Finite Group Databasebeta

← #27466332

Subgroups — #27466332

Order 13568.

790 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20278417C211yesmaximal, characteristic
2#20276277C211yesmaximal, characteristic, complemented
3#20276277C211yesmaximal, characteristic, complemented
4#20276123C211yesmaximal, characteristic, complemented
5#20275856C211yesmaximal, characteristic, complemented
6#20275675C211yesmaximal, characteristic, complemented
7#20275394C211yesmaximal, characteristic, complemented
8#20275394C211yesmaximal, characteristic, complemented
9#20275139C211yesmaximal, characteristic
10#20275023C211yesmaximal, characteristic
11#20274842C211yesmaximal, characteristic, complemented
12#20274805C211yesmaximal, characteristic
13C22⋊C4×#88322C211yesmaximal, characteristic
14C2×#16810385C211yesmaximal, characteristic
15C53×#2502C211yesmaximal, characteristic
16C23×#88322C2211yescharacteristic
17C2×#13133217C2211yescharacteristic
18C2×#13133194—21no—
19C22×C4⋊Dic53C2211yescharacteristic
20C2×#13133141C2211yescharacteristic
21C2×#13133141C2211yescharacteristic
22C2×#13133141C2211yescharacteristic
23C2×#13133141C2211yescharacteristic
24C2×C22⋊#88322C2211yescharacteristic
25C2×C4×#88322C2211yescharacteristic
26C2×C4×#88322C2211yescharacteristic
27#16810555—21no—
28#16810544—21no—
29#16810521C2211yescharacteristic
30#16810521C2211yescharacteristic
31#16810498—21no—
32#16810493C2211yescharacteristic
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×C22⋊Q8×C53C2211yescharacteristic
49C22×C4⋊C4×C53C2211yescharacteristic
50C53×#398—21no—
51C53×#398—21no—
52C53×#391C2211yescharacteristic
53C53×#391C2211yescharacteristic
54C53×#385—21no—
55C53×#385—21no—
56C53×#384C2211yescharacteristic
57C53×#384C2211yescharacteristic
58C4×C22⋊C4×C53C2211yescharacteristic
59C22×Dic106—21no—
60C23×Dic53—21no—
61C22×#88322—22no—
62C22×#88322—22no—
63C22×#88322—41no—
64C22×#88322C2311yescharacteristic
65C22×#88322C2311yescharacteristic
66C22×#88322C2311yescharacteristic
67C22×#88322C2311yescharacteristic
68#13133217—22no—
69#13133217—41no—
70#13133194—44no—
71C2×C4⋊Dic53—21no—
72C2×C4⋊Dic53—21no—
73C2×C4⋊Dic53—21no—
74C2×C4⋊Dic53—21no—
75C2×C4⋊Dic53—22no—
76C2×C4⋊Dic53—41no—
77C2×C4⋊Dic53C2311yescharacteristic
78C2×C4⋊Dic53C2311yescharacteristic
79C2×C4⋊Dic53C2311yescharacteristic
80C2×C4⋊Dic53C2311yescharacteristic
81C2×#11719919—21no—
82C2×#11719919—21no—
83C2×C22⋊Dic53—21no—
84#13133141—22no—
85#13133141—22no—
86#13133141—41no—
87#13133141—41no—
88C22⋊#88322—22no—
89C4×#88322—22no—
90C4×#88322—41no—
91#13133124—21no—
92#13133124—21no—
93#13133124—21no—
94C22×Q8×C53C2311yescharacteristic
95C23×C4×C53—21no—
96C23×C4×C53C2311yescharacteristic
97C22⋊Q8×C53—22no—
98C22⋊Q8×C53—22no—
99C2×C4⋊C4×C53—21no—
100C2×C4⋊C4×C53—21no—
101C2×C4⋊C4×C53—21no—
102C2×C4⋊C4×C53—21no—
103C2×C4⋊C4×C53—21no—
104C2×C4⋊C4×C53—21no—
105C2×C4⋊C4×C53—22no—
106C2×C4⋊C4×C53—22no—
107C2×C4⋊C4×C53C2311yescharacteristic
108C2×C4⋊C4×C53C2311yescharacteristic
109C2×C4⋊C4×C53C2311yescharacteristic
110C2×C22⋊C4×C53—21no—
111C2×C22⋊C4×C53D411yescharacteristic
112C2×C22⋊C4×C53D411yescharacteristic
113C2×C42×C53C2311yescharacteristic
114C2×C42×C53C2311yescharacteristic
115#95×C53—21no—
116#95×C53—21no—
117#95×C53—21no—
118#95×C53—21no—
119#95×C53D411yescharacteristic
