Finite Group Databasebeta

← #27350346

Subgroups — #27350346

Order 13568.

938 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
1C2×#16811341C211yesmaximal, characteristic
2#20283373C211yesmaximal, characteristic, complemented
3#20273085C212yesmaximal, complemented
4#20273082C212yesmaximal, complemented
5#20273080C211yesmaximal, characteristic, complemented
6#20273078C211yesmaximal, characteristic, complemented
7#20273058C212yesmaximal, complemented
8#20273051C211yesmaximal, characteristic
9#20273038C212yesmaximal, complemented
10#20273031C211yesmaximal, characteristic
11C53×#2924C211yesmaximal, characteristic, complemented
12#16811341—21no—
13#16811341—21no—
14#16811341—22no—
15#16811341—21no—
16#16811341—21no—
17#16811341—22no—
18C2×#13133216C411yescharacteristic
19C2×#13133211C2212yes—
20C2×#13133210C412yes—
21C2×#13133203C412yes—
22C2×#13133197C411yescharacteristic
23C2×#13133196C2211yescharacteristic
24C2×C4×Dic106C2212yes—
25C2×#13133182C411yescharacteristic
26C2×C4×C22⋊D53C2211yescharacteristic
27C2×#13133175C411yescharacteristic
28#16810684—22no—
29#16810680C2212yes—
30#16810676C2212yes—
31#16810650—21no—
32#16810587C2211yescharacteristic
33C2×C22⋊#88322C2211yescharacteristic
34#16810579—21no—
35#16810348C2211yescharacteristic
36#16810348C2211yescharacteristic
37#16810348C2212yes—
38#16810348C2212yes—
39#16810348C2212yes—
40#16810348C2212yes—
41#16810348C2212yes—
42#16810346C2211yescharacteristic
43#16810346C2211yescharacteristic
44#16810346C2212yes—
45C2×#117×C53C2211yescharacteristic
46C53×#433—22no—
47C53×#432C2212yes—
48C53×#431C2212yes—
49C53×#424—21no—
50C53×#407C2211yescharacteristic
51C2×C22⋊C8×C53C2211yescharacteristic
52C53×#405—21no—
53C22×Dic106C2×C412yes—
54C22×Dic106C2×C411yescharacteristic
55C22×C22⋊D53C2×C411yescharacteristic
56C22×C4×D53C2×C411yescharacteristic
57C2×#11719917—21no—
58C22×#88322—21no—
59#13133216—22no—
60#13133216—21no—
61#13133216—21no—
62#13133211—22no—
63#13133211—22no—
64#13133211—22no—
65#13133211—22no—
66#13133210—22no—
67#13133210—22no—
68#13133210—22no—
69#13133210—22no—
70#13133203—22no—
71#13133203—22no—
72#13133203—22no—
73#13133203—22no—
74#13133197—22no—
75#13133197—21no—
76#13133197—21no—
77#13133196—21no—
78#13133196—21no—
79#13133196—21no—
80#13133196—21no—
81C4×Dic106—22no—
82C4×Dic106—22no—
83C4×Dic106—22no—
84C4×Dic106—22no—
85#13133182—22no—
86#13133182—21no—
87#13133182—21no—
88C4×C22⋊D53—21no—
89C4×C22⋊D53—21no—
90C4×C22⋊D53—21no—
91C4×C22⋊D53—21no—
92#13133175—22no—
93#13133175—21no—
94#13133175—21no—
95C2×C4⋊Dic53C2×C411yescharacteristic
96C2×C4⋊Dic53C2×C412yes—
97C2×#11719919C2×C412yes—
98C2×#11719919C2×C412yes—
99C2×#11719919C2311yescharacteristic
100C2×#11719919C2312yes—
101C2×C22⋊Dic53C2×C411yescharacteristic
102C2×C22⋊Dic53C2×C412yes—
103C2×#11719921C2×C412yes—
104C2×#11719921C2×C412yes—
105C2×#11719921C2311yescharacteristic
106C2×C4×Dic53C2×C412yes—
