Finite Group Databasebeta

← #27315302

Subgroups — #27315302

Order 13568.

6494 conjugacy classes of subgroups.

The subgroup lattice has 694 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C24×C4×D53C211yesmaximal, characteristic
2C2×#16810498C211yesmaximal, characteristic, complemented
3C2×#16810497C214yesmaximal, complemented
4C2×#16810496C211yesmaximal, characteristic, complemented
5C2×#16810478C211yesmaximal, characteristic
6C2×#16810478C211yesmaximal, characteristic
7#20272937C211yesmaximal, characteristic, complemented
8#20272936C214yesmaximal, complemented
9C53×C22⋊#95C211yesmaximal, characteristic, complemented
10C25×D53C2211yescharacteristic
11C24×Dic53C2211yescharacteristic
12C23×C4×D53—24no—
13C23×C4×D53—24no—
14C23×C4×D53—216no—
15C23×C4×D53—24no—
16C23×C4×D53C2214yes—
17C22×C4⋊Dic53C2212yescomplemented
18C22×#11719919C2212yescomplemented
19C22×C22⋊Dic53C2211yescharacteristic
20C22×C22⋊Dic53C2211yescharacteristic
21C22×#11719921C2212yes—
22C22×#11719921C2212yes—
23C22×#11719921C2214yes—
24#16810498—24no—
25#16810497—216no—
26#16810496—24no—
27#16810478—24no—
28#16810478C418yescomplemented
29C2×#13133124C2212yescomplemented
30C2×#13133124C2212yescomplemented
31C2×#13133124C2214yescomplemented
32C24×C4×C53C2211yescharacteristic
33C22×C22⋊C4×C53C2211yescharacteristic
34C22×C22⋊C4×C53C2211yescharacteristic
35C2×#95×C53C2214yescomplemented
36C24×D53—22no—
37C24×D53—22no—
38C24×D53—22no—
39C24×D53—22no—
40C24×D53—28no—
41C24×D53—28no—
42C24×D53C2×C414yes—
43C24×D53D414yes—
44C24×D53Q814yes—
45C24×D53C2312yes—
46C23×Dic53—22no—
47C23×Dic53—22no—
48C23×Dic53—28no—
49C23×Dic53D414yes—
50C23×Dic53C2311yescharacteristic
51C23×Dic53C2311yescharacteristic
52C23×Dic53C2312yes—
53C23×Dic53C2312yes—
54C22×C4×D53—232no—
55C22×C4×D53—264no—
56C22×C4×D53—264no—
57C22×C4×D53—264no—
58C22×C4×D53—28no—
59C22×C4×D53—28no—
60C22×C4×D53—28no—
61C22×C4×D53—28no—
62C22×C4×D53—28no—
63C22×C4×D53—28no—
64C22×C4×D53D4116yes—
65C2×C4⋊Dic53—28no—
66C2×C4⋊Dic53—24no—
67C2×#11719919—24no—
68C2×#11719919—28no—
69C2×C22⋊Dic53—22no—
70C2×C22⋊Dic53—22no—
71C2×C22⋊Dic53—22no—
72C2×C22⋊Dic53—22no—
73C2×C22⋊Dic53—22no—
74C2×C22⋊Dic53C2×C414yescomplemented
75C2×#11719921—28no—
76C2×#11719921—28no—
77C2×#11719921—24no—
78C2×#11719921—24no—
79C2×#11719921—28no—
80C2×#11719921—28no—
81C2×#11719921C2×C4116yescomplemented
82#13133124—24no—
83#13133124—28no—
84#13133124—24no—
85C25×C53C2311yescharacteristic
86C23×C4×C53—22no—
87C23×C4×C53—22no—
88C23×C4×C53—28no—
89C23×C4×C53D414yes—
90C23×C4×C53C2312yes—
91C23×C4×C53C2312yes—
92C23×C4×C53C2312yes—
93C2×C22⋊C4×C53—22no—
94C2×C22⋊C4×C53—22no—
95C2×C22⋊C4×C53—22no—
96C2×C22⋊C4×C53—22no—
97C2×C22⋊C4×C53—22no—
98C2×C22⋊C4×C53C2×C414yescomplemented
99#95×C53—28no—
100C24×C53—24no—
101C24×C53—24no—
102C24×C53C22×C412yes—
103C24×C53C2×D412yes—
104C24×C53C2×D412yes—
105C24×C53C2×D412yes—
106C24×C53C2×Q812yes—
107C24×C53#4112yes—
