Finite Group Databasebeta

← #27435887

Subgroups — #27435887

Order 13568.

756 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811029C211yesmaximal, characteristic, complemented
2C2×#16810923C211yesmaximal, characteristic
3#20277809C211yesmaximal, characteristic, complemented
4#20276197C211yesmaximal, characteristic, complemented
5#20276197C211yesmaximal, characteristic, complemented
6#20276103C211yesmaximal, characteristic
7#20276089C211yesmaximal, characteristic
8#20275668C211yesmaximal, characteristic, complemented
9#20275481C211yesmaximal, characteristic, complemented
10#20275481C211yesmaximal, characteristic, complemented
11#20275439C211yesmaximal, characteristic, complemented
12#20275339C211yesmaximal, characteristic
13#20275332C211yesmaximal, characteristic
14#20274540C211yesmaximal, characteristic
15C53×#2563C211yesmaximal, characteristic
16C2×#13133209C2211yescharacteristic
17#16811029—24no—
18#16810923—24no—
19C2×#13133150C2211yescharacteristic
20C2×#13133150C2211yescharacteristic
21C2×#13133150C2211yescharacteristic
22C2×#13133148C2211yescharacteristic
23C2×#13133148C2211yescharacteristic
24C2×#13133148C2211yescharacteristic
25C2×C4⋊#88322C2211yescharacteristic
26C2×C4⋊#88322C2211yescharacteristic
27C2×C4⋊#88322C2211yescharacteristic
28C2×#13133138C2211yescharacteristic
29C2×#13133138C2211yescharacteristic
30C2×#13133138C2211yescharacteristic
31C2×#13133138C2211yescharacteristic
32C2×#13133128C2211yescharacteristic
33#16810521C2211yescharacteristic
34#16810521C2211yescharacteristic
35#16810518—21no—
36#16810505—21no—
37#16810504—21no—
38C4×C4⋊Dic53C2211yescharacteristic
39C4×C4⋊Dic53C2211yescharacteristic
40C4×C4⋊Dic53C2211yescharacteristic
41#16810474—21no—
42#16810385C2211yescharacteristic
43#16810385C2211yescharacteristic
44#16810385C2211yescharacteristic
45#16810385C2211yescharacteristic
46#16810385C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810378C2211yescharacteristic
49#16810378C2211yescharacteristic
50C2×C4⋊Q8×C53C2211yescharacteristic
51C2×C4×D4×C53C2211yescharacteristic
52C53×#393—21no—
53C53×#393—21no—
54C53×#390C2211yescharacteristic
55C53×#390C2211yescharacteristic
56C53×#382C2211yescharacteristic
57C53×#382C2211yescharacteristic
58C53×#380—21no—
59C53×#380—21no—
60C4×C4⋊C4×C53C2211yescharacteristic
61C22×#88322C2311yescharacteristic
62C22×#88322C2311yescharacteristic
63C22×#88322C2311yescharacteristic
64C22×#88322C2311yescharacteristic
65#13133209—22no—
66#13133209D414yes—
67C2×C4⋊Dic53C2311yescharacteristic
68C2×C4⋊Dic53C2311yescharacteristic
69C2×C4⋊Dic53C2311yescharacteristic
70C2×C4⋊Dic53C2311yescharacteristic
71C2×#11719919—21no—
72C2×#11719919—21no—
73C2×#11719919—21no—
74C2×#11719919—21no—
75C2×C4×Dic53—21no—
76C2×C4×Dic53—21no—
77C2×C4×Dic53—21no—
78#13133150—22no—
