Finite Group Databasebeta

← #27425455

Subgroups — #27425455

Order 13568.

746 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810683C211yesmaximal, characteristic
2#20280537C211yesmaximal, characteristic
3#20277391C211yesmaximal, characteristic
4#20276144C211yesmaximal, characteristic
5#20276139C211yesmaximal, characteristic
6#20275917C211yesmaximal, characteristic
7#20275910C211yesmaximal, characteristic
8#20275668C211yesmaximal, characteristic, complemented
9#20275485C211yesmaximal, characteristic, complemented
10#20275456C211yesmaximal, characteristic, complemented
11#20275446C211yesmaximal, characteristic, complemented
12#20275402C211yesmaximal, characteristic, complemented
13#20275388C211yesmaximal, characteristic, complemented
14#20274730C211yesmaximal, characteristic, complemented
15C2×C53×#477C211yesmaximal, characteristic, complemented
16C2×#13133193C2211yescharacteristic
17C2×#13133174C2211yescharacteristic
18#16810683—24no—
19C2×#13133151C2211yescharacteristic
20C2×#13133149C2211yescharacteristic
21C2×C4⋊#88322C2211yescharacteristic
22C2×#13133141C2211yescharacteristic
23C2×#13133141C2211yescharacteristic
24C2×#13133129C2211yescharacteristic
25C2×#13133129C2211yescharacteristic
26C2×#13133127C2211yescharacteristic
27C2×C8×Dic53C2211yescharacteristic
28#16810561—21no—
29#16810558—21no—
30#16810542—21no—
31#16810534—21no—
32#16810530C2211yescharacteristic
33#16810519C2211yescharacteristic
34#16810519C2211yescharacteristic
35#16810519C2211yescharacteristic
36#16810518—21no—
37#16810514—21no—
38#16810513—21no—
39#16810507—21no—
40#16810489C2211yescharacteristic
41#16810480C2211yescharacteristic
42#16810480C2211yescharacteristic
43#16810480C2211yescharacteristic
44C4×C4⋊Dic53C2211yescharacteristic
45#16810386C2211yescharacteristic
46#16810386C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810385C2211yescharacteristic
49#16810385C2211yescharacteristic
50#16810385C2211yescharacteristic
51#16810385C2211yescharacteristic
52#16810385C2211yescharacteristic
53C2×C4⋊Q8×C53C2211yescharacteristic
54C2×C4×Q8×C53C2211yescharacteristic
55C53×#477—24no—
56C2×#107×C53C2211yescharacteristic
57C2×#107×C53C2211yescharacteristic
58C2×C4⋊C8×C53C2211yescharacteristic
59C2×#103×C53C2211yescharacteristic
60C2×#103×C53C2211yescharacteristic
61C22×C4×D53C2311yescharacteristic
62C22×#88322C2311yescharacteristic
63C22×#88322C2311yescharacteristic
64#13133193—22no—
65#13133193—22no—
66#13133174—22no—
67#13133174—22no—
68C2×C4⋊Dic53C2311yescharacteristic
69C2×C4⋊Dic53C2311yescharacteristic
70C2×C4⋊Dic53C2311yescharacteristic
71C2×C4⋊Dic53C2311yescharacteristic
72C2×#11719919—21no—
73C2×#11719919—21no—
74C2×#11719919—21no—
75C2×#11719919—21no—
76C2×#11719919D411yescharacteristic
77C2×#11719919D411yescharacteristic
78C2×#11719921—21no—
79C2×#11719921—21no—
80C2×#11719921—21no—
81C2×#11719921—21no—