108C24×C53#4112yes—
109C24×C53C2411yescharacteristic
110C23×D53—216no—
111C23×D53—216no—
112C23×D53—232no—
113C23×D53—232no—
114C23×D53—216no—
115C23×D53—216no—
116C23×D53—28no—
117C23×D53—28no—
118C23×D53—28no—
119C23×D53—28no—
120C23×D53—28no—
121C23×D53—28no—
122C23×D53—24no—
123C23×D53—22no—
124C23×D53—22no—
125C23×D53—24no—
126C23×D53—22no—
127C23×D53—22no—
128C23×D53—232no—
129C23×D53—216no—
130C23×D53—232no—
131C23×D53—216no—
132C23×D53C4⋊C4116yescomplemented
133C23×D53C22×C414yes—
134C23×D53C2×D414yes—
135C23×D53C2×D418yes—
136C23×D53C2×Q814yes—
137C23×D53#4118yes—
138C22×C4×C53—216no—
139C22×C4×C53—216no—
140C22×C4×C53—28no—
141C22×C4×C53—22no—
142C22×C4×C53—24no—
143C22×C4×C53—22no—
144C22×C4×C53—216no—
145C22×C4×C53—24no—
146C22×C4×C53—22no—
147C22×C4×C53—22no—
148C22×C4×C53—24no—
149C22×C4×C53—24no—
150C22×C4×C53—22no—
151C22×C4×C53—22no—
152C22×C4×C53—22no—
153C22×C4×C53—24no—
154C22×C4×C53C22×C414yescomplemented
155C22×C4×C53C2×D414yes—
156C22×C4×C53C2×D418yes—
157C22×Dic53—216no—
158C22×Dic53—216no—
159C22×Dic53—28no—
160C22×Dic53—22no—
161C22×Dic53—24no—
162C22×Dic53—22no—
163C22×Dic53—216no—
164C22×Dic53—24no—
165C22×Dic53—22no—
166C22×Dic53—24no—
167C22×Dic53—22no—
168C22×Dic53—21no—
169C22×Dic53—21no—
170C22×Dic53—21no—
171C22×Dic53—21no—
172C22×Dic53—24no—
173C22×Dic53—21no—
174C22×Dic53—21no—
175C22×Dic53—24no—
176C22×Dic53C22×C414yescomplemented
177C22×Dic53C2×D414yes—
178C22×Dic53C2×D418yes—
179C2×C4×D53—2256no—
180C2×C4×D53—232no—
181C2×C4×D53—232no—
182C2×C4×D53—232no—
183C2×C4×D53—232no—
184C2×C4×D53—232no—
185C2×C4×D53—232no—
186C2×C4×D53—232no—
187C22⋊C4×C53—28no—
188C22⋊C4×C53—44no—
189C22⋊Dic53—28no—
190C22⋊Dic53—44no—
191#11719921—232no—
192#11719921—48no—
193#11719921—48no—
194C4⋊Dic53—48no—
195#11719919—48no—
196C23×C53—28no—
197C23×C53—24no—
198C23×C53—24no—
199C23×C53—24no—
200C23×C53—28no—
201C23×C53—24no—
202C23×C53—24no—
203C23×C53—24no—
204C23×C53—28no—
205C23×C53—28no—
206C23×C53C2×C4⋊C414yes—
207C23×C53C4×D414yes—
208C23×C53C4×D414yes—
209C23×C53#12014yes—
210C23×C53C22⋊Q814yes—
211C23×C53C22⋊Q814yes—
212C23×C53#12314yes—
213C23×C53C23×C411yescharacteristic
214C23×C53C22×D411yescharacteristic
215C23×C53C22×D414yes—
216C23×C53C22×D411yescharacteristic
217C23×C53C22×D411yescharacteristic
218C23×C53C22×Q811yescharacteristic
219C23×C53C2×#4114yes—
220C23×C53C2×#4111yescharacteristic
221C23×C53C2×#4111yescharacteristic
222C22×D53—232no—
223C22×D53—264no—
224C22×D53—264no—
225C22×D53—264no—
226C22×D53—232no—
227C22×D53—264no—
228C22×D53—264no—
229C22×D53—28no—
230C22×D53—24no—
231C22×D53—24no—
232C22×D53—28no—
233C22×D53—24no—
234C22×D53—28no—
235C22×D53—24no—
236C22×D53—28no—
237C22×D53—24no—
238C22×D53—24no—
239C22×D53—24no—
240C22×D53—24no—
241C22×D53—28no—
242C22×D53—28no—
243C22×D53—28no—
244C22×D53—28no—
245C22×D53—28no—
246C22×D53—28no—
247C22×D53—232no—
248C22×D53—264no—
249C22×D53C2×C4⋊C418yes—
250C22×D53C4×D4116yes—