120#95×C53D411yescharacteristic
121C24×C53C2×D411yescharacteristic
122C2×Q8×C53—21no—
123C2×Q8×C53—21no—
124C2×Q8×C53—21no—
125C2×Q8×C53—21no—
126C2×Q8×C53—21no—
127C2×Q8×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—42no—
137C22×C4×C53—81no—
138C22×C4×C53—81no—
139C22×C4×C53C2×D411yescharacteristic
140C22×C4×C53C2×D411yescharacteristic
141C22×C4×C53C2×D411yescharacteristic
142C22×C4×C53C2×D411yescharacteristic
143C22×C4×C53C2×D411yescharacteristic
144C22×C4×C53#4112yes—
145C22×C4×C53#4112yes—
146C22×C4×C53C2411yescharacteristic
147C22×Dic53—21no—
148C22×Dic53—21no—
149C22×Dic53—21no—
150C22×Dic53—21no—
151C22×Dic53—42no—
152C22×Dic53—42no—
153C22×Dic53—81no—
154C2×Dic106—42no—
155C2×Dic106—42no—
156C2×Dic106—81no—
157C4⋊C4×C53—21no—
158C4⋊C4×C53—21no—
159C4⋊C4×C53—21no—
160C4⋊C4×C53—21no—
161C4⋊C4×C53—21no—
162C4⋊C4×C53—21no—
163C4⋊C4×C53—42no—
164C4⋊C4×C53—42no—
165C4⋊C4×C53—81no—
166C4⋊C4×C53—81no—
167C4⋊C4×C53—81no—
168C4⋊C4×C53—81no—
169C22⋊C4×C53—42no—
170C22⋊C4×C53SD1614yes—
171C22⋊C4×C53Q1614yes—
172C42×C53—21no—
173C42×C53—21no—
174C42×C53—21no—
175C42×C53—21no—
176C22⋊Dic53—42no—
177C4⋊Dic53—21no—
178C4⋊Dic53—21no—
179C4⋊Dic53—21no—
180C4⋊Dic53—21no—
181C4⋊Dic53—21no—
182C4⋊Dic53—21no—
183C4⋊Dic53—21no—
184C4⋊Dic53—42no—
185C4⋊Dic53—42no—
186C4⋊Dic53—42no—
187C4⋊Dic53—81no—
188C4⋊Dic53#4112yes—
189#11719919—42no—
190#11719919—42no—
191C2×#88322—21no—
192C2×#88322—21no—
193C2×#88322—21no—
194C2×#88322—21no—
195C2×#88322—21no—
196C2×#88322—21no—
197C2×#88322—21no—
198C2×#88322—21no—
199C2×#88322—21no—
200C2×#88322—21no—
201C2×#88322—21no—
202C2×#88322—44no—
203C2×#88322#4112yes—
204C23×C53—21no—
205C23×C53—21no—
206C23×C53—21no—
207C23×C53—21no—
208C23×C53#12412yes—
209C23×C53C2×SD1612yes—
210C23×C53C2×Q1612yes—
211C23×C53C22×D411yescharacteristic
212Q8×C53—41no—
213Q8×C53—41no—
214Q8×C53—41no—
215Q8×C53—41no—
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—21no—
226C2×C4×C53—22no—
227C2×C4×C53—22no—
228C2×C4×C53—22no—
229C2×C4×C53—22no—
230C2×C4×C53—41no—
231C2×C4×C53—41no—
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—84no—
243C2×C4×C53#12412yes—
244C2×C4×C53C2×SD1612yes—
245C2×C4×C53C2×SD1612yes—
246C2×C4×C53C2×Q1612yes—
247C2×C4×C53C2×Q1612yes—
248C2×C4×C53C2×#4111yescharacteristic
249C2×C4×C53C2×#4111yescharacteristic
250C2×C4×C53C2×#4111yescharacteristic
251C2×C4×C53C2×#4111yescharacteristic
252C2×C4×C53H(5,𝔽2)11yescharacteristic
253C2×C4×C532−1+411yescharacteristic
254C2×Dic53—22no—
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—
267Dic106—84no—
268#88322—22no—
269#88322—41no—
270#88322—41no—
271#88322—41no—
272#27697—531nomaximal, Sylow
273C22×C53—22no—
274C22×C53—22no—
275C22×C53—22no—
276C22×C53—22no—
277C22×C53—22no—
278C22×C53—22no—
279C22×C53#48714yes—
280C22×C53C2×#12411yescharacteristic
281C22×C53#54611yescharacteristic
282C22×C53#54711yescharacteristic
283C22×C53C22×SD1611yescharacteristic
284C22×C53C22×Q1611yescharacteristic
285C22×C53#57611yescharacteristic
286C22×C53#57711yescharacteristic
287C4×C53—22no—
288C4×C53—22no—
289C4×C53—22no—
290C4×C53—22no—
291C4×C53—22no—
292C4×C53—81no—
293C4×C53—81no—
294C4×C53—81no—
295C4×C53—81no—
296C4×C53#52911yescharacteristic
297C4×C53#53911yescharacteristic
298C4×C53#55111yescharacteristic
