Finite Group Databasebeta

← #27306248

Subgroups — #27306248

Order 13568.

1962 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810495C212yesmaximal
2C2×#16810487C211yesmaximal, characteristic, complemented
3C2×C4×C22⋊Dic53C211yesmaximal, characteristic
4C22×#13133124C211yesmaximal, characteristic
5#20272973C212yesmaximal, complemented
6#20272970C212yesmaximal, complemented
7#20272943C211yesmaximal, characteristic, complemented
8#20272940C211yesmaximal, characteristic, complemented
9#20272939C211yesmaximal, characteristic, complemented
10#20272937C212yesmaximal
11C53×C22⋊#95C211yesmaximal, characteristic
12C24×Dic53C2212yes—
13C22×#11719919C2212yes—
14C22×#11719919C2212yes—
15C22×C22⋊Dic53C2211yescharacteristic
16C22×C22⋊Dic53C2211yescharacteristic
17C22×C4×Dic53C2211yescharacteristic
18C22×C4×Dic53C2211yescharacteristic
19#16810495—28no—
20#16810487—24no—
21C4×C22⋊Dic53C418yescomplemented
22C2×#13133124—24no—
23C2×#13133124C2212yes—
24C2×#13133124C2212yes—
25C2×#13133124C2211yescharacteristic
26C2×#13133124C2212yes—
27C2×#13133124C2212yes—
28C2×#13133124C2212yes—
29C2×#13133124C2212yes—
30C2×#13133124C2212yes—
31C2×#13133124C2211yescharacteristic
32C2×#13133124C2211yescharacteristic
33C2×#13133124C2211yescharacteristic
34C24×C4×C53C2211yescharacteristic
35C22×C22⋊C4×C53C2212yes—
36C2×#95×C53C2211yescharacteristic
37C2×#95×C53C2211yescharacteristic
38C2×#95×C53C2212yes—
39C23×Dic53—24no—
40C23×Dic53—24no—
41C23×Dic53—24no—
42C23×Dic53—24no—
43C23×Dic53—24no—
44C23×Dic53—42no—
45C23×Dic53—42no—
46C23×Dic53C2311yescharacteristic
47C23×Dic53C2311yescharacteristic
48C23×Dic53C2312yes—
49C23×Dic53C2312yes—
50C23×Dic53C2311yescharacteristic
51C23×Dic53C2311yescharacteristic
52C2×#11719919—24no—
53C2×#11719919—24no—
54C2×#11719919—24no—
55C2×#11719919—24no—
56C2×#11719919—42no—
57C2×#11719919—42no—
58C2×C22⋊Dic53—24no—
59C2×C22⋊Dic53C2×C414yes—
60C2×C22⋊Dic53C2×C414yes—
61C2×C22⋊Dic53D414yes—
62C2×C22⋊Dic53Q814yes—
63C2×C4×Dic53—22no—
64C2×C4×Dic53—22no—
65C2×C4×Dic53—42no—
66C2×C4×Dic53C2×C414yescomplemented
67C2×C4×Dic53C2×C414yescomplemented
68#13133124—22no—
69#13133124—24no—
70#13133124—24no—
71#13133124—24no—
72#13133124—24no—
73#13133124—24no—
74#13133124—42no—
75#13133124—42no—
76#13133124C2×C414yescomplemented
77#13133124C2×C414yescomplemented
78#13133124D414yes—
79C25×C53C2311yescharacteristic
80C23×C4×C53—24no—
81C23×C4×C53—22no—
82C23×C4×C53—22no—
83C23×C4×C53—42no—
84C23×C4×C53C2×C414yes—
85C23×C4×C53C2312yes—
86C23×C4×C53C2312yes—
87C23×C4×C53C2311yescharacteristic
88C23×C4×C53C2311yescharacteristic
89C2×C22⋊C4×C53—24no—
90C2×C22⋊C4×C53—24no—
91C2×C22⋊C4×C53—24no—
92#95×C53—22no—
93#95×C53—22no—
94#95×C53—42no—
95C24×C53—21no—
96C24×C53—21no—
97C24×C53—22no—
98C24×C53—22no—
99C24×C53—21no—
100C24×C53—21no—
101C24×C53C22×C412yes—
102C24×C53C2×D412yes—
103C24×C53C2×Q812yes—
104C24×C53#4114yes—
105C24×C53#4114yes—
