Finite Group Databasebeta

← #27393644

Subgroups — #27393644

Order 13568.

970 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811250C211yesmaximal, characteristic
2C2×#16810685C211yesmaximal, characteristic
3#20277325C211yesmaximal, characteristic
4#20275684C211yesmaximal, characteristic, complemented
5#20275679C211yesmaximal, characteristic, complemented
6#20275677C211yesmaximal, characteristic, complemented
7#20275671C211yesmaximal, characteristic, complemented
8#20275669C211yesmaximal, characteristic, complemented
9#20275665C211yesmaximal, characteristic, complemented
10#20275661C211yesmaximal, characteristic, complemented
11#20274809C211yesmaximal, characteristic
12#20274808C211yesmaximal, characteristic
13#20274804C211yesmaximal, characteristic
14#20274800C211yesmaximal, characteristic
15C53×#1993C211yesmaximal, characteristic, complemented
16#16811250—24no—
17#16811250C418yescomplemented
18C2×#13133216C2211yescharacteristic
19C2×#13133215—21no—
20C2×#13133209C2211yescharacteristic
21C2×#13133208C2211yescharacteristic
22C2×#13133207—21no—
23C2×#13133195—21no—
24C2×#13133193—21no—
25C2×#13133175C2211yescharacteristic
26C2×#13133174C2211yescharacteristic
27C2×C4⋊C4×D53C2211yescharacteristic
28#16810685—24no—
29C2×#13133146C2211yescharacteristic
30C2×#13133146C2211yescharacteristic
31C2×#13133129C2211yescharacteristic
32C2×#13133129C2211yescharacteristic
33C2×C4×#88322C2211yescharacteristic
34#16810515—21no—
35#16810515—21no—
36#16810510—21no—
37#16810510—21no—
38#16810493C2211yescharacteristic
39#16810480C2211yescharacteristic
40#16810480C2211yescharacteristic
41C4⋊C4×Dic53C2211yescharacteristic
42C4⋊C4×Dic53C2211yescharacteristic
43#16810386C2211yescharacteristic
44#16810386C2211yescharacteristic
45#16810386C2211yescharacteristic
46#16810386C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810385C2211yescharacteristic
49#16810385C2211yescharacteristic
50#16810385C2211yescharacteristic
51#16810385C2211yescharacteristic
52#16810385C2211yescharacteristic
53#16810385C2211yescharacteristic
54#16810385C2211yescharacteristic
55C2×C4⋊Q8×C53C2211yescharacteristic
56C2×#97×C53C2211yescharacteristic
57C53×#382C2211yescharacteristic
58C53×#340C2211yescharacteristic
59C53×#340C2211yescharacteristic
60C53×#340C2211yescharacteristic
61C53×#340C2211yescharacteristic
62C22×Dic106—21no—
63C22×C4×D53—21no—
64C22×C4×D53C2311yescharacteristic
65C22×#88322C2311yescharacteristic
66C22×#88322C2311yescharacteristic
67#13133216—22no—
68#13133216C2×C414yes—
69#13133215—28no—
70#13133209—22no—
71#13133209C2×C414yes—
72#13133208—22no—
73#13133208C2×C414yes—
74#13133207—28no—
75#13133195—28no—
76#13133193—28no—
77#13133175—22no—
78#13133175C2×C414yes—
79#13133174C2×C414yes—
80#13133174D414yes—
81C2×C4⋊Dic53—21no—
82C2×C4⋊Dic53—21no—
83C2×C4⋊Dic53—21no—
84C2×C4⋊Dic53C2311yescharacteristic
