Finite Group Databasebeta

← #27310311

Subgroups — #27310311

Order 13568.

1686 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×#16810536C211yesmaximal, characteristic
2C2×#16810528C211yesmaximal, characteristic, complemented
3C2×#16810470C211yesmaximal, characteristic
4#20272985C211yesmaximal, characteristic, complemented
5#20272957C211yesmaximal, characteristic
6#20272951C212yesmaximal
7#20272941C212yesmaximal, complemented
8#20272940C212yesmaximal, complemented
9#20272936C212yesmaximal
10#95×Dic53C211yesmaximal, characteristic, complemented
11C53×#1442C211yesmaximal, characteristic, complemented
12C23×Dic106C2211yescharacteristic
13C23×C4×D53C2211yescharacteristic
14C22×#11719921C2212yes—
15C22×#11719921C2211yescharacteristic
16C22×#11719921C2211yescharacteristic
17C22×#11719921C2212yes—
18C22×C4×Dic53C2211yescharacteristic
19C22×C4×Dic53C2212yes—
20C22×C4×Dic53C2211yescharacteristic
21#16810536C418yescomplemented
22#16810528—24no—
23#16810470—24no—
24C2×#13133124C2212yes—
25C2×#13133124C2211yescharacteristic
26C2×#13133124C2212yes—
27C2×#13133124C2211yescharacteristic
28C2×#13133124C2212yes—
29C2×#13133124C2212yes—
30C2×#13133124C2212yes—
31C2×#13133124C2212yes—
32C2×#13133124C2212yes—
33C22×C4⋊C4×C53C2211yescharacteristic
34C2×#95×C53C2212yes—
35C2×#95×C53C2211yescharacteristic
36C2×#95×C53C2212yes—
37C2×#95×C53C2211yescharacteristic
38C24×D53C2311yescharacteristic
39C22×Dic106—22no—
40C22×Dic106—22no—
41C22×Dic106—42no—
42C22×Dic106—42no—
43C22×Dic106C2×C414yescomplemented
44C23×Dic53C2311yescharacteristic
45C23×Dic53C2312yes—
46C23×Dic53C2312yes—
47C23×Dic53C2311yescharacteristic
48C23×Dic53C2311yescharacteristic
49C22×C4×D53—22no—
50C22×C4×D53—22no—
51C22×C4×D53—22no—
52C22×C4×D53—24no—
53C22×C4×D53—42no—
54C2×#11719921—24no—
55C2×#11719921—22no—
56C2×#11719921—22no—
57C2×#11719921—22no—
58C2×#11719921—22no—
59C2×#11719921—24no—
60C2×#11719921—42no—
61C2×#11719921—42no—
62C2×#11719921—42no—
63C2×#11719921C2×C414yes—
64C2×#11719921C2×C418yes—
65C2×#11719921Q814yescomplemented
66C2×C4×Dic53—22no—
67C2×C4×Dic53—24no—
68C2×C4×Dic53—24no—
69C2×C4×Dic53—24no—
70C2×C4×Dic53—42no—
71C2×C4×Dic53—42no—
72C2×C4×Dic53D414yes—
73#13133124—22no—
74#13133124—24no—
75#13133124—24no—
76#13133124—22no—
77#13133124—24no—
78#13133124—24no—
79#13133124—42no—
80#13133124—42no—
81#13133124C2×C418yescomplemented
82C23×C4×C53C2312yes—
83C23×C4×C53C2312yes—
84C23×C4×C53C2311yescharacteristic
85C23×C4×C53C2311yescharacteristic
86C23×C4×C53C2311yescharacteristic
87C2×C4⋊C4×C53—22no—
88C2×C4⋊C4×C53—42no—
89#95×C53—22no—
90#95×C53—24no—
91#95×C53—42no—
92#95×C53C2×C414yescomplemented
93C24×C53C2411yescharacteristic
94C23×D53—21no—
95C23×D53—21no—
96C23×D53—21no—
97C23×D53—22no—
98C23×D53—21no—
99C23×D53—22no—
100C23×D53—22no—
101C23×D53—22no—
102C23×D53C22×C412yes—