107C2×C4×Dic53C2×C411yescharacteristic
108C2×C4×Dic53C2×C412yes—
109C4⋊#88322—22no—
110C4⋊#88322—22no—
111C22⋊#88322—21no—
112C22⋊#88322—21no—
113C22⋊#88322C2311yescharacteristic
114C22⋊#88322C2311yescharacteristic
115C22⋊#88322C2312yes—
116#13133128—22no—
117C4×#88322—22no—
118C23×C4×C53C2311yescharacteristic
119C2×M16×C53—21no—
120C22×C8×C53—21no—
121#117×C53D411yescharacteristic
122#117×C53D411yescharacteristic
123#117×C53D412yes—
124#117×C53D411yescharacteristic
125#117×C53D411yescharacteristic
126#117×C53D412yes—
127C2×C4⋊C4×C53C2×C411yescharacteristic
128C2×C4⋊C4×C53C2×C411yescharacteristic
129C2×C22⋊C4×C53C2×C412yes—
130C2×C42×C53C2312yes—
131C4⋊C8×C53—22no—
132C4⋊C8×C53—22no—
133C22⋊C8×C53—21no—
134C22⋊C8×C53—21no—
135C22⋊C8×C53C2311yescharacteristic
136C22⋊C8×C53C2311yescharacteristic
137C22⋊C8×C53C2312yes—
138#97×C53—22no—
139C4×C8×C53—22no—
140C24×C53C22×C411yescharacteristic
141C23×D53C22×C411yescharacteristic
142C22×C4×C53—42no—
143C22×C4×C53C22×C412yes—
144C22×C4×C53C22×C411yescharacteristic
145C22×C4×C53C22×C412yes—
146C22×C4×C53C2×D411yescharacteristic
147C22×C4×C53C2×D411yescharacteristic
148C22×C4×C53C2×D411yescharacteristic
149C22×C4×C53C2×D411yescharacteristic
150C22×C4×C53C2411yescharacteristic
151C2×C22⋊D53—21no—
152C2×C22⋊D53—21no—
153C2×C22⋊D53—22no—
154C2×C22⋊D53—41no—
155C22×Dic53C22×C412yes—
156C22×Dic53C22×C411yescharacteristic
157C22×Dic53C22×C411yescharacteristic
158C22×Dic53C22×C412yes—
159C22×Dic53C22×C411yescharacteristic
160C2×C4×D53—21no—
161C2×C4×D53—21no—
162C2×C4×D53—42no—
163C2×Dic106—21no—
164C2×Dic106—21no—
165C2×Dic106—22no—
166C2×Dic106—22no—
167C2×Dic106—24no—
168C2×Dic106—42no—
169C2×Dic106—42no—
170M16×C53—42no—
171C2×C8×C53—21no—
172C2×C8×C53—22no—
173C2×C8×C53—21no—
174C2×C8×C53—42no—
175C4⋊C4×C53C22⋊C412yes—
176C4⋊C4×C53C22⋊C411yescharacteristic
177C4⋊C4×C53C22⋊C411yescharacteristic
178C4⋊C4×C53C22⋊C412yes—
179C4⋊C4×C53C22⋊C411yescharacteristic
180C4⋊C4×C53C22⋊C411yescharacteristic
181C22⋊C4×C53C22⋊C412yes—
182C22⋊C4×C53C22⋊C412yes—
183C22⋊C4×C53C22⋊C412yes—
184C22⋊C4×C53C22⋊C412yes—
185C42×C53C2×D412yes—
186C42×C53C2×D412yes—
187C42×C53C2×D412yes—
188C42×C53C2×D412yes—
189C22⋊Dic53—21no—
190C22⋊Dic53—21no—
191C22⋊Dic53—22no—
192C22⋊Dic53—22no—
193#11719921—22no—
194#11719921—22no—
195#11719921—22no—
196#11719921—22no—
197#11719921—21no—
198#11719921—21no—
199C4⋊Dic53—21no—
200C4⋊Dic53—21no—
201C4⋊Dic53—22no—
202C4⋊Dic53—22no—
203#11719919—22no—
204#11719919—22no—
205#11719919—22no—
206#11719919—22no—
207#11719919—21no—
208#11719919—21no—
209#11719919—22no—
210#11719919—22no—
211C4×Dic53—22no—
212C4×Dic53—22no—
213C4×Dic53—21no—
214C4×Dic53—21no—
215C4×Dic53—22no—
216C4×Dic53—22no—
217#11719917—42no—
218C2×#88322—21no—
219C2×#88322—22no—
220C2×#88322—21no—
221C2×#88322—42no—
222C23×C53—21no—
223C23×C53—21no—
224C23×C53—41no—
225C23×C53—41no—
226C23×C53C2×C22⋊C411yescharacteristic
