Finite Group Databasebeta

← #27310772

Subgroups — #27310772

Order 13568.

1782 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13133194C211yesmaximal, characteristic
2C2×#16810484C211yesmaximal, characteristic
3C2×C22⋊C4×Dic53C211yesmaximal, characteristic
4#20272985C212yesmaximal, complemented
5#20272981C212yesmaximal, complemented
6#20272964C211yesmaximal, characteristic
7#20272962C211yesmaximal, characteristic
8#20272962C211yesmaximal, characteristic
9#20272949C212yesmaximal, complemented
10#20272946C212yesmaximal, complemented
11C53×#1440C211yesmaximal, characteristic
12C23×Dic106C2211yescharacteristic
13C24×Dic53C2211yescharacteristic
14C2×#13133194—24no—
15C2×#13133194—24no—
16C2×#13133194C418yescomplemented
17C22×C4⋊Dic53C2211yescharacteristic
18C22×#11719919C2211yescharacteristic
19C22×#11719919C2211yescharacteristic
20C22×C22⋊Dic53C2211yescharacteristic
21C22×C22⋊Dic53C2211yescharacteristic
22C22×C22⋊Dic53C2211yescharacteristic
23C22×C4×Dic53C2212yes—
24#16810484—24no—
25C22⋊C4×Dic53—24no—
26C2×#13133124C2212yes—
27C2×#13133124C2212yes—
28C2×#13133124C2212yes—
29C2×#13133124C2212yes—
30C2×#13133124C2212yes—
31C2×#13133124C2212yes—
32C2×#13133124C2212yes—
33C2×#13133124C2212yes—
34C2×#13133124C2212yes—
35C22×C22⋊C4×C53C2211yescharacteristic
36C22×C22⋊C4×C53C2211yescharacteristic
37C22×C22⋊C4×C53C2211yescharacteristic
38C2×#95×C53C2212yes—
39C2×#95×C53C2212yes—
40C22×Dic106—22no—
41C22×Dic106—22no—
42C22×Dic106—42no—
43C22×Dic106—42no—
44C22×Dic106C2×C414yes—
45C23×Dic53—42no—
46C23×Dic53—42no—
47C23×Dic53C2×C414yes—
48C23×Dic53D414yes—
49C23×Dic53D414yes—
50C23×Dic53C2312yes—
51C23×Dic53C2311yescharacteristic
52C23×Dic53C2311yescharacteristic
53C23×Dic53C2312yes—
54C23×Dic53C2311yescharacteristic
55C23×Dic53C2311yescharacteristic
56#13133194—232no—
57C2×C4⋊Dic53—22no—
58C2×C4⋊Dic53—22no—
59C2×C4⋊Dic53C2×C414yes—
60C2×#11719919—22no—
61C2×#11719919—22no—
62C2×#11719919—22no—
63C2×#11719919—22no—
64C2×#11719919C2×C414yes—
65C2×#11719919C2×C414yes—
66C2×C22⋊Dic53—22no—
67C2×C22⋊Dic53—22no—
68C2×C22⋊Dic53—22no—
69C2×C22⋊Dic53—22no—
70C2×C22⋊Dic53—24no—
71C2×C22⋊Dic53—42no—
72C2×C22⋊Dic53C2×C414yes—
73C2×C4×Dic53—24no—
74C2×C4×Dic53—42no—
75C2×C4×Dic53—42no—
76#13133124—24no—
77#13133124—24no—
78#13133124—24no—
79#13133124—24no—
80#13133124—24no—
81#13133124—24no—
82#13133124—42no—
83#13133124—42no—
84#13133124—42no—
85C25×C53C2311yescharacteristic
86C23×C4×C53C2312yes—
87C23×C4×C53C2311yescharacteristic
88C23×C4×C53C2311yescharacteristic
89C23×C4×C53C2312yes—
90C2×C22⋊C4×C53—22no—
91C2×C22⋊C4×C53—22no—
92C2×C22⋊C4×C53—22no—
93C2×C22⋊C4×C53—22no—
94C2×C22⋊C4×C53—24no—
95C2×C22⋊C4×C53—42no—
96C2×C22⋊C4×C53C2×C414yes—
97#95×C53—24no—
98#95×C53—42no—
99C24×C53—22no—
100C24×C53—22no—
101C24×C53—22no—
102C24×C53—22no—
103C24×C53—22no—
104C24×C53—22no—
105C24×C53C22×C412yes—
106C24×C53C2×D412yes—
107C24×C53C2×D412yes—
108C24×C53C2411yescharacteristic
109C22×C4×C53—22no—
110C22×C4×C53—21no—
111C22×C4×C53—21no—
112C22×C4×C53—21no—
113C22×C4×C53—21no—
114C22×C4×C53—22no—
115C22×C4×C53—22no—
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×C53—22no—