103C23×D53C2×Q812yes—
104C23×D53#4112yes—
105C22×C4×C53—22no—
106C22×C4×C53—22no—
107C22×C4×C53—21no—
108C22×C4×C53—21no—
109C22×C4×C53—22no—
110C22×C4×C53—22no—
111C22×C4×C53—22no—
112C22×C4×C53—22no—
113C22×C4×C53—22no—
114C22×C4×C53—21no—
115C22×C4×C53—21no—
116C22×C4×C53—22no—
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—22no—
126C22×C4×C53—22no—
127C22×C4×C53—22no—
128C22×C4×C53C22×C414yes—
129C22×C4×C53C22×C412yes—
130C22×C4×C53C2×D412yes—
131C22×C4×C53C2×Q812yes—
132C22×C4×C53#4112yes—
133C22×C4×C53#4112yes—
134C22×Dic53—22no—
135C22×Dic53—21no—
136C22×Dic53—21no—
137C22×Dic53—22no—
138C22×Dic53—22no—
139C22×Dic53—22no—
140C22×Dic53—21no—
141C22×Dic53—22no—
142C22×Dic53—22no—
143C22×Dic53—22no—
144C22×Dic53—22no—
145C22×Dic53—22no—
146C22×Dic53—21no—
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×Dic53C22×C412yes—
156C22×Dic53C22×C414yes—
157C22×Dic53C2×D412yes—
158C22×Dic53C2×D412yes—
159C22×Dic53C2×Q812yes—
160C22×Dic53#4114yes—
161C22×Dic53#4112yes—
162C22×Dic53#4114yes—
163C2×C4×D53—28no—
164C2×C4×D53—48no—
165C2×C4×D53—48no—
166C2×C4×D53—44no—
167C2×C4×D53—44no—
168C2×Dic106—28no—
169C2×Dic106—48no—
170C2×Dic106—44no—
171C2×Dic106—44no—
172C2×Dic106—48no—
173C4⋊C4×C53—44no—
174#11719921—28no—
175#11719921—28no—
176#11719921—48no—
177#11719921—48no—
178C4×Dic53—44no—
179C4×Dic53—48no—
180C4×Dic53—44no—
181C23×C53C23×C411yescharacteristic
182C23×C53C22×D411yescharacteristic
183C23×C53C22×Q811yescharacteristic
184C23×C53C2×#4111yescharacteristic
185C23×C53C2×#4111yescharacteristic
186C23×C53C2×#4112yes—
187C23×C53C2×#4112yes—
188C23×C53C2×#4111yescharacteristic
189C23×C53C2×#4111yescharacteristic
190C23×C53H(5,𝔽2)12yes—
191C23×C53H(5,𝔽2)12yes—
192C22×D53—22no—
193C22×D53—22no—
194C22×D53—24no—
195C22×D53—24no—
196C22×D53—22no—
197C22×D53—22no—
198C22×D53—22no—
199C22×D53—22no—
200C22×D53—24no—
201C22×D53—24no—
202C22×D53—24no—
203C22×D53—24no—
204C22×D53—41no—
205C22×D53—42no—
206C22×D53—41no—
207C22×D53—41no—
208C22×D53—41no—
209C22×D53—42no—
210C22×D53—42no—
211C22×D53—42no—
212C22×D53—42no—
213C22×D53—42no—
214C22×D53C4×Q814yes—
215C2×C4×C53—24no—
216C2×C4×C53—22no—
217C2×C4×C53—22no—
218C2×C4×C53—24no—
219C2×C4×C53—24no—
220C2×C4×C53—24no—
221C2×C4×C53—24no—
222C2×C4×C53—22no—
223C2×C4×C53—22no—
224C2×C4×C53—24no—
225C2×C4×C53—24no—
226C2×C4×C53—41no—
227C2×C4×C53—41no—
228C2×C4×C53—41no—
229C2×C4×C53—41no—
230C2×C4×C53—42no—
231C2×C4×C53—42no—
232C2×C4×C53—42no—
233C2×C4×C53—42no—
234C2×C4×C53—42no—
235C2×C4×C53—42no—
236C2×C4×C53—42no—
237C2×C4×C53—42no—
238C2×C4×C53—42no—
239C2×C4×C53—42no—
240C2×C4×C53—42no—
241C2×C4×C53—42no—
242C2×C4×C53—42no—
243C2×C4×C53C4×D414yes—
244C2×Dic53—22no—
245C2×Dic53—22no—
246C2×Dic53—22no—
247C2×Dic53—22no—
248C2×Dic53—24no—
249C2×Dic53—24no—
250C2×Dic53—24no—
251C2×Dic53—24no—
252C2×Dic53—24no—