82C2×#11719921D411yescharacteristic
83C2×#11719921D411yescharacteristic
84C2×C4×Dic53—21no—
85C2×C4×Dic53C2311yescharacteristic
86#13133151—22no—
87#13133149—22no—
88C4⋊#88322—22no—
89#13133141—22no—
90#13133141—22no—
91#13133129—22no—
92#13133129—22no—
93#13133127—22no—
94C8×Dic53—22no—
95#13133124—21no—
96#13133124—21no—
97#13133124—21no—
98#13133124—21no—
99#13133124—21no—
100#13133124—21no—
101#13133124—21no—
102#13133124—21no—
103#13133124—21no—
104#13133124—21no—
105#13133124—21no—
106C22×Q8×C53—21no—
107C22×Q8×C53C2311yescharacteristic
108C22×C8×C53C2311yescharacteristic
109C22×C8×C53C2311yescharacteristic
110C4⋊Q8×C53—22no—
111C4⋊Q8×C53—22no—
112C4×Q8×C53—22no—
113C4×Q8×C53—22no—
114C2×C4⋊C4×C53—21no—
115C2×C4⋊C4×C53—21no—
116C2×C4⋊C4×C53C2311yescharacteristic
117C2×C4⋊C4×C53C2311yescharacteristic
118C2×C4⋊C4×C53C2311yescharacteristic
119C2×C42×C53—21no—
120C2×C42×C53C2311yescharacteristic
121#107×C53—22no—
122#107×C53—22no—
123C4⋊C8×C53D414yes—
124#103×C53—22no—
125#103×C53—22no—
126C23×D53C2×D411yescharacteristic
127C2×Q8×C53—21no—
128C2×Q8×C53—21no—
129C2×Q8×C53—21no—
130C2×Q8×C53—21no—
131C2×Q8×C53—21no—
132C2×Q8×C53—42no—
133C2×Q8×C53—81no—
134C2×Q8×C53—81no—
135C2×Q8×C53#4112yes—
136C22×C4×C53—21no—
137C22×C4×C53—21no—
138C22×C4×C53—21no—
139C22×C4×C53—21no—
140C22×C4×C53C2×D411yescharacteristic
141C22×C4×C53C2×D411yescharacteristic
142C22×C4×C53C2411yescharacteristic
143C22×Dic53—21no—
144C22×Dic53—21no—
145C22×Dic53—21no—
146C22×Dic53—21no—
147C22×Dic53C2×D411yescharacteristic
148C22×Dic53C2×D411yescharacteristic
149C22×Dic53C2×D411yescharacteristic
150C2×C4×D53—21no—
151C2×C4×D53—21no—
152C2×C4×D53—21no—
153C2×C4×D53—21no—
154C2×C4×D53—21no—
155C2×C4×D53—21no—
156C2×C8×C53—21no—
157C2×C8×C53—21no—
158C2×C8×C53—21no—
159C2×C8×C53—21no—
160C2×C8×C53C2×D412yes—
161C2×C8×C53C2×D412yes—
162C4⋊C4×C53—21no—
163C4⋊C4×C53—21no—
164C4⋊C4×C53—21no—
165C4⋊C4×C53—21no—
166C4⋊C4×C53—21no—
167C4⋊C4×C53—21no—
168C4⋊C4×C53—42no—
169C4⋊C4×C53—42no—
170C42×C53—42no—
171C42×C53C2×D412yes—
172C42×C53#4112yes—
173#11719921—24no—
174#11719921—22no—
175#11719921—22no—
176#11719921—81no—
177#11719921—81no—
178#11719921—81no—
179C4⋊Dic53—21no—
180C4⋊Dic53—21no—
181C4⋊Dic53—21no—
182C4⋊Dic53—21no—
183C4⋊Dic53—21no—
184C4⋊Dic53—21no—
185C4⋊Dic53—21no—
186C4⋊Dic53#4112yes—
187#11719919—24no—
188#11719919—22no—
189#11719919—22no—
190#11719919—81no—
191#11719919—81no—
192#11719919—81no—
193C4×Dic53—21no—
194C4×Dic53—21no—
195C4×Dic53—81no—
196C2×#88322—21no—
197C2×#88322—21no—
198C2×#88322—21no—
199C2×#88322—21no—
200C2×#88322—21no—
201C2×#88322—21no—
202C23×C53C22×D411yescharacteristic
203C22×D53—21no—
204C22×D53—21no—
205C22×D53—21no—
206C22×D53—21no—
207C22×D53—21no—
208C22×D53—21no—
209C22×D53—21no—
210Q8×C53—22no—
211Q8×C53—22no—
212Q8×C53—41no—
213Q8×C53—41no—
214Q8×C53—84no—
215C2×C4×C53—21no—
216C2×C4×C53—21no—
