Finite Group Databasebeta

← #27320737

Subgroups — #27320737

Order 13568.

1814 conjugacy classes of subgroups.

The subgroup lattice has 676 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810558C212yesmaximal
2C2×#16810518C212yesmaximal, complemented
3C2×#16810492C211yesmaximal, characteristic, complemented
4C2×#16810482C212yesmaximal
5C2×#16810470C211yesmaximal, characteristic
6#20272981C212yesmaximal, complemented
7#20272979C211yesmaximal, characteristic, complemented
8#20272961C211yesmaximal, characteristic
9#20272957C211yesmaximal, characteristic
10#20272943C211yesmaximal, characteristic, complemented
11C53×#1444C211yesmaximal, characteristic, complemented
12C23×C4×D53C2211yescharacteristic
13C22×C4⋊Dic53C2211yescharacteristic
14C22×#11719919C2212yes—
15C22×#11719919C2212yes—
16C22×#11719919C2212yes—
17C22×#11719919C2212yes—
18C22×#11719921C2212yes—
19C22×#11719921C2212yes—
20C22×#11719921C2211yescharacteristic
21C22×#11719921C2211yescharacteristic
22C22×C4×Dic53C2211yescharacteristic
23C22×C4×Dic53C2211yescharacteristic
24#16810558—28no—
25#16810518—28no—
26#16810492—24no—
27#16810482—28no—
28#16810470—24no—
29C2×#13133124C2211yescharacteristic
30C2×#13133124C2212yes—
31C2×#13133124C2211yescharacteristic
32C2×#13133124C2212yes—
33C2×#13133124C2212yes—
34C2×#13133124C2212yes—
35C22×C4⋊C4×C53C2212yes—
36C22×C4⋊C4×C53C2211yescharacteristic
37C2×#95×C53C2211yescharacteristic
38C2×#95×C53C2211yescharacteristic
39C2×#95×C53C2212yes—
40C24×D53C2311yescharacteristic
41C23×Dic53C2312yes—
42C23×Dic53C2312yes—
43C23×Dic53C2311yescharacteristic
44C23×Dic53C2311yescharacteristic
45C23×Dic53C2311yescharacteristic
46C22×C4×D53—22no—
47C22×C4×D53—24no—
48C22×C4×D53—24no—
49C22×C4×D53—22no—
50C22×C4×D53Q814yes—
51C2×C4⋊Dic53—22no—
52C2×C4⋊Dic53—42no—
53C2×#11719919—24no—
54C2×#11719919—24no—
55C2×#11719919—24no—
56C2×#11719919—24no—
57C2×#11719919—24no—
58C2×#11719919—24no—
59C2×#11719919—24no—
60C2×#11719919—24no—
61C2×#11719919—24no—
62C2×#11719919—24no—
63C2×#11719919—42no—
64C2×#11719919D418yes—
65C2×#11719921—24no—
66C2×#11719921—24no—
67C2×#11719921—22no—
68C2×#11719921—24no—
69C2×#11719921—22no—
70C2×#11719921—22no—
71C2×#11719921—24no—
72C2×#11719921—22no—
73C2×#11719921—42no—
74C2×#11719921—42no—
75C2×#11719921D414yes—
76C2×#11719921Q814yes—
77C2×C4×Dic53—24no—
78C2×C4×Dic53—22no—
79C2×C4×Dic53—24no—
80C2×C4×Dic53D414yes—
81#13133124—24no—
82#13133124—22no—
83#13133124—24no—
84#13133124—24no—
85#13133124—22no—
86#13133124—24no—
87C23×C4×C53C2312yes—
88C23×C4×C53C2311yescharacteristic
89C23×C4×C53C2312yes—
90C23×C4×C53C2311yescharacteristic
91C23×C4×C53C2311yescharacteristic
92C2×C4⋊C4×C53—24no—
93C2×C4⋊C4×C53—24no—
94C2×C4⋊C4×C53—22no—
95C2×C4⋊C4×C53—42no—
96C2×C4⋊C4×C53—42no—
97#95×C53—22no—
98#95×C53—24no—
99#95×C53—22no—
100C24×C53C2411yescharacteristic