106C24×C53C2411yescharacteristic
107C22×C4×C53—22no—
108C22×C4×C53—22no—
109C22×C4×C53—28no—
110C22×C4×C53—21no—
111C22×C4×C53—22no—
112C22×C4×C53—21no—
113C22×C4×C53—22no—
114C22×C4×C53—28no—
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—44no—
130C22×C4×C53—48no—
131C22×C4×C53—48no—
132C22×C4×C53C22×C412yes—
133C22×C4×C53C22×C412yes—
134C22×C4×C53C2×D412yes—
135C22×C4×C53#4112yes—
136C22×Dic53—21no—
137C22×Dic53—21no—
138C22×Dic53—22no—
139C22×Dic53—22no—
140C22×Dic53—21no—
141C22×Dic53—21no—
142C22×Dic53—216no—
143C22×Dic53—21no—
144C22×Dic53—22no—
145C22×Dic53—22no—
146C22×Dic53—22no—
147C22×Dic53—22no—
148C22×Dic53—21no—
149C22×Dic53—22no—
150C22×Dic53—22no—
151C22×Dic53—22no—
152C22×Dic53—22no—
153C22×Dic53—22no—
154C22×Dic53—22no—
155C22×Dic53—22no—
156C22×Dic53—21no—
157C22×Dic53—22no—
158C22×Dic53—22no—
159C22×Dic53—22no—
160C22×Dic53—22no—
161C22×Dic53—22no—
162C22×Dic53—21no—
163C22×Dic53—48no—
164C22×Dic53—48no—
165C22×Dic53—48no—
166C22×Dic53—48no—
167C22×Dic53—48no—
168C22×Dic53—48no—
169C22×Dic53C22×C412yes—
170C22×Dic53C22×C412yes—
171C22×Dic53C22×C412yes—
172C22×Dic53C22×C412yes—
173C22×Dic53C2×D414yes—
174C22×Dic53C2×D412yes—
175C22×Dic53C2×D412yes—
176C22×Dic53C2×Q812yes—
177C22×Dic53C2×Q812yes—
178C22×Dic53#4114yes—
179C22⋊C4×C53—48no—
180C22⋊Dic53—28no—
181C22⋊Dic53C4⋊C4116yes—
182#11719919—48no—
183#11719919—48no—
184C4×Dic53—28no—
185C4×Dic53—44no—
186C23×C53—22no—
187C23×C53—22no—
188C23×C53—22no—
189C23×C53—22no—
190C23×C53—24no—
191C23×C53—24no—
192C23×C53—24no—
193C23×C53—24no—
194C23×C53—22no—
195C23×C53—24no—
196C23×C53—24no—
197C23×C53—22no—
198C23×C53—22no—
199C23×C53—24no—
200C23×C53—24no—
201C23×C53—24no—
202C23×C53—24no—
203C23×C53—22no—
204C23×C53C2×C4⋊C414yes—
205C23×C53#11714yes—
206C23×C53#11714yes—
207C23×C53#12418yes—
208C23×C53#12518yes—
209C23×C53C23×C411yescharacteristic
210C23×C53C22×D411yescharacteristic
211C23×C53C22×D411yescharacteristic
212C23×C53C22×Q811yescharacteristic
213C23×C53C2×#4112yes—
214C23×C53C2×#4112yes—
215C23×C53C2×#4111yescharacteristic
216C23×C53C2×#4112yes—
217C23×C53H(5,𝔽2)12yes—
218C23×C53H(5,𝔽2)11yescharacteristic
219C23×C532−1+411yescharacteristic
220C2×C4×C53—22no—
221C2×C4×C53—22no—
222C2×C4×C53—24no—
223C2×C4×C53—24no—
224C2×C4×C53—22no—
225C2×C4×C53—22no—
226C2×C4×C53—22no—
227C2×C4×C53—416no—
228C2×C4×C53—42no—
229C2×C4×C53—42no—
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×C53—42no—
244C2×C4×C53C4×D414yes—
245C2×Dic53—22no—
246C2×Dic53—22no—
247C2×Dic53—22no—
248C2×Dic53—22no—
249C2×Dic53—22no—
250C2×Dic53—24no—
251C2×Dic53—22no—
252C2×Dic53—22no—
253C2×Dic53—24no—
254C2×Dic53—22no—
255C2×Dic53—22no—
256C2×Dic53—22no—
257C2×Dic53—22no—
258C2×Dic53—22no—
259C2×Dic53—22no—
260C2×Dic53—22no—
261C2×Dic53—24no—
262C2×Dic53—24no—
263C2×Dic53—24no—
264C2×Dic53—24no—
265C2×Dic53—24no—
266C2×Dic53—24no—
267C2×Dic53—432no—
268C2×Dic53—42no—
269C2×Dic53—42no—
270C2×Dic53—42no—
