Finite Group Databasebeta

← #27311074

Subgroups — #27311074

Order 13568.

1622 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810535C211yesmaximal, characteristic
2C2×#16810529C211yesmaximal, characteristic, complemented
3#20272985C211yesmaximal, characteristic, complemented
4#20272979C211yesmaximal, characteristic, complemented
5#20272976C212yesmaximal, complemented
6#20272960C211yesmaximal, characteristic
7#20272958C212yesmaximal
8#20272948C212yesmaximal, complemented
9#20272942C211yesmaximal, characteristic, complemented
10#20272938C212yesmaximal
11C2×C4×C22⋊C4×C53C211yesmaximal, characteristic
12C23×Dic106C2211yescharacteristic
13C22×C4⋊Dic53C2211yescharacteristic
14C22×#11719919C2212yes—
15C22×C22⋊Dic53C2212yes—
16C22×C22⋊Dic53C2212yes—
17#16810535C418yescomplemented
18#16810529—24no—
19C2×#13133124C2212yes—
20C2×#13133124C2212yes—
21C2×#13133124C2211yescharacteristic
22C2×#13133124C2212yes—
23C2×#13133124C2212yes—
24C2×#13133124C2212yes—
25C2×#13133124C2212yes—
26C2×#13133124C2212yes—
27C2×#13133124C2212yes—
28C2×#13133124C2211yescharacteristic
29C2×#13133124C2212yes—
30C24×C4×C53C2211yescharacteristic
31C22×C22⋊C4×C53C2211yescharacteristic
32C22×C22⋊C4×C53C2211yescharacteristic
33C22×C42×C53C2211yescharacteristic
34C22×C42×C53C2211yescharacteristic
35C4×C22⋊C4×C53—24no—
36C2×#95×C53C2211yescharacteristic
37C2×#95×C53C2211yescharacteristic
38C22×Dic106—22no—
39C22×Dic106—22no—
40C22×Dic106—42no—
41C22×Dic106—42no—
42C22×Dic106C2×C414yescomplemented
43C23×Dic53C2312yes—
44C23×Dic53C2312yes—
45C23×Dic53C2312yes—
46C23×Dic53C2312yes—
47C2×C4⋊Dic53—22no—
48C2×C4⋊Dic53—42no—
49C2×#11719919—24no—
50C2×#11719919—42no—
51C2×#11719919—42no—
52C2×C22⋊Dic53—24no—
53C2×C22⋊Dic53—42no—
54C2×C22⋊Dic53—42no—
55C2×C22⋊Dic53—42no—
56C2×C22⋊Dic53—42no—
57C2×C22⋊Dic53C2×C418yes—
58#13133124—24no—
59#13133124—24no—
60#13133124—24no—
61#13133124—22no—
62#13133124—42no—
63#13133124—42no—
64#13133124—42no—
65#13133124—42no—
66#13133124—42no—
67#13133124C2×C414yescomplemented
68#13133124C2×C418yescomplemented
69C25×C53C2311yescharacteristic
70C23×C4×C53—22no—
71C23×C4×C53—22no—
72C23×C4×C53—42no—
73C23×C4×C53—42no—
74C23×C4×C53Q814yescomplemented
75C23×C4×C53C2311yescharacteristic
76C23×C4×C53C2311yescharacteristic
77C23×C4×C53C2311yescharacteristic
78C23×C4×C53C2311yescharacteristic
79C23×C4×C53C2311yescharacteristic
80C23×C4×C53C2311yescharacteristic
81C2×C22⋊C4×C53—22no—
82C2×C22⋊C4×C53—22no—
83C2×C22⋊C4×C53—22no—
84C2×C22⋊C4×C53—22no—
85C2×C22⋊C4×C53—22no—
86C2×C22⋊C4×C53C2×C414yes—
87C2×C42×C53—22no—
88C2×C42×C53—42no—
89C2×C42×C53—42no—
90C2×C42×C53D414yes—
91#95×C53—22no—
92#95×C53—22no—
93C24×C53—21no—
94C24×C53—21no—
95C24×C53—21no—
96C24×C53—21no—
97C24×C53—22no—
98C24×C53—22no—
99C24×C53—22no—
100C24×C53—22no—
101C24×C53C22×C412yes—
102C24×C53C2×Q812yes—
103C24×C53#4112yes—
104C24×C53C2411yescharacteristic
