Finite Group Databasebeta

← #27416801

Subgroups — #27416801

Order 13568.

1428 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810961C212yesmaximal, complemented
2#20277098C211yesmaximal, characteristic
3#20275915C212yesmaximal, complemented
4#20275908C212yesmaximal, complemented
5#20275028C212yesmaximal, complemented
6#20274908C211yesmaximal, characteristic, complemented
7#20274903C211yesmaximal, characteristic, complemented
8#20274650C211yesmaximal, characteristic, complemented
9#20274650C211yesmaximal, characteristic, complemented
10#20274625C211yesmaximal, characteristic
11C22×C22⋊Q8×C53C211yesmaximal, characteristic
12C2×#13133176C2211yescharacteristic
13C2×#13133176C2211yescharacteristic
14#16810961—28no—
15C2×#13133150C2212yescomplemented
16C2×#13133148C2212yescomplemented
17C2×#13133141C2212yescomplemented
18C2×#13133141C2212yescomplemented
19C2×#13133141C2212yescomplemented
20C2×#13133141C2212yescomplemented
21C2×C22⋊#88322C2212yes—
22C2×C22⋊#88322C2211yescharacteristic
23C2×C22⋊#88322C2211yescharacteristic
24C2×C22⋊#88322C2212yes—
25#16810486—21no—
26#16810483C2211yescharacteristic
27#16810483C2211yescharacteristic
28#16810483C2211yescharacteristic
29#16810483C2211yescharacteristic
30#16810479—21no—
31C4×C22⋊Dic53—21no—
32C4×C22⋊Dic53—21no—
33#16810385C2212yescomplemented
34#16810385C2212yescomplemented
35C23×Q8×C53C2211yescharacteristic
36C24×C4×C53C2211yescharacteristic
37C2×C22⋊Q8×C53—21no—
38C2×C22⋊Q8×C53—21no—
39C2×C22⋊Q8×C53—28no—
40C2×C22⋊Q8×C53C2212yes—
41C2×C22⋊Q8×C53C2212yes—
42C22×C4⋊C4×C53—21no—
43C22×C4⋊C4×C53C2211yescharacteristic
44C22×C22⋊C4×C53—21no—
45C22×#88322C2312yes—
46C22×#88322C2312yes—
47#13133176—24no—
48#13133176—42no—
49C2×C4⋊Dic53C2311yescharacteristic
50C2×C4⋊Dic53C2311yescharacteristic
51C2×C4⋊Dic53C2311yescharacteristic
52C2×C4⋊Dic53C2311yescharacteristic
53C2×C22⋊Dic53—21no—
54C2×C22⋊Dic53—21no—
55C2×C22⋊Dic53—21no—
56C2×C22⋊Dic53—21no—
57C2×C22⋊Dic53—21no—
58C2×C22⋊Dic53—21no—
59C2×C4×Dic53—21no—
60C2×C4×Dic53—21no—
61#13133150—24no—
62#13133148—24no—
63#13133141—24no—
64#13133141—24no—
65#13133141—42no—
66#13133141—42no—
67C22⋊#88322—24no—
68C22⋊#88322—22no—
69C22⋊#88322—22no—
70C22⋊#88322—42no—
71#13133124—21no—
72#13133124—21no—
73#13133124—21no—
74#13133124—21no—
75C25×C53D411yescharacteristic
76C22×Q8×C53—24no—
77C22×Q8×C53—21no—
78C22×Q8×C53—22no—
79C22×Q8×C53—22no—
80C22×Q8×C53—42no—
81C22×Q8×C53C2312yes—
82C23×C4×C53—21no—
83C23×C4×C53—21no—
84C23×C4×C53—24no—
85C23×C4×C53—22no—
86C23×C4×C53—22no—
87C23×C4×C53—42no—
88C23×C4×C53D411yescharacteristic
89C23×C4×C53D411yescharacteristic
90C23×C4×C53D411yescharacteristic
91C23×C4×C53C2311yescharacteristic
92C23×C4×C53C2311yescharacteristic
93C23×C4×C53C2311yescharacteristic
94C22⋊Q8×C53—28no—