271C2×Dic53—42no—
272C2×Dic53—42no—
273C2×Dic53—42no—
274C2×Dic53—42no—
275C2×Dic53—42no—
276C2×Dic53—42no—
277C2×Dic53—42no—
278C2×Dic53—42no—
279C2×Dic53—42no—
280C2×Dic53C2×C4⋊C414yes—
281C2×Dic53C2×C4⋊C414yes—
282#13896—531nomaximal, Sylow
283C22×C53—24no—
284C22×C53—24no—
285C22×C53—24no—
286C22×C53—24no—
287C22×C53—24no—
288C22×C53—28no—
289C22×C53—28no—
290C22×C53—28no—
291C22×C53—28no—
292C22×C53—24no—
293C22×C53#38218yescomplemented
294C22×C53C22×C4⋊C411yescharacteristic
295C22×C53C2×#11711yescharacteristic
296C22×C53C2×#11711yescharacteristic
297C22×C53C2×C4×D411yescharacteristic
298C22×C53C4×#4111yescharacteristic
299C22×C53#51811yescharacteristic
300C22×C53#52011yescharacteristic
301C22×C53#52512yes—
302C22×C53C2×#12412yes—
303C22×C53C2×#12512yes—
304C22×C53#52912yes—
305C22×C53#53812yes—
306C22×C53#53912yes—
307C22×C53D4211yescharacteristic
308C22×C53#54612yes—
309C22×C53#54612yes—
310C22×C53#54711yescharacteristic
311C22×C53D4×Q811yescharacteristic
312C22×C53#55112yes—
313C22×C53#55112yes—
314C22×C53#55212yes—
315C22×C53#55412yes—
316C22×C53#55411yescharacteristic
317C4×C53—24no—
318C4×C53—42no—
319C4×C53—82no—
320C4×C53—82no—
321Dic53—24no—
322Dic53—24no—
323Dic53—42no—
324Dic53—44no—
325Dic53—44no—
326Dic53—42no—
327C2×#385—532no—
328C2×#382—531no—
329C2×C4×C22⋊C4—531no—
330C22×#95—531no—
331#1442—532no—
332#1442—532no—
333C4⋊#95—531no—
334#1434—531no—
335#1434—531no—
336C22⋊#95—532no—
337C22⋊#95D5311yescharacteristic
338C2×C53—28no—
339C2×C53C2×#38211yescharacteristic
340C2×C53#229011yescharacteristic
341C2×C53#231811yescharacteristic
342C2×C53#234211yescharacteristic
343C2×C53#234311yescharacteristic
344C2×C53D4×C4⋊C411yescharacteristic
345C2×C53#234911yescharacteristic
346C2×C53#247212yes—
347C2×C53#247312yes—
348C2×C53#248812yes—
349C2×C53#250512yes—
350C24×C4—532no—
351C24×C4C2×D5311yescharacteristic
352C22×C4⋊C4—532no—
353C22×C4⋊C4—532no—
354C22×C22⋊C4—531no—
355C22×C22⋊C4—531no—
356C22×C22⋊C4C2×D5312yes—
357C22×C42—531no—
358C22×C42—531no—
359#385—1068no—
360#382—1064no—
361C4×C22⋊C4—538no—
362C2×#95—531no—
363C2×#95—531no—
364C2×#95—531no—
365C2×#95—532no—
366C2×#95—532no—
367C2×#95—531no—
368C2×#95—532no—
369C2×#95—532no—
370C2×#95—532no—
371C2×#95—532no—
372C2×#95—532no—
373C2×#95—1064no—
374C2×#95C2×D5311yescharacteristic
375C2×#95C2×D5311yescharacteristic
376C2×#95C2×D5312yes—
377C53#1389611yescharacteristic, complemented, Sylow
378C25C22×D5311yescharacteristic
379C23×C4—24no—
380C23×C4—22no—
381C23×C4—22no—
382C23×C4—42no—
383C23×C4—531no—
384C23×C4—531no—
385C23×C4—531no—
386C23×C4—531no—
387C23×C4—532no—
388C23×C4—532no—
389C23×C4—1064no—
390C23×C4—1064no—
391C23×C4—1064no—
392C23×C4—1064no—
393C23×C4—1064no—
394C23×C4—2122no—
395C23×C4—2122no—
396C23×C4C4×D5314yes—
397C23×C4C22×D5312yes—
398C23×C4C22×D5312yes—
399C23×C4C22×D5311yescharacteristic
400C23×C4C22×D5311yescharacteristic
401C2×C4⋊C4—1064no—