105C22×C4×C53—22no—
106C22×C4×C53—22no—
107C22×C4×C53—22no—
108C22×C4×C53—22no—
109C22×C4×C53—21no—
110C22×C4×C53—21no—
111C22×C4×C53—21no—
112C22×C4×C53—21no—
113C22×C4×C53—21no—
114C22×C4×C53—21no—
115C22×C4×C53—21no—
116C22×C4×C53—21no—
117C22×C4×C53—22no—
118C22×C4×C53—22no—
119C22×C4×C53—22no—
120C22×C4×C53—22no—
121C22×C4×C53—22no—
122C22×C4×C53—22no—
123C22×C4×C53—22no—
124C22×C4×C53—22no—
125C22×C4×C53—28no—
126C22×C4×C53—48no—
127C22×C4×C53—44no—
128C22×C4×C53—44no—
129C22×C4×C53—48no—
130C22×C4×C53C22×C412yes—
131C22×C4×C53C22×C412yes—
132C22×C4×C53C2×D412yes—
133C22×C4×C53C2×D412yes—
134C22×C4×C53C2×D412yes—
135C22×C4×C53C2×Q812yes—
136C22×C4×C53C2×Q812yes—
137C22×C4×C53#4112yes—
138C22×C4×C53#4112yes—
139C22×C4×C53#4112yes—
140C22×Dic53—22no—
141C22×Dic53—22no—
142C22×Dic53—22no—
143C22×Dic53—22no—
144C22×Dic53—22no—
145C22×Dic53—22no—
146C22×Dic53—22no—
147C22×Dic53—22no—
148C22×Dic53—22no—
149C22×Dic53—22no—
150C22×Dic53—22no—
151C22×Dic53—22no—
152C22×Dic53—22no—
153C22×Dic53—22no—
154C22×Dic53—22no—
155C22×Dic53—22no—
156C22×Dic53—22no—
157C22×Dic53—22no—
158C22×Dic53—22no—
159C22×Dic53—22no—
160C22×Dic53—22no—
161C22×Dic53—22no—
162C22×Dic53—22no—
163C22×Dic53—22no—
164C22×Dic53—22no—
165C22×Dic53—22no—
166C22×Dic53C22×C414yes—
167C22×Dic53C22×C414yes—
168C2×Dic106—28no—
169C2×Dic106—48no—
170C2×Dic106—44no—
171C2×Dic106—44no—
172C2×Dic106—48no—
173C22⋊C4×C53—28no—
174C22⋊C4×C53—28no—
175C42×C53—44no—
176C42×C53—44no—
177C22⋊Dic53—48no—
178C22⋊Dic53—48no—
179C4⋊Dic53—44no—
180#11719919—48no—
181C23×C53—22no—
182C23×C53—22no—
183C23×C53—24no—
184C23×C53—24no—
185C23×C53—22no—
186C23×C53—22no—
187C23×C53—22no—
188C23×C53—22no—
189C23×C53—24no—
190C23×C53—24no—
191C23×C53—24no—
192C23×C53—24no—
193C23×C53—41no—
194C23×C53—41no—
195C23×C53—41no—
196C23×C53—41no—
197C23×C53—42no—
198C23×C53—42no—
199C23×C53—42no—
200C23×C53—42no—
201C23×C53—42no—
202C23×C53—42no—
203C23×C53C4×Q814yes—
204C23×C53C23×C411yescharacteristic
205C23×C53C22×D411yescharacteristic
206C23×C53C22×Q811yescharacteristic
207C23×C53C2×#4111yescharacteristic
208C23×C53C2×#4111yescharacteristic
209C23×C53C2×#4111yescharacteristic
210C23×C53C2×#4111yescharacteristic
211C23×C53H(5,𝔽2)12yes—
212C23×C53H(5,𝔽2)12yes—
213C23×C53H(5,𝔽2)12yes—
214C23×C532−1+412yes—
215C2×C4×C53—24no—
216C2×C4×C53—22no—
217C2×C4×C53—22no—
218C2×C4×C53—22no—
219C2×C4×C53—22no—
220C2×C4×C53—24no—
221C2×C4×C53—24no—
222C2×C4×C53—24no—
223C2×C4×C53—22no—
224C2×C4×C53—24no—
225C2×C4×C53—22no—
226C2×C4×C53—22no—
227C2×C4×C53—22no—
228C2×C4×C53—22no—
229C2×C4×C53—22no—
230C2×C4×C53—24no—
231C2×C4×C53—24no—
232C2×C4×C53—416no—
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—42no—
240C2×C4×C53—42no—
241C2×C4×C53—42no—
242C2×C4×C53—42no—
243C2×C4×C53—42no—
244C2×C4×C53C4×D414yes—
245C2×C4×C53C22⋊Q814yes—
246C2×Dic53—24no—
247C2×Dic53—24no—
248C2×Dic53—24no—
249C2×Dic53—24no—