129C22×C4×C53—22no—
130C22×C4×C53C22×C412yes—
131C22×C4×C53C22×C412yes—
132C22×C4×C53#4114yes—
133C22×Dic53—22no—
134C22×Dic53—22no—
135C22×Dic53—22no—
136C22×Dic53—22no—
137C22×Dic53—22no—
138C22×Dic53—21no—
139C22×Dic53—21no—
140C22×Dic53—21no—
141C22×Dic53—21no—
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—48no—
159C22×Dic53—44no—
160C22×Dic53—44no—
161C22×Dic53—48no—
162C22×Dic53C22⋊C4116yes—
163C22×Dic53C22×C412yes—
164C22×Dic53C22×C412yes—
165C22×Dic53C22×C412yes—
166C22×Dic53C22×C412yes—
167C22×Dic53C2×D412yes—
168C22×Dic53C2×D412yes—
169C22×Dic53C2×D412yes—
170C22×Dic53C2×D412yes—
171C22×Dic53#4114yes—
172C2×Dic106—28no—
173C2×Dic106—48no—
174C2×Dic106—48no—
175C2×Dic106—44no—
176C2×Dic106—44no—
177C22⋊C4×C53—28no—
178C22⋊C4×C53—28no—
179C22⋊C4×C53—44no—
180C22⋊Dic53—28no—
181C22⋊Dic53—28no—
182C22⋊Dic53—44no—
183C4⋊Dic53—28no—
184#11719919—28no—
185#11719919—28no—
186C4×Dic53—48no—
187C23×C53—24no—
188C23×C53—24no—
189C23×C53—22no—
190C23×C53—22no—
191C23×C53—22no—
192C23×C53—22no—
193C23×C53—24no—
194C23×C53—24no—
195C23×C53—22no—
196C23×C53—22no—
197C23×C53—24no—
198C23×C53—24no—
199C23×C53—42no—
200C23×C53—42no—
201C23×C53—42no—
202C23×C53—42no—
203C23×C53—42no—
204C23×C53—42no—
205C23×C53—42no—
206C23×C53—42no—
207C23×C53C2×C22⋊C414yes—
208C23×C53C23×C411yescharacteristic
209C23×C53C22×D411yescharacteristic
210C23×C53C22×D411yescharacteristic
211C23×C53C2×#4112yes—
212C23×C53C2×#4112yes—
213C23×C53H(5,𝔽2)12yes—
214C23×C53H(5,𝔽2)12yes—
215C23×C532−1+412yes—
216C23×C532−1+412yes—
217C2×C4×C53—22no—
218C2×C4×C53—22no—
219C2×C4×C53—24no—
220C2×C4×C53—24no—
221C2×C4×C53—24no—
222C2×C4×C53—24no—
223C2×C4×C53—24no—
224C2×C4×C53—41no—
225C2×C4×C53—41no—
226C2×C4×C53—42no—
227C2×C4×C53—42no—
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×Dic53—24no—
242C2×Dic53—24no—
243C2×Dic53—24no—
244C2×Dic53—22no—
245C2×Dic53—22no—
246C2×Dic53—22no—
247C2×Dic53—22no—
248C2×Dic53—24no—
249C2×Dic53—22no—
250C2×Dic53—22no—
251C2×Dic53—24no—
252C2×Dic53—24no—
253C2×Dic53—24no—
254C2×Dic53—24no—
255C2×Dic53—24no—
256C2×Dic53—416no—
257C2×Dic53—41no—
258C2×Dic53—41no—
259C2×Dic53—42no—
260C2×Dic53—42no—
261C2×Dic53—41no—
262C2×Dic53—41no—
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—
274C2×Dic53C2×C22⋊C414yes—
275C2×Dic53C2×C22⋊C414yes—
276Dic106—416no—
277#14292—531nomaximal, Sylow
278C22×C53—24no—
279C22×C53—24no—
280C22×C53—24no—
281C22×C53—44no—
282C22×C53—44no—
283C22×C53—44no—
284C22×C53—44no—
285C22×C53—44no—
286C22×C53—44no—
287C22×C53C22×C22⋊C411yescharacteristic
288C22×C53C4×#4112yes—
289C22×C53#51812yes—
290C22×C53#51911yescharacteristic
291C22×C53#51911yescharacteristic
292C22×C53#52511yescharacteristic
293C22×C53#52511yescharacteristic
294C22×C53#53212yes—
295C22×C53#53412yes—
296C22×C53#53411yescharacteristic
297C22×C53#53411yescharacteristic
298C22×C53#53612yes—
299C22×C53#53612yes—
300C22×C53#53912yes—
301C22×C53#53912yes—
302C22×C53#53912yes—
303C22×C53#53912yes—
304C22×C53#54212yes—
305C22×C53#54212yes—
306C22×C53#54212yes—
307C22×C53#54212yes—
308C4×C53—42no—
309C4×C53—44no—
310C4×C53—42no—
311C4×C53—82no—
312Dic53—24no—
313Dic53—24no—
314Dic53—42no—