79#13133150—22no—
80#13133150—22no—
81#13133148—22no—
82#13133148—22no—
83#13133148—22no—
84C4⋊#88322—22no—
85C4⋊#88322—22no—
86C4⋊#88322—22no—
87#13133138—22no—
88#13133138—22no—
89#13133138—22no—
90#13133138Q814yescomplemented
91#13133128—22no—
92#13133124—21no—
93#13133124—21no—
94#13133124—21no—
95#13133124—21no—
96#13133124—21no—
97C22×Q8×C53—21no—
98C22×D4×C53C2311yescharacteristic
99C23×C4×C53—21no—
100C4⋊Q8×C53—22no—
101C4⋊Q8×C53—22no—
102C4×D4×C53—22no—
103C4×D4×C53—22no—
104C2×C4⋊C4×C53—21no—
105C2×C4⋊C4×C53D411yescharacteristic
106C2×C4⋊C4×C53D411yescharacteristic
107C2×C4⋊C4×C53C2311yescharacteristic
108C2×C4⋊C4×C53C2311yescharacteristic
109C2×C4⋊C4×C53C2311yescharacteristic
110C2×C22⋊C4×C53—21no—
111C2×C22⋊C4×C53—21no—
112C2×C22⋊C4×C53—21no—
113C2×C22⋊C4×C53—21no—
114C2×C22⋊C4×C53—21no—
115C2×C42×C53C2311yescharacteristic
116C2×C42×C53C2311yescharacteristic
117C2×C42×C53C2311yescharacteristic
118#95×C53—21no—
119#95×C53—21no—
120#95×C53—21no—
121#95×C53—21no—
122#95×C53D411yescharacteristic
123#95×C53D411yescharacteristic
124C24×C53—21no—
125C2×Q8×C53—42no—
126C2×Q8×C53—81no—
127C2×Q8×C53—81no—
128C2×D4×C53—21no—
129C2×D4×C53—21no—
130C2×D4×C53—21no—
131C2×D4×C53—21no—
132C2×D4×C53—21no—
133C2×D4×C53C2×Q812yes—
134C22×C4×C53—21no—
135C22×C4×C53—21no—
136C22×C4×C53—21no—
137C22×C4×C53—42no—
138C22×C4×C53—42no—
139C22×C4×C53—42no—
140C22×C4×C53C2×D411yescharacteristic
141C22×C4×C53C2×D411yescharacteristic
142C22×C4×C53C2×D411yescharacteristic
143C22×C4×C53C2×D411yescharacteristic
144C22×C4×C53C2×D411yescharacteristic
145C22×C4×C53C2×D411yescharacteristic
146C22×C4×C53C2411yescharacteristic
147C22×Dic53—21no—
148C22×Dic53—21no—
149C22×Dic53—21no—
150C22×Dic53—21no—
151C4⋊C4×C53—21no—
152C4⋊C4×C53—21no—
153C4⋊C4×C53—21no—
154C4⋊C4×C53—21no—
155C4⋊C4×C53—21no—
156C4⋊C4×C53—21no—
157C4⋊C4×C53—41no—
158C4⋊C4×C53—41no—
159C4⋊C4×C53—42no—
160C22⋊C4×C53—42no—
161C22⋊C4×C53—81no—
162C22⋊C4×C53—81no—
163C22⋊C4×C53—81no—
164C22⋊C4×C53—81no—
165C42×C53—21no—
166C42×C53—21no—
167C42×C53—21no—
168C42×C53—21no—
169C42×C53C2×D412yes—
170C42×C53#4112yes—
171C4⋊Dic53—21no—
172C4⋊Dic53—21no—
173C4⋊Dic53—21no—
174C4⋊Dic53—21no—
175C4⋊Dic53—21no—
176C4⋊Dic53C2×D412yes—
177C4⋊Dic53C2×D412yes—
178C4⋊Dic53C2×Q812yes—
179#11719919—24no—
180#11719919—24no—
181#11719919—42no—
182#11719919—81no—
183C4×Dic53—42no—
184C4×Dic53—81no—
185C4×Dic53—81no—
186C2×#88322—21no—
187C2×#88322—21no—
188C2×#88322—21no—
189C2×#88322—21no—
190C2×#88322—21no—
191C2×#88322—21no—
192C2×#88322—21no—
193C2×#88322—21no—
194C2×#88322—21no—
195C2×#88322—21no—
196C2×#88322C2×Q812yes—
197C2×#88322#4112yes—
198C23×C53—22no—
199C23×C53—41no—
200C23×C53—41no—
201C23×C53—41no—
202C23×C53—41no—
203C23×C53—41no—
204C23×C53—41no—
205C23×C53C22×D411yescharacteristic
