Finite Group Databasebeta

← #27416834

Subgroups — #27416834

Order 13568.

1468 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810939C211yesmaximal, characteristic, complemented
2C2×#16810934C211yesmaximal, characteristic, complemented
3#20277307C211yesmaximal, characteristic, complemented
4#20275901C212yesmaximal, complemented
5#20275894C212yesmaximal, complemented
6#20275038C211yesmaximal, characteristic, complemented
7#20275038C211yesmaximal, characteristic, complemented
8#20274914C212yesmaximal, complemented
9#20274646C212yesmaximal, complemented
10#20274627C211yesmaximal, characteristic, complemented
11C2×C53×#535C211yesmaximal, characteristic
12C2×#13133176C2211yescharacteristic, complemented
13C22×C4⋊Dic53C2211yescharacteristic, complemented
14#16810939—24no—
15#16810934—24no—
16C2×#13133150C2212yescomplemented
17C2×#13133148C2212yescomplemented
18C2×#13133138C2212yes—
19C2×#13133138C2212yes—
20C2×#13133138C2212yes—
21C2×#13133138C2212yes—
22C2×C22⋊#88322C2212yes—
23C2×C22⋊#88322C2211yescharacteristic, complemented
24C2×C22⋊#88322C2211yescharacteristic, complemented
25C2×C22⋊#88322C2212yescomplemented
26#16810530—21no—
27#16810526—21no—
28#16810503—21no—
29#16810503—21no—
30#16810483C2212yes—
31#16810483C2212yescomplemented
32#16810385C2212yescomplemented
33#16810385C2212yescomplemented
34C22×#41×C53C2211yescharacteristic
35C23×D4×C53C2211yescharacteristic
36C53×#535—24no—
37C53×#535—24no—
38C2×#127×C53—21no—
39C2×#124×C53—21no—
40C2×#121×C53C2211yescharacteristic
41C2×#121×C53C2211yescharacteristic
42C2×#121×C53C2211yescharacteristic
43C2×#121×C53C2211yescharacteristic
44C2×#120×C53—21no—
45C2×#120×C53—21no—
46C2×#117×C53C2211yescharacteristic
47C23×Dic53—21no—
48C22×#88322C2312yes—
49C22×#88322C2312yes—
50#13133176—42no—
51C2×C4⋊Dic53—21no—
52C2×C4⋊Dic53—22no—
53C2×C4⋊Dic53—22no—
54C2×C4⋊Dic53C2312yes—
55C2×C4⋊Dic53C2312yes—
56C2×C22⋊Dic53—21no—
57C2×C22⋊Dic53—22no—
58C2×C22⋊Dic53—22no—
59C2×C4×Dic53—21no—
60#13133150—24no—
61#13133148—24no—
62#13133138—24no—
63#13133138—24no—
64#13133138—42no—
65#13133138—42no—
66C22⋊#88322—22no—
67C22⋊#88322—22no—
68C22⋊#88322—24no—
69C22⋊#88322—42no—
70#13133124—22no—
71#13133124—22no—
72C25×C53—21no—
73C2×#41×C53—22no—
74C2×#41×C53—22no—
75C2×#41×C53—24no—
76C2×#41×C53—42no—
77C22×Q8×C53D411yescharacteristic
78C22×D4×C53—21no—
79C22×D4×C53—21no—
80C22×D4×C53—21no—
81C22×D4×C53—21no—
82C22×D4×C53—22no—
83C22×D4×C53—22no—
84C22×D4×C53—21no—
85C22×D4×C53—24no—
86C22×D4×C53—42no—
87C22×D4×C53D411yescharacteristic
88C22×D4×C53D411yescharacteristic
89C22×D4×C53D411yescharacteristic
90C22×D4×C53C2311yescharacteristic
91C22×D4×C53C2311yescharacteristic
92C23×C4×C53C2311yescharacteristic
93C23×C4×C53C2311yescharacteristic
94C23×C4×C53C2311yescharacteristic
95#127×C53—28no—
96#124×C53—28no—
97#121×C53—22no—
98#121×C53—22no—
99#121×C53—22no—
100#121×C53—22no—
101#121×C53—22no—
102#121×C53—22no—
