Finite Group Databasebeta

← #27440630

Subgroups — #27440630

Order 13568.

1302 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810982C211yesmaximal, characteristic, complemented
2C2×#16810908C211yesmaximal, characteristic, complemented
3#20279888C211yesmaximal, characteristic, complemented
4#20275908C212yesmaximal, complemented
5#20275901C212yesmaximal, complemented
6#20275648C211yesmaximal, characteristic, complemented
7#20275514C211yesmaximal, characteristic, complemented
8#20274655C212yesmaximal, complemented
9#20274650C212yesmaximal, complemented
10#20274545C211yesmaximal, characteristic, complemented
11C2×C53×#548C211yesmaximal, characteristic
12C2×#13133217C2211yescharacteristic
13#16810982—24no—
14#16810908—24no—
15C2×#13133148C2212yes—
16C2×C4⋊#88322C2211yescharacteristic
17C2×#13133141C2212yes—
18C2×#13133141C2212yes—
19C2×#13133141C2212yes—
20C2×#13133141C2212yes—
21C2×#13133138C2212yes—
22C2×#13133138C2212yes—
23C2×C22⋊#88322C2212yes—
24C2×C22⋊#88322C2212yes—
25C2×#13133128C2211yescharacteristic
26#16810535—21no—
27#16810535—21no—
28#16810526—21no—
29#16810488—21no—
30#16810483C2212yes—
31#16810483C2212yes—
32C4×C4⋊Dic53C2211yescharacteristic
33#16810385C2212yes—
34C22×#41×C53C2211yescharacteristic
35C23×Q8×C53C2211yescharacteristic
36C53×#548—24no—
37C53×#548—24no—
38C2×#124×C53—21no—
39C2×#124×C53C2211yescharacteristic
40C2×C22⋊Q8×C53—21no—
41C2×C22⋊Q8×C53C2211yescharacteristic
42C2×#121×C53—21no—
43C2×#121×C53C2211yescharacteristic
44C2×C4×Q8×C53C2211yescharacteristic
45C2×C4×D4×C53—21no—
46C2×C4×D4×C53C2211yescharacteristic
47C22×Dic106—21no—
48C22×#88322C2312yes—
49C22×#88322C2312yes—
50#13133217—22no—
51#13133217—22no—
52C2×C4⋊Dic53—21no—
53C2×C4⋊Dic53C2312yes—
54C2×C4⋊Dic53C2312yes—
55C2×C22⋊Dic53—22no—
56C2×C22⋊Dic53—22no—
57C2×C4×Dic53—21no—
58#13133148—24no—
59C4⋊#88322—22no—
60#13133141—24no—
61#13133141—24no—
62#13133141—24no—
63#13133141—42no—
64#13133138—24no—
65#13133138—42no—
66C22⋊#88322—42no—
67C22⋊#88322—42no—
68#13133128—22no—
69#13133124—21no—
70#13133124—22no—
71#13133124—22no—
72C2×#41×C53—22no—
73C2×#41×C53—22no—
74C2×#41×C53—24no—
75C2×#41×C53—42no—
76C22×Q8×C53—22no—
77C22×Q8×C53—22no—
78C22×Q8×C53—21no—
79C22×Q8×C53—24no—
80C22×Q8×C53—42no—
81C22×Q8×C53C2311yescharacteristic
82C22×Q8×C53C2311yescharacteristic
83C22×D4×C53—21no—
84C22×D4×C53—21no—
85C22×D4×C53D411yescharacteristic
86C22×D4×C53C2311yescharacteristic
87C23×C4×C53—21no—
88C23×C4×C53—21no—
89C23×C4×C53C2311yescharacteristic
90C23×C4×C53C2311yescharacteristic
91#124×C53—28no—
92#124×C53—22no—
93#124×C53D414yes—
94C22⋊Q8×C53—28no—
95C22⋊Q8×C53—22no—
96C22⋊Q8×C53—22no—