206Q8×C53—84no—
207D4×C53—22no—
208D4×C53—22no—
209D4×C53—41no—
210D4×C53—41no—
211C2×C4×C53—21no—
212C2×C4×C53—21no—
213C2×C4×C53—21no—
214C2×C4×C53—21no—
215C2×C4×C53—21no—
216C2×C4×C53—21no—
217C2×C4×C53—21no—
218C2×C4×C53—21no—
219C2×C4×C53—21no—
220C2×C4×C53—21no—
221C2×C4×C53—21no—
222C2×C4×C53—21no—
223C2×C4×C53—21no—
224C2×C4×C53—21no—
225C2×C4×C53—21no—
226C2×C4×C53—21no—
227C2×C4×C53—41no—
228C2×C4×C53—41no—
229C2×C4×C53—41no—
230C2×C4×C53—41no—
231C2×C4×C53—41no—
232C2×C4×C53—41no—
233C2×C4×C53—41no—
234C2×C4×C53—41no—
235C2×C4×C53—41no—
236C2×C4×C53—84no—
237C2×C4×C53#12112yes—
238C2×C4×C53#12112yes—
239C2×C4×C53C22×D411yescharacteristic
240C2×C4×C53C22×Q811yescharacteristic
241C2×C4×C53C2×#4111yescharacteristic
242C2×C4×C53C2×#4111yescharacteristic
243C2×C4×C532−1+411yescharacteristic
244C2×C4×C532−1+411yescharacteristic
245C2×Dic53—22no—
246C2×Dic53—22no—
247C2×Dic53—22no—
248C2×Dic53—41no—
249C2×Dic53—41no—
250C2×Dic53—41no—
251C2×Dic53—41no—
252C2×Dic53—41no—
253C2×Dic53—41no—
254C2×Dic53—41no—
255C2×Dic53—41no—
256C2×Dic53—41no—
257#88322—22no—
258#88322—22no—
259#88322—41no—
260#88322—41no—
261#25466—531nomaximal, Sylow
262C22×C53—42no—
263C22×C53—42no—
264C22×C53—42no—
265C22×C53—81no—
266C22×C53—81no—
267C22×C53—81no—
268C22×C53—81no—
269C22×C53C2×#12111yescharacteristic
270C22×C53#54711yescharacteristic
271C22×C53D4×Q811yescharacteristic
272C22×C53#57511yescharacteristic
273C22×C53#57511yescharacteristic
274C22×C53#57611yescharacteristic
275C22×C53#57711yescharacteristic
276C4×C53—22no—
277C4×C53—22no—
278C4×C53—41no—
279C4×C53—41no—
280C4×C53—41no—
281C4×C53—41no—
282C4×C53—81no—
283C4×C53—81no—
284C4×C53—81no—
285C4×C53#53311yescharacteristic
286C4×C53#54111yescharacteristic
287C4×C53#54811yescharacteristic
288C4×C53D4×Q811yescharacteristic
289Dic53—42no—
290Dic53—42no—
291Dic53—42no—
292Dic53—81no—
293C2×#501—531no—
294C2×#478—531no—
295#2563D5311yescharacteristic
296#2370—531no—
297#2085—531no—
298#2085—531no—
299#2067—531no—
300#2065—531no—
301#1988—531no—
302#1949—531no—
303#1949—531no—
304#1940—531no—
305#1926—531no—
306#1925—531no—
307#1762—531no—
308C2×C53—82no—
309C2×C53#242911yescharacteristic
310C2×C53#304011yescharacteristic
311C2×C53#304511yescharacteristic
312C2×C53#334711yescharacteristic
313C2×C53#334811yescharacteristic
314C2×C53#338811yescharacteristic
315C2×C53#338911yescharacteristic
316C2×C4⋊Q8C2×D5311yescharacteristic
317C2×#125—531no—
318C2×C4×D4C2×D5311yescharacteristic
319#501—1064no—
320#478—1064no—
321C2×#107—531no—
322C2×#107—531no—
323C2×#107—531no—
324C2×#106—531no—
325C2×#106—531no—
326C2×#106—531no—
327C2×C4⋊C8—531no—
328C2×C4⋊C8—531no—
329C2×C4⋊C8—531no—
330C2×#102—531no—
331C2×#102—531no—
332C2×#102—531no—
333C2×#102—531no—
334C2×#97—531no—