103#121×C53D414yes—
104#121×C53D414yes—
105#120×C53—28no—
106#120×C53—28no—
107#117×C53—22no—
108#117×C53—22no—
109C2×C4⋊C4×C53C2311yescharacteristic
110C2×C4⋊C4×C53C2311yescharacteristic
111C2×C22⋊C4×C53—21no—
112C2×C22⋊C4×C53—21no—
113C2×C22⋊C4×C53—21no—
114C2×C22⋊C4×C53—21no—
115C2×C22⋊C4×C53—21no—
116C2×C42×C53—21no—
117C24×C53—24no—
118C24×C53—21no—
119C24×C53—21no—
120C24×C53—42no—
121C24×C53—42no—
122C24×C53—82no—
123C24×C53—82no—
124C24×C53C2×D411yescharacteristic
125C24×C53C2×D411yescharacteristic
126C24×C53C2×D411yescharacteristic
127#41×C53—48no—
128#41×C53—48no—
129C2×Q8×C53—22no—
130C2×Q8×C53—22no—
131C2×Q8×C53—41no—
132C2×D4×C53—24no—
133C2×D4×C53—22no—
134C2×D4×C53—22no—
135C2×D4×C53—22no—
136C2×D4×C53—22no—
137C2×D4×C53—24no—
138C2×D4×C53—24no—
139C2×D4×C53—24no—
140C2×D4×C53—24no—
141C2×D4×C53—22no—
142C2×D4×C53—22no—
143C2×D4×C53—21no—
144C2×D4×C53—21no—
145C2×D4×C53—22no—
146C2×D4×C53—22no—
147C2×D4×C53—22no—
148C2×D4×C53—22no—
149C2×D4×C53—48no—
150C2×D4×C53—48no—
151C2×D4×C53—41no—
152C2×D4×C53—41no—
153C2×D4×C53—41no—
154C2×D4×C53—42no—
155C2×D4×C53—42no—
156C2×D4×C53—82no—
157C2×D4×C53—82no—
158C2×D4×C53—82no—
159C2×D4×C53—82no—
160C2×D4×C53C2×D412yes—
161C2×D4×C53C2×D412yes—
162C22×C4×C53—21no—
163C22×C4×C53—21no—
164C22×C4×C53—22no—
165C22×C4×C53—22no—
166C22×C4×C53—22no—
167C22×C4×C53—21no—
168C22×C4×C53—21no—
169C22×C4×C53—21no—
170C22×C4×C53—21no—
171C22×C4×C53—22no—
172C22×C4×C53—22no—
173C22×C4×C53C2×D411yescharacteristic
174C22×C4×C53C2×D411yescharacteristic
175C22×C4×C53C2×D412yes—
176C22×C4×C53C2×D412yes—
177C22×C4×C53C2×D411yescharacteristic
178C22×C4×C53#4114yes—
179C22×C4×C53C2411yescharacteristic
180C22×Dic53—22no—
181C22×Dic53—22no—
182C22×Dic53—42no—
183C22×Dic53—82no—
184C4⋊C4×C53—21no—
185C4⋊C4×C53—21no—
186C4⋊C4×C53C2×D412yes—
187C4⋊C4×C53C2×D412yes—
188C22⋊C4×C53—24no—
189C22⋊C4×C53—24no—
190C22⋊C4×C53—24no—
191C22⋊C4×C53—24no—
192C22⋊C4×C53—24no—
193C42×C53—24no—
194C22⋊Dic53—42no—
195C22⋊Dic53—82no—
196C22⋊Dic53—82no—
197C4⋊Dic53—22no—
198C4⋊Dic53—22no—
199C4⋊Dic53—22no—
200C4⋊Dic53—22no—
201C4⋊Dic53—42no—
202C4×Dic53—42no—
203C2×#88322—22no—
204C2×#88322—22no—
205C2×#88322—22no—
206C2×#88322—22no—
207C2×#88322—22no—
208C2×#88322—22no—
209C23×C53—21no—
210C23×C53—21no—
211C23×C53—21no—
212C23×C53—21no—
213C23×C53—22no—
214C23×C53—22no—
215C23×C53—22no—
216C23×C53—22no—
217C23×C53—21no—
218C23×C53—22no—
219C23×C53—22no—
220C23×C53—21no—
221C23×C53—21no—
222C23×C53—22no—
223C23×C53—22no—
224C23×C53—48no—
225C23×C53—41no—
226C23×C53—41no—
227C23×C53—41no—
228C23×C53—41no—
229C23×C53—42no—
230C23×C53—42no—
231C23×C53—42no—
232C23×C53—42no—
233C23×C53—88no—
234C23×C53—84no—
235C23×C53—84no—
236C23×C53—88no—
237C23×C53#12012yes—
238C23×C53#12312yes—
239C23×C53C22×D411yescharacteristic
240Q8×C53—44no—
241D4×C53—24no—
242D4×C53—24no—
243D4×C53—44no—
244D4×C53—44no—
245D4×C53—48no—
246D4×C53—44no—