227C23×C53C2×C22⋊C411yescharacteristic
228C23×C53C23×C411yescharacteristic
229C22×D53—21no—
230C22×D53—21no—
231C22×D53—21no—
232C22×D53—41no—
233C22×D53—41no—
234C2×C4×C53—22no—
235C2×C4×C53—22no—
236C2×C4×C53—22no—
237C2×C4×C53—22no—
238C2×C4×C53—22no—
239C2×C4×C53—41no—
240C2×C4×C53—41no—
241C2×C4×C53—41no—
242C2×C4×C53—42no—
243C2×C4×C53C2×C22⋊C411yescharacteristic
244C2×C4×C53C2×C22⋊C412yes—
245C2×C4×C53C2×C22⋊C412yes—
246C2×C4×C53C2×C22⋊C412yes—
247C2×C4×C53C2×C22⋊C412yes—
248C2×C4×C53C2×C22⋊C411yescharacteristic
249C2×C4×C53C22×D411yescharacteristic
250C2×C4×C53C22×D411yescharacteristic
251C22⋊D53—42no—
252C22⋊D53—42no—
253C2×Dic53—21no—
254C2×Dic53—21no—
255C2×Dic53—22no—
256C2×Dic53—22no—
257C2×Dic53—21no—
258C2×Dic53—21no—
259C2×Dic53—22no—
260C2×Dic53—22no—
261C2×Dic53—42no—
262C2×Dic53—41no—
263C2×Dic53—41no—
264C2×Dic53#13112yes—
265C4×D53—42no—
266C4×D53—42no—
267Dic106—44no—
268Dic106—44no—
269Dic106—42no—
270Dic106—42no—
271C8×C53—41no—
272C8×C53—41no—
273C8×C53—42no—
274#88322—41no—
275#88322—41no—
276#88322—42no—
277#17420—531nomaximal, Sylow
278C22×C53—21no—
279C22×C53—21no—
280C22×C53—21no—
281C22×C53—21no—
282C22×C53—21no—
283C22×C53—21no—
284C22×C53—21no—
285C22×C53—21no—
286C22×C53—81no—
287C22×C53—81no—
288C22×C53C22×C22⋊C411yescharacteristic
289C22×C53C2×#13111yescharacteristic
290C22×C53#56811yescharacteristic
291C2×D53—41no—
292C2×D53—41no—
293C2×D53—41no—
294C2×D53—81no—
295C2×D53—81no—
296C4×C53—22no—
297C4×C53—22no—
298C4×C53—21no—
299C4×C53—21no—
300C4×C53—21no—
301C4×C53—22no—
302C4×C53—22no—
303C4×C53#41012yes—
304Dic53—22no—
305Dic53—42no—
306Dic53—42no—
307Dic53—41no—
308Dic53—41no—
309C2×#556—531no—
310#2924—531no—
311#2924D5311yescharacteristic, complemented
312#1473—532no—
313#1473—532no—
314#1472—531no—
315#1472—531no—
316#1467—532no—
317#1465—531no—
318#1461—532no—
319#1459—531no—
320C2×C53—21no—
321C2×C53—21no—
322C2×C53—41no—
323C2×C53—41no—
324C2×C53#288011yescharacteristic
325C2×C53C2×#41011yescharacteristic
326C2×C53#288911yescharacteristic
327D53—81no—
328#556—1061no—
329#556—1061no—
330#556—1062no—
331#556—1061no—
332#556—1061no—
333#556—1062no—
334C2×C4⋊Q8—531no—
335C2×#126—532no—
336C2×#126—532no—
337C2×#124—532no—
338C2×C22⋊Q8—531no—
339C2×C22⋊Q8—531no—
340C2×C4×Q8—531no—
341C2×C4×Q8—532no—
342C2×C4×D4—531no—
343C2×#117—531no—
344C2×#117C2×D5311yescharacteristic
345#433—22no—
346#433—1062no—
347#432—532no—
348#432C2×D5312yes—
349#431—532no—
350#431C2×D5312yes—
351#424—21no—
352#424—1061no—
353#407—531no—
354#407C2×D5311yescharacteristic
355C2×C22⋊C8—531no—
356C2×C22⋊C8C2×D5311yescharacteristic
357#405—21no—
358#405—1061no—
359#324—531no—
360#324—531no—
361#324—532no—
362#324—532no—
363#324—532no—
364#324—532no—
365#324—532no—
366#323—531no—
367#323—531no—
368#323—532no—
369C53#1742011yescharacteristic, complemented, Sylow
370C22×Q8—532no—
371C22×Q8—531no—
372C22×D4—531no—