217C2×C4×C53—21no—
218C2×C4×C53—21no—
219C2×C4×C53—22no—
220C2×C4×C53—41no—
221C2×C4×C53—41no—
222C2×C4×C53—41no—
223C2×C4×C53—41no—
224C2×C4×C53—41no—
225C2×C4×C53—41no—
226C2×C4×C53—41no—
227C2×C4×C53—41no—
228C2×C4×C53—41no—
229C2×C4×C53—41no—
230C2×C4×C53—41no—
231C2×C4×C53#12112yes—
232C2×C4×C53#12112yes—
233C2×C4×C53C22×D411yescharacteristic
234C2×C4×C53C2×#4111yescharacteristic
235C2×C4×C53C2×#4111yescharacteristic
236C2×C4×C53C2×#4111yescharacteristic
237C2×C4×C532−1+411yescharacteristic
238C2×C4×C532−1+411yescharacteristic
239C2×Dic53—21no—
240C2×Dic53—21no—
241C2×Dic53—21no—
242C2×Dic53—21no—
243C2×Dic53—22no—
244C2×Dic53—21no—
245C2×Dic53—21no—
246C2×Dic53—21no—
247C2×Dic53—21no—
248C2×Dic53—21no—
249C2×Dic53—41no—
250C2×Dic53—41no—
251C2×Dic53—41no—
252C2×Dic53—41no—
253C2×Dic53—41no—
254C2×Dic53—41no—
255C2×Dic53—41no—
256C2×Dic53—41no—
257C2×Dic53—41no—
258C2×Dic53—41no—
259C2×Dic53—41no—
260C4×D53—41no—
261C4×D53—41no—
262C4×D53—41no—
263C4×D53—41no—
264C8×C53—22no—
265C8×C53—22no—
266#88322—41no—
267#88322—41no—
268#24714—531nomaximal, Sylow
269C22×C53C2×#12111yescharacteristic
270C22×C53#52511yescharacteristic
271C22×C53#53611yescharacteristic
272C22×C53#57611yescharacteristic
273C22×C53#57711yescharacteristic
274C22×C53#57811yescharacteristic
275C22×C53#57811yescharacteristic
276C2×D53—41no—
277C2×D53—41no—
278C2×D53—41no—
279C2×D53—41no—
280C2×D53—41no—
281C2×D53—41no—
282C2×D53—41no—
283C4×C53—22no—
284C4×C53—22no—
285C4×C53—42no—
286C4×C53—81no—
287C4×C53—81no—
288C4×C53—81no—
289C4×C53#54711yescharacteristic
290C4×C53#54811yescharacteristic
291C4×C53#55211yescharacteristic
292C4×C53#55611yescharacteristic
293Dic53—42no—
294Dic53—41no—
295Dic53—41no—
296Dic53—41no—
297Dic53—81no—
298Dic53—81no—
299Dic53—81no—
300C2×#477D5311yescharacteristic, complemented
301C2×#433—531no—
302#2628—531no—
303#2323—531no—
304#2076—531no—
305#2075—531no—
306#2036—531no—
307#2035—531no—
308#1988—531no—
309#1949—531no—
310#1944—531no—
311#1942—531no—
312#1935—531no—
313#1933—531no—
314#1805—531no—
315C2×C53#241511yescharacteristic
316C2×C53#305911yescharacteristic
317C2×C53#306211yescharacteristic
318C2×C53#312111yescharacteristic
319C2×C53#312411yescharacteristic
320C2×C53#324211yescharacteristic
321C2×C53#324311yescharacteristic
322D53—81no—
323C2×C4⋊Q8C2×D5311yescharacteristic
324C2×C22⋊Q8—531no—
325C2×C4×Q8C2×D5311yescharacteristic
326C2×#117—531no—
327#477—24no—
328#433—1064no—
329C2×#107—531no—
330C2×#107C2×D5311yescharacteristic
331C2×#107C2×D5311yescharacteristic
332C2×#106—531no—
333C2×C4⋊C8—531no—
334C2×C4⋊C8C2×D5311yescharacteristic
335C2×#103—531no—
336C2×#103—531no—
337C2×#103C2×D5311yescharacteristic
338C2×#103C2×D5311yescharacteristic
339C2×C22⋊C8—531no—
340C2×C22⋊C8—531no—
341C2×#97—531no—
342C2×C4×C8—531no—
343#400—1061no—
344#399—1061no—
345#395—1061no—
346#393—1061no—