247D4×C53—42no—
248D4×C53—42no—
249D4×C53—84no—
250D4×C53—84no—
251D4×C53—84no—
252D4×C53—84no—
253C2×C4×C53—21no—
254C2×C4×C53—21no—
255C2×C4×C53—21no—
256C2×C4×C53—21no—
257C2×C4×C53—24no—
258C2×C4×C53—24no—
259C2×C4×C53—24no—
260C2×C4×C53—24no—
261C2×C4×C53—21no—
262C2×C4×C53—22no—
263C2×C4×C53—22no—
264C2×C4×C53—21no—
265C2×C4×C53—21no—
266C2×C4×C53—42no—
267C2×C4×C53—42no—
268C2×C4×C53—42no—
269C2×C4×C53—42no—
270C2×C4×C53#12012yes—
271C2×C4×C53#12312yes—
272C2×C4×C53C22×D411yescharacteristic
273C2×C4×C53C22×D411yescharacteristic
274C2×C4×C53C2×#4112yes—
275C2×C4×C53H(5,𝔽2)12yes—
276C2×Dic53—42no—
277C2×Dic53—42no—
278C2×Dic53—42no—
279C2×Dic53—42no—
280C2×Dic53—42no—
281C2×Dic53—42no—
282C2×Dic53—84no—
283#88322—42no—
284#88322—42no—
285#24013—531nomaximal, Sylow
286C22×C53—24no—
287C22×C53—22no—
288C22×C53—22no—
289C22×C53—22no—
290C22×C53—22no—
291C22×C53—42no—
292C22×C53—42no—
293C22×C53—42no—
294C22×C53—41no—
295C22×C53—41no—
296C22×C53—42no—
297C22×C53—41no—
298C22×C53—42no—
299C22×C53—42no—
300C22×C53—42no—
301C22×C53—41no—
302C22×C53—42no—
303C22×C53—42no—
304C22×C53—41no—
305C22×C53—42no—
306C22×C53—42no—
307C22×C53—816no—
308C22×C53—82no—
309C22×C53—82no—
310C22×C53—82no—
311C22×C53—82no—
312C22×C53C2×#12011yescharacteristic
313C22×C53C2×#12311yescharacteristic
314C22×C53#53411yescharacteristic
315C22×C53#57612yes—
316C22×C53#57712yes—
317C4×C53—22no—
318C4×C53—22no—
319C4×C53—41no—
320C4×C53—42no—
321C4×C53—42no—
322C4×C53#54612yes—
323C4×C53#54612yes—
324Dic53—82no—
325Dic53—82no—
326C2×#535D5311yescharacteristic
327C2×#481—531no—
328C2×#480—531no—
329#2311—531no—
330#2034—532no—
331#2033—532no—
332#1875—531no—
333#1875—531no—
334#1854—532no—
335#1786—532no—
336#1780—531no—
337C2×C53—42no—
338C2×C53—42no—
339C2×C53—81no—
340C2×C53—82no—
341C2×C53—82no—
342C2×C53#240511yescharacteristic
343C2×C53#300811yescharacteristic
344C2×C53#300911yescharacteristic
345C2×C53#309612yes—
346C2×C53#319812yes—
347C22×#41C2×D5311yescharacteristic
348C23×D4C2×D5311yescharacteristic
349#535—24no—
350#535—24no—
351C2×#127—21no—
352C2×#124—21no—
353C2×#121C2×D5311yescharacteristic
354C2×#121C2×D5311yescharacteristic
355C2×#121C2×D5311yescharacteristic
356C2×#121C2×D5311yescharacteristic
357C2×#120—21no—
358C2×#120—21no—
359C2×#117—531no—
360C2×#117C2×D5311yescharacteristic
361C22×C4⋊C4—531no—
362#481—1064no—
363#480—1064no—
364C2×#107—532no—
365C2×#106—532no—
366C2×#102—532no—
367C2×#102—532no—
368C2×#102—532no—
369C2×#102—532no—
370C2×C22⋊C8—531no—
371C2×C22⋊C8—531no—
372C2×C22⋊C8—532no—
373C2×C22⋊C8—532no—
374#391—1061no—
375#390—1061no—
376#386—1061no—
377#386—1061no—
378#380—532no—
379#380—532no—
380#340—532no—
381#340—532no—
382C53#2401311yescharacteristic, complemented, Sylow
383C25—21no—
384C2×#41—24no—
385C2×#41—22no—
386C2×#41—22no—
387C2×#41—42no—
388C22×Q8C22⋊D5311yescharacteristic
389C22×D4—21no—
390C22×D4—21no—
391C22×D4—21no—