250C2×Dic53—24no—
251C2×Dic53—24no—
252C2×Dic53—42no—
253C2×Dic53—42no—
254C2×Dic53—42no—
255C2×Dic53—42no—
256C2×Dic53—42no—
257C2×Dic53—42no—
258C2×Dic53—42no—
259C2×Dic53—42no—
260C2×Dic53—42no—
261C2×Dic53—42no—
262C2×Dic53—42no—
263C2×Dic53—42no—
264C2×Dic53—42no—
265C2×Dic53—42no—
266C2×Dic53—42no—
267C2×Dic53—42no—
268C2×Dic53—42no—
269C2×Dic53—42no—
270C2×Dic53—42no—
271C2×Dic53—42no—
272C2×Dic53—42no—
273C2×Dic53—42no—
274Dic106—416no—
275#14319—531nomaximal, Sylow
276C22×C53—28no—
277C22×C53—24no—
278C22×C53—42no—
279C22×C53—42no—
280C22×C53—44no—
281C22×C53—42no—
282C22×C53—42no—
283C22×C53—42no—
284C22×C53—42no—
285C22×C53—44no—
286C22×C53—44no—
287C22×C53C2×C4×D411yescharacteristic
288C22×C53C2×C4×Q811yescharacteristic
289C22×C53C4×#4111yescharacteristic
290C22×C53#51812yes—
291C22×C53#52012yes—
292C22×C53C2×C22⋊Q811yescharacteristic
293C22×C53#52511yescharacteristic
294C22×C53#52911yescharacteristic
295C22×C53#53311yescharacteristic
296C22×C53#53411yescharacteristic
297C22×C53#53511yescharacteristic
298C22×C53#53511yescharacteristic
299C22×C53#53611yescharacteristic
300C22×C53#53812yes—
301C22×C53#53812yes—
302C22×C53#53812yes—
303C22×C53#53812yes—
304C22×C53#53912yes—
305C22×C53#54212yes—
306C22×C53#54212yes—
307C22×C53#54212yes—
308C22×C53#54312yes—
309C22×C53#54412yes—
310C4×C53—24no—
311C4×C53—24no—
312C4×C53—42no—
313C4×C53—42no—
314C4×C53—42no—
315C4×C53—42no—
316Dic53—44no—
317Dic53—44no—
318Dic53—82no—
319Dic53—82no—
320C2×#393—531no—
321C2×#391—531no—
322C2×C4×C22⋊C4D5311yescharacteristic
323#1446—531no—
324#1444—531no—
325#1443—532no—
326#1439—532no—
327#1439—531no—
328#1436—532no—
329#1434—531no—
330C22⋊#95—532no—
331C2×C53—44no—
332C2×C53C4×C22⋊Q811yescharacteristic
333C2×C53#232111yescharacteristic
334C2×C53#232611yescharacteristic
335C2×C53#232912yes—
336C2×C53#233712yes—
337C2×C53#262611yescharacteristic
338C2×C53#262811yescharacteristic
339C2×C53#263111yescharacteristic
340C2×C53#263811yescharacteristic
341C2×C53#265312yes—
342C2×C53#266512yes—
343C23×Q8—531no—
344C24×C4C2×D5311yescharacteristic
345C22×C4⋊C4—532no—
346C22×C4⋊C4—531no—
347C22×C22⋊C4—532no—
348C22×C22⋊C4—532no—
349C22×C22⋊C4C2×D5311yescharacteristic
350C22×C22⋊C4C2×D5311yescharacteristic
351C22×C42C2×D5311yescharacteristic
352C22×C42C2×D5311yescharacteristic
353#393—538no—
354#391—1064no—
355C4×C22⋊C4—24no—
356C2×#95—531no—
357C2×#95—532no—
358C2×#95—532no—
359C2×#95—532no—
360C2×#95—531no—
361C2×#95—532no—
362C2×#95—532no—
363C2×#95—532no—
364C2×#95—532no—
365C2×#95—532no—
366C2×#95—532no—
367C2×#95C2×D5311yescharacteristic
368C2×#95C2×D5311yescharacteristic
369C53#1431911yescharacteristic, complemented, Sylow
370C25C22×D5311yescharacteristic
371C22×Q8—534no—
372C22×Q8—1062no—
373C22×Q8—1062no—
374C22×Q8—2122no—
375C22×Q8—2122no—
376C23×C4—22no—
377C23×C4—22no—
378C23×C4—42no—
379C23×C4—42no—
380C23×C4—532no—
381C23×C4—532no—