253C2×Dic53—22no—
254C2×Dic53—24no—
255C2×Dic53—24no—
256C2×Dic53—24no—
257C2×Dic53—24no—
258C2×Dic53—24no—
259C2×Dic53—24no—
260C2×Dic53—41no—
261C2×Dic53—42no—
262C2×Dic53—42no—
263C2×Dic53—41no—
264C2×Dic53—42no—
265C2×Dic53—42no—
266C2×Dic53—42no—
267C2×Dic53—42no—
268C2×Dic53—42no—
269C2×Dic53—42no—
270C2×Dic53—42no—
271C2×Dic53C22⋊Q814yes—
272C2×Dic53#12414yes—
273C4×D53—416no—
274Dic106—416no—
275#14243—531nomaximal, Sylow
276C22×C53C2×C4×D411yescharacteristic
277C22×C53C2×C4×Q811yescharacteristic
278C22×C53C4×#4111yescharacteristic
279C22×C53C4×#4112yes—
280C22×C53#51812yes—
281C22×C53C2×C22⋊Q811yescharacteristic
282C22×C53#52511yescharacteristic
283C22×C53C2×#12411yescharacteristic
284C22×C53#52911yescharacteristic
285C22×C53#52912yes—
286C22×C53#53211yescharacteristic
287C22×C53#53311yescharacteristic
288C22×C53#53411yescharacteristic
289C22×C53#53511yescharacteristic
290C22×C53#53812yes—
291C22×C53#55112yes—
292C22×C53#55112yes—
293C22×C53#55112yes—
294C22×C53#55112yes—
295C22×C53#55412yes—
296C22×C53#55412yes—
297C22×C53#55512yes—
298C22×C53#55512yes—
299C2×D53—28no—
300C2×D53—24no—
301C2×D53—42no—
302C2×D53—42no—
303C2×D53—42no—
304C2×D53—44no—
305C2×D53—44no—
306C2×D53—42no—
307C2×D53—42no—
308C2×D53—44no—
309C2×D53—42no—
310C4×C53—24no—
311C4×C53—42no—
312C4×C53—42no—
313C4×C53—44no—
314C4×C53—82no—
315Dic53—24no—
316Dic53—24no—
317Dic53—44no—
318Dic53—44no—
319Dic53—42no—
320C2×#393—531no—
321C2×#391—531no—
322C2×C4×C22⋊C4—531no—
323#1446—531no—
324#1442D5311yescharacteristic, complemented
325#1439—531no—
326#1437—532no—
327#1434—532no—
328#1434—532no—
329C22⋊#95—532no—
330C4×#95—531no—
331C2×C53C4×C22⋊Q811yescharacteristic
332C2×C53C4×#12411yescharacteristic
333C2×C53#232111yescharacteristic
334C2×C53#236812yes—
335C2×C53#237212yes—
336C2×C53#256111yescharacteristic
337C2×C53#256311yescharacteristic
338C2×C53#256411yescharacteristic
339C2×C53#257212yes—
340C2×C53#257412yes—
341C2×C53#261011yescharacteristic
342D53—44no—
343C23×Q8—531no—
344C24×C4—531no—
345C22×C4⋊C4C2×D5311yescharacteristic
346C22×C22⋊C4—531no—
347C22×C22⋊C4—531no—
348C22×C22⋊C4—532no—
349C22×C22⋊C4—532no—
350C22×C42—531no—
351C22×C42—531no—
352C22×C42—532no—
353#393—538no—
354#391—1064no—
355C4×C22⋊C4—1064no—
356C2×#95—532no—
357C2×#95—532no—
358C2×#95—531no—
359C2×#95—532no—
360C2×#95—532no—
361C2×#95—531no—
362C2×#95—532no—
363C2×#95—532no—
364C2×#95—532no—
365C2×#95C2×D5312yes—
366C2×#95C2×D5311yescharacteristic
367C2×#95C2×D5312yes—
368C2×#95C2×D5311yescharacteristic
369C53#1424311yescharacteristic, complemented, Sylow
370C25—531no—
371C22×Q8—534no—
372C22×Q8—1062no—
373C22×Q8—1062no—
374C22×Q8—2122no—
375C22×Q8—2122no—
376C23×C4—531no—
377C23×C4—531no—
378C23×C4—532no—
379C23×C4—531no—
380C23×C4—532no—
381C23×C4—1062no—
382C23×C4—1064no—
383C23×C4—1062no—