101C23×D53—21no—
102C23×D53—22no—
103C23×D53—21no—
104C23×D53—21no—
105C23×D53—21no—
106C23×D53—22no—
107C23×D53—22no—
108C23×D53—22no—
109C23×D53C2×D412yes—
110C23×D53C2×Q812yes—
111C23×D53C2×Q812yes—
112C22×C4×C53—22no—
113C22×C4×C53—22no—
114C22×C4×C53—21no—
115C22×C4×C53—22no—
116C22×C4×C53—21no—
117C22×C4×C53—22no—
118C22×C4×C53—22no—
119C22×C4×C53—22no—
120C22×C4×C53—21no—
121C22×C4×C53—22no—
122C22×C4×C53—22no—
123C22×C4×C53—21no—
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×C53—22no—
131C22×C4×C53—22no—
132C22×C4×C53—22no—
133C22×C4×C53—22no—
134C22×C4×C53—22no—
135C22×C4×C53C2×D414yes—
136C22×C4×C53C2×D412yes—
137C22×C4×C53C2×D412yes—
138C22×C4×C53C2×Q812yes—
139C22×C4×C53C2×Q812yes—
140C22×C4×C53#4112yes—
141C22×Dic53—22no—
142C22×Dic53—22no—
143C22×Dic53—22no—
144C22×Dic53—21no—
145C22×Dic53—22no—
146C22×Dic53—22no—
147C22×Dic53—21no—
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—22no—
159C22×Dic53—22no—
160C22×Dic53C2×D414yes—
161C22×Dic53C2×D412yes—
162C22×Dic53C2×D412yes—
163C22×Dic53C2×D412yes—
164C22×Dic53C2×D414yes—
165C22×Dic53C2×Q812yes—
166C22×Dic53C2×Q812yes—
167C22×Dic53#4112yes—
168C22×Dic53#4114yes—
169C22×Dic53#4112yes—
170C2×C4×D53—28no—
171C2×C4×D53—216no—
172C2×C4×D53—44no—
173C2×C4×D53—48no—
174C2×C4×D53—44no—
175C4⋊C4×C53—48no—
176C4⋊C4×C53—44no—
177#11719921—28no—
178#11719921—28no—
179#11719921—48no—
180#11719921—48no—
181C4⋊Dic53—44no—
182#11719919—216no—
183#11719919—48no—
184#11719919—48no—
185#11719919—48no—
186C4×Dic53—28no—
187C4×Dic53—44no—
188C23×C53C22×D411yescharacteristic
189C23×C53C22×D411yescharacteristic
190C23×C53C22×D412yes—
191C23×C53C22×Q811yescharacteristic
192C23×C53C22×Q811yescharacteristic
193C23×C53C2×#4111yescharacteristic
194C23×C53C2×#4111yescharacteristic
195C23×C53C2×#4112yes—
196C23×C53C2×#4111yescharacteristic
197C23×C53H(5,𝔽2)12yes—
198C23×C532−1+412yes—
199C22×D53—22no—
200C22×D53—22no—
201C22×D53—24no—
202C22×D53—22no—
203C22×D53—22no—
204C22×D53—22no—
205C22×D53—22no—
206C22×D53—24no—
207C22×D53—24no—
208C22×D53—24no—
209C22×D53—24no—
210C22×D53—24no—
211C22×D53—42no—
212C22×D53—41no—
213C22×D53—41no—
214C22×D53—41no—
215C22×D53—41no—
216C22×D53—42no—
217C22×D53—42no—
218C22×D53—42no—
219C22×D53—42no—
220C22×D53—42no—
221C22×D53C4⋊Q814yes—
222C2×C4×C53—22no—
223C2×C4×C53—22no—
224C2×C4×C53—24no—
225C2×C4×C53—24no—
226C2×C4×C53—22no—
227C2×C4×C53—22no—
228C2×C4×C53—24no—
229C2×C4×C53—24no—
230C2×C4×C53—24no—
231C2×C4×C53—24no—
232C2×C4×C53—24no—
233C2×C4×C53—42no—
234C2×C4×C53—41no—
235C2×C4×C53—41no—
236C2×C4×C53—41no—
237C2×C4×C53—42no—
238C2×C4×C53—41no—
239C2×C4×C53—42no—
240C2×C4×C53—42no—
241C2×C4×C53—42no—
242C2×C4×C53—42no—
243C2×C4×C53—42no—
244C2×C4×C53—42no—
245C2×C4×C53—42no—
246C2×C4×C53—42no—
247C2×C4×C53—42no—
248C2×C4×C53—42no—
249C2×C4×C53—42no—
250C2×C4×C53C22⋊Q814yes—