347#391—531no—
348#389—531no—
349#389—531no—
350#389—531no—
351#389—1061no—
352#388—1061no—
353#388—1061no—
354#387—1061no—
355#382—531no—
356#380—531no—
357#380—531no—
358#380—531no—
359C4×C4⋊C4—531no—
360#340—531no—
361#340—531no—
362#340—531no—
363#340—531no—
364#340—531no—
365#340—531no—
366#340—531no—
367#340—531no—
368C53#2471411yescharacteristic, complemented, Sylow
369C22×Q8—21no—
370C22×Q8C22×D5311yescharacteristic
371C23×C4—531no—
372C22×C8—531no—
373C22×C8—531no—
374C22×C8C22×D5311yescharacteristic
375C22×C8C22×D5311yescharacteristic
376C4⋊Q8—22no—
377C4⋊Q8—22no—
378C22⋊Q8—1062no—
379C22⋊Q8—1062no—
380C4×Q8—22no—
381C4×Q8—22no—
382#117—1062no—
383#117—1062no—
384C2×C4⋊C4—21no—
385C2×C4⋊C4—21no—
386C2×C4⋊C4—531no—
387C2×C4⋊C4—531no—
388C2×C4⋊C4—531no—
389C2×C4⋊C4—531no—
390C2×C4⋊C4—531no—
391C2×C4⋊C4—531no—
392C2×C4⋊C4—1061no—
393C2×C4⋊C4—1061no—
394C2×C4⋊C4—1061no—
395C2×C4⋊C4—1061no—
396C2×C4⋊C4C22×D5311yescharacteristic
397C2×C4⋊C4C22×D5311yescharacteristic
398C2×C4⋊C4C22×D5311yescharacteristic
399C2×C22⋊C4—531no—
400C2×C22⋊C4—531no—
401C2×C22⋊C4—1061no—
402C2×C22⋊C4—1061no—
403C2×C22⋊C4—1061no—
404C2×C22⋊C4—1061no—
405C2×C42—21no—
406C2×C42—531no—
407C2×C42—1061no—
408C2×C42C22×D5311yescharacteristic
409#107—22no—
410#107—22no—
411#107—1062no—
412#106—1062no—
413C4⋊C8—1062no—
414C4⋊C8C22⋊D5314yes—
415#103—22no—
416#103—22no—
417#103—1062no—
418#103—1062no—
419C22⋊C8—1062no—
420C22⋊C8—1062no—
421#97—1062no—
422C4×C8—1062no—
423#95—1061no—
424#95—1061no—
425#95—1061no—
426#95—1061no—
427#95—1061no—
428#95—1061no—
429#95—1061no—
430#95—1061no—
431#95—1061no—
432#95—1061no—
433#95—1061no—
434C24—531no—
435C2×Q8—21no—
436C2×Q8—21no—
437C2×Q8—21no—
438C2×Q8—21no—
439C2×Q8—21no—
440C2×Q8—42no—
441C2×Q8—81no—
442C2×Q8—81no—
443C2×Q8#1171994912yes—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—531no—
449C22×C4—531no—
450C22×C4—531no—
451C22×C4—1061no—
452C22×C4—1061no—
453C22×C4—1061no—
454C22×C4—1061no—
455C22×C4—1061no—
456C22×C4—1061no—
457C22×C4—1061no—
458C22×C4—1061no—
459C22×C4—1061no—
460C22×C4—1061no—
461C22×C4D4×D5311yescharacteristic
462C22×C4D4×D5311yescharacteristic
463C22×C4C23×D5311yescharacteristic
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—1061no—
469C2×C8—1061no—
470C2×C8—1061no—
471C2×C8—1061no—
472C2×C8—1061no—
473C2×C8—1061no—
474C2×C8C2×C22⋊D5312yes—
475C2×C8C2×C22⋊D5312yes—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—42no—
483C4⋊C4—42no—
484C4⋊C4—532no—
485C4⋊C4—1064no—
486C4⋊C4—1062no—
487C4⋊C4—1062no—
488C4⋊C4—1061no—
489C4⋊C4—1061no—
490C4⋊C4—1061no—
491C4⋊C4—1061no—
492C4⋊C4—1061no—
493C4⋊C4—1061no—
494C4⋊C4—1061no—
495C4⋊C4—4241no—
496C4⋊C4—4241no—
497C4⋊C4—4241no—
498C22⋊C4—1064no—
499C22⋊C4—1062no—
500C22⋊C4—1062no—
501C22⋊C4—4241no—
502C22⋊C4—4241no—
503C22⋊C4—4241no—
504C42—42no—
505C42—1061no—
506C42—1061no—
507C42—4241no—