85C2×C4⋊Dic53C2311yescharacteristic
86C2×C4⋊Dic53C2311yescharacteristic
87C2×C4⋊Dic53C2311yescharacteristic
88C2×#11719919—21no—
89C2×#11719919—21no—
90C2×#11719919—21no—
91C2×#11719919—21no—
92C2×#11719919—21no—
93C2×#11719919D411yescharacteristic
94C2×#11719919D411yescharacteristic
95C4⋊C4×D53—22no—
96C4⋊C4×D53C2×C414yes—
97C2×#11719921—21no—
98C2×#11719921—21no—
99C2×#11719921—21no—
100C2×#11719921D411yescharacteristic
101C2×#11719921D411yescharacteristic
102C2×C4×Dic53—21no—
103C2×C4×Dic53—21no—
104C2×C4×Dic53—21no—
105C2×C4×Dic53C2311yescharacteristic
106#13133146—22no—
107#13133146—22no—
108#13133129—22no—
109#13133129—22no—
110C4×#88322—22no—
111#13133124—21no—
112#13133124—21no—
113#13133124—21no—
114#13133124—21no—
115C22×Q8×C53—21no—
116C22×C8×C53C2311yescharacteristic
117C22×C8×C53C2311yescharacteristic
118C4⋊Q8×C53—22no—
119C4⋊Q8×C53C2×C414yes—
120C2×C4⋊C4×C53—21no—
121C2×C4⋊C4×C53C2311yescharacteristic
122C2×C4⋊C4×C53C2311yescharacteristic
123C2×C4⋊C4×C53C2311yescharacteristic
124C2×C4⋊C4×C53C2311yescharacteristic
125C2×C42×C53C2311yescharacteristic
126#97×C53—22no—
127#95×C53—21no—
128#95×C53—21no—
129C23×D53C2×D411yescharacteristic
130C2×Q8×C53—24no—
131C2×Q8×C53—81no—
132C2×Q8×C53—81no—
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×C53C2411yescharacteristic
140C22×Dic53—21no—
141C22×Dic53—21no—
142C22×Dic53—21no—
143C22×Dic53—21no—
144C22×Dic53C2×D411yescharacteristic
145C22×Dic53C2×D411yescharacteristic
146C22×Dic53C2×D411yescharacteristic
147C2×C4×D53—21no—
148C2×C4×D53—21no—
149C2×C4×D53—21no—
150C2×C4×D53—21no—
151C2×C4×D53—24no—
152C2×C4×D53—81no—
153C2×C4×D53—81no—
154C2×C4×D53C22×C412yes—
155C2×C4×D53C2×D412yes—
156C2×Dic106—24no—
157C2×Dic106—81no—
158C2×Dic106—81no—
159C2×C8×C53—21no—
160C2×C8×C53—21no—
161C2×C8×C53—21no—
162C2×C8×C53—21no—
163C2×C8×C53—21no—
164C2×C8×C53—21no—
165C4⋊C4×C53—24no—
166C4⋊C4×C53—21no—
167C4⋊C4×C53—21no—
168C4⋊C4×C53—21no—
169C4⋊C4×C53—21no—
170C4⋊C4×C53—21no—
171C4⋊C4×C53—21no—
172C4⋊C4×C53C22×C412yes—
173C4⋊C4×C53C22×C412yes—
174C42×C53C22×C412yes—
175C42×C53C2×D412yes—
176#11719921—24no—
177#11719921—81no—
178#11719921—81no—
179#11719921C22⋊C414yes—
180#11719921C22⋊C414yes—
181C4⋊Dic53—21no—
182C4⋊Dic53—21no—
183C4⋊Dic53—24no—
184C4⋊Dic53—21no—
185C4⋊Dic53—21no—
186C4⋊Dic53—21no—
187C4⋊Dic53—21no—
188C4⋊Dic53—81no—
189C4⋊Dic53—81no—
190C4⋊Dic53C22×C412yes—
191C4⋊Dic53C22×C412yes—
192#11719919—24no—
193#11719919—24no—
194#11719919—24no—
195#11719919—24no—
196#11719919—24no—
197#11719919C22⋊C414yes—
198#11719919C22⋊C414yes—
199C4×Dic53—24no—
200C4×Dic53—81no—
201C4×Dic53—81no—
202C4×Dic53C22×C412yes—
203C4×Dic53C2×D412yes—
204C2×#88322—21no—
205C2×#88322—21no—
206C2×#88322—21no—
207C2×#88322—21no—
208C2×#88322—21no—
209C2×#88322—21no—
210C23×C53C22×D411yescharacteristic
211C22×D53—21no—