392C22×D4—21no—
393C22×D4—22no—
394C22×D4—22no—
395C22×D4—24no—
396C22×D4—21no—
397C22×D4—42no—
398C22×D4C22⋊D5311yescharacteristic
399C22×D4C22⋊D5311yescharacteristic
400C22×D4C22⋊D5311yescharacteristic
401C22×D4C22×D5311yescharacteristic
402C22×D4C22×D5311yescharacteristic
403C23×C4—1061no—
404C23×C4C22×D5311yescharacteristic
405C23×C4C22×D5311yescharacteristic
406C23×C4C22×D5311yescharacteristic
407C22×C8—532no—
408C22×C8—532no—
409#127—28no—
410#124—28no—
411#121—22no—
412#121—22no—
413#121—22no—
414#121—22no—
415#121—22no—
416#121—22no—
417#121C22⋊D5314yes—
418#121C22⋊D5314yes—
419#120—28no—
420#120—28no—
421#117—22no—
422#117—22no—
423#117—2122no—
424C2×C4⋊C4—532no—
425C2×C4⋊C4—532no—
426C2×C4⋊C4—1062no—
427C2×C4⋊C4—1062no—
428C2×C4⋊C4—1061no—
429C2×C4⋊C4C22×D5311yescharacteristic
430C2×C4⋊C4C22×D5311yescharacteristic
431C2×C22⋊C4—21no—
432C2×C22⋊C4—21no—
433C2×C22⋊C4—21no—
434C2×C22⋊C4—21no—
435C2×C22⋊C4—21no—
436C2×C22⋊C4—1062no—
437C2×C22⋊C4—1062no—
438C2×C22⋊C4—1061no—
439C2×C42—21no—
440C2×C42—1061no—
441#107—1064no—
442#106—1064no—
443#102—1064no—
444#102—1064no—
445#102—2122no—
446#102—2122no—
447C22⋊C8—1064no—
448C22⋊C8—1062no—
449C22⋊C8—1062no—
450C22⋊C8—2122no—
451#95—1062no—
452#95—1062no—
453C24—24no—
454C24—21no—
455C24—21no—
456C24—42no—
457C24—42no—
458C24—82no—
459C24—82no—
460C24C2×C22⋊D5311yescharacteristic
461C24C2×C22⋊D5311yescharacteristic
462C24C2×C22⋊D5311yescharacteristic
463#41—48no—
464#41—48no—
465C2×Q8—22no—
466C2×Q8—22no—
467C2×Q8—41no—
468C2×D4—24no—
469C2×D4—22no—
470C2×D4—22no—
471C2×D4—22no—
472C2×D4—22no—
473C2×D4—24no—
474C2×D4—24no—
475C2×D4—24no—
476C2×D4—24no—
477C2×D4—22no—
478C2×D4—22no—
479C2×D4—21no—
480C2×D4—21no—
481C2×D4—22no—
482C2×D4—22no—
483C2×D4—22no—
484C2×D4—22no—
485C2×D4—48no—
486C2×D4—48no—
487C2×D4—41no—
488C2×D4—41no—
489C2×D4—41no—
490C2×D4—42no—
491C2×D4—42no—
492C2×D4—82no—
493C2×D4—82no—
494C2×D4—82no—
495C2×D4—82no—
496C2×D4C2×C22⋊D5312yes—
497C2×D4C2×C22⋊D5312yes—
498C22×C4—21no—
499C22×C4—21no—
500C22×C4—22no—
501C22×C4—22no—
502C22×C4—22no—
503C22×C4—21no—
504C22×C4—21no—
505C22×C4—21no—
506C22×C4—21no—
507C22×C4—22no—
508C22×C4—22no—
509C22×C4—1062no—
510C22×C4—1062no—
511C22×C4—4242no—
512C22×C4—2122no—
513C22×C4#1171994714yes—
514C22×C4C2×C22⋊D5311yescharacteristic
515C22×C4C2×C22⋊D5311yescharacteristic
516C22×C4C2×C22⋊D5312yes—
517C22×C4C2×C22⋊D5312yes—
518C22×C4C2×C22⋊D5311yescharacteristic
519C22×C4C23×D5311yescharacteristic
520C2×C8—1062no—
521C2×C8—1062no—
522C2×C8—1062no—
523C2×C8—1062no—
524C2×C8—1062no—
525C2×C8—1062no—
526C4⋊C4—21no—
527C4⋊C4—21no—
528C4⋊C4—1062no—
529C4⋊C4—1062no—
530C4⋊C4—1062no—
531C4⋊C4—1062no—
532C4⋊C4—2122no—
533C4⋊C4C2×C22⋊D5312yes—
534C4⋊C4C2×C22⋊D5312yes—
535C22⋊C4—24no—
536C22⋊C4—24no—
537C22⋊C4—24no—
538C22⋊C4—24no—
539C22⋊C4—24no—
540C22⋊C4—4242no—
541C22⋊C4—4242no—
542C22⋊C4—2122no—
543C42—24no—
544C42—2122no—