299C4×C53#55211yescharacteristic
300Dic53—42no—
301Dic53—81no—
302Dic53—81no—
303Dic53—81no—
304#2502D5311yescharacteristic
305#2440—531no—
306#2100—531no—
307#2100—531no—
308#2072—531no—
309#2024—531no—
310#1990—531no—
311#1933—531no—
312#1933—531no—
313#1892—531no—
314#1872—531no—
315#1834—531no—
316#1824—531no—
317C8×C22⋊C4—531no—
318C2×#340—531no—
319C2×C53—24no—
320C2×C53#248811yescharacteristic
321C2×C53C2×#48711yescharacteristic
322C2×C53#313611yescharacteristic
323C2×C53#329511yescharacteristic
324C2×C53#329911yescharacteristic
325C2×C53#333511yescharacteristic
326C2×C53#333811yescharacteristic
327C23×C8—531no—
328C2×C4⋊Q8—531no—
329C2×C22⋊Q8—1061no—
330C2×C22⋊Q8C2×D5311yescharacteristic
331C22×C4⋊C4—531no—
332C22×C4⋊C4C2×D5311yescharacteristic
333C2×#103—531no—
334C2×#103—531no—
335C2×#103—531no—
336C2×#103—531no—
337C2×C22⋊C8—531no—
338C2×C4×C8—531no—
339C2×C4×C8—531no—
340#398—21no—
341#398—21no—
342#398—1061no—
343#395—1061no—
344#391C2×D5311yescharacteristic
345#391C2×D5311yescharacteristic
346#389—531no—
347#389—531no—
348#385—21no—
349#385—21no—
350#385—1061no—
351#384—531no—
352#384C2×D5311yescharacteristic
353#384C2×D5311yescharacteristic
354#380—531no—
355C4×C22⋊C4C2×D5311yescharacteristic
356#340—531no—
357#340—531no—
358#340—531no—
359#340—531no—
360#340—531no—
361#340—531no—
362#340—531no—
363#340—531no—
364#340—531no—
365#340—531no—
366#340—531no—
367#340—531no—
368#336—531no—
369#336—531no—
370C53#2769711yescharacteristic, complemented, Sylow
371C22×Q8—1061no—
372C22×Q8C22×D5311yescharacteristic
373C23×C4—21no—
374C23×C4—1061no—
375C23×C4C22×D5311yescharacteristic
376C22×C8—531no—
377C22×C8—531no—
378C22×C8—531no—
379C22×C8—531no—
380C22×C8—1062no—
381C22×C8—1062no—
382C22×C8—2121no—
383C4⋊Q8—1062no—
384C4⋊Q8—2121no—
385C22⋊Q8—22no—
386C22⋊Q8—22no—
387C22⋊Q8—2124no—
388C2×C4⋊C4—22no—
389C2×C4⋊C4—22no—
390C2×C4⋊C4—21no—
391C2×C4⋊C4—21no—
392C2×C4⋊C4—21no—
393C2×C4⋊C4—21no—
394C2×C4⋊C4—21no—
395C2×C4⋊C4—21no—
396C2×C4⋊C4—531no—
397C2×C4⋊C4—531no—
398C2×C4⋊C4—531no—
399C2×C4⋊C4—531no—
400C2×C4⋊C4—1062no—
401C2×C4⋊C4—1061no—
402C2×C4⋊C4—1061no—
403C2×C4⋊C4—1061no—
404C2×C4⋊C4—1061no—
405C2×C4⋊C4—1061no—
406C2×C4⋊C4—1061no—
407C2×C4⋊C4—2121no—
408C2×C4⋊C4C22×D5311yescharacteristic
409C2×C4⋊C4C22×D5311yescharacteristic
410C2×C4⋊C4C22×D5311yescharacteristic
411C2×C22⋊C4—21no—
412C2×C22⋊C4—1061no—
413C2×C22⋊C4C22⋊D5311yescharacteristic
414C2×C22⋊C4C22⋊D5311yescharacteristic
415C2×C42C22×D5311yescharacteristic
416C2×C42C22×D5311yescharacteristic
417#103—1062no—
418#103—1062no—
419#103—2121no—
420#103—2121no—
421C22⋊C8—1062no—
422C4×C8—1062no—
423C4×C8—2121no—
424#95—21no—
425#95—21no—
426#95—21no—
427#95—21no—
428#95—1061no—
429#95—1061no—
430#95—1061no—
431#95C22⋊D5311yescharacteristic
432#95C22⋊D5311yescharacteristic
433C24C2×C22⋊D5311yescharacteristic
434C2×Q8—21no—
435C2×Q8—21no—
436C2×Q8—21no—
437C2×Q8—21no—
438C2×Q8—21no—
439C2×Q8—21no—
440C2×Q8—4241no—
441C2×Q8—2122no—
442C2×Q8—2122no—
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—42no—
452C22×C4—81no—
453C22×C4—81no—
454C22×C4—1061no—
455C22×C4—1061no—
456C22×C4—1061no—
457C22×C4—1061no—
458C22×C4—4241no—
459C22×C4—2122no—
460C22×C4—2122no—
461C22×C4#1171994712yes—
462C22×C4#1171994712yes—
463C22×C4C2×C22⋊D5311yescharacteristic