251C2×Dic53—22no—
252C2×Dic53—22no—
253C2×Dic53—24no—
254C2×Dic53—24no—
255C2×Dic53—22no—
256C2×Dic53—24no—
257C2×Dic53—24no—
258C2×Dic53—24no—
259C2×Dic53—24no—
260C2×Dic53—41no—
261C2×Dic53—42no—
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×Dic53#12114yes—
275C2×Dic53C22⋊Q814yes—
276C2×Dic53C22⋊Q818yes—
277C2×Dic53#12314yes—
278C2×Dic53#12318yes—
279C2×Dic53#12714yes—
280C4×D53—416no—
281#15135—531nomaximal, Sylow
282C22×C53C2×#12111yescharacteristic
283C22×C53C2×C22⋊Q811yescharacteristic
284C22×C53C2×C22⋊Q812yes—
285C22×C53C2×C22⋊Q811yescharacteristic
286C22×C53C2×#12312yes—
287C22×C53C2×#12311yescharacteristic
288C22×C53C2×#12711yescharacteristic
289C22×C53C2×C4⋊Q811yescharacteristic
290C22×C53#53211yescharacteristic
291C22×C53#53311yescharacteristic
292C22×C53#53311yescharacteristic
293C22×C53#53511yescharacteristic
294C22×C53#53611yescharacteristic
295C22×C53#54212yes—
296C22×C53#54412yes—
297C22×C53D4212yes—
298C22×C53#54712yes—
299C22×C53#54712yes—
300C22×C53D4×Q812yes—
301C22×C53#55012yes—
302C22×C53#55212yes—
303C22×C53#55312yes—
304C22×C53#55412yes—
305C2×D53—24no—
306C2×D53—28no—
307C2×D53—42no—
308C2×D53—42no—
309C2×D53—44no—
310C2×D53—44no—
311C2×D53—44no—
312C2×D53—42no—
313C2×D53—42no—
314C2×D53—42no—
315C2×D53—42no—
316C4×C53—24no—
317C4×C53—42no—
318C4×C53—44no—
319C4×C53—42no—
320C4×C53—82no—
321Dic53—24no—
322Dic53—44no—
323Dic53—42no—
324Dic53—82no—
325Dic53#39918yescomplemented
326C2×#399—532no—
327C2×#389—532no—
328C2×#384—531no—
329C2×#380—532no—
330C2×C4×C22⋊C4—531no—
331#1445—532no—
332#1444—531no—
333#1444D5311yescharacteristic, complemented
334#1439—531no—
335#1439—531no—
336C4⋊#95—531no—
337C2×C53C2×#39911yescharacteristic
338C2×C53#242212yes—
339C2×C53#242312yes—
340C2×C53#244012yes—
341C2×C53#244112yes—
342C2×C53#248611yescharacteristic
343C2×C53#248711yescharacteristic
344C2×C53#254111yescharacteristic
345C2×C53#254211yescharacteristic
346C2×C53#262811yescharacteristic
347C2×C53#285711yescharacteristic
348D53—44no—
349C24×C4—531no—
350C22×C4⋊C4—532no—
351C22×C4⋊C4—532no—
352C22×C4⋊C4—532no—
353C22×C4⋊C4—532no—
354C22×C4⋊C4—531no—
355C22×C4⋊C4C2×D5312yes—
356C22×C4⋊C4C2×D5311yescharacteristic
357C22×C22⋊C4—531no—
358C22×C22⋊C4—531no—
359C22×C22⋊C4—532no—
360C22×C22⋊C4—532no—
361C22×C42—531no—
362C22×C42—531no—
363#399—1068no—
364#389—1068no—
365#384—1064no—
366#380—1068no—
367C4×C22⋊C4—1064no—
368C2×#95—532no—
369C2×#95—531no—
370C2×#95—532no—
371C2×#95—532no—
372C2×#95—532no—
373C2×#95—531no—
374C2×#95C2×D5311yescharacteristic
375C2×#95C2×D5311yescharacteristic
376C2×#95C2×D5312yes—
377C53#1513511yescharacteristic, complemented, Sylow
378C25—531no—
379C23×C4—531no—
380C23×C4—531no—
381C23×C4—532no—
382C23×C4—531no—
383C23×C4—534no—
384C23×C4—532no—
385C23×C4—1062no—
386C23×C4—1064no—
387C23×C4—1064no—
388C23×C4—1062no—