251C22×D53#12018yes—
252C22×D53C22⋊Q8116yes—
253C22×D53#12318yes—
254C2×C4×C53—232no—
255C2×C4×C53—28no—
256C2×C4×C53—28no—
257C2×C4×C53—24no—
258C2×C4×C53—24no—
259C2×C4×C53—28no—
260C2×C4×C53—24no—
261C2×C4×C53—42no—
262C2×C4×C53—42no—
263C2×C4×C53—44no—
264C2×C4×C53—42no—
265C2×C4×C53—42no—
266C2×C4×C53—42no—
267C2×C4×C53—44no—
268C2×C4×C53—44no—
269C2×C4×C53#12018yes—
270C2×Dic53—232no—
271C2×Dic53—24no—
272C2×Dic53—28no—
273C2×Dic53—28no—
274C2×Dic53—24no—
275C2×Dic53—28no—
276C2×Dic53—24no—
277C2×Dic53—42no—
278C2×Dic53—44no—
279C2×Dic53—42no—
280C2×Dic53—42no—
281C2×Dic53—42no—
282C2×Dic53—42no—
283C2×Dic53—41no—
284C2×Dic53—41no—
285C2×Dic53—42no—
286C2×Dic53—42no—
287C2×Dic53—42no—
288C2×Dic53#12018yes—
289C4×D53—2128no—
290#14693—531nomaximal, Sylow
291C22×C53—24no—
292C22×C53—28no—
293C22×C53—28no—
294C22×C53—24no—
295C22×C53—28no—
296C22×C53—28no—
297C22×C53—28no—
298C22×C53—28no—
299C22×C53#38518yescomplemented
300C22×C53C22×C4⋊C411yescharacteristic
301C22×C53C2×C4×D414yes—
302C22×C53C2×C4×D411yescharacteristic
303C22×C53C2×C4×D411yescharacteristic
304C22×C53C2×#12012yes—
305C22×C53C2×#12012yes—
306C22×C53C2×#12012yes—
307C22×C53C2×#12011yescharacteristic
308C22×C53C2×C22⋊Q814yes—
309C22×C53C2×C22⋊Q811yescharacteristic
310C22×C53C2×C22⋊Q811yescharacteristic
311C22×C53C2×#12312yes—
312C22×C53C2×#12311yescharacteristic
313C22×C53#52512yes—
314C22×C53#52514yes—
315C22×C53#52514yes—
316C22×C53#52512yes—
317C2×D53—2128no—
318C2×D53—28no—
319C2×D53—216no—
320C2×D53—216no—
321C2×D53—28no—
322C2×D53—28no—
323C2×D53—28no—
324C2×D53—28no—
325C2×D53—28no—
326C2×D53—216no—
327C2×D53—216no—
328C2×D53#385116yescomplemented
329C4×C53—28no—
330C4×C53—44no—
331C4×C53—82no—
332Dic53—28no—
333Dic53—44no—
334Dic53—81no—
335Dic53—81no—
336C25×C4—531no—
337C2×#385—534no—
338C2×#385—531no—
339C2×#385—531no—
340C2×#379—531no—
341C2×#379—531no—
342C22⋊#95—534no—
343C22⋊#95—531no—
344C22⋊#95D5311yescharacteristic, complemented
345C2×C53—28no—
346C2×C53C2×#38512yes—
347C2×C53C2×#38511yescharacteristic
348C2×C53C4×#12012yes—
349C2×C53C4×#12012yes—
350C2×C53#284611yescharacteristic
351C2×C53#284812yes—
352C2×C53#284812yes—
353C2×C53#284912yes—
354C2×C53#284911yescharacteristic
355D53—216no—
356C26—531no—
357C24×C4—531no—
358C24×C4—534no—
359C24×C4—1064no—
360C24×C4—10616no—
361C24×C4—1064no—
362C24×C4—1064no—
363C24×C4C2×D5311yescharacteristic
364C22×C4⋊C4—532no—
365C22×C4⋊C4—532no—
366C22×C22⋊C4—531no—
367C22×C22⋊C4—534no—
368C22×C22⋊C4—531no—
369C22×C22⋊C4—532no—
370C22×C22⋊C4—532no—
371C22×C22⋊C4C2×D5311yescharacteristic
372C22×C22⋊C4C2×D5311yescharacteristic
373#385—10616no—
374#385—1064no—
375#385—1064no—
376#379—538no—
377#379—1064no—
378C2×#95—534no—
379C2×#95—532no—
380C2×#95—532no—
381C2×#95C2×D5314yescomplemented
382C53#1469311yescharacteristic, complemented, Sylow
383C25—532no—
384C25—534no—
385C25—534no—
386C25—534no—
387C25—1068no—
388C25—1068no—
389C25—1062no—