373C23×C4—531no—
374C23×C4C22×D5311yescharacteristic
375C2×M16—21no—
376C2×M16—1061no—
377C22×C8—21no—
378C22×C8—1061no—
379C4⋊Q8—1062no—
380C4⋊Q8—1061no—
381C4⋊Q8—1061no—
382#126—1062no—
383#126—1062no—
384#126—1062no—
385#126—1062no—
386#126—1062no—
387#126—1062no—
388#126—1062no—
389#126—1062no—
390#124—1062no—
391#124—1062no—
392#124—1062no—
393#124—1062no—
394C22⋊Q8—1061no—
395C22⋊Q8—1061no—
396C22⋊Q8—1061no—
397C22⋊Q8—1061no—
398C22⋊Q8—1062no—
399C22⋊Q8—1061no—
400C22⋊Q8—1061no—
401C4×Q8—1062no—
402C4×Q8—1061no—
403C4×Q8—1061no—
404C4×Q8—1062no—
405C4×Q8—1062no—
406C4×Q8—1062no—
407C4×Q8—1062no—
408C4×D4—1061no—
409C4×D4—1061no—
410C4×D4—1061no—
411C4×D4—1061no—
412#117—1062no—
413#117—1061no—
414#117—1061no—
415#117D21211yescharacteristic
416#117D21211yescharacteristic
417#117D21212yes—
418#117C22⋊D5311yescharacteristic
419#117C22⋊D5311yescharacteristic
420#117C22⋊D5312yes—
421C2×C4⋊C4—531no—
422C2×C4⋊C4—532no—
423C2×C4⋊C4—532no—
424C2×C4⋊C4—532no—
425C2×C4⋊C4—531no—
426C2×C4⋊C4—532no—
427C2×C4⋊C4C4×D5311yescharacteristic
428C2×C4⋊C4C4×D5311yescharacteristic
429C2×C22⋊C4—531no—
430C2×C22⋊C4—532no—
431C2×C22⋊C4—532no—
432C2×C22⋊C4—532no—
433C2×C22⋊C4—531no—
434C2×C22⋊C4C4×D5312yes—
435C2×C42—532no—
436C2×C42—531no—
437C2×C42—532no—
438C2×C42C22×D5312yes—
439C4⋊C8—22no—
440C4⋊C8—22no—
441C4⋊C8—1062no—
442C4⋊C8—1062no—
443C22⋊C8—21no—
444C22⋊C8—21no—
445C22⋊C8—531no—
446C22⋊C8—531no—
447C22⋊C8—532no—
448C22⋊C8—1061no—
449C22⋊C8—1061no—
450C22⋊C8C22×D5311yescharacteristic
451C22⋊C8C22×D5311yescharacteristic
452C22⋊C8C22×D5312yes—
453#97—22no—
454#97—1062no—
455C4×C8—22no—
456C4×C8—1062no—
457C24—531no—
458C24C2×C4×D5311yescharacteristic
459C2×Q8—1064no—
460C2×Q8—1061no—
461C2×Q8—1061no—
462C2×Q8—1062no—
463C2×Q8—1062no—
464C2×Q8—2122no—
465C2×Q8—2122no—
466C2×D4—1062no—
467C2×D4—1061no—
468C2×D4—1061no—
469C2×D4—2121no—
470C22×C4—42no—
471C22×C4—531no—
472C22×C4—532no—
473C22×C4—531no—
474C22×C4—531no—
475C22×C4—532no—
476C22×C4—1061no—
477C22×C4—1061no—
478C22×C4—2122no—
479C22×C4C2×C4×D5311yescharacteristic
480C22×C4C2×C4×D5312yes—
481C22×C4C2×C4×D5312yes—
482C22×C4C2×D21211yescharacteristic
483C22×C4C2×D21211yescharacteristic
484C22×C4C2×C22⋊D5311yescharacteristic
485C22×C4C2×C22⋊D5311yescharacteristic
486C22×C4C23×D5311yescharacteristic
487M16—42no—
488M16—2122no—
489C2×C8—21no—
490C2×C8—22no—
491C2×C8—21no—
492C2×C8—42no—
493C2×C8—1061no—
494C2×C8—1062no—
495C2×C8—1061no—
496C2×C8—2122no—
497C4⋊C4—1061no—
498C4⋊C4—1061no—
499C4⋊C4—1062no—
500C4⋊C4—1062no—
501C4⋊C4—1062no—
502C4⋊C4—1062no—
503C4⋊C4—1062no—
504C4⋊C4—1062no—
505C4⋊C4—1061no—
506C4⋊C4—1062no—
507C4⋊C4—1061no—
508C4⋊C4—1062no—
509C4⋊C4#1171992112yes—
510C4⋊C4#1171992111yescharacteristic
511C4⋊C4#1171992111yescharacteristic
512C4⋊C4#1171992112yes—
513C4⋊C4#1171992111yescharacteristic
514C4⋊C4#1171992111yescharacteristic
515C22⋊C4—1061no—
516C22⋊C4—1061no—