95C22⋊Q8×C53—28no—
96C22⋊Q8×C53—24no—
97C22⋊Q8×C53—24no—
98C22⋊Q8×C53—24no—
99C22⋊Q8×C53D418yes—
100C2×C4⋊C4×C53—28no—
101C2×C4⋊C4×C53—24no—
102C2×C4⋊C4×C53—22no—
103C2×C4⋊C4×C53—22no—
104C2×C4⋊C4×C53—21no—
105C2×C4⋊C4×C53C2312yes—
106C2×C22⋊C4×C53—28no—
107C2×C22⋊C4×C53—22no—
108C2×C22⋊C4×C53—22no—
109C24×C53—22no—
110C24×C53—22no—
111C24×C53—22no—
112C24×C53—22no—
113C24×C53—22no—
114C24×C53—41no—
115C24×C53—41no—
116C24×C53C2×D411yescharacteristic
117C24×C53C2×D411yescharacteristic
118C24×C53C2×D411yescharacteristic
119C2×Q8×C53—24no—
120C2×Q8×C53—22no—
121C2×Q8×C53—22no—
122C2×Q8×C53—22no—
123C2×Q8×C53—22no—
124C2×Q8×C53—22no—
125C2×Q8×C53—48no—
126C2×Q8×C53—48no—
127C2×Q8×C53—42no—
128C2×Q8×C53—42no—
129C2×Q8×C53C2×D414yes—
130C22×C4×C53—22no—
131C22×C4×C53—21no—
132C22×C4×C53—21no—
133C22×C4×C53—22no—
134C22×C4×C53—22no—
135C22×C4×C53—24no—
136C22×C4×C53—24no—
137C22×C4×C53—22no—
138C22×C4×C53—22no—
139C22×C4×C53—22no—
140C22×C4×C53—22no—
141C22×C4×C53—22no—
142C22×C4×C53—22no—
143C22×C4×C53—22no—
144C22×C4×C53—22no—
145C22×C4×C53—22no—
146C22×C4×C53—21no—
147C22×C4×C53—21no—
148C22×C4×C53—22no—
149C22×C4×C53—41no—
150C22×C4×C53—41no—
151C22×C4×C53—41no—
152C22×C4×C53—48no—
153C22×C4×C53—48no—
154C22×C4×C53—42no—
155C22×C4×C53—42no—
156C22×C4×C53—42no—
157C22×C4×C53—42no—
158C22×C4×C53C2×D411yescharacteristic
159C22×C4×C53C2×D411yescharacteristic
160C22×C4×C53C2×D414yes—
161C22×C4×C53C2×D411yescharacteristic
162C22×C4×C53#4112yes—
163C22×C4×C53#4112yes—
164C22×C4×C53C2411yescharacteristic
165C22×Dic53—21no—
166C22×Dic53—21no—
167C22×Dic53—21no—
168C22×Dic53—21no—
169C4⋊C4×C53—24no—
170C4⋊C4×C53—28no—
171C4⋊C4×C53—28no—
172C4⋊C4×C53—22no—
173C4⋊C4×C53C2×D414yes—
174C22⋊C4×C53—28no—
175C22⋊C4×C53—28no—
176C22⋊Dic53—42no—
177C22⋊Dic53—42no—
178C22⋊Dic53—42no—
179C22⋊Dic53—42no—
180C22⋊Dic53—42no—
181C22⋊Dic53—42no—
182C4⋊Dic53—22no—
183C4⋊Dic53—22no—
184C4⋊Dic53—22no—
185C4⋊Dic53—22no—
186C4×Dic53—42no—
187C4×Dic53—42no—
188C2×#88322—22no—
189C2×#88322—22no—
190C2×#88322—22no—
191C2×#88322—22no—
192C2×#88322—22no—
193C2×#88322—22no—
194C23×C53—21no—
195C23×C53—21no—
196C23×C53—21no—
197C23×C53—21no—
198C23×C53—28no—
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—44no—
212C23×C53—44no—
213C23×C53—44no—
214C23×C53—44no—
215C23×C53—44no—
216C23×C53—44no—
217C23×C53#12012yes—
218C23×C53#12312yes—
219C23×C53#13512yes—
220C23×C53#13712yes—
221C23×C53#13712yes—
222C23×C53C22×D411yescharacteristic
223Q8×C53—24no—
224Q8×C53—24no—
225Q8×C53—48no—
226Q8×C53—42no—
227Q8×C53—42no—
228C2×C4×C53—21no—
229C2×C4×C53—21no—
230C2×C4×C53—21no—
231C2×C4×C53—21no—
232C2×C4×C53—24no—
233C2×C4×C53—24no—
234C2×C4×C53—24no—
235C2×C4×C53—24no—
236C2×C4×C53—21no—
237C2×C4×C53—24no—
238C2×C4×C53—24no—
