Finite Group Databasebeta

← #27416823

Subgroups — #27416823

Order 13568.

1340 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810939C211yesmaximal, characteristic, complemented
2C2×#16810951C211yesmaximal, characteristic, complemented
3#20277307C211yesmaximal, characteristic, complemented
4#20275908C212yesmaximal, complemented
5#20275901C212yesmaximal, complemented
6#20275038C211yesmaximal, characteristic, complemented
7#20275028C211yesmaximal, characteristic, complemented
8#20274908C212yesmaximal, complemented
9#20274655C212yesmaximal, complemented
10#20274627C211yesmaximal, characteristic, complemented
11C2×C53×#537C211yesmaximal, characteristic
12C2×#13133176C2211yescharacteristic
13C22×C4⋊Dic53C2211yescharacteristic
14#16810951—24no—
15#16810939—24no—
16C2×#13133148C2212yes—
17C2×#13133148C2212yes—
18C2×#13133141C2212yes—
19C2×#13133141C2212yes—
20C2×#13133138C2212yes—
21C2×#13133138C2212yes—
22C2×C22⋊#88322C2211yescharacteristic
23C2×C22⋊#88322C2211yescharacteristic
24C2×C22⋊#88322C2212yes—
25C2×C22⋊#88322C2212yes—
26#16810530—21no—
27#16810526—21no—
28#16810503—21no—
29#16810503—21no—
30#16810483C2212yes—
31#16810483C2212yes—
32#16810385C2212yes—
33#16810385C2212yes—
34C22×#41×C53C2211yescharacteristic
35C22×#41×C53C2211yescharacteristic
36C53×#537—24no—
37C53×#537—24no—
38C2×C22⋊Q8×C53—21no—
39C2×C22⋊Q8×C53C2211yescharacteristic
40C2×C22⋊Q8×C53C2211yescharacteristic
41C2×#121×C53—21no—
42C2×#121×C53—21no—
43C2×#121×C53—21no—
44C2×#121×C53C2211yescharacteristic
45C2×#121×C53C2211yescharacteristic
46C22×C4⋊C4×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#13133148—24no—
61#13133148—24no—
62#13133141—24no—
63#13133141—42no—
64#13133138—24no—
65#13133138—42no—
66C22⋊#88322—22no—
67C22⋊#88322—22no—
68C22⋊#88322—24no—
69C22⋊#88322—42no—
70#13133124—22no—
71#13133124—22no—
72C2×#41×C53—22no—
73C2×#41×C53—22no—
74C2×#41×C53—22no—
75C2×#41×C53—22no—
76C2×#41×C53—24no—
77C2×#41×C53—24no—
78C2×#41×C53—42no—
79C2×#41×C53—42no—
80C22×Q8×C53D411yescharacteristic
81C22×Q8×C53C2311yescharacteristic
82C22×D4×C53—21no—
83C22×D4×C53—21no—
84C22×D4×C53—21no—
85C22×D4×C53D411yescharacteristic
86C22×D4×C53D411yescharacteristic
87C22×D4×C53D411yescharacteristic
88C22×D4×C53C2311yescharacteristic
89C23×C4×C53—21no—
90C23×C4×C53—21no—
91C23×C4×C53C2311yescharacteristic
92C23×C4×C53C2311yescharacteristic
93C23×C4×C53C2311yescharacteristic
94C22⋊Q8×C53—28no—
95C22⋊Q8×C53—22no—
96C22⋊Q8×C53—22no—
97C22⋊Q8×C53—22no—
98C22⋊Q8×C53D414yes—
99#121×C53—28no—
100#121×C53—28no—
101#121×C53—28no—
102#121×C53—22no—
103#121×C53—22no—
104#121×C53—22no—
105#121×C53D414yes—
106C2×C4⋊C4×C53—22no—
107C2×C4⋊C4×C53—22no—
108C2×C4⋊C4×C53—21no—