384C23×C4—1062no—
385C23×C4—2122no—
386C23×C4C22×D5312yes—
387C23×C4C22×D5312yes—
388C23×C4C22×D5311yescharacteristic
389C23×C4C22×D5311yescharacteristic
390C23×C4C22×D5311yescharacteristic
391C2×C4⋊C4—22no—
392C2×C4⋊C4—42no—
393C2×C22⋊C4—534no—
394C2×C22⋊C4—534no—
395C2×C22⋊C4—538no—
396C2×C22⋊C4—1062no—
397C2×C22⋊C4—1062no—
398C2×C22⋊C4—1062no—
399C2×C22⋊C4—1062no—
400C2×C22⋊C4—1064no—
401C2×C22⋊C4—1064no—
402C2×C22⋊C4—2122no—
403C2×C22⋊C4—2122no—
404C2×C22⋊C4—2122no—
405C2×C42—534no—
406C2×C42—1062no—
407C2×C42—1064no—
408C2×C42—1064no—
409C2×C42—1064no—
410C2×C42—2122no—
411C2×C42—2122no—
412#95—22no—
413#95—24no—
414#95—42no—
415#95—538no—
416#95—1064no—
417#95—1064no—
418#95—1062no—
419#95—1064no—
420#95—1064no—
421#95—1062no—
422#95—2122no—
423#95—2122no—
424#95C4×D5314yescomplemented
425C24—532no—
426C24—532no—
427C24—532no—
428C24—1061no—
429C24—1062no—
430C24—1061no—
431C24—1062no—
432C24—1061no—
433C24—1061no—
434C24—1062no—
435C24—1062no—
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—21no—
445C22×C4—21no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—21no—
451C22×C4—22no—
452C22×C4—21no—
453C22×C4—22no—
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—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—532no—
466C22×C4—532no—
467C22×C4—532no—
468C22×C4—534no—
469C22×C4—532no—
470C22×C4—532no—
471C22×C4—534no—
472C22×C4—534no—
473C22×C4—1068no—
474C22×C4—1062no—
475C22×C4—1061no—
476C22×C4—1061no—
477C22×C4—1062no—
478C22×C4—1062no—
479C22×C4—1062no—
480C22×C4—1061no—
481C22×C4—1062no—
482C22×C4—1062no—
483C22×C4—1062no—
484C22×C4—1062no—
485C22×C4—1062no—
486C22×C4—1062no—
487C22×C4—1061no—
488C22×C4—1062no—
489C22×C4—1062no—
490C22×C4—1062no—
491C22×C4—1062no—
492C22×C4—1062no—
493C22×C4—1062no—
494C22×C4—1062no—
495C22×C4—2128no—
496C22×C4—2124no—
497C22×C4—2128no—
498C22×C4—2124no—
499C22×C4C2×C4×D5314yes—
500C22×C4C2×C4×D5312yes—
501C22×C4D4×D5312yes—
502C22×C4#1171994712yes—
503C22×C4Q8×D5312yes—
504C22×C4#1171994912yes—
505C4⋊C4—44no—
506C22⋊C4—1068no—
507C22⋊C4—1068no—
508C22⋊C4—2128no—
509C22⋊C4—2128no—
510C42—2124no—
511C42—2124no—
512C42—2128no—
513C23—534no—
514C23—1064no—
515C23—1064no—
516C23—1064no—
517C23—1064no—
518C23—1062no—
519C23—1062no—
520C23—1064no—
521C23—1064no—
522C23—1062no—
523C23—1062no—
524C23—1062no—
525C23—1062no—
526C23—2122no—
527C23—2122no—
528C23—2122no—
529C23—2122no—
530C23—2121no—
531C23—2122no—
532C23—2121no—
533C23—2121no—
534C23—2121no—
535C23—2122no—
536C23C22×C4×D5311yescharacteristic
537C23C2×D4×D5311yescharacteristic
538C23C2×Q8×D5311yescharacteristic
539C23#41×D5312yes—
540C23#41×D5311yescharacteristic
541C23#41×D5311yescharacteristic
542C23#41×D5312yes—
543C23C2×#1171994711yescharacteristic
544C23C2×#1171994911yescharacteristic
545C23#1313326212yes—
546C23#1313326312yes—