464C22×C4C2×C22⋊D5311yescharacteristic
465C22×C4C2×C22⋊D5311yescharacteristic
466C22×C4C2×C22⋊D5311yescharacteristic
467C22×C4C2×C22⋊D5311yescharacteristic
468C22×C4C23×D5311yescharacteristic
469C2×C8—532no—
470C2×C8—1061no—
471C2×C8—1061no—
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—2124no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—42no—
489C4⋊C4—42no—
490C4⋊C4—81no—
491C4⋊C4—81no—
492C4⋊C4—81no—
493C4⋊C4—81no—
494C4⋊C4—532no—
495C4⋊C4—1061no—
496C4⋊C4—1061no—
497C4⋊C4—1061no—
498C4⋊C4—1061no—
499C4⋊C4—1061no—
500C4⋊C4—1061no—
501C4⋊C4—1061no—
502C4⋊C4—4241no—
503C4⋊C4—2122no—
504C4⋊C4—2122no—
505C4⋊C4—2122no—
506C4⋊C4—2122no—
507C4⋊C4—2122no—
508C22⋊C4—42no—
509C22⋊C4—2122no—
510C22⋊C4#1171992414yes—
511C22⋊C4#1171992514yes—
512C42—21no—
513C42—21no—
514C42—21no—
515C42—21no—
516C23—21no—
517C23—21no—
518C23—21no—
519C23—21no—
520C23#1313320412yes—
521C23C2×#1171992412yes—
522C23C2×#1171992512yes—
523C23C22×C22⋊D5311yescharacteristic
524Q8—41no—
525Q8—41no—
526Q8—41no—
527Q8—41no—
528Q8—4244no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—84no—
556C2×C4—1062no—
557C2×C4—4244no—
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#1313320412yes—
570C2×C4C2×#1171992412yes—
571C2×C4C2×#1171992412yes—
572C2×C4C2×#1171992512yes—
573C2×C4C2×#1171992512yes—
574C2×C4#41×D5311yescharacteristic
575C2×C4#41×D5311yescharacteristic
576C2×C4C2×#1171994711yescharacteristic
577C2×C4C2×#1171994711yescharacteristic
578C2×C4#1313326311yescharacteristic
579C2×C4#1313326411yescharacteristic
580C8—1062no—
581C8—2121no—
582C8—2121no—
583C8—2121no—
584C22—22no—
585C22—22no—
586C22—22no—
587C22—22no—
588C22—22no—
589C22—22no—
590C22#1681097314yes—
591C22C2×#1313320411yescharacteristic
592C22#1681126111yescharacteristic
593C22#1681127211yescharacteristic
594C22C22×#1171992411yescharacteristic
595C22C22×#1171992511yescharacteristic
596C22#1681142611yescharacteristic
597C22#1681143411yescharacteristic
598C4—22no—
599C4—22no—
600C4—22no—
601C4—22no—
602C4—22no—
603C4—81no—
604C4—81no—
605C4—81no—
606C4—81no—
607C4—4241no—
608C4—4241no—
609C4—4241no—
610C4—2122no—
611C4#1681115011yescharacteristic
612C4#1681121711yescharacteristic
613C4#1681129611yescharacteristic
614C4#1681131011yescharacteristic
615C2—24no—
616C2#2027904511yescharacteristic
617C2C2×#1681097311yescharacteristic
618C2#2028506911yescharacteristic
619C2#2028653311yescharacteristic
620C2#2028658511yescharacteristic
621C2#2028706611yescharacteristic
622C2#2028710311yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2502C2
socleC23×C53C22×D4
prime radicalC24×C53C2×D4
hypercenter#2502D53
hypocenterC53#27697
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2502

Complements

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

NH ≅ G/Ntype
2. #20276277C2semidirect
3. #20276277C2semidirect
4. #20276123C2semidirect
5. #20275856C2semidirect
6. #20275675C2semidirect
7. #20275394C2semidirect
8. #20275394C2semidirect
11. #20274842C2semidirect
370. C53#27697semidirect

Maximal subgroups

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

subgroupindex
#202784172
#202762772
#202761232
#202758562
#202756752
#202753942
#202751392
#202750232
#202748422
#202748052
C22⋊C4×#883222
C2×#168103852
C53×#25022
#2769753