Finite Group Databasebeta

← #27306087

Subgroups — #27306087

Order 13568.

2222 conjugacy classes of subgroups.

The subgroup lattice has 700 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#11719919C211yesmaximal, characteristic
2C2×#16810507C212yesmaximal, complemented
3C2×#16810481C211yesmaximal, characteristic
4#20272981C212yesmaximal, complemented
5#20272966C212yesmaximal, complemented
6#20272948C212yesmaximal, complemented
7#20272938C211yesmaximal, characteristic
8#20272937C211yesmaximal, characteristic
9#20272937C211yesmaximal, characteristic
10#20272937C211yesmaximal, characteristic
11C2×C4×C22⋊C4×C53C211yesmaximal, characteristic
12C24×Dic53C2211yescharacteristic
13C24×Dic53C2211yescharacteristic
14C22×#11719919—28no—
15C22×#11719919C418yescomplemented
16C22×#11719919C2212yes—
17C22×#11719919C2211yescharacteristic
18C22×#11719919C2211yescharacteristic
19C22×#11719919C2211yescharacteristic
20C22×#11719919C2211yescharacteristic
21C22×C22⋊Dic53C2211yescharacteristic
22C22×C22⋊Dic53C2211yescharacteristic
23#16810507—28no—
24#16810481—24no—
25C2×#13133124C2212yes—
26C2×#13133124C2212yes—
27C2×#13133124C2212yes—
28C2×#13133124C2212yes—
29C2×#13133124C2212yes—
30C2×#13133124C2212yes—
31C2×#13133124C2212yes—
32C2×#13133124C2212yes—
33C2×#13133124C2212yes—
34C24×C4×C53C2211yescharacteristic
35C22×C22⋊C4×C53C2211yescharacteristic
36C22×C22⋊C4×C53C2211yescharacteristic
37C22×C42×C53C2212yes—
38C4×C22⋊C4×C53—24no—
39C2×#95×C53C2212yes—
40C23×Dic53—22no—
41C23×Dic53—22no—
42C23×Dic53—24no—
43C23×Dic53—24no—
44C23×Dic53—22no—
45C23×Dic53—22no—
46C23×Dic53—24no—
47C23×Dic53—42no—
48C23×Dic53C2×C414yes—
49C23×Dic53C2×C414yes—
50C23×Dic53C2312yes—
51C23×Dic53C2312yes—
52C23×Dic53C2311yescharacteristic
53C23×Dic53C2311yescharacteristic
54C23×Dic53C2311yescharacteristic
55C23×Dic53C2311yescharacteristic
56C2×#11719919—232no—
57C2×#11719919—24no—
58C2×#11719919—24no—
59C2×#11719919—24no—
60C2×#11719919—24no—
61C2×#11719919—24no—
62C2×#11719919—42no—
63C2×#11719919C2×C414yes—
64C2×#11719919C2×C414yes—
65C2×#11719919C2×C414yes—
66C2×#11719919C2×C414yes—
67C2×#11719919D418yes—
68C2×C22⋊Dic53—22no—
69C2×C22⋊Dic53—22no—
70C2×C22⋊Dic53—22no—
71C2×C22⋊Dic53—22no—
72C2×C22⋊Dic53—22no—
73C2×C22⋊Dic53—22no—
74#13133124—24no—
75#13133124—24no—
76#13133124—24no—
77#13133124—24no—
78#13133124—24no—
79#13133124—24no—
80#13133124—24no—
81#13133124—42no—
82#13133124—42no—
83C25×C53C2311yescharacteristic
84C23×C4×C53—22no—
85C23×C4×C53—22no—
86C23×C4×C53—24no—
87C23×C4×C53—42no—
88C23×C4×C53C2×C414yes—