239C2×C4×C53—21no—
240C2×C4×C53—21no—
241C2×C4×C53—44no—
242C2×C4×C53—44no—
243C2×C4×C53—44no—
244C2×C4×C53—48no—
245C2×C4×C53—48no—
246C2×C4×C53—42no—
247C2×C4×C53—42no—
248C2×C4×C53—42no—
249C2×C4×C53—42no—
250C2×C4×C53—42no—
251C2×C4×C53—42no—
252C2×C4×C53#12012yes—
253C2×C4×C53#12312yes—
254C2×C4×C53C22×D412yes—
255C2×C4×C53C2×#4111yescharacteristic
256C2×C4×C53C2×#4111yescharacteristic
257C2×C4×C53H(5,𝔽2)12yes—
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—41no—
267C2×Dic53—41no—
268C2×Dic53—41no—
269C2×Dic53—41no—
270#88322—42no—
271#88322—42no—
272#24010—531nomaximal, Sylow
273C22×C53—22no—
274C22×C53—22no—
275C22×C53—22no—
276C22×C53—22no—
277C22×C53—22no—
278C22×C53—22no—
279C22×C53—22no—
280C22×C53—22no—
281C22×C53—22no—
282C22×C53—22no—
283C22×C53—22no—
284C22×C53—22no—
285C22×C53—416no—
286C22×C53—41no—
287C22×C53—41no—
288C22×C53—41no—
289C22×C53—41no—
290C22×C53—41no—
291C22×C53—41no—
292C22×C53—41no—
293C22×C53—41no—
294C22×C53#48514yescomplemented
295C22×C53C2×#12011yescharacteristic
296C22×C53C2×#12311yescharacteristic
297C22×C53#53411yescharacteristic
298C22×C53C2×#13511yescharacteristic
299C22×C53C2×#13711yescharacteristic
300C22×C53C2×#13711yescharacteristic
301C22×C53#57511yescharacteristic
302C4×C53—22no—
303C4×C53—22no—
304C4×C53—41no—
305C4×C53—44no—
306C4×C53—44no—
307C4×C53#54612yes—
308C4×C53#54612yes—
309Dic53—81no—
310Dic53—81no—
311Dic53—81no—
312Dic53—81no—
313C22×C22⋊Q8D5311yescharacteristic
314C2×#485—532no—
315#2280—531no—
316#2036—532no—
317#2035—532no—
318#1873—532no—
319#1853—531no—
320#1852—531no—
321#1787—531no—
322#1787—531no—
323#1779—531no—
324C2×C53—24no—
325C2×C53—42no—
326C2×C53—42no—
327C2×C53#240511yescharacteristic
328C2×C53#300211yescharacteristic
329C2×C53#300411yescharacteristic
330C2×C53C2×#48511yescharacteristic
331C2×C53#309511yescharacteristic
332C2×C53#319011yescharacteristic
333C2×C53#319611yescharacteristic
334C23×Q8C2×D5311yescharacteristic
335C24×C4C2×D5311yescharacteristic
336C2×C22⋊Q8—21no—
337C2×C22⋊Q8—21no—
338C2×C22⋊Q8—28no—
339C2×C22⋊Q8C2×D5312yes—
340C2×C22⋊Q8C2×D5312yes—
341C2×#117—531no—
342C2×#117—531no—
343C22×C4⋊C4—21no—
344C22×C4⋊C4C2×D5311yescharacteristic
345C22×C22⋊C4—21no—
346#485—1068no—
347C2×#107—532no—
348C2×#106—532no—
349C2×#103—532no—
350C2×#103—532no—
351C2×#103—532no—
352C2×#103—532no—
353C2×C22⋊C8—532no—
354C2×C22⋊C8—531no—
355C2×C22⋊C8—531no—
356C2×C22⋊C8—532no—
357#381—1061no—
358#380—531no—
359#380—531no—
360#380—531no—
361#380—531no—
362#379—1061no—
363C4×C22⋊C4—1061no—
364C4×C22⋊C4—1061no—
365#340—532no—
366#340—532no—
367C53#2401011yescharacteristic, complemented, Sylow
368C25C22⋊D5311yescharacteristic
369C22×Q8—24no—
370C22×Q8—21no—
371C22×Q8—22no—
372C22×Q8—22no—
373C22×Q8—42no—