109C2×C4⋊C4×C53—21no—
110C2×C4⋊C4×C53—21no—
111C2×C4⋊C4×C53C2311yescharacteristic
112C2×C4⋊C4×C53C2311yescharacteristic
113C2×C22⋊C4×C53—21no—
114C2×C22⋊C4×C53—21no—
115C2×C22⋊C4×C53—21no—
116C2×C22⋊C4×C53—21no—
117C24×C53—21no—
118C24×C53C2×D411yescharacteristic
119C24×C53C2×D411yescharacteristic
120C24×C53C2×D411yescharacteristic
121#41×C53—48no—
122#41×C53—48no—
123#41×C53—48no—
124#41×C53—48no—
125C2×Q8×C53—22no—
126C2×Q8×C53—22no—
127C2×Q8×C53—21no—
128C2×Q8×C53—22no—
129C2×Q8×C53—22no—
130C2×Q8×C53—41no—
131C2×Q8×C53C2×D412yes—
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—22no—
140C2×D4×C53—22no—
141C2×D4×C53—21no—
142C2×D4×C53—22no—
143C2×D4×C53—22no—
144C2×D4×C53—41no—
145C2×D4×C53—41no—
146C2×D4×C53—41no—
147C2×D4×C53—42no—
148C2×D4×C53—42no—
149C2×D4×C53—82no—
150C2×D4×C53—82no—
151C2×D4×C53C2×D412yes—
152C22×C4×C53—21no—
153C22×C4×C53—21no—
154C22×C4×C53—22no—
155C22×C4×C53—22no—
156C22×C4×C53—22no—
157C22×C4×C53—24no—
158C22×C4×C53—21no—
159C22×C4×C53—21no—
160C22×C4×C53—21no—
161C22×C4×C53—24no—
162C22×C4×C53—21no—
163C22×C4×C53—21no—
164C22×C4×C53—22no—
165C22×C4×C53—22no—
166C22×C4×C53—42no—
167C22×C4×C53—42no—
168C22×C4×C53—82no—
169C22×C4×C53C2×D411yescharacteristic
170C22×C4×C53C2×D411yescharacteristic
171C22×C4×C53C2×D412yes—
172C22×C4×C53C2×D412yes—
173C22×C4×C53C2×D411yescharacteristic
174C22×C4×C53#4114yes—
175C22×C4×C53C2411yescharacteristic
176C22×Dic53—22no—
177C22×Dic53—22no—
178C22×Dic53—42no—
179C22×Dic53—82no—
180C4⋊C4×C53—24no—
181C4⋊C4×C53—24no—
182C4⋊C4×C53—24no—
183C4⋊C4×C53—21no—
184C4⋊C4×C53—21no—
185C4⋊C4×C53C2×D412yes—
186C4⋊C4×C53C2×D412yes—
187C22⋊C4×C53—24no—
188C22⋊C4×C53—24no—
189C22⋊C4×C53—24no—
190C22⋊C4×C53—24no—
191C22⋊Dic53—42no—
192C22⋊Dic53—82no—
193C22⋊Dic53—82no—
194C4⋊Dic53—22no—
195C4⋊Dic53—22no—
196C4⋊Dic53—22no—
197C4⋊Dic53—22no—
198C4⋊Dic53—42no—
199C4×Dic53—42no—
200C2×#88322—22no—
201C2×#88322—22no—
202C2×#88322—22no—
203C2×#88322—22no—
204C2×#88322—22no—
205C2×#88322—22no—
206C23×C53—21no—
207C23×C53—21no—
208C23×C53—21no—
209C23×C53—21no—
210C23×C53—22no—
211C23×C53—22no—
212C23×C53—22no—
213C23×C53—22no—
214C23×C53—21no—
215C23×C53—22no—
216C23×C53—21no—
217C23×C53—21no—
218C23×C53—22no—
219C23×C53—22no—
220C23×C53—41no—
221C23×C53—41no—
222C23×C53—42no—
223C23×C53—42no—
224C23×C53#12012yes—
225C23×C53#12312yes—
226C23×C53C22×D411yescharacteristic
227Q8×C53—24no—
228Q8×C53—44no—
229Q8×C53—42no—
230D4×C53—24no—
231D4×C53—44no—
232D4×C53—44no—
233D4×C53—44no—
234D4×C53—48no—
235D4×C53—42no—
236D4×C53—84no—
237D4×C53—84no—
238C2×C4×C53—21no—
239C2×C4×C53—21no—
240C2×C4×C53—21no—
241C2×C4×C53—21no—
242C2×C4×C53—24no—
243C2×C4×C53—24no—
244C2×C4×C53—24no—