89C23×C4×C53C2311yescharacteristic
90C23×C4×C53C2312yes—
91C23×C4×C53C2312yes—
92C23×C4×C53C2311yescharacteristic
93C2×C22⋊C4×C53—22no—
94C2×C22⋊C4×C53—22no—
95C2×C22⋊C4×C53—22no—
96C2×C22⋊C4×C53—22no—
97C2×C22⋊C4×C53D414yes—
98C2×C22⋊C4×C53Q814yes—
99C2×C42×C53—24no—
100C2×C42×C53—24no—
101C2×C42×C53—42no—
102#95×C53—24no—
103C24×C53—22no—
104C24×C53—22no—
105C24×C53—22no—
106C24×C53—22no—
107C24×C53C22×C412yes—
108C24×C53C2×D412yes—
109C24×C53C2×Q812yes—
110C24×C53#4112yes—
111C24×C53#4112yes—
112C24×C53#4112yes—
113C24×C53#4112yes—
114C24×C53C2411yescharacteristic
115C22×C4×C53—21no—
116C22×C4×C53—21no—
117C22×C4×C53—22no—
118C22×C4×C53—28no—
119C22×C4×C53—21no—
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—21no—
128C22×C4×C53—22no—
129C22×C4×C53—22no—
130C22×C4×C53—22no—
131C22×C4×C53—22no—
132C22×C4×C53—22no—
133C22×C4×C53—22no—
134C22×C4×C53—22no—
135C22×C4×C53—28no—
136C22×C4×C53—44no—
137C22×C4×C53—48no—
138C22×C4×C53—48no—
139C22×C4×C53C22×C412yes—
140C22×C4×C53C22×C412yes—
141C22×C4×C53C2×D414yes—
142C22×C4×C53C2×D414yes—
143C22×C4×C53C2×Q814yes—
144C22×Dic53—216no—
145C22×Dic53—216no—
146C22×Dic53—28no—
147C22×Dic53—22no—
148C22×Dic53—21no—
149C22×Dic53—22no—
150C22×Dic53—21no—
151C22×Dic53—21no—
152C22×Dic53—21no—
153C22×Dic53—21no—
154C22×Dic53—21no—
155C22×Dic53—22no—
156C22×Dic53—28no—
157C22×Dic53—28no—
158C22×Dic53—28no—
159C22×Dic53—21no—
160C22×Dic53—22no—
161C22×Dic53—21no—
162C22×Dic53—22no—
163C22×Dic53—28no—
164C22×Dic53—22no—
165C22×Dic53—22no—
166C22×Dic53—22no—
167C22×Dic53—22no—
168C22×Dic53—22no—
169C22×Dic53—22no—
170C22×Dic53—22no—
171C22×Dic53—22no—
172C22×Dic53—22no—
173C22×Dic53—22no—
174C22×Dic53—22no—
175C22×Dic53—22no—
176C22×Dic53—22no—
177C22×Dic53—22no—
178C22×Dic53—44no—
179C22×Dic53—48no—
180C22×Dic53—48no—
181C22×Dic53C22×C412yes—
182C22×Dic53C22×C412yes—
183C22×Dic53C22×C412yes—
184C22×Dic53C22×C412yes—
185C22×Dic53C2×D414yes—
186C22×Dic53C2×D414yes—
187C22×Dic53#4114yes—
188C22⋊C4×C53—28no—
189C22⋊C4×C53—28no—
190C42×C53—48no—
191C22⋊Dic53—44no—
192C22⋊Dic53—44no—
193#11719919—28no—
194#11719919—28no—
195#11719919—28no—
196#11719919—48no—
197#11719919C22⋊C4116yescomplemented
198C23×C53—22no—
199C23×C53—22no—
200C23×C53—22no—
201C23×C53—22no—
202C23×C53—24no—
203C23×C53—24no—
204C23×C53—24no—
205C23×C53—24no—
206C23×C53—24no—
207C23×C53—24no—
208C23×C53—24no—
209C23×C53—24no—
210C23×C53—24no—
211C23×C53—24no—
212C23×C53—24no—
213C23×C53—24no—
214C23×C53#11714yes—
215C23×C53C4×D414yes—
216C23×C53C4×Q814yes—
217C23×C53C22⋊Q814yes—