97#121×C53—28no—
98#121×C53—22no—
99#121×C53—22no—
100C4×Q8×C53—22no—
101C4×Q8×C53D414yes—
102C4×D4×C53—28no—
103C4×D4×C53—22no—
104C4×D4×C53—22no—
105C2×C4⋊C4×C53—21no—
106C2×C4⋊C4×C53—21no—
107C2×C4⋊C4×C53D411yescharacteristic
108C2×C4⋊C4×C53C2311yescharacteristic
109C2×C22⋊C4×C53—21no—
110C2×C22⋊C4×C53—21no—
111C2×C22⋊C4×C53—21no—
112C2×C22⋊C4×C53—21no—
113C2×C22⋊C4×C53D411yescharacteristic
114C2×C22⋊C4×C53D411yescharacteristic
115C2×C42×C53—21no—
116C2×C42×C53C2311yescharacteristic
117C24×C53—21no—
118C24×C53C2×D411yescharacteristic
119C24×C53C2×D411yescharacteristic
120#41×C53—48no—
121#41×C53—44no—
122#41×C53—44no—
123C2×Q8×C53—24no—
124C2×Q8×C53—21no—
125C2×Q8×C53—21no—
126C2×Q8×C53—22no—
127C2×Q8×C53—22no—
128C2×Q8×C53—22no—
129C2×Q8×C53—44no—
130C2×Q8×C53—48no—
131C2×Q8×C53—44no—
132C2×Q8×C53—44no—
133C2×Q8×C53C2×D412yes—
134C2×Q8×C53C2×D412yes—
135C2×Q8×C53#4114yes—
136C2×D4×C53—24no—
137C2×D4×C53—24no—
138C2×D4×C53—22no—
139C2×D4×C53—22no—
140C2×D4×C53—21no—
141C2×D4×C53—22no—
142C2×D4×C53—22no—
143C2×D4×C53—41no—
144C2×D4×C53—44no—
145C2×D4×C53—42no—
146C2×D4×C53—42no—
147C2×D4×C53C2×D412yes—
148C22×C4×C53—21no—
149C22×C4×C53—24no—
150C22×C4×C53—24no—
151C22×C4×C53—21no—
152C22×C4×C53—21no—
153C22×C4×C53—21no—
154C22×C4×C53—21no—
155C22×C4×C53—21no—
156C22×C4×C53—21no—
157C22×C4×C53—22no—
158C22×C4×C53—22no—
159C22×C4×C53—22no—
160C22×C4×C53—22no—
161C22×C4×C53—44no—
162C22×C4×C53—44no—
163C22×C4×C53C2×D411yescharacteristic
164C22×C4×C53C2×D411yescharacteristic
165C22×C4×C53C2×D411yescharacteristic
166C22×C4×C53C2×D411yescharacteristic
167C22×C4×C53C2411yescharacteristic
168C22×Dic53—22no—
169C22×Dic53—22no—
170C2×Dic106—42no—
171C2×Dic106—82no—
172C4⋊C4×C53—24no—
173C4⋊C4×C53—24no—
174C4⋊C4×C53—22no—
175C4⋊C4×C53—21no—
176C4⋊C4×C53C2×D412yes—
177C22⋊C4×C53—24no—
178C22⋊C4×C53—24no—
179C22⋊C4×C53—24no—
180C22⋊C4×C53—24no—
181C22⋊C4×C53—22no—
182C22⋊C4×C53—22no—
183C42×C53—24no—
184C42×C53C2×D412yes—
185C42×C53C2×D412yes—
186C22⋊Dic53—82no—
187C22⋊Dic53—82no—
188C4⋊Dic53—22no—
189C4⋊Dic53—22no—
190C4⋊Dic53—22no—
191C4⋊Dic53—22no—
192C4⋊Dic53—42no—
193C4×Dic53—42no—
194C2×#88322—22no—
195C2×#88322—22no—
196C2×#88322—22no—
197C2×#88322—22no—
198C2×#88322—22no—
199C2×#88322—22no—
200C23×C53—21no—
201C23×C53—22no—
202C23×C53—21no—
203C23×C53—22no—
204C23×C53—21no—
205C23×C53—22no—
206C23×C53—22no—
207C23×C53—21no—
208C23×C53—21no—
209C23×C53—22no—
210C23×C53—42no—
211C23×C53—41no—
212C23×C53—41no—
213C23×C53—42no—
214C23×C53#13712yes—
215C23×C53C22×D411yescharacteristic
216Q8×C53—22no—
217Q8×C53—24no—
218Q8×C53—24no—
219Q8×C53—48no—
220Q8×C53—41no—
221Q8×C53—41no—
222Q8×C53#12414yes—