245C2×C4×C53—24no—
246C2×C4×C53—22no—
247C2×C4×C53—22no—
248C2×C4×C53—22no—
249C2×C4×C53—21no—
250C2×C4×C53—21no—
251C2×C4×C53—21no—
252C2×C4×C53—48no—
253C2×C4×C53—41no—
254C2×C4×C53—41no—
255C2×C4×C53—42no—
256C2×C4×C53—42no—
257C2×C4×C53—42no—
258C2×C4×C53—42no—
259C2×C4×C53—84no—
260C2×C4×C53#12012yes—
261C2×C4×C53#12312yes—
262C2×C4×C53C22×D411yescharacteristic
263C2×C4×C53C22×D411yescharacteristic
264C2×C4×C53C2×#4112yes—
265C2×C4×C53H(5,𝔽2)12yes—
266C2×Dic53—42no—
267C2×Dic53—42no—
268C2×Dic53—42no—
269C2×Dic53—42no—
270C2×Dic53—42no—
271C2×Dic53—42no—
272C2×Dic53—84no—
273#88322—42no—
274#88322—42no—
275#24012—531nomaximal, Sylow
276C22×C53—24no—
277C22×C53—22no—
278C22×C53—22no—
279C22×C53—22no—
280C22×C53—22no—
281C22×C53—42no—
282C22×C53—42no—
283C22×C53—42no—
284C22×C53—41no—
285C22×C53—41no—
286C22×C53—42no—
287C22×C53—42no—
288C22×C53—41no—
289C22×C53—42no—
290C22×C53—41no—
291C22×C53—42no—
292C22×C53—41no—
293C22×C53—42no—
294C22×C53—82no—
295C22×C53—82no—
296C22×C53C2×#12011yescharacteristic
297C22×C53C2×#12311yescharacteristic
298C22×C53#53411yescharacteristic
299C22×C53#57712yes—
300C22×C53#57712yes—
301C4×C53—22no—
302C4×C53—22no—
303C4×C53—41no—
304C4×C53—42no—
305C4×C53—42no—
306C4×C53—42no—
307C4×C53#54612yes—
308C4×C53#54612yes—
309Dic53—82no—
310Dic53—82no—
311C2×#537D5311yescharacteristic
312C2×#481—531no—
313C2×#483—531no—
314#2311—531no—
315#2035—532no—
316#2034—532no—
317#1875—531no—
318#1873—531no—
319#1853—532no—
320#1788—532no—
321#1780—531no—
322C2×C53—42no—
323C2×C53—42no—
324C2×C53—81no—
325C2×C53—82no—
326C2×C53#240511yescharacteristic
327C2×C53#300911yescharacteristic
328C2×C53#301011yescharacteristic
329C2×C53#309912yes—
330C2×C53#320412yes—
331C22×#41C2×D5311yescharacteristic
332C22×#41C2×D5311yescharacteristic
333#537—24no—
334#537—24no—
335C2×C22⋊Q8—21no—
336C2×C22⋊Q8C2×D5311yescharacteristic
337C2×C22⋊Q8C2×D5311yescharacteristic
338C2×#121—21no—
339C2×#121—21no—
340C2×#121—21no—
341C2×#121C2×D5311yescharacteristic
342C2×#121C2×D5311yescharacteristic
343C2×#117—531no—
344C22×C4⋊C4—531no—
345C22×C4⋊C4C2×D5311yescharacteristic
346#483—1064no—
347#481—1064no—
348C2×#106—532no—
349C2×#106—532no—
350C2×#103—532no—
351C2×#103—532no—
352C2×#102—532no—
353C2×#102—532no—
354C2×C22⋊C8—531no—
355C2×C22⋊C8—531no—
356C2×C22⋊C8—532no—
357C2×C22⋊C8—532no—
358#391—1061no—
359#390—1061no—
360#386—1061no—
361#386—1061no—
362#380—532no—
363#380—532no—
364#340—532no—
365#340—532no—
366C53#2401211yescharacteristic, complemented, Sylow
367C2×#41—22no—
368C2×#41—22no—
369C2×#41—24no—
370C2×#41—22no—
371C2×#41—22no—
372C2×#41—24no—
373C2×#41—42no—
374C2×#41—42no—
375C22×Q8C22⋊D5311yescharacteristic