218C23×C53#12314yes—
219C23×C53#12514yes—
220C23×C53#12614yes—
221C23×C53C23×C411yescharacteristic
222C23×C53C22×D412yes—
223C23×C53C22×D411yescharacteristic
224C23×C53C22×Q811yescharacteristic
225C23×C53C2×#4112yes—
226C23×C53C2×#4111yescharacteristic
227C23×C53C2×#4111yescharacteristic
228C23×C53C2×#4111yescharacteristic
229C23×C53C2×#4111yescharacteristic
230C23×C53H(5,𝔽2)12yes—
231C23×C532−1+412yes—
232C2×C4×C53—22no—
233C2×C4×C53—22no—
234C2×C4×C53—24no—
235C2×C4×C53—24no—
236C2×C4×C53—22no—
237C2×C4×C53—22no—
238C2×C4×C53—22no—
239C2×C4×C53—24no—
240C2×C4×C53—24no—
241C2×C4×C53—24no—
242C2×C4×C53—24no—
243C2×C4×C53—24no—
244C2×C4×C53—24no—
245C2×C4×C53—416no—
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—42no—
253C2×C4×C53—42no—
254C2×C4×C53—42no—
255C2×C4×C53—42no—
256C2×C4×C53C2×C22⋊C414yes—
257C2×Dic53—232no—
258C2×Dic53—22no—
259C2×Dic53—22no—
260C2×Dic53—22no—
261C2×Dic53—22no—
262C2×Dic53—22no—
263C2×Dic53—22no—
264C2×Dic53—24no—
265C2×Dic53—24no—
266C2×Dic53—24no—
267C2×Dic53—24no—
268C2×Dic53—24no—
269C2×Dic53—24no—
270C2×Dic53—22no—
271C2×Dic53—22no—
272C2×Dic53—22no—
273C2×Dic53—416no—
274C2×Dic53—42no—
275C2×Dic53—42no—
276C2×Dic53—42no—
277C2×Dic53—42no—
278C2×Dic53—42no—
279C2×Dic53—42no—
280C2×Dic53—42no—
281C2×Dic53—42no—
282C2×Dic53—42no—
283C2×Dic53—42no—
284C2×Dic53—42no—
285C2×Dic53—42no—
286C2×Dic53—42no—
287C2×Dic53—42no—
288C2×Dic53—42no—
289C2×Dic53—42no—
290C2×Dic53C2×C22⋊C414yes—
291C2×Dic53C2×C22⋊C414yes—
292C2×Dic53#11714yes—
293#13882—531nomaximal, Sylow
294C22×C53—24no—
295C22×C53—24no—
296C22×C53—24no—
297C22×C53—24no—
298C22×C53—24no—
299C22×C53—28no—
300C22×C53—28no—
301C22×C53—28no—
302C22×C53—28no—
303C22×C53—24no—
304C22×C53#38718yescomplemented
305C22×C53C22×C22⋊C411yescharacteristic
306C22×C53C2×#11711yescharacteristic
307C22×C53C2×#11711yescharacteristic
308C22×C53C2×C4×D411yescharacteristic
309C22×C53C2×C4×Q811yescharacteristic
310C22×C53#51811yescharacteristic
311C22×C53#51911yescharacteristic
312C22×C53C2×C22⋊Q811yescharacteristic
313C22×C53C2×#12311yescharacteristic
314C22×C53#52512yes—
315C22×C53#52512yes—
316C22×C53C2×#12511yescharacteristic
317C22×C53C2×#12611yescharacteristic
318C22×C53#53912yes—
319C22×C53#53912yes—
320C22×C53D4212yes—
321C22×C53#54612yes—
322C22×C53#54712yes—
323C22×C53#54812yes—
324C22×C53D4×Q812yes—
325C22×C53#55212yes—
326C22×C53#55312yes—
327C22×C53#55412yes—
328C4×C53—24no—
329C4×C53—42no—
330C4×C53—44no—
331C4×C53—44no—
332Dic53—24no—
333Dic53—24no—
334Dic53—24no—
335Dic53—42no—