223D4×C53—24no—
224D4×C53—44no—
225D4×C53—48no—
226D4×C53—48no—
227D4×C53—41no—
228D4×C53—41no—
229C2×C4×C53—22no—
230C2×C4×C53—24no—
231C2×C4×C53—21no—
232C2×C4×C53—21no—
233C2×C4×C53—22no—
234C2×C4×C53—21no—
235C2×C4×C53—22no—
236C2×C4×C53—21no—
237C2×C4×C53—22no—
238C2×C4×C53—22no—
239C2×C4×C53—22no—
240C2×C4×C53—48no—
241C2×C4×C53—48no—
242C2×C4×C53—42no—
243C2×C4×C53—41no—
244C2×C4×C53—41no—
245C2×C4×C53—41no—
246C2×C4×C53—41no—
247C2×C4×C53—41no—
248C2×C4×C53—41no—
249C2×C4×C53—42no—
250C2×C4×C53—42no—
251C2×C4×C53#12012yes—
252C2×C4×C53#12012yes—
253C2×C4×C53C22×D411yescharacteristic
254C2×C4×C53C22×D411yescharacteristic
255C2×C4×C53C2×#4112yes—
256C2×C4×C53H(5,𝔽2)12yes—
257C2×Dic53—42no—
258C2×Dic53—42no—
259C2×Dic53—42no—
260C2×Dic53—42no—
261C2×Dic53—42no—
262C2×Dic53—42no—
263Dic106—84no—
264#88322—42no—
265#88322—42no—
266#25895—531nomaximal, Sylow
267C22×C53—22no—
268C22×C53—24no—
269C22×C53—41no—
270C22×C53—41no—
271C22×C53—42no—
272C22×C53—42no—
273C22×C53—41no—
274C22×C53—41no—
275C22×C53—42no—
276C22×C53—41no—
277C22×C53—42no—
278C22×C53—42no—
279C22×C53—42no—
280C22×C53—82no—
281C22×C53—82no—
282C22×C53C2×#12011yescharacteristic
283C22×C53#54612yes—
284C22×C53C2×#13711yescharacteristic
285C22×C53#57511yescharacteristic
286C22×C53#57712yes—
287C4×C53—24no—
288C4×C53—22no—
289C4×C53—22no—
290C4×C53—41no—
291C4×C53—41no—
292C4×C53—42no—
293C4×C53—42no—
294C4×C53C2×#12411yescharacteristic
295C4×C53#53511yescharacteristic
296C4×C53#54612yes—
297Dic53—82no—
298Dic53—82no—
299C2×#548D5311yescharacteristic
300C2×#489—531no—
301C2×#475—531no—
302#2563—531no—
303#2035—532no—
304#2034—532no—
305#1983—531no—
306#1955—531no—
307#1788—532no—
308#1787—532no—
309#1763—531no—
310C2×C53—42no—
311C2×C53—81no—
312C2×C53—82no—
313C2×C53#243511yescharacteristic
314C2×C53#300711yescharacteristic
315C2×C53#301011yescharacteristic
316C2×C53#331612yes—
317C2×C53#332112yes—
318C22×#41C2×D5311yescharacteristic
319C23×Q8C2×D5311yescharacteristic
320#548—24no—
321#548—24no—
322C2×C4⋊Q8—531no—
323C2×#124—21no—
324C2×#124C2×D5311yescharacteristic
325C2×C22⋊Q8—21no—
326C2×C22⋊Q8C2×D5311yescharacteristic
327C2×#121—21no—
328C2×#121C2×D5311yescharacteristic
329C2×C4×Q8C2×D5311yescharacteristic
330C2×C4×D4—21no—
331C2×C4×D4C2×D5311yescharacteristic
332#489—1064no—
333#475—1064no—
334C2×#106—532no—
335C2×C4⋊C8—531no—
336C2×#103—532no—
337C2×#103—532no—
338C2×#103—532no—
339C2×#103—532no—
340C2×#102—532no—
341C2×#102—532no—
342C2×C22⋊C8—532no—
343C2×C22⋊C8—532no—
344C2×#97—531no—
345#393—1061no—
346#393—1061no—
347#390—1061no—
348#382—1061no—
349#380—532no—
350#380—532no—
351C4×C4⋊C4—531no—