376C22×Q8C22×D5311yescharacteristic
377C22×D4—21no—
378C22×D4—21no—
379C22×D4—21no—
380C22×D4C22⋊D5311yescharacteristic
381C22×D4C22⋊D5311yescharacteristic
382C22×D4C22⋊D5311yescharacteristic
383C22×D4C22×D5311yescharacteristic
384C23×C4—21no—
385C23×C4—21no—
386C23×C4—1061no—
387C23×C4C22×D5311yescharacteristic
388C23×C4C22×D5311yescharacteristic
389C23×C4C22×D5311yescharacteristic
390C22×C8—532no—
391C22×C8—532no—
392C22⋊Q8—28no—
393C22⋊Q8—22no—
394C22⋊Q8—22no—
395C22⋊Q8—22no—
396C22⋊Q8C22⋊D5314yes—
397#121—28no—
398#121—28no—
399#121—28no—
400#121—22no—
401#121—22no—
402#121—22no—
403#121C22⋊D5314yes—
404#117—2122no—
405C2×C4⋊C4—21no—
406C2×C4⋊C4—21no—
407C2×C4⋊C4—21no—
408C2×C4⋊C4—22no—
409C2×C4⋊C4—22no—
410C2×C4⋊C4—532no—
411C2×C4⋊C4—532no—
412C2×C4⋊C4—1062no—
413C2×C4⋊C4—1062no—
414C2×C4⋊C4—1061no—
415C2×C4⋊C4C22×D5311yescharacteristic
416C2×C4⋊C4C22×D5311yescharacteristic
417C2×C22⋊C4—21no—
418C2×C22⋊C4—21no—
419C2×C22⋊C4—21no—
420C2×C22⋊C4—21no—
421C2×C22⋊C4—1062no—
422C2×C22⋊C4—1062no—
423C2×C22⋊C4—1061no—
424C2×C42—1061no—
425#106—1064no—
426#106—1064no—
427#103—1064no—
428#103—2122no—
429#102—1064no—
430#102—2122no—
431C22⋊C8—1064no—
432C22⋊C8—1062no—
433C22⋊C8—1062no—
434C22⋊C8—2122no—
435#95—1062no—
436#95—1062no—
437C24—21no—
438C24C2×C22⋊D5311yescharacteristic
439C24C2×C22⋊D5311yescharacteristic
440C24C2×C22⋊D5311yescharacteristic
441#41—48no—
442#41—48no—
443#41—48no—
444#41—48no—
445C2×Q8—22no—
446C2×Q8—22no—
447C2×Q8—21no—
448C2×Q8—22no—
449C2×Q8—22no—
450C2×Q8—41no—
451C2×Q8C2×C22⋊D5312yes—
452C2×D4—24no—
453C2×D4—22no—
454C2×D4—22no—
455C2×D4—22no—
456C2×D4—22no—
457C2×D4—24no—
458C2×D4—24no—
459C2×D4—22no—
460C2×D4—22no—
461C2×D4—21no—
462C2×D4—22no—
463C2×D4—22no—
464C2×D4—41no—
465C2×D4—41no—
466C2×D4—41no—
467C2×D4—42no—
468C2×D4—42no—
469C2×D4—82no—
470C2×D4—82no—
471C2×D4C2×C22⋊D5312yes—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—24no—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—24no—
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—42no—
487C22×C4—42no—
488C22×C4—82no—
489C22×C4—1062no—
490C22×C4—1062no—
491C22×C4—4242no—
492C22×C4—2122no—
493C22×C4#1171994714yes—
494C22×C4C2×C22⋊D5311yescharacteristic
495C22×C4C2×C22⋊D5311yescharacteristic
496C22×C4C2×C22⋊D5312yes—
497C22×C4C2×C22⋊D5312yes—
498C22×C4C2×C22⋊D5311yescharacteristic
499C22×C4C23×D5311yescharacteristic
500C2×C8—1062no—
501C2×C8—1062no—
502C2×C8—1062no—
503C2×C8—1062no—
504C2×C8—1062no—
505C2×C8—1062no—
506C4⋊C4—24no—
507C4⋊C4—24no—
508C4⋊C4—24no—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—1062no—
512C4⋊C4—1062no—
513C4⋊C4—1062no—
514C4⋊C4—1062no—
515C4⋊C4—2122no—
516C4⋊C4C2×C22⋊D5312yes—