390C25—1062no—
391C25—1062no—
392C25—1062no—
393C25C22×D5311yescharacteristic
394C23×C4—22no—
395C23×C4—22no—
396C23×C4—28no—
397C23×C4—532no—
398C23×C4—534no—
399C23×C4—532no—
400C23×C4—5316no—
401C23×C4—531no—
402C23×C4—531no—
403C23×C4—1068no—
404C23×C4—1068no—
405C23×C4—1068no—
406C23×C4—1068no—
407C23×C4—10664no—
408C23×C4—10664no—
409C23×C4—10632no—
410C23×C4—10664no—
411C23×C4—1068no—
412C23×C4—1062no—
413C23×C4—1062no—
414C23×C4—1068no—
415C23×C4—1068no—
416C23×C4D21214yes—
417C23×C4C22×D5312yes—
418C23×C4C22×D5312yes—
419C23×C4C22×D5312yes—
420C2×C4⋊C4—1068no—
421C2×C4⋊C4—1064no—
422C2×C4⋊C4—1064no—
423C2×C4⋊C4—1068no—
424C2×C22⋊C4—22no—
425C2×C22⋊C4—22no—
426C2×C22⋊C4—22no—
427C2×C22⋊C4—22no—
428C2×C22⋊C4—22no—
429C2×C22⋊C4—534no—
430C2×C22⋊C4—5316no—
431C2×C22⋊C4—1068no—
432C2×C22⋊C4—1068no—
433C2×C22⋊C4—1062no—
434C2×C22⋊C4—1062no—
435C2×C22⋊C4—1068no—
436C2×C22⋊C4—1064no—
437C2×C22⋊C4—1064no—
438C2×C22⋊C4—1068no—
439C2×C22⋊C4—1062no—
440C2×C22⋊C4—1062no—
441C2×C22⋊C4—1062no—
442C2×C22⋊C4C4×D5314yescomplemented
443#95—28no—
444#95—1064no—
445#95—1068no—
446#95—1064no—
447C24—24no—
448C24—24no—
449C24—534no—
450C24—538no—
451C24—534no—
452C24—534no—
453C24—538no—
454C24—5316no—
455C24—10616no—
456C24—10632no—
457C24—10632no—
458C24—10616no—
459C24—10616no—
460C24—10616no—
461C24—10616no—
462C24—10632no—
463C24—10632no—
464C24—10616no—
465C24—1068no—
466C24—1068no—
467C24—1068no—
468C24—1068no—
469C24—1068no—
470C24—1068no—
471C24—1064no—
472C24—1062no—
473C24—1062no—
474C24—1064no—
475C24—1062no—
476C24—1062no—
477C24C2×C4×D5312yes—
478C24C2×D21212yes—
479C24#1171994512yes—
480C24D4×D5312yes—
481C24D4×D5312yes—
482C24#1171994712yes—
483C24Q8×D5312yes—
484C24C23×D5311yescharacteristic
485C22×C4—216no—
486C22×C4—216no—
487C22×C4—28no—
488C22×C4—22no—
489C22×C4—24no—
490C22×C4—22no—
491C22×C4—216no—
492C22×C4—24no—
493C22×C4—22no—
494C22×C4—24no—
495C22×C4—22no—
496C22×C4—24no—
497C22×C4—22no—
498C22×C4—24no—
499C22×C4—22no—
500C22×C4—22no—
501C22×C4—538no—
502C22×C4—534no—
503C22×C4—534no—
504C22×C4—10632no—
505C22×C4—10632no—
506C22×C4—106256no—
507C22×C4—10632no—
508C22×C4—10632no—
509C22×C4—10632no—
510C22×C4—10632no—
511C22×C4—10616no—
512C22×C4—10616no—
513C22×C4—10616no—
514C22×C4—1068no—
515C22×C4—1062no—
516C22×C4—1064no—
517C22×C4—1062no—
518C22×C4—10632no—
519C22×C4—1064no—
520C22×C4—1062no—
521C22×C4—1064no—
522C22×C4—1062no—
523C22×C4—1061no—
524C22×C4—1061no—
525C22×C4—1061no—
526C22×C4—1064no—
527C22×C4—1061no—
528C22×C4—1061no—
529C22×C4—1061no—
530C22×C4—1064no—
531C22×C4C2×C4×D5314yescomplemented
532C22×C4C2×D21214yes—
533C22×C4D4×D5318yes—
534C4⋊C4—2128no—
535C4⋊C4—2128no—
536C22⋊C4—28no—
537C22⋊C4—44no—
538C22⋊C4—10632no—
539C22⋊C4—1068no—
540C22⋊C4—2128no—
541C22⋊C4—2128no—
542C22⋊C4—2124no—
543C23—28no—
544C23—24no—
545C23—24no—
546C23—24no—
547C23—28no—
548C23—24no—
549C23—24no—
550C23—28no—
551C23—28no—
552C23—24no—