545C23—21no—
546C23—21no—
547C23—22no—
548C23—22no—
549C23—22no—
550C23—22no—
551C23—21no—
552C23—21no—
553C23—22no—
554C23—22no—
555C23—21no—
556C23—21no—
557C23—22no—
558C23—22no—
559C23—21no—
560C23—48no—
561C23—41no—
562C23—41no—
563C23—42no—
564C23—42no—
565C23—42no—
566C23—42no—
567C23—41no—
568C23—41no—
569C23—88no—
570C23—84no—
571C23—84no—
572C23—88no—
573C23#1313318612yes—
574C23#1313319812yes—
575C23C22×C22⋊D5311yescharacteristic
576Q8—44no—
577D4—24no—
578D4—24no—
579D4—44no—
580D4—44no—
581D4—48no—
582D4—44no—
583D4—42no—
584D4—42no—
585D4—84no—
586D4—84no—
587D4—84no—
588D4—84no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—4244no—
607C2×C4—2122no—
608C2×C4—2122no—
609C2×C4—2122no—
610C2×C4—2122no—
611C2×C4—2122no—
612C2×C4—2122no—
613C2×C4#1313318612yes—
614C2×C4#1313319812yes—
615C2×C4C22×C22⋊D5311yescharacteristic
616C2×C4C22×C22⋊D5311yescharacteristic
617C2×C4C2×#1171994712yes—
618C2×C4#1313326212yes—
619C8—2122no—
620C8—2122no—
621C22—22no—
622C22—22no—
623C22—22no—
624C22—22no—
625C22—24no—
626C22—41no—
627C22—41no—
628C22—42no—
629C22—42no—
630C22—42no—
631C22—41no—
632C22—42no—
633C22—42no—
634C22—41no—
635C22—42no—
636C22—42no—
637C22—41no—
638C22—42no—
639C22—42no—
640C22—42no—
641C22—42no—
642C22—816no—
643C22—82no—
644C22—82no—
645C22—82no—
646C22—82no—
647C22C2×#1313318611yescharacteristic
648C22C2×#1313319811yescharacteristic
649C22#1681118211yescharacteristic
650C22#1681142612yes—
651C22#1681143412yes—
652C4—22no—
653C4—22no—
654C4—41no—
655C4—42no—
656C4—42no—
657C4—4242no—
658C4—4242no—
659C4#1681126712yes—
660C4#1681126712yes—
661C2—42no—
662C2—42no—
663C2—81no—
664C2—82no—
665C2—82no—
666C2#2027803811yescharacteristic
667C2#2028398311yescharacteristic
668C2#2028399211yescharacteristic
669C2#2028466712yes—
670C2#2028556712yes—
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
Fitting, prime radicalC2×C53×#535C2
socleC23×C53C22×D4
hypercenterC2×#535D53
hypocenterC53#24013
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C22×#41 ◁ C2×#535

Complements

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

NH ≅ G/Ntype
1. C2×#16810939C2semidirect
2. C2×#16810934C2semidirect
3. #20277307C2semidirect
4. #20275901C2semidirect
5. #20275894C2semidirect
6. #20275038C2semidirect
7. #20275038C2semidirect
8. #20274914C2semidirect
9. #20274646C2semidirect
10. #20274627C2semidirect
12. C2×#13133176C22semidirect
13. C22×C4⋊Dic53C22semidirect
16. C2×#13133150C22semidirect
17. C2×#13133148C22semidirect
23. C2×C22⋊#88322C22semidirect
24. C2×C22⋊#88322C22semidirect
25. C2×C22⋊#88322C22semidirect
31. #16810483C22semidirect
32. #16810385C22semidirect
33. #16810385C22semidirect
382. C53#24013semidirect

Maximal subgroups

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

subgroupindex
C2×#168109392
C2×#168109342
#202773072
#202759012
#202758942
#202750382
#202749142
#202746462
#202746272
C2×C53×#5352
#2401353