315Dic53—44no—
316Dic53—42no—
317Dic53—82no—
318C22×C22⋊Q8—531no—
319C2×#381—531no—
320C2×C4×C22⋊C4—531no—
321#1446—532no—
322#1445—532no—
323#1440—531no—
324#1440—531no—
325#1440—531no—
326#1440D5311yescharacteristic
327#1436—532no—
328#1436—532no—
329C2×C53—44no—
330C2×C53#229711yescharacteristic
331C2×C53#230811yescharacteristic
332C2×C53#231011yescharacteristic
333C2×C53#233212yes—
334C2×C53#233212yes—
335C2×C53#263812yes—
336C2×C53#263812yes—
337C2×C53#266312yes—
338C2×C53#266312yes—
339C23×Q8—531no—
340C24×C4—531no—
341C2×C22⋊Q8—538no—
342C2×C22⋊Q8—1064no—
343C2×C22⋊Q8—1064no—
344C22×C4⋊C4—531no—
345C22×C4⋊C4—531no—
346C22×C4⋊C4—531no—
347C22×C22⋊C4—531no—
348C22×C22⋊C4—531no—
349C22×C22⋊C4—531no—
350C22×C22⋊C4C2×D5311yescharacteristic
351C22×C22⋊C4C2×D5311yescharacteristic
352C22×C22⋊C4C2×D5311yescharacteristic
353C22×C42—532no—
354#381—1064no—
355C4×C22⋊C4—1064no—
356C2×#95—532no—
357C2×#95—532no—
358C2×#95—532no—
359C2×#95—532no—
360C2×#95—532no—
361C2×#95—532no—
362C2×#95—532no—
363C2×#95—532no—
364C2×#95—532no—
365C2×#95C2×D5312yes—
366C2×#95C2×D5312yes—
367C53#1429211yescharacteristic, complemented, Sylow
368C25C22×D5311yescharacteristic
369C22×Q8—534no—
370C22×Q8—1062no—
371C22×Q8—1062no—
372C22×Q8—2122no—
373C22×Q8—2122no—
374C23×C4—531no—
375C23×C4—531no—
376C23×C4—534no—
377C23×C4—534no—
378C23×C4—534no—
379C23×C4—531no—
380C23×C4—531no—
381C23×C4—532no—
382C23×C4—532no—
383C23×C4—2122no—
384C23×C4—2122no—
385C23×C4C22×D5312yes—
386C23×C4C22×D5311yescharacteristic
387C23×C4C22×D5311yescharacteristic
388C23×C4C22×D5312yes—
389C22⋊Q8—10632no—
390C2×C4⋊C4—534no—
391C2×C4⋊C4—534no—
392C2×C4⋊C4—534no—
393C2×C4⋊C4—1062no—
394C2×C4⋊C4—1062no—
395C2×C4⋊C4—1062no—
396C2×C4⋊C4—1062no—
397C2×C4⋊C4—1062no—
398C2×C4⋊C4—1062no—
399C2×C22⋊C4—22no—
400C2×C22⋊C4—22no—
401C2×C22⋊C4—22no—
402C2×C22⋊C4—24no—
403C2×C22⋊C4—22no—
404C2×C22⋊C4—42no—
405C2×C22⋊C4—534no—
406C2×C22⋊C4—1064no—
407C2×C22⋊C4—1062no—
408C2×C22⋊C4—1062no—
409C2×C22⋊C4—1062no—
410C2×C22⋊C4—1062no—
411C2×C22⋊C4—2122no—
412C2×C22⋊C4C4×D5314yes—
413C2×C42—1064no—
414C2×C42—2122no—
415C2×C42—2122no—
416#95—24no—
417#95—42no—
418#95—1064no—
419#95—1064no—
420#95—1064no—
421#95—1064no—
422#95—1064no—
423#95—1064no—
424#95—2122no—
425#95—2122no—
426#95—2122no—
427C24—22no—
428C24—22no—
429C24—22no—
430C24—22no—
431C24—22no—
432C24—22no—
433C24C2×C4×D5312yes—
434C24D4×D5312yes—
435C24D4×D5312yes—
436C24C23×D5311yescharacteristic
437C2×Q8—1068no—
438C2×Q8—2128no—
439C2×Q8—2128no—
440C2×Q8—2124no—
441C2×Q8—2124no—
442C22×C4—22no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
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—532no—
464C22×C4—532no—
465C22×C4—532no—
466C22×C4—532no—
467C22×C4—532no—
468C22×C4—532no—
469C22×C4—5316no—
470C22×C4—532no—
471C22×C4—532no—
472C22×C4—534no—
473C22×C4—1062no—
474C22×C4—1062no—
475C22×C4—1062no—
476C22×C4—1062no—
477C22×C4—1062no—
478C22×C4—1061no—
479C22×C4—1061no—
480C22×C4—1061no—
481C22×C4—1061no—
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×C4—1062no—
496C22×C4—1062no—
497C22×C4—1062no—
498C22×C4—2128no—
499C22×C4—2128no—
500C22×C4—2124no—