389C23×C4C22×D5312yes—
390C23×C4C22×D5311yescharacteristic
391C23×C4C22×D5312yes—
392C23×C4C22×D5311yescharacteristic
393C23×C4C22×D5311yescharacteristic
394C2×C4⋊C4—22no—
395C2×C4⋊C4—24no—
396C2×C4⋊C4—24no—
397C2×C4⋊C4—42no—
398C2×C4⋊C4—42no—
399C2×C4⋊C4—538no—
400C2×C4⋊C4—1064no—
401C2×C4⋊C4—1064no—
402C2×C4⋊C4—1064no—
403C2×C4⋊C4—1064no—
404C2×C4⋊C4—1064no—
405C2×C4⋊C4—1064no—
406C2×C4⋊C4—1064no—
407C2×C4⋊C4—1064no—
408C2×C4⋊C4—1064no—
409C2×C4⋊C4—1064no—
410C2×C4⋊C4—1062no—
411C2×C4⋊C4—2122no—
412C2×C4⋊C4—2122no—
413C2×C22⋊C4—534no—
414C2×C22⋊C4—534no—
415C2×C22⋊C4—1062no—
416C2×C22⋊C4—1062no—
417C2×C22⋊C4—1062no—
418C2×C22⋊C4—1062no—
419C2×C22⋊C4—1064no—
420C2×C22⋊C4—1064no—
421C2×C22⋊C4—1064no—
422C2×C22⋊C4—1064no—
423C2×C22⋊C4—2122no—
424C2×C22⋊C4—2122no—
425C2×C42—534no—
426C2×C42—1064no—
427C2×C42—1062no—
428C2×C42—1064no—
429#95—22no—
430#95—22no—
431#95—24no—
432#95—1062no—
433#95—1064no—
434#95—1064no—
435#95—1064no—
436#95—1062no—
437#95—1064no—
438C24—532no—
439C24—532no—
440C24—532no—
441C24—1062no—
442C24—1061no—
443C24—1061no—
444C24—1062no—
445C24—1061no—
446C24—1061no—
447C24—1062no—
448C24—1062no—
449C24C23×D5311yescharacteristic
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—21no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—21no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—532no—
474C22×C4—534no—
475C22×C4—532no—
476C22×C4—534no—
477C22×C4—532no—
478C22×C4—532no—
479C22×C4—532no—
480C22×C4—532no—
481C22×C4—534no—
482C22×C4—532no—
483C22×C4—1068no—
484C22×C4—10616no—
485C22×C4—1062no—
486C22×C4—1062no—
487C22×C4—1062no—
488C22×C4—1061no—
489C22×C4—1062no—
490C22×C4—1062no—
491C22×C4—1062no—
492C22×C4—1062no—
493C22×C4—1061no—
494C22×C4—1062no—
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—2124no—
505C22×C4—2128no—
506C22×C4—2124no—
507C22×C4#1171994512yes—
508C22×C4D4×D5314yes—
509C22×C4D4×D5312yes—
510C22×C4D4×D5312yes—
511C22×C4Q8×D5312yes—
512C22×C4Q8×D5312yes—
513C4⋊C4—48no—
514C4⋊C4—44no—
515C4⋊C4—10616no—
516C4⋊C4—2128no—
517C4⋊C4—2124no—
518C4⋊C4—2128no—
519C4⋊C4—2128no—
520C22⋊C4—1068no—
521C22⋊C4—1068no—
522C22⋊C4—2128no—
523C22⋊C4—2128no—
524C42—1068no—
525C42—2124no—
526C23—534no—
527C23—1064no—
528C23—1064no—
529C23—1064no—
530C23—1062no—
531C23—1064no—
532C23—1062no—
533C23—1062no—
534C23—1062no—
535C23—1062no—
536C23—1062no—
537C23—1064no—
538C23—1064no—
539C23—2122no—
540C23—2122no—
541C23—2122no—
542C23—2122no—
543C23—2122no—
544C23—2121no—
545C23—2121no—
546C23—2121no—
547C23—2121no—
548C23—2122no—
549C23C2×D4×D5312yes—
550C23C2×D4×D5311yescharacteristic
551C23C2×D4×D5311yescharacteristic
552C23C2×Q8×D5311yescharacteristic
553C23C2×Q8×D5311yescharacteristic
554C23#41×D5312yes—
555C23#41×D5311yescharacteristic
556C23#41×D5311yescharacteristic
557C23C2×#1171994511yescharacteristic