352#340—532no—
353C53#2589511yescharacteristic, complemented, Sylow
354C2×#41—22no—
355C2×#41—22no—
356C2×#41—24no—
357C2×#41—42no—
358C22×Q8—22no—
359C22×Q8—22no—
360C22×Q8—21no—
361C22×Q8—24no—
362C22×Q8—42no—
363C22×Q8—1061no—
364C22×Q8C22×D5311yescharacteristic
365C22×Q8C22×D5311yescharacteristic
366C22×D4—21no—
367C22×D4—21no—
368C22×D4C22⋊D5311yescharacteristic
369C22×D4C22×D5311yescharacteristic
370C23×C4—21no—
371C23×C4—21no—
372C23×C4C22×D5311yescharacteristic
373C23×C4C22×D5311yescharacteristic
374C22×C8—532no—
375C22×C8—532no—
376C4⋊Q8—1062no—
377C4⋊Q8—1062no—
378#124—28no—
379#124—22no—
380#124C22⋊D5314yes—
381C22⋊Q8—28no—
382C22⋊Q8—22no—
383C22⋊Q8—22no—
384#121—28no—
385#121—22no—
386#121—22no—
387C4×Q8—22no—
388C4×Q8C22⋊D5314yes—
389C4×D4—28no—
390C4×D4—22no—
391C4×D4—22no—
392C2×C4⋊C4—21no—
393C2×C4⋊C4—21no—
394C2×C4⋊C4—532no—
395C2×C4⋊C4—532no—
396C2×C4⋊C4—1061no—
397C2×C4⋊C4C22⋊D5311yescharacteristic
398C2×C4⋊C4C22×D5311yescharacteristic
399C2×C22⋊C4—21no—
400C2×C22⋊C4—21no—
401C2×C22⋊C4—21no—
402C2×C22⋊C4—21no—
403C2×C22⋊C4—1062no—
404C2×C22⋊C4—1062no—
405C2×C22⋊C4C22⋊D5311yescharacteristic
406C2×C22⋊C4C22⋊D5311yescharacteristic
407C2×C42—21no—
408C2×C42—1061no—
409C2×C42C22×D5311yescharacteristic
410#106—1064no—
411C4⋊C8—1062no—
412#103—1064no—
413#103—1064no—
414#103—1064no—
415#103—2122no—
416#102—1064no—
417#102—2122no—
418C22⋊C8—2122no—
419C22⋊C8—2122no—
420#97—1062no—
421#95—1061no—
422#95—1062no—
423#95—1062no—
424C24—21no—
425C24C2×C22⋊D5311yescharacteristic
426C24C2×C22⋊D5311yescharacteristic
427#41—48no—
428#41—44no—
429#41—44no—
430C2×Q8—24no—
431C2×Q8—21no—
432C2×Q8—21no—
433C2×Q8—22no—
434C2×Q8—22no—
435C2×Q8—22no—
436C2×Q8—44no—
437C2×Q8—48no—
438C2×Q8—44no—
439C2×Q8—44no—
440C2×Q8—4242no—
441C2×Q8—2122no—
442C2×Q8#1171994714yes—
443C2×Q8C2×C22⋊D5312yes—
444C2×Q8C2×C22⋊D5312yes—
445C2×D4—24no—
446C2×D4—24no—
447C2×D4—22no—
448C2×D4—22no—
449C2×D4—21no—
450C2×D4—22no—
451C2×D4—22no—
452C2×D4—41no—
453C2×D4—44no—
454C2×D4—42no—
455C2×D4—42no—
456C2×D4C2×C22⋊D5312yes—
457C22×C4—21no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—44no—
471C22×C4—44no—
472C22×C4—1062no—
473C22×C4—1062no—
474C22×C4C2×C22⋊D5311yescharacteristic
475C22×C4C2×C22⋊D5311yescharacteristic
476C22×C4C2×C22⋊D5311yescharacteristic
477C22×C4C2×C22⋊D5311yescharacteristic
478C22×C4C23×D5311yescharacteristic
479C2×C8—1062no—
480C2×C8—1062no—
481C2×C8—1062no—
482C2×C8—1062no—
483C2×C8—1062no—
484C2×C8—1062no—
485C4⋊C4—24no—
486C4⋊C4—24no—
487C4⋊C4—22no—
488C4⋊C4—21no—
489C4⋊C4—1062no—
490C4⋊C4—1062no—
491C4⋊C4—1062no—
492C4⋊C4—1062no—
493C4⋊C4—2122no—
494C4⋊C4C2×C22⋊D5312yes—