374C22×Q8C22×D5312yes—
375C23×C4—21no—
376C23×C4—21no—
377C23×C4—22no—
378C23×C4—22no—
379C23×C4—24no—
380C23×C4—42no—
381C23×C4C22⋊D5311yescharacteristic
382C23×C4C22⋊D5311yescharacteristic
383C23×C4C22⋊D5311yescharacteristic
384C23×C4C22×D5311yescharacteristic
385C23×C4C22×D5311yescharacteristic
386C23×C4C22×D5311yescharacteristic
387C22×C8—532no—
388C22×C8—532no—
389C22⋊Q8—28no—
390C22⋊Q8—28no—
391C22⋊Q8—24no—
392C22⋊Q8—24no—
393C22⋊Q8—24no—
394C22⋊Q8C22⋊D5318yes—
395#117—1064no—
396#117—2122no—
397C2×C4⋊C4—28no—
398C2×C4⋊C4—22no—
399C2×C4⋊C4—22no—
400C2×C4⋊C4—21no—
401C2×C4⋊C4—24no—
402C2×C4⋊C4—531no—
403C2×C4⋊C4—531no—
404C2×C4⋊C4—531no—
405C2×C4⋊C4—531no—
406C2×C4⋊C4C22×D5312yes—
407C2×C22⋊C4—28no—
408C2×C22⋊C4—22no—
409C2×C22⋊C4—22no—
410C2×C22⋊C4—1061no—
411C2×C22⋊C4—1061no—
412C2×C22⋊C4—1061no—
413C2×C22⋊C4—1061no—
414C2×C22⋊C4—1061no—
415C2×C22⋊C4—1061no—
416C2×C42—1061no—
417C2×C42—1061no—
418#107—1064no—
419#106—1064no—
420#103—1064no—
421#103—1064no—
422#103—2122no—
423#103—2122no—
424C22⋊C8—1062no—
425C22⋊C8—1062no—
426C22⋊C8—1064no—
427C22⋊C8—2122no—
428#95—1061no—
429#95—1061no—
430#95—1061no—
431#95—1061no—
432C24—22no—
433C24—22no—
434C24—22no—
435C24—22no—
436C24—22no—
437C24—41no—
438C24—41no—
439C24C2×C22⋊D5311yescharacteristic
440C24C2×C22⋊D5311yescharacteristic
441C24C2×C22⋊D5311yescharacteristic
442C2×Q8—24no—
443C2×Q8—22no—
444C2×Q8—22no—
445C2×Q8—22no—
446C2×Q8—22no—
447C2×Q8—22no—
448C2×Q8—48no—
449C2×Q8—48no—
450C2×Q8—42no—
451C2×Q8—42no—
452C2×Q8C2×C22⋊D5314yes—
453C22×C4—22no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—22no—
472C22×C4—41no—
473C22×C4—41no—
474C22×C4—41no—
475C22×C4—48no—
476C22×C4—48no—
477C22×C4—42no—
478C22×C4—42no—
479C22×C4—42no—
480C22×C4—42no—
481C22×C4—1061no—
482C22×C4—1061no—
483C22×C4—1061no—
484C22×C4—1061no—
485C22×C4#1171994712yes—
486C22×C4#1171994712yes—
487C22×C4C2×C22⋊D5311yescharacteristic
488C22×C4C2×C22⋊D5311yescharacteristic
489C22×C4C2×C22⋊D5314yes—
490C22×C4C2×C22⋊D5311yescharacteristic
491C22×C4C23×D5311yescharacteristic
492C2×C8—1062no—
493C2×C8—1062no—
494C2×C8—1062no—
495C2×C8—1062no—
496C2×C8—1062no—
497C2×C8—1062no—
498C4⋊C4—24no—
499C4⋊C4—28no—
500C4⋊C4—28no—
501C4⋊C4—22no—
502C4⋊C4—1062no—
503C4⋊C4—1062no—
504C4⋊C4—1062no—
505C4⋊C4—1062no—
506C4⋊C4C2×C22⋊D5314yes—
507C22⋊C4—28no—
508C22⋊C4—28no—
509C22⋊C4—2122no—
510C22⋊C4—2122no—
511C22⋊C4—2122no—
512C22⋊C4—2122no—
513C22⋊C4—2122no—
514C22⋊C4—2122no—
515C42—2122no—
516C42—2122no—
517C23—21no—
518C23—21no—
519C23—21no—
520C23—21no—
521C23—28no—
522C23—21no—
523C23—21no—
524C23—21no—
525C23—21no—
526C23—21no—
527C23—21no—
528C23—21no—
529C23—21no—
530C23—21no—
531C23—21no—
532C23—21no—
533C23—21no—
534C23—44no—
535C23—44no—
536C23—44no—
537C23—44no—
538C23—44no—