558C23#1313326212yes—
559C23#1313326512yes—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—24no—
565C2×C4—22no—
566C2×C4—24no—
567C2×C4—22no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—42no—
579C2×C4—41no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—538no—
589C2×C4—534no—
590C2×C4—534no—
591C2×C4—538no—
592C2×C4—534no—
593C2×C4—534no—
594C2×C4—1064no—
595C2×C4—1064no—
596C2×C4—1062no—
597C2×C4—1062no—
598C2×C4—1064no—
599C2×C4—1064no—
600C2×C4—1062no—
601C2×C4—1064no—
602C2×C4—1064no—
603C2×C4—21216no—
604C2×C4—2122no—
605C2×C4—2122no—
606C2×C4—2122no—
607C2×C4—2122no—
608C2×C4—2122no—
609C2×C4—2122no—
610C2×C4—2122no—
611C2×C4—2122no—
612C2×C4—2122no—
613C2×C4—2122no—
614C2×C4—2121no—
615C2×C4—2122no—
616C2×C4—2121no—
617C2×C4—2122no—
618C2×C4#1313319514yes—
619C22—1068no—
620C22—1064no—
621C22—2122no—
622C22—2122no—
623C22—2124no—
624C22—2122no—
625C22—2124no—
626C22—2122no—
627C22—2124no—
628C22—2122no—
629C22—2122no—
630C22#121×D5311yescharacteristic
631C22C22⋊Q8×D5312yes—
632C22C22⋊Q8×D5311yescharacteristic
633C22C2×#1313319511yescharacteristic
634C22#123×D5312yes—
635C22#123×D5311yescharacteristic
636C22#127×D5311yescharacteristic
637C22C4⋊Q8×D5311yescharacteristic
638C22#1681117011yescharacteristic
639C22#1681117711yescharacteristic
640C22#1681117911yescharacteristic
641C22#1681118811yescharacteristic
642C22#1681119811yescharacteristic
643C22#1681124312yes—
644C22#1681125012yes—
645C22#1681125512yes—
646C22#1681127412yes—
647C22#1681127512yes—
648C22#1681128812yes—
649C22#1681129112yes—
650C22#1681130512yes—
651C22#1681132212yes—
652C22#1681132712yes—
653C4—24no—
654C4—42no—
655C4—42no—
656C4—44no—
657C4—82no—
658C4—538no—
659C4—1064no—
660C4—4242no—
661C4—2122no—
662C4—2124no—
663C2—2124no—
664C2#399×D5311yescharacteristic
665C2#2027822812yes—
666C2#2027823512yes—
667C2#2027841012yes—
668C2#2027842512yes—
669C2#2027901811yescharacteristic
670C2#2027903111yescharacteristic
671C2#2027965611yescharacteristic
672C2#2027966711yescharacteristic
673C2#2028053811yescharacteristic
674C2#2028306611yescharacteristic
subgroupG/H
center, FrattiniC24C23×D53
commutator, socle, cosocleC24×C53C24
FittingC53×#1444C2
prime radicalC24×D53C23
hypercenter#1444D53
hypocenterC53#15135
LCSG ▷ C24×C53 ▷ C53
UCSC1 ◁ C24 ◁ #1444

Complements

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

NH ≅ G/Ntype
2. C2×#16810518C2semidirect
3. C2×#16810492C2semidirect
6. #20272981C2semidirect
7. #20272979C2semidirect
10. #20272943C2semidirect
11. C53×#1444C2semidirect
325. Dic53#399semidirect
333. #1444D53semidirect
377. C53#15135semidirect

Maximal subgroups

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

subgroupindex
C2×#168105582
C2×#168105182
C2×#168104922
C2×#168104822
C2×#168104702
#202729812
#202729792
#202729612
#202729572
#202729432
C53×#14442
#1513553