517C22⋊C4—1062no—
518C22⋊C4—1062no—
519C22⋊C4—1062no—
520C22⋊C4—1062no—
521C22⋊C4—1062no—
522C22⋊C4—1062no—
523C22⋊C4—1061no—
524C22⋊C4—1061no—
525C22⋊C4#1171992112yes—
526C22⋊C4#1171992112yes—
527C22⋊C4#1171992112yes—
528C22⋊C4#1171992112yes—
529C42—1062no—
530C42—1062no—
531C42—1061no—
532C42—1061no—
533C42—1062no—
534C42—1062no—
535C42C2×D21212yes—
536C42C2×D21212yes—
537C42C2×C22⋊D5312yes—
538C42C2×C22⋊D5312yes—
539C23—21no—
540C23—21no—
541C23—41no—
542C23—41no—
543C23—1061no—
544C23—1061no—
545C23—1061no—
546C23—2121no—
547C23—2121no—
548C23C2×#1171992111yescharacteristic
549C23C2×#1171992111yescharacteristic
550C23C22×C4×D5311yescharacteristic
551Q8—2124no—
552Q8—2124no—
553Q8—2122no—
554Q8—2122no—
555D4—2122no—
556D4—2122no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—42no—
566C2×C4—532no—
567C2×C4—1061no—
568C2×C4—1061no—
569C2×C4—1061no—
570C2×C4—1061no—
571C2×C4—1062no—
572C2×C4—1062no—
573C2×C4—1062no—
574C2×C4—1062no—
575C2×C4—2122no—
576C2×C4—2122no—
577C2×C4—2121no—
578C2×C4—2121no—
579C2×C4—2122no—
580C2×C4C2×#1171992112yes—
581C2×C4C2×#1171992112yes—
582C2×C4C2×#1171992111yescharacteristic
583C2×C4C2×#1171992112yes—
584C2×C4C2×#1171992112yes—
585C2×C4C2×#1171992111yescharacteristic
586C2×C4C22×C22⋊D5311yescharacteristic
587C2×C4C22×D21211yescharacteristic
588C8—41no—
589C8—41no—
590C8—42no—
591C8—2121no—
592C8—2121no—
593C8—2122no—
594C22—21no—
595C22—21no—
596C22—21no—
597C22—21no—
598C22—21no—
599C22—21no—
600C22—21no—
601C22—21no—
602C22—81no—
603C22—81no—
604C22—4241no—
605C22—4241no—
606C22—2121no—
607C22—2121no—
608C22—2121no—
609C22C22×#1171992111yescharacteristic
610C22#131×D5311yescharacteristic
611C22#1681138811yescharacteristic
612C4—22no—
613C4—22no—
614C4—21no—
615C4—21no—
616C4—21no—
617C4—22no—
618C4—22no—
619C4—1062no—
620C4—2122no—
621C4—2121no—
622C4—2122no—
623C4—2121no—
624C4#1681059612yes—
625C2—21no—
626C2—21no—
627C2—41no—
628C2—41no—
629C2—4241no—
630C2#2028316211yescharacteristic
631C2C2×#1681059611yescharacteristic
632C2#2028319711yescharacteristic
subgroupG/H
centerC22C22×#11719921
commutatorC23×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#2924C2
socleC22×C53C22×C22⋊C4
prime radicalC22×C22⋊D53C2×C4
cosocleC22×C4×C53C24
hypercenter#2924D53
hypocenterC53#17420
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×#117 ◁ #2924

Complements

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

NH ≅ G/Ntype
2. #20283373C2semidirect
3. #20273085C2semidirect
4. #20273082C2semidirect
5. #20273080C2semidirect
6. #20273078C2semidirect
7. #20273058C2semidirect
9. #20273038C2semidirect
11. C53×#2924C2semidirect
311. #2924D53semidirect
369. C53#17420semidirect

Maximal subgroups

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

subgroupindex
C2×#168113412
#202833732
#202730852
#202730822
#202730802
#202730782
#202730582
#202730512
#202730382
#202730312
C53×#29242
#1742053