212C22×D53—21no—
213C22×D53—21no—
214C22×D53—21no—
215C22×D53—21no—
216C22×D53—21no—
217C22×D53C2×C22⋊C412yes—
218Q8×C53—84no—
219C2×C4×C53—21no—
220C2×C4×C53—21no—
221C2×C4×C53—21no—
222C2×C4×C53—21no—
223C2×C4×C53—22no—
224C2×C4×C53—22no—
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—41no—
234C2×C4×C53—41no—
235C2×C4×C53C2×C22⋊C412yes—
236C2×C4×C53C2×C22⋊C412yes—
237C2×C4×C53C23×C411yescharacteristic
238C2×C4×C53C22×D411yescharacteristic
239C2×C4×C532−1+411yescharacteristic
240C2×C4×C532−1+411yescharacteristic
241C2×C4×C532−1+411yescharacteristic
242C2×C4×C532−1+411yescharacteristic
243C2×Dic53—21no—
244C2×Dic53—21no—
245C2×Dic53—21no—
246C2×Dic53—21no—
247C2×Dic53—21no—
248C2×Dic53—21no—
249C2×Dic53—22no—
250C2×Dic53—22no—
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×Dic53C2×C22⋊C412yes—
262C2×Dic53C2×C22⋊C412yes—
263C2×Dic53C2×C22⋊C412yes—
264C4×D53—22no—
265C4×D53—22no—
266C4×D53—22no—
267C4×D53—22no—
268C4×D53—84no—
269Dic106—84no—
270C8×C53—41no—
271C8×C53—41no—
272#88322—41no—
273#88322—41no—
274#21843—531nomaximal, Sylow
275C22×C53C22×C22⋊C411yescharacteristic
276C22×C53#53611yescharacteristic
277C22×C53#53611yescharacteristic
278C22×C53#57611yescharacteristic
279C22×C53#57711yescharacteristic
280C22×C53#57711yescharacteristic
281C22×C53#57811yescharacteristic
282C2×D53—22no—
283C2×D53—22no—
284C2×D53—22no—
285C2×D53—41no—
286C2×D53—41no—
287C2×D53—41no—
288C2×D53—41no—
289C4×C53—22no—
290C4×C53—22no—
291C4×C53—42no—
292C4×C53—42no—
293C4×C53—81no—
294C4×C53—81no—
295C4×C53#51911yescharacteristic
296C4×C53#51911yescharacteristic
297C4×C53#53611yescharacteristic
298C4×C53#53611yescharacteristic
299Dic53—22no—
300Dic53—22no—
301Dic53—22no—
302Dic53—42no—
303Dic53—42no—
304Dic53—81no—
305Dic53—81no—
306C2×#544—531no—
307C2×#433—531no—
308#2314—531no—
309#1993D5311yescharacteristic, complemented
310#1992—531no—
311#1991—531no—
312#1990—531no—
313#1989—531no—
314#1989—531no—
315#1988—531no—
316#1987—531no—
317#1825—531no—
318#1825—531no—
319#1824—531no—
320#1823—531no—
321C2×C53#231011yescharacteristic
322C2×C53#289711yescharacteristic
323C2×C53#289811yescharacteristic
324C2×C53#324111yescharacteristic
325C2×C53#324211yescharacteristic
326C2×C53#324311yescharacteristic
327C2×C53#324411yescharacteristic
328D53—42no—
329#544—538no—
330#544—1064no—
331C2×C4⋊Q8—531no—
332C2×C4⋊Q8—1061no—
333C2×C4⋊Q8C2×D5311yescharacteristic
334C2×#125—531no—
335C2×#125—531no—
336C2×#125—1061no—
337C2×C22⋊Q8—1061no—
338C2×C22⋊Q8—1061no—
339C2×#117—531no—
340C2×#117—531no—
341C22×C4⋊C4—531no—
342#433—1064no—
343C2×C4⋊C8—531no—
344C2×C4⋊C8—531no—
345C2×C22⋊C8—531no—
346C2×C22⋊C8—531no—
347C2×#97C2×D5311yescharacteristic
348C2×C4×C8—531no—
349#388—1061no—
350#388—1061no—
351#387—1061no—
352#387—1061no—
353#384—531no—
354#382C2×D5311yescharacteristic
355#380—531no—