335#393—21no—
336#393—21no—
337#390C2×D5311yescharacteristic
338#390C2×D5311yescharacteristic
339#389—531no—
340#389—531no—
341#389—1061no—
342#387—1061no—
343#387—1061no—
344#382C2×D5311yescharacteristic
345#382C2×D5311yescharacteristic
346#380—21no—
347#380—21no—
348C4×C4⋊C4—531no—
349C4×C4⋊C4—531no—
350C4×C4⋊C4—531no—
351C4×C4⋊C4—1061no—
352C4×C4⋊C4C2×D5311yescharacteristic
353#340—531no—
354#340—531no—
355#340—531no—
356#340—531no—
357#340—531no—
358#340—531no—
359#336—531no—
360#336—531no—
361C53#2546611yescharacteristic, complemented, Sylow
362C22×Q8—21no—
363C22×D4C22×D5311yescharacteristic
364C23×C4—21no—
365C22×C8—531no—
366C22×C8—531no—
367C22×C8—531no—
368C22×C8—531no—
369C4⋊Q8—22no—
370C4⋊Q8—22no—
371#125—534no—
372#125—1062no—
373C4×D4—22no—
374C4×D4—22no—
375C2×C4⋊C4—21no—
376C2×C4⋊C4—531no—
377C2×C4⋊C4—531no—
378C2×C4⋊C4—531no—
379C2×C4⋊C4—531no—
380C2×C4⋊C4—1061no—
381C2×C4⋊C4—1061no—
382C2×C4⋊C4—1061no—
383C2×C4⋊C4—1061no—
384C2×C4⋊C4C22⋊D5311yescharacteristic
385C2×C4⋊C4C22⋊D5311yescharacteristic
386C2×C4⋊C4C22×D5311yescharacteristic
387C2×C4⋊C4C22×D5311yescharacteristic
388C2×C4⋊C4C22×D5311yescharacteristic
389C2×C22⋊C4—21no—
390C2×C22⋊C4—21no—
391C2×C22⋊C4—21no—
392C2×C22⋊C4—21no—
393C2×C22⋊C4—21no—
394C2×C42—1061no—
395C2×C42—1061no—
396C2×C42—1061no—
397C2×C42C22×D5311yescharacteristic
398C2×C42C22×D5311yescharacteristic
399C2×C42C22×D5311yescharacteristic
400#107—1062no—
401#107—1062no—
402#107—1062no—
403#106—1062no—
404#106—1062no—
405#106—1062no—
406C4⋊C8—1062no—
407C4⋊C8—1062no—
408C4⋊C8—1062no—
409#102—534no—
410#102—1062no—
411#102—1062no—
412#102—1062no—
413#97—1062no—
414#95—21no—
415#95—21no—
416#95—21no—
417#95—21no—
418#95—1061no—
419#95—1061no—
420#95—1061no—
421#95—1061no—
422#95—1061no—
423#95C22⋊D5311yescharacteristic
424#95C22⋊D5311yescharacteristic
425C24—21no—
426C2×Q8—42no—
427C2×Q8—81no—
428C2×Q8—81no—
429C2×D4—21no—
430C2×D4—21no—
431C2×D4—21no—
432C2×D4—21no—
433C2×D4—21no—
434C2×D4Q8×D5312yes—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—1061no—
442C22×C4—1061no—
443C22×C4—1061no—
444C22×C4—1061no—
445C22×C4C2×C22⋊D5311yescharacteristic
446C22×C4C2×C22⋊D5311yescharacteristic
447C22×C4C2×C22⋊D5311yescharacteristic
448C22×C4C2×C22⋊D5311yescharacteristic
449C22×C4C2×C22⋊D5311yescharacteristic
450C22×C4C2×C22⋊D5311yescharacteristic
451C22×C4C23×D5311yescharacteristic
452C2×C8—532no—
453C2×C8—532no—
454C2×C8—1061no—
455C2×C8—1061no—
456C2×C8—1061no—
457C2×C8—1061no—
458C2×C8—1061no—
459C2×C8—1061no—
460C2×C8—1061no—
461C2×C8—1061no—
462C2×C8—1061no—
463C2×C8—1061no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—41no—
471C4⋊C4—41no—
472C4⋊C4—42no—
473C4⋊C4—532no—
474C4⋊C4—532no—
475C4⋊C4—532no—
476C4⋊C4—1064no—
477C4⋊C4—1064no—
478C4⋊C4—1061no—