508C42#1171994712yes—
509C42C2×C22⋊D5312yes—
510C23—1061no—
511C23—1061no—
512C23—1061no—
513C23—1061no—
514C23—1061no—
515C23—1061no—
516C23—1061no—
517C23C2×D4×D5311yescharacteristic
518Q8—22no—
519Q8—22no—
520Q8—41no—
521Q8—41no—
522Q8—84no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—22no—
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—41no—
538C2×C4—41no—
539C2×C4—1061no—
540C2×C4—1061no—
541C2×C4—1061no—
542C2×C4—1062no—
543C2×C4—1061no—
544C2×C4—1061no—
545C2×C4—1061no—
546C2×C4—1061no—
547C2×C4—1061no—
548C2×C4—1061no—
549C2×C4—2121no—
550C2×C4—2121no—
551C2×C4—2121no—
552C2×C4—2121no—
553C2×C4—2121no—
554C2×C4—2121no—
555C2×C4—2121no—
556C2×C4—2121no—
557C2×C4—2121no—
558C2×C4—2121no—
559C2×C4—2121no—
560C2×C4—2121no—
561C2×C4—2121no—
562C2×C4—2121no—
563C2×C4—2121no—
564C2×C4#1313318812yes—
565C2×C4#1313318812yes—
566C2×C4C22×C22⋊D5311yescharacteristic
567C2×C4#41×D5311yescharacteristic
568C2×C4C2×#1171994711yescharacteristic
569C2×C4C2×#1171994911yescharacteristic
570C2×C4#1313326411yescharacteristic
571C2×C4#1313326511yescharacteristic
572C8—22no—
573C8—22no—
574C8—2121no—
575C8—2121no—
576C22—2121no—
577C22—2121no—
578C22—2121no—
579C22—2121no—
580C22—2121no—
581C22—2121no—
582C22—2121no—
583C22C2×#1313318811yescharacteristic
584C22#1681113611yescharacteristic
585C22#1681119611yescharacteristic
586C22#1681142711yescharacteristic
587C22#1681143311yescharacteristic
588C22#1681143611yescharacteristic
589C22#1681143611yescharacteristic
590C4—22no—
591C4—22no—
592C4—42no—
593C4—81no—
594C4—81no—
595C4—81no—
596C4—4241no—
597C4—4241no—
598C4—4241no—
599C4—2122no—
600C4—2121no—
601C4—2121no—
602C4—2121no—
603C4#1681127211yescharacteristic
604C4#1681128411yescharacteristic
605C4#1681131511yescharacteristic
606C4#1681134411yescharacteristic
607C2—4241no—
608C2#2027814011yescharacteristic
609C2#2028436611yescharacteristic
610C2#2028438911yescharacteristic
611C2#2028494911yescharacteristic
612C2#2028498411yescharacteristic
613C2#2028598211yescharacteristic
614C2#2028600311yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC2×C53×#477C2
socleC23×C53C22×D4
prime radicalC23×D53C2×D4
hypercenterC2×#477D53
hypocenterC53#24714
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#477

Complements

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

NH ≅ G/Ntype
8. #20275668C2semidirect
9. #20275485C2semidirect
10. #20275456C2semidirect
11. #20275446C2semidirect
12. #20275402C2semidirect
13. #20275388C2semidirect
14. #20274730C2semidirect
15. C2×C53×#477C2semidirect
300. C2×#477D53semidirect
368. C53#24714semidirect

Maximal subgroups

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

subgroupindex
C2×#168106832
#202805372
#202773912
#202761442
#202761392
#202759172
#202759102
#202756682
#202754852
#202754562
#202754462
#202754022
#202753882
#202747302
C2×C53×#4772
#2471453