382C23×C4—532no—
383C23×C4—532no—
384C23×C4Dic10614yescomplemented
385C23×C4C22×D5311yescharacteristic
386C23×C4C22×D5311yescharacteristic
387C23×C4C22×D5311yescharacteristic
388C23×C4C22×D5311yescharacteristic
389C23×C4C22×D5311yescharacteristic
390C23×C4C22×D5311yescharacteristic
391C2×C4⋊C4—1064no—
392C2×C4⋊C4—1062no—
393C2×C4⋊C4—2122no—
394C2×C4⋊C4—2122no—
395C2×C4⋊C4—2122no—
396C2×C22⋊C4—22no—
397C2×C22⋊C4—22no—
398C2×C22⋊C4—22no—
399C2×C22⋊C4—22no—
400C2×C22⋊C4—22no—
401C2×C22⋊C4—538no—
402C2×C22⋊C4—1064no—
403C2×C22⋊C4—2122no—
404C2×C22⋊C4—2122no—
405C2×C22⋊C4—2122no—
406C2×C22⋊C4—2122no—
407C2×C22⋊C4C4×D5314yes—
408C2×C42—22no—
409C2×C42—42no—
410C2×C42—42no—
411C2×C42C22⋊D5314yes—
412#95—22no—
413#95—22no—
414#95—534no—
415#95—538no—
416#95—1064no—
417#95—1062no—
418#95—1064no—
419#95—1064no—
420#95—2122no—
421#95—2122no—
422#95—2122no—
423#95—2122no—
424#95—2122no—
425C24—21no—
426C24—21no—
427C24—21no—
428C24—21no—
429C24—22no—
430C24—22no—
431C24—22no—
432C24—22no—
433C24C2×Dic10612yes—
434C24C2×C4×D5312yes—
435C24#1171994512yes—
436C24C23×D5311yescharacteristic
437C2×Q8—1068no—
438C2×Q8—2128no—
439C2×Q8—2124no—
440C2×Q8—2124no—
441C2×Q8—2128no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
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—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—28no—
463C22×C4—48no—
464C22×C4—44no—
465C22×C4—44no—
466C22×C4—48no—
467C22×C4—534no—
468C22×C4—534no—
469C22×C4—1062no—
470C22×C4—1062no—
471C22×C4—1062no—
472C22×C4—1062no—
473C22×C4—1062no—
474C22×C4—1062no—
475C22×C4—1062no—
476C22×C4—1062no—
477C22×C4—1062no—
478C22×C4—1062no—
479C22×C4—1062no—
480C22×C4—1062no—
481C22×C4—1062no—
482C22×C4—1062no—
483C22×C4—1062no—
484C22×C4—1062no—
485C22×C4—1062no—
486C22×C4—1062no—
487C22×C4—1062no—
488C22×C4—1062no—
489C22×C4—1062no—
490C22×C4—1062no—
491C22×C4—1062no—
492C22×C4—1062no—
493C22×C4—1062no—
494C22×C4—1062no—
495C22×C4C2×Dic10612yes—
496C22×C4C2×Dic10612yes—
497C22×C4C2×C4×D5312yes—
498C22×C4C2×C4×D5312yes—
499C22×C4#1171994512yes—
500C22×C4#1171994512yes—
501C22×C4#1171994512yes—
502C22×C4C2×C22⋊D5312yes—
503C22×C4C2×C22⋊D5312yes—
504C22×C4C2×C22⋊D5312yes—
505C4⋊C4—2128no—
506C4⋊C4—2124no—
507C22⋊C4—28no—
508C22⋊C4—28no—
509C22⋊C4—2128no—
510C22⋊C4—2128no—
511C42—44no—
512C42—44no—
513C23—22no—
514C23—22no—
515C23—24no—
516C23—24no—
517C23—22no—
518C23—22no—
519C23—22no—
520C23—22no—
521C23—24no—
522C23—24no—
523C23—24no—
524C23—24no—
525C23—41no—
526C23—41no—
527C23—41no—
528C23—41no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23C4×Dic10614yes—
536C23C22×C4×D5311yescharacteristic
537C23C22×C22⋊D5311yescharacteristic
538C23C22×Dic10611yescharacteristic
539C23C2×#1171994511yescharacteristic
540C23C2×#1171994511yescharacteristic
541C23C2×#1171994511yescharacteristic
542C23C2×#1171994511yescharacteristic
543C23#1313326212yes—