402C2×C4⋊C4—1064no—
403C2×C4⋊C4—1064no—
404C2×C4⋊C4—1064no—
405C2×C4⋊C4—2122no—
406C2×C4⋊C4—2122no—
407C2×C22⋊C4—24no—
408C2×C22⋊C4—24no—
409C2×C22⋊C4—24no—
410C2×C22⋊C4—534no—
411C2×C22⋊C4—534no—
412C2×C22⋊C4—534no—
413C2×C22⋊C4—534no—
414C2×C22⋊C4—1064no—
415C2×C42—534no—
416C2×C42—534no—
417C2×C42—1062no—
418C2×C42—1062no—
419C2×C42—2122no—
420#95—22no—
421#95—22no—
422#95—42no—
423#95—534no—
424#95—534no—
425#95—534no—
426#95—1064no—
427#95—1064no—
428#95—1064no—
429#95—1064no—
430#95—1064no—
431#95—1062no—
432#95—2122no—
433#95—2122no—
434C24—21no—
435C24—22no—
436C24—21no—
437C24—22no—
438C24—21no—
439C24—21no—
440C24C2×C4×D5312yes—
441C24#1171994514yes—
442C24D4×D5312yes—
443C24#1171994714yes—
444C24Q8×D5312yes—
445C24C23×D5311yescharacteristic
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—28no—
449C22×C4—21no—
450C22×C4—22no—
451C22×C4—21no—
452C22×C4—22no—
453C22×C4—28no—
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—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—44no—
469C22×C4—48no—
470C22×C4—48no—
471C22×C4—532no—
472C22×C4—532no—
473C22×C4—532no—
474C22×C4—532no—
475C22×C4—532no—
476C22×C4—532no—
477C22×C4—532no—
478C22×C4—532no—
479C22×C4—534no—
480C22×C4—534no—
481C22×C4—1061no—
482C22×C4—1062no—
483C22×C4—1061no—
484C22×C4—1062no—
485C22×C4—1061no—
486C22×C4—1061no—
487C22×C4—10616no—
488C22×C4—1061no—
489C22×C4—1062no—
490C22×C4—1062no—
491C22×C4—1062no—
492C22×C4—1062no—
493C22×C4—1062no—
494C22×C4—1061no—
495C22×C4—1062no—
496C22×C4—1062no—
497C22×C4—1062no—
498C22×C4—1062no—
499C22×C4—1062no—
500C22×C4—1062no—
501C22×C4—1062no—
502C22×C4—1062no—
503C22×C4—1062no—
504C22×C4—1061no—
505C22×C4—1062no—
506C22×C4—1062no—
507C22×C4—1061no—
508C22×C4—2128no—
509C22×C4—2128no—
510C22×C4—2128no—
511C22×C4—2128no—
512C22×C4—2128no—
513C22×C4—2128no—
514C22×C4C2×C4×D5312yes—
515C22×C4C2×C4×D5312yes—
516C22×C4D4×D5312yes—
517C22×C4#1171994712yes—
518C4⋊C4—2128no—
519C4⋊C4—2128no—
520C22⋊C4—48no—
521C22⋊C4—5316no—
522C22⋊C4—1068no—
523C42—1068no—
524C42—2124no—
525C23—22no—
526C23—22no—
527C23—24no—
528C23—24no—
529C23—24no—
530C23—24no—
531C23—22no—
532C23—22no—
533C23—22no—
534C23—22no—
535C23—22no—
536C23—24no—
537C23—24no—
538C23—24no—
539C23—24no—
540C23—22no—
541C23—24no—
542C23—24no—
543C23C4⋊C4×D5314yes—
544C23#1313317314yes—
545C23#1313317414yes—
546C23#1313320318yes—
547C23#1313320718yes—
548C23C22×C4×D5311yescharacteristic
549C23C2×D4×D5311yescharacteristic
550C23C2×D4×D5311yescharacteristic
551C23C2×Q8×D5311yescharacteristic
552C23#41×D5312yes—
553C23C2×#1171994711yescharacteristic
554C23C2×#1171994712yes—
555C23C2×#1171994512yes—
556C23#1313326212yes—
557C23#1313326311yescharacteristic
558C23#1313326411yescharacteristic
559C2×C4—24no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—24no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—416no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—534no—