495C22⋊C4—24no—
496C22⋊C4—24no—
497C22⋊C4—24no—
498C22⋊C4—24no—
499C22⋊C4—22no—
500C22⋊C4—22no—
501C22⋊C4—4242no—
502C22⋊C4—4242no—
503C42—24no—
504C42—2122no—
505C42C2×C22⋊D5312yes—
506C42C2×C22⋊D5312yes—
507C23—22no—
508C23—21no—
509C23—21no—
510C23—22no—
511C23—22no—
512C23—21no—
513C23—22no—
514C23—21no—
515C23—21no—
516C23—22no—
517C23—42no—
518C23—41no—
519C23—41no—
520C23—42no—
521C23#1313324712yes—
522C23C22×C22⋊D5311yescharacteristic
523Q8—24no—
524Q8—22no—
525Q8—24no—
526Q8—48no—
527Q8—41no—
528Q8—41no—
529Q8—4244no—
530Q8#1313320414yes—
531D4—24no—
532D4—44no—
533D4—48no—
534D4—48no—
535D4—41no—
536D4—41no—
537C2×C4—24no—
538C2×C4—22no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—48no—
549C2×C4—48no—
550C2×C4—42no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—2122no—
560C2×C4—2122no—
561C2×C4—2122no—
562C2×C4—2122no—
563C2×C4—2122no—
564C2×C4—2122no—
565C2×C4#1313318612yes—
566C2×C4#1313318612yes—
567C2×C4C22×C22⋊D5311yescharacteristic
568C2×C4C22×C22⋊D5311yescharacteristic
569C2×C4C2×#1171994712yes—
570C2×C4#1313326212yes—
571C8—2122no—
572C8—2122no—
573C22—22no—
574C22—24no—
575C22—41no—
576C22—41no—
577C22—42no—
578C22—41no—
579C22—42no—
580C22—42no—
581C22—41no—
582C22—41no—
583C22—42no—
584C22—42no—
585C22—42no—
586C22—82no—
587C22—82no—
588C22C2×#1313318611yescharacteristic
589C22#1681126712yes—
590C22C2×#1313324711yescharacteristic
591C22#1681142111yescharacteristic
592C22#1681143112yes—
593C4—22no—
594C4—24no—
595C4—22no—
596C4—42no—
597C4—41no—
598C4—41no—
599C4—42no—
600C4—4242no—
601C4—4242no—
602C4C2×#1313320411yescharacteristic
603C4#1681119011yescharacteristic
604C4#1681126712yes—
605C2—42no—
606C2—81no—
607C2—82no—
608C2#2027836211yescharacteristic
609C2#2028397511yescharacteristic
610C2#2028400111yescharacteristic
611C2#2028682612yes—
612C2#2028690112yes—
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
Fitting, prime radicalC2×C53×#548C2
socleC23×C53C22×D4
hypercenterC2×#548D53
hypocenterC53#25895
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#548

Complements

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

NH ≅ G/Ntype
1. C2×#16810982C2semidirect
2. C2×#16810908C2semidirect
3. #20279888C2semidirect
4. #20275908C2semidirect
5. #20275901C2semidirect
6. #20275648C2semidirect
7. #20275514C2semidirect
8. #20274655C2semidirect
9. #20274650C2semidirect
10. #20274545C2semidirect
353. C53#25895semidirect

Maximal subgroups

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

subgroupindex
C2×#168109822
C2×#168109082
#202798882
#202759082
#202759012
#202756482
#202755142
#202746552
#202746502
#202745452
C2×C53×#5482
#2589553