544C23#1313326212yes—
545C23#1313326312yes—
546C23#1313326412yes—
547Q8—21216no—
548C2×C4—24no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—22no—
557C2×C4—24no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—416no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—1064no—
578C2×C4—1064no—
579C2×C4—1064no—
580C2×C4—1064no—
581C2×C4—1064no—
582C2×C4—1064no—
583C2×C4—2122no—
584C2×C4—2122no—
585C2×C4—2122no—
586C2×C4—2122no—
587C2×C4—2122no—
588C2×C4—2122no—
589C2×C4—2122no—
590C2×C4—2122no—
591C2×C4—2122no—
592C2×C4—2122no—
593C2×C4—2122no—
594C2×C4—2122no—
595C2×C4—2122no—
596C2×C4—2122no—
597C2×C4—2122no—
598C2×C4—2122no—
599C2×C4—2122no—
600C2×C4—2122no—
601C2×C4—2122no—
602C2×C4—2122no—
603C2×C4—2122no—
604C2×C4—2122no—
605C2×C4C4×C22⋊D5314yes—
606C2×C4#1313319614yes—
607C22—24no—
608C22—28no—
609C22—44no—
610C22—44no—
611C22—42no—
612C22—42no—
613C22—42no—
614C22—44no—
615C22—42no—
616C22—42no—
617C22—42no—
618C22C2×C4×C22⋊D5311yescharacteristic
619C22C2×C4×Dic10611yescharacteristic
620C22C4×#1171994511yescharacteristic
621C22#1681109512yes—
622C22#1681110212yes—
623C22C2×#1313319611yescharacteristic
624C22#1681113411yescharacteristic
625C22#1681115711yescharacteristic
626C22#1681118011yescharacteristic
627C22#1681118211yescharacteristic
628C22#1681118711yescharacteristic
629C22#1681119011yescharacteristic
630C22#1681119411yescharacteristic
631C22#1681120612yes—
632C22#1681120612yes—
633C22#1681120612yes—
634C22#1681121112yes—
635C22#1681121512yes—
636C22#1681123812yes—
637C22#1681123812yes—
638C22#1681124412yes—
639C22#1681124912yes—
640C22#1681125112yes—
641C4—24no—
642C4—24no—
643C4—42no—
644C4—42no—
645C4—42no—
646C4—42no—
647C4—4242no—
648C4—4242no—
649C4—2124no—
650C4—2124no—
651C2—44no—
652C2C4×#1313319611yescharacteristic
653C2#2027736911yescharacteristic
654C2#2027742711yescharacteristic
655C2#2027744312yes—
656C2#2027752312yes—
657C2#2028052011yescharacteristic
658C2#2028053611yescharacteristic
659C2#2028056711yescharacteristic
660C2#2028063211yescharacteristic
661C2#2028077212yes—
662C2#2028088512yes—
subgroupG/H
center, FrattiniC24C23×D53
commutatorC23×C53C23×C4
FittingC2×C4×C22⋊C4×C53C2
socle, cosocleC24×C53C24
prime radicalC25×C53C23
hypercenterC2×C4×C22⋊C4D53
hypocenterC53#14319
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×#16810529C2semidirect
3. #20272985C2semidirect
4. #20272979C2semidirect
5. #20272976C2semidirect
8. #20272948C2semidirect
9. #20272942C2semidirect
17. #16810535C4semidirect
42. C22×Dic106C2×C4semidirect
67. #13133124C2×C4semidirect
68. #13133124C2×C4semidirect
74. C23×C4×C53Q8semidirect
369. C53#14319semidirect
384. C23×C4Dic106semidirect

Maximal subgroups

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

subgroupindex
C2×#168105352
C2×#168105292
#202729852
#202729792
#202729762
#202729602
#202729582
#202729482
#202729422
#202729382
C2×C4×C22⋊C4×C532
#1431953