553C23—538no—
554C23—5316no—
555C23—538no—
556C23—5316no—
557C23—538no—
558C23—10664no—
559C23—10632no—
560C23—10664no—
561C23—10664no—
562C23—10664no—
563C23—10632no—
564C23—10664no—
565C23—10664no—
566C23—10632no—
567C23—1068no—
568C23—1064no—
569C23—1064no—
570C23—1068no—
571C23—1064no—
572C23—1068no—
573C23—1064no—
574C23—1068no—
575C23—1064no—
576C23—1064no—
577C23—1064no—
578C23—1064no—
579C23—1068no—
580C23—1068no—
581C23—1068no—
582C23—1068no—
583C23—1068no—
584C23—1068no—
585C23C4⋊C4×D5314yes—
586C23#1313317714yes—
587C23C4×D21214yes—
588C23#1313318714yes—
589C23#1313319514yes—
590C23#1313319714yes—
591C23#1313319914yes—
592C23C22×C4×D5311yescharacteristic
593C23C2×D4×D5311yescharacteristic
594C23C2×D4×D5314yes—
595C23C2×D4×D5311yescharacteristic
596C23C22×D21211yescharacteristic
597C23C2×Q8×D5311yescharacteristic
598C23#41×D5314yes—
599C23C2×#1171994711yescharacteristic
600C23C2×#1171994511yescharacteristic
601C2×C4—232no—
602C2×C4—28no—
603C2×C4—28no—
604C2×C4—24no—
605C2×C4—24no—
606C2×C4—28no—
607C2×C4—24no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—44no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—538no—
617C2×C4—106128no—
618C2×C4—10632no—
619C2×C4—1064no—
620C2×C4—1068no—
621C2×C4—1068no—
622C2×C4—1064no—
623C2×C4—1068no—
624C2×C4—1064no—
625C2×C4—2122no—
626C2×C4—2122no—
627C2×C4—2121no—
628C2×C4—2121no—
629C2×C4—2122no—
630C2×C4—2122no—
631C2×C4—2122no—
632C2×C4—2122no—
633C2×C4—2122no—
634C2×C4—2124no—
635C2×C4—2122no—
636C2×C4#1313318718yes—
637C22—24no—
638C22—28no—
639C22—28no—
640C22—28no—
641C22—24no—
642C22—28no—
643C22—28no—
644C22—28no—
645C22—5316no—
646C22—106128no—
647C22—1068no—
648C22—10616no—
649C22—10616no—
650C22—1068no—
651C22—1068no—
652C22—1068no—
653C22—1068no—
654C22—1068no—
655C22—10616no—
656C22—10616no—
657C22#1681049718yescomplemented
658C22C2×C4⋊C4×D5311yescharacteristic
659C22C4×D4×D5314yes—
660C22C2×#1313317711yescharacteristic
661C22C2×C4×D21211yescharacteristic
662C22#120×D5312yes—
663C22#120×D5312yes—
664C22C2×#1313318712yes—
665C22C2×#1313318711yescharacteristic
666C22C22⋊Q8×D5314yes—
667C22C2×#1313319511yescharacteristic
668C22C2×#1313319711yescharacteristic
669C22#123×D5312yes—
670C22C2×#1313319911yescharacteristic
671C22#1681113112yes—
672C22#1681113214yes—
673C22#1681113314yes—
674C22#1681113512yes—
675C4—28no—
676C4—44no—
677C4—82no—
678C4—1068no—
679C4—4241no—
680C4—4241no—
681C4—2124no—
682C2—28no—
683C2—10616no—
684C2#385×D5312yes—
685C2C2×#1681049711yescharacteristic
686C2C4×#1313318712yes—
687C2#2027722012yes—
688C2#2028299311yescharacteristic
689C2#2028300112yes—
690C2#2028300312yes—
691C2#2028300612yes—
692C2#2028300811yescharacteristic
subgroupG/H
center, FrattiniC24C23×D53
commutatorC23×C53C23×C4
FittingC53×C22⋊#95C2
socle, cosocleC24×C53C24
prime radicalC25×D53C22
hypercenterC22⋊#95D53
hypocenterC53#14693
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C24 ◁ C22⋊#95