356#380—531no—
357C4×C4⋊C4—531no—
358C4×C4⋊C4—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#340—531no—
369#340—531no—
370#340—531no—
371#340C2×D5311yescharacteristic
372#340C2×D5311yescharacteristic
373#340C2×D5311yescharacteristic
374#340C2×D5311yescharacteristic
375C53#2184311yescharacteristic, complemented, Sylow
376C22×Q8—21no—
377C22×Q8—1061no—
378C23×C4—531no—
379C23×C4—1061no—
380C22×C8—531no—
381C22×C8—531no—
382C22×C8C22×D5311yescharacteristic
383C22×C8C22×D5311yescharacteristic
384C4⋊Q8—22no—
385C4⋊Q8—534no—
386C4⋊Q8—1062no—
387C4⋊Q8—1068no—
388C4⋊Q8C4×D5314yes—
389#125—534no—
390#125—534no—
391#125—1062no—
392#125—1068no—
393#125—1062no—
394C22⋊Q8—1068no—
395C22⋊Q8—1068no—
396#117—534no—
397#117—534no—
398#117—534no—
399#117—1062no—
400C2×C4⋊C4—21no—
401C2×C4⋊C4—531no—
402C2×C4⋊C4—531no—
403C2×C4⋊C4—531no—
404C2×C4⋊C4—531no—
405C2×C4⋊C4—534no—
406C2×C4⋊C4—531no—
407C2×C4⋊C4—531no—
408C2×C4⋊C4—1062no—
409C2×C4⋊C4—1061no—
410C2×C4⋊C4—1061no—
411C2×C4⋊C4—1061no—
412C2×C4⋊C4—1061no—
413C2×C4⋊C4—1061no—
414C2×C4⋊C4—1061no—
415C2×C4⋊C4—1061no—
416C2×C4⋊C4—1061no—
417C2×C4⋊C4C22×D5311yescharacteristic
418C2×C4⋊C4C22×D5311yescharacteristic
419C2×C4⋊C4C22×D5311yescharacteristic
420C2×C4⋊C4C22×D5311yescharacteristic
421C2×C22⋊C4—531no—
422C2×C22⋊C4—531no—
423C2×C22⋊C4—1061no—
424C2×C22⋊C4—1061no—
425C2×C22⋊C4—1061no—
426C2×C42—531no—
427C2×C42—1061no—
428C2×C42—1061no—
429C2×C42—1061no—
430C2×C42C22×D5311yescharacteristic
431C4⋊C8—1062no—
432C4⋊C8—1062no—
433C22⋊C8—1062no—
434C22⋊C8—1062no—
435#97—22no—
436C4×C8—1062no—
437#95—21no—
438#95—21no—
439#95—1061no—
440#95—1061no—
441#95—1061no—
442#95—1061no—
443C24—531no—
444C2×Q8—24no—
445C2×Q8—81no—
446C2×Q8—81no—
447C2×Q8—1064no—
448C2×Q8—4241no—
449C2×Q8—4241no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—531no—
455C22×C4—531no—
456C22×C4—531no—
457C22×C4—532no—
458C22×C4—532no—
459C22×C4—1064no—
460C22×C4—1061no—
461C22×C4—1061no—
462C22×C4—1061no—
463C22×C4—1061no—
464C22×C4—1061no—
465C22×C4—1061no—
466C22×C4—1061no—
467C22×C4—1061no—
468C22×C4—4241no—
469C22×C4—4241no—
470C22×C4D4×D5311yescharacteristic
471C22×C4D4×D5311yescharacteristic
472C22×C4C23×D5311yescharacteristic
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—21no—
479C2×C8—1061no—
480C2×C8—1061no—
481C2×C8—1061no—
482C2×C8—1061no—
483C2×C8—1061no—
484C2×C8—1061no—
485C4⋊C4—24no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—534no—
493C4⋊C4—534no—
494C4⋊C4—532no—
495C4⋊C4—532no—
496C4⋊C4—1064no—
497C4⋊C4—1064no—
498C4⋊C4—1064no—
499C4⋊C4—1064no—
500C4⋊C4—1064no—
501C4⋊C4—1064no—
502C4⋊C4—1061no—
503C4⋊C4—1061no—
504C4⋊C4—1061no—
505C4⋊C4—1061no—
506C4⋊C4—1061no—
507C4⋊C4—1061no—
508C4⋊C4—4241no—
509C4⋊C4—4241no—
510C4⋊C4C2×C4×D5312yes—
511C4⋊C4C2×C4×D5312yes—
512C22⋊C4—534no—
513C22⋊C4—534no—
514C22⋊C4—1064no—