547Q8—21216no—
548C2×C4—24no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—534no—
577C2×C4—534no—
578C2×C4—1064no—
579C2×C4—1064no—
580C2×C4—1064no—
581C2×C4—1064no—
582C2×C4—1062no—
583C2×C4—1062no—
584C2×C4—1062no—
585C2×C4—1062no—
586C2×C4—1064no—
587C2×C4—1064no—
588C2×C4—1062no—
589C2×C4—1064no—
590C2×C4—1064no—
591C2×C4—1064no—
592C2×C4—1064no—
593C2×C4—1064no—
594C2×C4—21216no—
595C2×C4—2122no—
596C2×C4—2122no—
597C2×C4—2121no—
598C2×C4—2122no—
599C2×C4—2122no—
600C2×C4—2121no—
601C2×C4—2122no—
602C2×C4—2122no—
603C2×C4—2122no—
604C2×C4—2122no—
605C2×C4—2122no—
606C2×C4#1313317914yes—
607C22—1068no—
608C22—1064no—
609C22—2122no—
610C22—2122no—
611C22—2124no—
612C22—2122no—
613C22—2122no—
614C22—2122no—
615C22—2122no—
616C22—2124no—
617C22—2124no—
618C22C2×#1313317911yescharacteristic
619C22C4×Q8×D5311yescharacteristic
620C22C4×#1171994711yescharacteristic
621C22#1681109112yes—
622C22#1681109512yes—
623C22C22⋊Q8×D5311yescharacteristic
624C22#1681113311yescharacteristic
625C22#124×D5311yescharacteristic
626C22#1681115812yes—
627C22#1681115811yescharacteristic
628C22#1681117211yescharacteristic
629C22#1681117811yescharacteristic
630C22#1681118411yescharacteristic
631C22#1681118811yescharacteristic
632C22#1681121312yes—
633C22#1681129612yes—
634C22#1681129812yes—
635C22#1681129812yes—
636C22#1681130112yes—
637C22#1681132712yes—
638C22#1681132912yes—
639C22#1681133512yes—
640C22#1681133712yes—
641C4—24no—
642C4—42no—
643C4—44no—
644C4—42no—
645C4—82no—
646C4—1064no—
647C4—1064no—
648C4—2122no—
649C4—2124no—
650C4—2124no—
651C2—2124no—
652C2#2027724511yescharacteristic
653C2#2027725411yescharacteristic
654C2#2027737211yescharacteristic
655C2#2027779212yes—
656C2#2027782212yes—
657C2#2027986711yescharacteristic
658C2#2027988311yescharacteristic
659C2#2027989811yescharacteristic
660C2#2027996512yes—
661C2#2027999212yes—
662C2#2028040511yescharacteristic
subgroupG/H
center, FrattiniC24C23×D53
commutatorC23×C53C23×C4
FittingC53×#1442C2
socle, cosocleC24×C53C24
prime radicalC24×D53C23
hypercenter#1442D53
hypocenterC53#14243
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C24 ◁ #1442

Complements

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

NH ≅ G/Ntype
2. C2×#16810528C2semidirect
4. #20272985C2semidirect
7. #20272941C2semidirect
8. #20272940C2semidirect
10. #95×Dic53C2semidirect
11. C53×#1442C2semidirect
21. #16810536C4semidirect
43. C22×Dic106C2×C4semidirect
65. C2×#11719921Q8semidirect
81. #13133124C2×C4semidirect
92. #95×C53C2×C4semidirect
324. #1442D53semidirect
369. C53#14243semidirect
424. #95C4×D53semidirect

Maximal subgroups

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

subgroupindex
C2×#168105362
C2×#168105282
C2×#168104702
#202729852
#202729572
#202729512
#202729412
#202729402
#202729362
#95×Dic532
C53×#14422
#1424353