336Dic53—82no—
337Dic53—82no—
338C23×C4⋊C4—531no—
339C2×#387—532no—
340C2×#380—531no—
341C2×C4×C22⋊C4D5311yescharacteristic
342#1445—532no—
343#1441—532no—
344#1436—532no—
345C22⋊#95—531no—
346C22⋊#95—531no—
347C22⋊#95—531no—
348C22⋊#95—531no—
349C2×C53—28no—
350C2×C53#228011yescharacteristic
351C2×C53C2×#38711yescharacteristic
352C2×C53#231311yescharacteristic
353C2×C53D4×C22⋊C411yescharacteristic
354C2×C53Q8×C22⋊C411yescharacteristic
355C2×C53#234411yescharacteristic
356C2×C53#234511yescharacteristic
357C2×C53#244312yes—
358C2×C53#244912yes—
359C2×C53#247312yes—
360C2×C53#248112yes—
361C24×C4—531no—
362C24×C4—531no—
363C24×C4C2×D5311yescharacteristic
364C22×C4⋊C4—531no—
365C22×C4⋊C4—538no—
366C22×C4⋊C4—531no—
367C22×C4⋊C4—531no—
368C22×C4⋊C4—531no—
369C22×C4⋊C4—532no—
370C22×C4⋊C4—1068no—
371C22×C22⋊C4—531no—
372C22×C22⋊C4—531no—
373C22×C22⋊C4C2×D5311yescharacteristic
374C22×C22⋊C4C2×D5311yescharacteristic
375C22×C42C2×D5312yes—
376#387—1068no—
377#380—1064no—
378C4×C22⋊C4—24no—
379C2×#95—532no—
380C2×#95—532no—
381C2×#95—532no—
382C2×#95—532no—
383C2×#95—532no—
384C2×#95—532no—
385C2×#95—532no—
386C2×#95—532no—
387C2×#95—532no—
388C2×#95C2×D5312yes—
389C53#1388211yescharacteristic, complemented, Sylow
390C25C22×D5311yescharacteristic
391C23×C4—22no—
392C23×C4—22no—
393C23×C4—24no—
394C23×C4—42no—
395C23×C4—531no—
396C23×C4—531no—
397C23×C4—531no—
398C23×C4—534no—
399C23×C4—531no—
400C23×C4—534no—
401C23×C4—532no—
402C23×C4—532no—
403C23×C4—1064no—
404C23×C4—1064no—
405C23×C4—1064no—
406C23×C4—1062no—
407C23×C4—1062no—
408C23×C4—1062no—
409C23×C4—1062no—
410C23×C4—2122no—
411C23×C4C4×D5314yes—
412C23×C4C22×D5312yes—
413C23×C4C22×D5311yescharacteristic
414C23×C4C22×D5312yes—
415C23×C4C22×D5311yescharacteristic
416C2×C4⋊C4—534no—
417C2×C4⋊C4—538no—
418C2×C4⋊C4—534no—
419C2×C4⋊C4—534no—
420C2×C4⋊C4—534no—
421C2×C4⋊C4—10632no—
422C2×C4⋊C4—1064no—
423C2×C4⋊C4—1064no—
424C2×C4⋊C4—1064no—
425C2×C4⋊C4—1064no—
426C2×C4⋊C4—1064no—
427C2×C4⋊C4—2122no—
428C2×C22⋊C4—22no—
429C2×C22⋊C4—22no—
430C2×C22⋊C4—22no—
431C2×C22⋊C4—22no—
432C2×C22⋊C4—1062no—
433C2×C22⋊C4—1062no—
434C2×C22⋊C4—1062no—
435C2×C22⋊C4—1062no—
436C2×C22⋊C4—1062no—
437C2×C22⋊C4—1062no—
438C2×C22⋊C4Dic10614yes—
439C2×C22⋊C4C22⋊D5314yes—
440C2×C42—24no—
441C2×C42—24no—
442C2×C42—42no—
443#95—24no—
444#95—1064no—
445#95—1064no—
446#95—1064no—
447#95—1064no—
448#95—1064no—
449#95—1064no—
450#95—1064no—
451#95—2122no—
452#95—2122no—
453C24—22no—
454C24—22no—
455C24—22no—
456C24—22no—
457C24C2×Dic10612yes—
458C24C2×C4×D5312yes—