515C22⋊C4—4241no—
516C22⋊C4—4241no—
517C42—532no—
518C42—532no—
519C42—1064no—
520C42—4241no—
521C42—4241no—
522C42C2×C4×D5312yes—
523C42D4×D5312yes—
524C23—532no—
525C23—1061no—
526C23—1061no—
527C23—1061no—
528C23—1061no—
529C23—1061no—
530C23—1061no—
531C23C2×D4×D5311yescharacteristic
532Q8—84no—
533Q8—4244no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—41no—
541C2×C4—41no—
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—532no—
551C2×C4—532no—
552C2×C4—532no—
553C2×C4—1061no—
554C2×C4—1061no—
555C2×C4—1061no—
556C2×C4—1061no—
557C2×C4—1061no—
558C2×C4—1061no—
559C2×C4—1062no—
560C2×C4—1062no—
561C2×C4—1062no—
562C2×C4—1062no—
563C2×C4—1062no—
564C2×C4—1062no—
565C2×C4—4244no—
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×C4C22⋊C4×D5312yes—
577C2×C4C22⋊C4×D5312yes—
578C2×C4C22×C4×D5311yescharacteristic
579C2×C4C2×D4×D5311yescharacteristic
580C2×C4#1313326411yescharacteristic
581C2×C4#1313326411yescharacteristic
582C2×C4#1313326511yescharacteristic
583C2×C4#1313326511yescharacteristic
584C8—41no—
585C8—41no—
586C8—2121no—
587C8—2121no—
588C22—1062no—
589C22—1062no—
590C22—1062no—
591C22—2121no—
592C22—2121no—
593C22—2121no—
594C22—2121no—
595C22C2×C22⋊C4×D5311yescharacteristic
596C22#1681119311yescharacteristic
597C22#1681119811yescharacteristic
598C22#1681142711yescharacteristic
599C22#1681143211yescharacteristic
600C22#1681143311yescharacteristic
601C22#1681143911yescharacteristic
602C4—22no—
603C4—22no—
604C4—42no—
605C4—42no—
606C4—81no—
607C4—81no—
608C4—1062no—
609C4—1062no—
610C4—1062no—
611C4—4241no—
612C4—4241no—
613C4—2122no—
614C4—2122no—
615C4#1681109911yescharacteristic
616C4#1681109911yescharacteristic
617C4#1681119611yescharacteristic
618C4#1681119611yescharacteristic
619C2—2122no—
620C2#2027729811yescharacteristic
621C2#2028324011yescharacteristic
622C2#2028324711yescharacteristic
623C2#2028596911yescharacteristic
624C2#2028598411yescharacteristic
625C2#2028599311yescharacteristic
626C2#2028600811yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1993C2
socleC23×C53C22×D4
prime radicalC23×D53C2×D4
cosocleC22×C4×C53C24
hypercenter#1993D53
hypocenterC53#21843
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1993

Complements

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

NH ≅ G/Ntype
4. #20275684C2semidirect
5. #20275679C2semidirect
6. #20275677C2semidirect
7. #20275671C2semidirect
8. #20275669C2semidirect
9. #20275665C2semidirect
10. #20275661C2semidirect
15. C53×#1993C2semidirect
17. #16811250C4semidirect
309. #1993D53semidirect
375. C53#21843semidirect

Maximal subgroups

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

subgroupindex
C2×#168112502
C2×#168106852
#202773252
#202756842
#202756792
#202756772
#202756712
#202756692
#202756652
#202756612
#202748092
#202748082
#202748042
#202748002
C53×#19932
#2184353