479C4⋊C4—1061no—
480C4⋊C4—1061no—
481C4⋊C4—1061no—
482C4⋊C4—1061no—
483C4⋊C4—4241no—
484C4⋊C4—2122no—
485C22⋊C4—42no—
486C22⋊C4—81no—
487C22⋊C4—81no—
488C22⋊C4—81no—
489C22⋊C4—81no—
490C42—21no—
491C42—21no—
492C42—21no—
493C42—21no—
494C42—4241no—
495C42—4241no—
496C42—2122no—
497C42D4×D5312yes—
498C42#1171994712yes—
499C23—22no—
500C23—41no—
501C23—41no—
502C23—41no—
503C23—41no—
504C23—41no—
505C23—41no—
506C23C22×C22⋊D5311yescharacteristic
507Q8—84no—
508D4—22no—
509D4—22no—
510D4—41no—
511D4—41no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—84no—
538C2×C4—1062no—
539C2×C4—1062no—
540C2×C4—1062no—
541C2×C4—2121no—
542C2×C4—2121no—
543C2×C4—2121no—
544C2×C4—2121no—
545C2×C4—2121no—
546C2×C4—2121no—
547C2×C4—2121no—
548C2×C4—2121no—
549C2×C4—2121no—
550C2×C4#1313318812yes—
551C2×C4#1313318812yes—
552C2×C4C2×D4×D5311yescharacteristic
553C2×C4C2×Q8×D5311yescharacteristic
554C2×C4#41×D5311yescharacteristic
555C2×C4C2×#1171994711yescharacteristic
556C2×C4#1313326411yescharacteristic
557C2×C4#1313326511yescharacteristic
558C8—1062no—
559C8—1062no—
560C8—2121no—
561C8—2121no—
562C22—42no—
563C22—42no—
564C22—42no—
565C22—81no—
566C22—81no—
567C22—81no—
568C22—81no—
569C22C2×#1313318811yescharacteristic
570C22#1681127211yescharacteristic
571C22Q8×C22⋊D5311yescharacteristic
572C22#1681142111yescharacteristic
573C22#1681142111yescharacteristic
574C22#1681142611yescharacteristic
575C22#1681143411yescharacteristic
576C4—22no—
577C4—22no—
578C4—41no—
579C4—41no—
580C4—41no—
581C4—41no—
582C4—81no—
583C4—81no—
584C4—81no—
585C4—4241no—
586C4—2122no—
587C4—2122no—
588C4—2122no—
589C4#1681117511yescharacteristic
590C4#1681123311yescharacteristic
591C4#1681128311yescharacteristic
592C4#1681128611yescharacteristic
593C2—82no—
594C2#2027831011yescharacteristic
595C2#2028420511yescharacteristic
596C2#2028425011yescharacteristic
597C2#2028721511yescharacteristic
598C2#2028723011yescharacteristic
599C2#2028766011yescharacteristic
600C2#2028766911yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2563C2
socleC23×C53C22×D4
prime radicalC22×D4×C53C23
hypercenter#2563D53
hypocenterC53#25466
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2563

Complements

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

NH ≅ G/Ntype
1. C2×#16811029C2semidirect
3. #20277809C2semidirect
4. #20276197C2semidirect
5. #20276197C2semidirect
8. #20275668C2semidirect
9. #20275481C2semidirect
10. #20275481C2semidirect
11. #20275439C2semidirect
90. #13133138Q8semidirect
361. C53#25466semidirect

Maximal subgroups

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

subgroupindex
C2×#168110292
C2×#168109232
#202778092
#202761972
#202761032
#202760892
#202756682
#202754812
#202754392
#202753392
#202753322
#202745402
C53×#25632
#2546653