Complements

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

NH ≅ G/Ntype
2. C2×#16810498C2semidirect
3. C2×#16810497C2semidirect
4. C2×#16810496C2semidirect
7. #20272937C2semidirect
8. #20272936C2semidirect
9. C53×C22⋊#95C2semidirect
17. C22×C4⋊Dic53C22semidirect
18. C22×#11719919C22semidirect
28. #16810478C4semidirect
29. C2×#13133124C22semidirect
30. C2×#13133124C22semidirect
31. C2×#13133124C22semidirect
35. C2×#95×C53C22semidirect
74. C2×C22⋊Dic53C2×C4semidirect
81. C2×#11719921C2×C4semidirect
98. C2×C22⋊C4×C53C2×C4semidirect
132. C23×D53C4⋊C4semidirect
154. C22×C4×C53C22×C4semidirect
176. C22×Dic53C22×C4semidirect
299. C22×C53#385semidirect
328. C2×D53#385semidirect
344. C22⋊#95D53semidirect
381. C2×#95C2×D53semidirect
382. C53#14693semidirect
442. C2×C22⋊C4C4×D53semidirect
531. C22×C4C2×C4×D53semidirect
657. C22#16810497semidirect

Maximal subgroups

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

subgroupindex
C24×C4×D532
C2×#168104982
C2×#168104972
C2×#168104962
C2×#168104782
#202729372
#202729362
C53×C22⋊#952
#1469353