584C2×C4—534no—
585C2×C4—1062no—
586C2×C4—1062no—
587C2×C4—1064no—
588C2×C4—1062no—
589C2×C4—1062no—
590C2×C4—1062no—
591C2×C4—1062no—
592C2×C4—1062no—
593C2×C4—1062no—
594C2×C4—1064no—
595C2×C4—1064no—
596C2×C4—1062no—
597C2×C4—1062no—
598C2×C4—1062no—
599C2×C4—1062no—
600C2×C4—1062no—
601C2×C4—1062no—
602C2×C4—1064no—
603C2×C4—1064no—
604C2×C4—1064no—
605C2×C4—1064no—
606C2×C4—1064no—
607C2×C4—21232no—
608C2×C4—2122no—
609C2×C4—2122no—
610C2×C4—2122no—
611C2×C4—2122no—
612C2×C4—2122no—
613C2×C4—2122no—
614C2×C4—2122no—
615C2×C4—2122no—
616C2×C4—2122no—
617C2×C4—2122no—
618C2×C4—2122no—
619C2×C4—2122no—
620C2×C4#1313317714yes—
621C22—28no—
622C22—28no—
623C22—24no—
624C22—24no—
625C22—28no—
626C22—24no—
627C22—24no—
628C22—24no—
629C22—24no—
630C22—28no—
631C22#1681048718yescomplemented
632C22C2×C4⋊C4×D5311yescharacteristic
633C22C2×#1313317311yescharacteristic
634C22C2×#1313317411yescharacteristic
635C22C2×#1313317711yescharacteristic
636C22#1681109111yescharacteristic
637C22#1681109611yescharacteristic
638C22#1681110311yescharacteristic
639C22#1681113212yes—
640C22C2×#1313320312yes—
641C22C2×#1313320712yes—
642C22#1681115312yes—
643C22#1681120612yes—
644C22#1681121512yes—
645C22#1681125711yescharacteristic
646C22#1681126012yes—
647C22#1681126212yes—
648C22#1681127311yescharacteristic
649C22#1681128611yescharacteristic
650C22#1681129612yes—
651C22#1681130412yes—
652C22#1681131012yes—
653C22#1681132712yes—
654C22#1681133111yescharacteristic
655C4—24no—
656C4—42no—
657C4—82no—
658C4—82no—
659C4—1064no—
660C4—1064no—
661C4—2124no—
662C4—2124no—
663C4—2122no—
664C4—2122no—
665C2—28no—
666C2C2×#1681048711yescharacteristic
667C2#2027715411yescharacteristic
668C2#2027734211yescharacteristic
669C2#2027756611yescharacteristic
670C2#2027757711yescharacteristic
671C2#2027761111yescharacteristic
672C2#2027761911yescharacteristic
673C2#2027884312yes—
674C2#2027885812yes—
675C2#2027904312yes—
676C2#2027921812yes—
subgroupG/H
center, FrattiniC24C23×D53
commutatorC23×C53C23×C4
FittingC53×C22⋊#95C2
socle, cosocleC24×C53C24
prime radicalC25×C53C23
hypercenterC22⋊#95D53
hypocenterC53#13896
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C24 ◁ C22⋊#95

Complements

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

NH ≅ G/Ntype
2. C2×#16810487C2semidirect
5. #20272973C2semidirect
6. #20272970C2semidirect
7. #20272943C2semidirect
8. #20272940C2semidirect
9. #20272939C2semidirect
21. C4×C22⋊Dic53C4semidirect
66. C2×C4×Dic53C2×C4semidirect
67. C2×C4×Dic53C2×C4semidirect
76. #13133124C2×C4semidirect
77. #13133124C2×C4semidirect
293. C22×C53#382semidirect
377. C53#13896semidirect
631. C22#16810487semidirect

Maximal subgroups

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

subgroupindex
C2×#168104952
C2×#168104872
C2×C4×C22⋊Dic532
C22×#131331242
#202729732
#202729702
#202729432
#202729402
#202729392
#202729372
C53×C22⋊#952
#1389653