517C4⋊C4C2×C22⋊D5312yes—
518C22⋊C4—24no—
519C22⋊C4—24no—
520C22⋊C4—24no—
521C22⋊C4—24no—
522C22⋊C4—4242no—
523C22⋊C4—4242no—
524C22⋊C4—2122no—
525C42—2122no—
526C23—21no—
527C23—21no—
528C23—22no—
529C23—22no—
530C23—22no—
531C23—22no—
532C23—21no—
533C23—21no—
534C23—21no—
535C23—22no—
536C23—21no—
537C23—22no—
538C23—22no—
539C23—21no—
540C23—41no—
541C23—42no—
542C23—42no—
543C23—41no—
544C23#1313318612yes—
545C23#1313319812yes—
546C23C22×C22⋊D5311yescharacteristic
547Q8—24no—
548Q8—44no—
549Q8—42no—
550D4—24no—
551D4—44no—
552D4—44no—
553D4—44no—
554D4—48no—
555D4—42no—
556D4—84no—
557D4—84no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—48no—
573C2×C4—41no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—41no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—84no—
580C2×C4—4244no—
581C2×C4—2122no—
582C2×C4—2122no—
583C2×C4—2122no—
584C2×C4—2122no—
585C2×C4—2122no—
586C2×C4—2122no—
587C2×C4#1313318612yes—
588C2×C4#1313319812yes—
589C2×C4C22×C22⋊D5311yescharacteristic
590C2×C4C22×C22⋊D5311yescharacteristic
591C2×C4C2×#1171994712yes—
592C2×C4#1313326212yes—
593C8—2122no—
594C8—2122no—
595C22—22no—
596C22—22no—
597C22—24no—
598C22—22no—
599C22—22no—
600C22—42no—
601C22—41no—
602C22—41no—
603C22—42no—
604C22—42no—
605C22—42no—
606C22—41no—
607C22—42no—
608C22—41no—
609C22—42no—
610C22—41no—
611C22—42no—
612C22—42no—
613C22—82no—
614C22—82no—
615C22C2×#1313318611yescharacteristic
616C22C2×#1313319811yescharacteristic
617C22#1681118211yescharacteristic
618C22#1681143112yes—
619C22#1681143412yes—
620C4—22no—
621C4—22no—
622C4—41no—
623C4—42no—
624C4—42no—
625C4—42no—
626C4—4242no—
627C4—4242no—
628C4#1681126712yes—
629C4#1681126712yes—
630C2—42no—
631C2—42no—
632C2—81no—
633C2—82no—
634C2#2027803811yescharacteristic
635C2#2028399211yescharacteristic
636C2#2028400111yescharacteristic
637C2#2028470012yes—
638C2#2028562112yes—
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
Fitting, prime radicalC2×C53×#537C2
socleC23×C53C22×D4
hypercenterC2×#537D53
hypocenterC53#24012
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C22×#41 ◁ C2×#537

Complements

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

NH ≅ G/Ntype
1. C2×#16810939C2semidirect
2. C2×#16810951C2semidirect
3. #20277307C2semidirect
4. #20275908C2semidirect
5. #20275901C2semidirect
6. #20275038C2semidirect
7. #20275028C2semidirect
8. #20274908C2semidirect
9. #20274655C2semidirect
10. #20274627C2semidirect
366. C53#24012semidirect

Maximal subgroups

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

subgroupindex
C2×#168109392
C2×#168109512
#202773072
#202759082
#202759012
#202750382
#202750282
#202749082
#202746552
#202746272
C2×C53×#5372
#2401253