459C24#1171994512yes—
460C24#1171994512yes—
461C24#1171994512yes—
462C24#1171994512yes—
463C24C2×C22⋊D5312yes—
464C24C23×D5311yescharacteristic
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—22no—
468C22×C4—28no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—21no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—21no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—28no—
486C22×C4—44no—
487C22×C4—48no—
488C22×C4—48no—
489C22×C4—532no—
490C22×C4—532no—
491C22×C4—532no—
492C22×C4—534no—
493C22×C4—534no—
494C22×C4—534no—
495C22×C4—532no—
496C22×C4—1068no—
497C22×C4—1068no—
498C22×C4—1068no—
499C22×C4—10616no—
500C22×C4—10616no—
501C22×C4—1068no—
502C22×C4—1062no—
503C22×C4—1061no—
504C22×C4—1062no—
505C22×C4—1061no—
506C22×C4—1061no—
507C22×C4—1061no—
508C22×C4—1061no—
509C22×C4—1061no—
510C22×C4—1062no—
511C22×C4—1068no—
512C22×C4—1062no—
513C22×C4—1061no—
514C22×C4—1061no—
515C22×C4—1062no—
516C22×C4—1062no—
517C22×C4—1062no—
518C22×C4—1062no—
519C22×C4—1062no—
520C22×C4—1062no—
521C22×C4—1062no—
522C22×C4—1062no—
523C22×C4—1062no—
524C22×C4—1062no—
525C22×C4—1062no—
526C22×C4—1062no—
527C22×C4—1062no—
528C22×C4—1062no—
529C22×C4—1062no—
530C22×C4—2124no—
531C22×C4—2128no—
532C22×C4—2128no—
533C22×C4C2×Dic10614yes—
534C22×C4C2×C4×D5312yes—
535C22×C4C2×C4×D5312yes—
536C22×C4D4×D5314yes—
537C22×C4C2×C22⋊D5314yes—
538C4⋊C4—5316no—
539C4⋊C4—1068no—
540C4⋊C4—1068no—
541C4⋊C4—1068no—
542C4⋊C4—2128no—
543C22⋊C4—28no—
544C22⋊C4—28no—
545C22⋊C4—2124no—
546C22⋊C4—2124no—
547C42—48no—
548C23—24no—
549C23—24no—
550C23—24no—
551C23—24no—
552C23—24no—
553C23—22no—
554C23—22no—
555C23—24no—
556C23—22no—
557C23—22no—
558C23—24no—
559C23—24no—
560C23—24no—
561C23—24no—
562C23—24no—
563C23—24no—
564C23#1313317414yes—
565C23C4×C22⋊D5314yes—
566C23C4×Dic10614yes—
567C23#1313319614yes—
568C23#1313320014yes—
569C23#1313320914yes—
570C23#1313321114yes—
571C23C22×C4×D5311yescharacteristic
572C23C2×D4×D5312yes—
573C23C22×C22⋊D5311yescharacteristic
574C23C22×Dic10611yescharacteristic
575C23#41×D5312yes—
576C23C2×#1171994511yescharacteristic
577C23C2×#1171994511yescharacteristic
578C23C2×#1171994511yescharacteristic
579C23C2×#1171994511yescharacteristic
580C23#1313326212yes—
581C23#1313326412yes—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—22no—
589C2×C4—24no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—416no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—534no—
607C2×C4—534no—
608C2×C4—534no—
609C2×C4—10632no—
610C2×C4—1062no—
611C2×C4—1062no—
612C2×C4—1062no—