501C22×C4—2124no—
502C22×C4C2×C4×D5312yes—
503C22×C4C2×C4×D5312yes—
504C22×C4#1171994714yes—
505C4⋊C4—1068no—
506C4⋊C4—1068no—
507C4⋊C4—1068no—
508C22⋊C4—28no—
509C22⋊C4—28no—
510C22⋊C4—44no—
511C22⋊C4—1068no—
512C22⋊C4—1068no—
513C22⋊C4—2124no—
514C42—2128no—
515C23—24no—
516C23—24no—
517C23—22no—
518C23—22no—
519C23—22no—
520C23—22no—
521C23—24no—
522C23—24no—
523C23—22no—
524C23—22no—
525C23—24no—
526C23—24no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23C22⋊C4×D5314yes—
536C23C22×C4×D5311yescharacteristic
537C23C2×D4×D5311yescharacteristic
538C23C2×D4×D5311yescharacteristic
539C23#41×D5312yes—
540C23C2×#1171994712yes—
541C23#1313326212yes—
542C23#1313326312yes—
543C23#1313326412yes—
544C23#1313326412yes—
545Q8—21216no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—534no—
571C2×C4—534no—
572C2×C4—1064no—
573C2×C4—1064no—
574C2×C4—1064no—
575C2×C4—1064no—
576C2×C4—1062no—
577C2×C4—1062no—
578C2×C4—1062no—
579C2×C4—1062no—
580C2×C4—1062no—
581C2×C4—1062no—
582C2×C4—1064no—
583C2×C4—1064no—
584C2×C4—1064no—
585C2×C4—1064no—
586C2×C4—1064no—
587C2×C4—21216no—
588C2×C4—2122no—
589C2×C4—2122no—
590C2×C4—2122no—
591C2×C4—2122no—
592C2×C4—2122no—
593C2×C4—2122no—
594C2×C4—2122no—
595C2×C4—2121no—
596C2×C4—2121no—
597C2×C4—2122no—
598C2×C4—2122no—
599C2×C4—2122no—
600C2×C4—2121no—
601C2×C4—2121no—
602C2×C4—2122no—
603C2×C4—2122no—
604C2×C4—2122no—
605C22—24no—
606C22—24no—
607C22—24no—
608C22—44no—
609C22—44no—
610C22—44no—
611C22—44no—
612C22—44no—
613C22—44no—
614C22C2×C22⋊C4×D5311yescharacteristic
615C22C4×#1171994712yes—
616C22#1681109512yes—
617C22#1681110011yescharacteristic
618C22#1681110011yescharacteristic
619C22#1681112911yescharacteristic
620C22#1681113111yescharacteristic
621C22#1681116812yes—
622C22#1681118112yes—
623C22#1681118311yescharacteristic
624C22#1681118411yescharacteristic
625C22#1681119512yes—
626C22#1681119512yes—
627C22#1681121412yes—
628C22#1681121612yes—
629C22#1681121712yes—
630C22#1681121812yes—
631C22#1681123712yes—
632C22#1681123912yes—
633C22#1681124012yes—
634C22#1681124212yes—
635C4—42no—
636C4—44no—
637C4—42no—
638C4—82no—
639C4—1064no—
640C4—1064no—
641C4—4242no—
642C4—2122no—
643C4—2124no—
644C4—2122no—
645C2—44no—
646C2#2027720011yescharacteristic
647C2#2027728911yescharacteristic
648C2#2027730011yescharacteristic
649C2#2027746512yes—
650C2#2027747212yes—
651C2#2028063112yes—
652C2#2028063412yes—
653C2#2028086212yes—
654C2#2028086412yes—
subgroupG/H
center, FrattiniC24C23×D53
commutatorC23×C53C23×C4
FittingC53×#1440C2
socle, cosocleC24×C53C24
prime radicalC25×C53C23
hypercenter#1440D53
hypocenterC53#14292
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C24 ◁ #1440

Complements

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

NH ≅ G/Ntype
4. #20272985C2semidirect
5. #20272981C2semidirect
9. #20272949C2semidirect
10. #20272946C2semidirect
16. C2×#13133194C4semidirect
367. C53#14292semidirect

Maximal subgroups

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

subgroupindex
C22×#131331942
C2×#168104842
C2×C22⋊C4×Dic532
#202729852
#202729812
#202729642
#202729622
#202729492
#202729462
C53×#14402
#1429253