1061no—
353#382—531no—
354#380—531no—
355#380—531no—
356#380—531no—
357#380—531no—
358#380—531no—
359#380—531no—
360C4×C4⋊C4—531no—
361C4×C4⋊C4—531no—
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×D4D21211yescharacteristic
377C23×C4—1061no—
378C23×C4C22×D5311yescharacteristic
379C23×C4C22×D5311yescharacteristic
380C22×C8C22×D5311yescharacteristic
381C22×C8C22×D5311yescharacteristic
382C22×C8C22×D5311yescharacteristic
383C22×C8C22×D5311yescharacteristic
384#125—1062no—
385#125—1062no—
386C4×D4—22no—
387C4×D4—22no—
388#117—1062no—
389#117—2121no—
390C2×C4⋊C4—531no—
391C2×C4⋊C4—531no—
392C2×C4⋊C4—531no—
393C2×C4⋊C4—531no—
394C2×C4⋊C4—531no—
395C2×C4⋊C4—531no—
396C2×C4⋊C4—531no—
397C2×C4⋊C4—531no—
398C2×C4⋊C4—1062no—
399C2×C4⋊C4—1061no—
400C2×C4⋊C4—1061no—
401C2×C4⋊C4—1061no—
402C2×C4⋊C4—2121no—
403C2×C4⋊C4D21211yescharacteristic
404C2×C22⋊C4—1061no—
405C2×C22⋊C4—1061no—
406C2×C22⋊C4—1061no—
407C2×C22⋊C4—1061no—
408C2×C22⋊C4—1061no—
409C2×C22⋊C4—1061no—
410C2×C22⋊C4—1061no—
411C2×C22⋊C4D21211yescharacteristic
412C2×C22⋊C4D21211yescharacteristic
413C2×C42—1061no—
414C2×C42—1061no—
415C2×C42—1061no—
416C2×C42C22×D5311yescharacteristic
417#107—2121no—
418#106—1062no—
419#106—1062no—
420#106—2121no—
421C4⋊C8—22no—
422C22⋊C8—22no—
423C22⋊C8—22no—
424C22⋊C8—41no—
425C22⋊C8—41no—
426C4×C8—22no—
427#95—1061no—
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—
437C24C2×D21211yescharacteristic
438C24C2×D21211yescharacteristic
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—1061no—
455C22×C4—1061no—
456C22×C4—1061no—
457C22×C4—1061no—
458C22×C4—1061no—
459C22×C4—1061no—
460C22×C4—1061no—
461C22×C4—1061no—
462C22×C4—4241no—
463C22×C4—2122no—
464C22×C4—2122no—
465C22×C4C2×D21211yescharacteristic
466C22×C4C2×D21211yescharacteristic
467C22×C4C2×D21211yescharacteristic
468C22×C4C2×D21211yescharacteristic
469C22×C4C23×D5311yescharacteristic
470C2×C8—21no—
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#1171994712yes—
482C4⋊C4—22no—
483C4⋊C4—532no—
484C4⋊C4—1061no—
485C4⋊C4—1061no—
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—2122no—
500C4⋊C4—2122no—
501C4⋊C4—2122no—
502C22⋊C4—22no—
503C22⋊C4—22no—
504C22⋊C4—4241no—
505C22⋊C4—4241no—
506C22⋊C4—4241no—
507C22⋊C4—2122no—
508C22⋊C4—2122no—
509C22⋊C4—2122no—
510C22⋊C4—2122no—
511C42—4241no—
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#1313323912yes—
530C23C22×D21211yescharacteristic
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—1062no—
551C2×C4—4244no—
552C2×C4—2121no—
553C2×C4—2121no—
554C2×C4—2121no—
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#1313320112yes—
576C2×C4#1313320112yes—
577C2×C4#41×D5311yescharacteristic
578C2×C4C2×#1171994711yescharacteristic
579C2×C4C2×#1171994711yescharacteristic
580C2×C4C2×#1171994711yescharacteristic
581C2×C4#1313326211yescharacteristic
582C2×C4#1313326411yescharacteristic
583C8—22no—
584C8—41no—
585C8—41no—
586C8—41no—
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×#1313320111yescharacteristic
602C22#1681126811yescharacteristic
603C22#1681127711yescharacteristic
604C22C2×#1313323911yescharacteristic
605C22#1681142311yescharacteristic
606C22#1681143511yescharacteristic
607C22#1681143511yescharacteristic
608C4—22no—
609C4—22no—
610C4—41no—
611C4—41no—
612C4—4241no—
613C4—4241no—
614C4—4241no—
615C4—4241no—
616C4—4241no—
617C4—4241no—
618C4—4241no—
619C4—2122no—
620C4#1681115411yescharacteristic
621C4#1681124011yescharacteristic
622C4#1681134211yescharacteristic
623C4#1681135911yescharacteristic
624C2—42no—
625C2—81no—
626C2#2027850011yescharacteristic
627C2#2028463011yescharacteristic
628C2#2028470611yescharacteristic
629C2#2028661911yescharacteristic
630C2#2028693311yescharacteristic
631C2#2028709611yescharacteristic
632C2#2028727711yescharacteristic
subgroupG/H
centerC23C22×D212
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#1954C2
socleC23×C53C22×D4
prime radicalC22×D4×C53D4
hypercenter#1954D53
hypocenterC53#26304
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #1954

Complements

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

NH ≅ G/Ntype
3. #20276148C2semidirect
4. #20276148C2semidirect
5. #20276148C2semidirect
6. #20276138C2semidirect
7. #20275680C2semidirect
8. #20275670C2semidirect
9. #20275485C2semidirect
10. #20274916C2semidirect
11. #20274910C2semidirect
12. #20274811C2semidirect
13. #20274806C2semidirect
14. C4×#13133149C2semidirect
375. C53#26304semidirect

Maximal subgroups

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

subgroupindex
#202798812
#202776442
#202761482
#202761382
#202756802
#202756702
#202754852
#202749162
#202749102
#202748112
#202748062
C4×#131331492
C53×#19542
#2630453