613C2×C4—1062no—
614C2×C4—1062no—
615C2×C4—1062no—
616C2×C4—1064no—
617C2×C4—1064no—
618C2×C4—1064no—
619C2×C4—1064no—
620C2×C4—1064no—
621C2×C4—1064no—
622C2×C4—1062no—
623C2×C4—1062no—
624C2×C4—1062no—
625C2×C4—21216no—
626C2×C4—2122no—
627C2×C4—2122no—
628C2×C4—2122no—
629C2×C4—2122no—
630C2×C4—2122no—
631C2×C4—2122no—
632C2×C4—2122no—
633C2×C4—2122no—
634C2×C4—2122no—
635C2×C4—2122no—
636C2×C4—2122no—
637C2×C4—2122no—
638C2×C4—2122no—
639C2×C4—2122no—
640C2×C4—2122no—
641C2×C4—2122no—
642C2×C4C22⋊C4×D5314yes—
643C22—24no—
644C22—24no—
645C22—24no—
646C22—28no—
647C22—24no—
648C22—24no—
649C22—24no—
650C22—28no—
651C22—28no—
652C22—28no—
653C22#1681050718yescomplemented
654C22C2×C22⋊C4×D5311yescharacteristic
655C22#117×D5311yescharacteristic
656C22C2×#1313317411yescharacteristic
657C22C2×C4×C22⋊D5311yescharacteristic
658C22C2×C4×Dic10611yescharacteristic
659C22#1681109611yescharacteristic
660C22#1681110011yescharacteristic
661C22C2×#1313319611yescharacteristic
662C22C2×#1313320011yescharacteristic
663C22#1681113212yes—
664C22#1681113212yes—
665C22C2×#1313320911yescharacteristic
666C22C2×#1313321111yescharacteristic
667C22#1681121512yes—
668C22#1681121512yes—
669C22D4×C22⋊D5312yes—
670C22#1681126312yes—
671C22#1681127212yes—
672C22#1681128212yes—
673C22D4×Dic10612yes—
674C22#1681130612yes—
675C22#1681132212yes—
676C22#1681133212yes—
677C4—24no—
678C4—42no—
679C4—44no—
680C4—44no—
681C4—1064no—
682C4—1064no—
683C4—1064no—
684C4—4242no—
685C4—4242no—
686C4—2122no—
687C2—28no—
688C2#2027709611yescharacteristic
689C2C2×#1681050711yescharacteristic
690C2#2027731511yescharacteristic
691C2C22⋊C4×C22⋊D5311yescharacteristic
692C2C22⋊C4×Dic10611yescharacteristic
693C2#2027758511yescharacteristic
694C2#2027759311yescharacteristic
695C2#2027844412yes—
696C2#2027851812yes—
697C2#2027885612yes—
698C2#2027897612yes—
subgroupG/H
center, FrattiniC24C23×D53
commutatorC23×C53C23×C4
FittingC2×C4×C22⋊C4×C53C2
socle, cosocleC24×C53C24
prime radicalC25×C53C23
hypercenterC2×C4×C22⋊C4D53
hypocenterC53#13882
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×#16810507C2semidirect
4. #20272981C2semidirect
5. #20272966C2semidirect
6. #20272948C2semidirect
15. C22×#11719919C4semidirect
197. #11719919C22⋊C4semidirect
304. C22×C53#387semidirect
389. C53#13882semidirect
653. C22#16810507semidirect

Maximal subgroups

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

subgroupindex
C23×#117199192
C2×#168105072
C2×#168104812
#202729812
#202729662
#202729482
#202729382
#202729372
C2×C4×C22⋊C4×C532
#1388253