539C23—44no—
540C23#1313318612yes—
541C23#1313319812yes—
542C23#1313323712yes—
543C23#1313324812yes—
544C23#1313324812yes—
545C23C22×C22⋊D5311yescharacteristic
546Q8—24no—
547Q8—24no—
548Q8—48no—
549Q8—42no—
550Q8—42no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—48no—
568C2×C4—48no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—2121no—
576C2×C4—2121no—
577C2×C4—2121no—
578C2×C4—2121no—
579C2×C4—2121no—
580C2×C4—2121no—
581C2×C4—2121no—
582C2×C4—2121no—
583C2×C4—2121no—
584C2×C4—2121no—
585C2×C4—2121no—
586C2×C4—2121no—
587C2×C4#1313318612yes—
588C2×C4#1313319812yes—
589C2×C4C22×C22⋊D5312yes—
590C2×C4C2×#1171994711yescharacteristic
591C2×C4C2×#1171994711yescharacteristic
592C2×C4#1313326212yes—
593C8—2122no—
594C8—2122no—
595C22—22no—
596C22—22no—
597C22—22no—
598C22—22no—
599C22—22no—
600C22—22no—
601C22—22no—
602C22—22no—
603C22—22no—
604C22—22no—
605C22—22no—
606C22—22no—
607C22—416no—
608C22—41no—
609C22—41no—
610C22—41no—
611C22—41no—
612C22—41no—
613C22—41no—
614C22—41no—
615C22—41no—
616C22#1681096114yescomplemented
617C22C2×#1313318611yescharacteristic
618C22C2×#1313319811yescharacteristic
619C22#1681118211yescharacteristic
620C22C2×#1313323711yescharacteristic
621C22C2×#1313324811yescharacteristic
622C22C2×#1313324811yescharacteristic
623C22#1681142111yescharacteristic
624C4—22no—
625C4—22no—
626C4—41no—
627C4—44no—
628C4—44no—
629C4—4241no—
630C4—4241no—
631C4—4241no—
632C4—4241no—
633C4#1681126712yes—
634C4#1681126712yes—
635C2—24no—
636C2—42no—
637C2—42no—
638C2#2027803811yescharacteristic
639C2#2028393311yescharacteristic
640C2#2028394911yescharacteristic
641C2C2×#1681096111yescharacteristic
642C2#2028465911yescharacteristic
643C2#2028549611yescharacteristic
644C2#2028554511yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC22×C22⋊Q8×C53C2
socleC23×C53C22×D4
prime radicalC25×C53D4
hypercenterC22×C22⋊Q8D53
hypocenterC53#24010
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C24×C4 ◁ C22×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×#16810961C2semidirect
3. #20275915C2semidirect
4. #20275908C2semidirect
5. #20275028C2semidirect
6. #20274908C2semidirect
7. #20274903C2semidirect
8. #20274650C2semidirect
9. #20274650C2semidirect
15. C2×#13133150C22semidirect
16. C2×#13133148C22semidirect
17. C2×#13133141C22semidirect
18. C2×#13133141C22semidirect
19. C2×#13133141C22semidirect
20. C2×#13133141C22semidirect
33. #16810385C22semidirect
34. #16810385C22semidirect
294. C22×C53#485semidirect
367. C53#24010semidirect
616. C22#16810961semidirect

Maximal subgroups

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

subgroupindex
C2×#168109612
#202770982
#202759152
#202759082
#202750282
#202749082
#202749032
#202746502
#202746252
C22×C22⋊Q8×C532
#2401053