Finite Group Databasebeta

← #27425459

Subgroups — #27425459

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×#16810918C211yesmaximal, characteristic, complemented
2C2×#16810684C211yesmaximal, characteristic
3#20280536C211yesmaximal, characteristic
4#20276146C211yesmaximal, characteristic
5#20276141C211yesmaximal, characteristic
6#20275915C211yesmaximal, characteristic
7#20275908C211yesmaximal, characteristic
8#20275667C211yesmaximal, characteristic, complemented
9#20275483C211yesmaximal, characteristic, complemented
10#20275455C211yesmaximal, characteristic, complemented
11#20275445C211yesmaximal, characteristic, complemented
12#20275408C211yesmaximal, characteristic, complemented
13#20275394C211yesmaximal, characteristic, complemented
14#20274730C211yesmaximal, characteristic, complemented
15C53×#2323C211yesmaximal, characteristic
16C2×#13133217C2211yescharacteristic
17#16810918—24no—
18#16810684—24no—
19C2×#13133150C2211yescharacteristic
20C2×#13133150C2211yescharacteristic
21C2×#13133150C2211yescharacteristic
22C2×#13133148C2211yescharacteristic
23C2×C4⋊#88322C2211yescharacteristic
24C2×C4⋊#88322C2211yescharacteristic
25C2×#13133141C2211yescharacteristic
26C2×#13133141C2211yescharacteristic
27C2×#13133141C2211yescharacteristic
28C2×#13133141C2211yescharacteristic
29C2×C22⋊#88322C2211yescharacteristic
30C2×C22⋊#88322C2211yescharacteristic
31C2×#13133128C2211yescharacteristic
32C2×C4×#88322C2211yescharacteristic
33#16810562—21no—
34#16810559—21no—
35#16810543—21no—
36#16810535—21no—
37#16810521C2211yescharacteristic
38#16810521C2211yescharacteristic
39#16810489C2211yescharacteristic
40#16810483C2211yescharacteristic
41#16810483C2211yescharacteristic
42#16810385C2211yescharacteristic
43#16810385C2211yescharacteristic
44#16810385C2211yescharacteristic
45#16810385C2211yescharacteristic
46#16810385C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810385C2211yescharacteristic
49#16810385C2211yescharacteristic
50C2×C22⋊Q8×C53C2211yescharacteristic
51C2×C4×Q8×C53C2211yescharacteristic
52C2×#117×C53C2211yescharacteristic
53C53×#391C2211yescharacteristic
54C53×#389—21no—
55C53×#389C2211yescharacteristic
56C53×#388—21no—
57C53×#388—21no—
58C53×#387—21no—
59C53×#380C2211yescharacteristic
60C4×C4⋊C4×C53C2211yescharacteristic
61C22×Dic106—21no—
62C22×#88322C2311yescharacteristic
63C22×#88322C2311yescharacteristic
64C22×#88322C2311yescharacteristic
65C22×#88322C2311yescharacteristic
66#13133217—22no—
67#13133217—22no—
68C2×C4⋊Dic53—21no—
69C2×C4⋊Dic53C2311yescharacteristic
70C2×C4⋊Dic53C2311yescharacteristic
71C2×C4⋊Dic53C2311yescharacteristic
72C2×C4⋊Dic53C2311yescharacteristic
73C2×#11719919—21no—
74C2×#11719919—21no—
75C2×C22⋊Dic53—21no—
76C2×C22⋊Dic53—21no—
77#13133150—22no—
78#13133150—22no—
79#13133150—22no—
80#13133148—22no—
81C4⋊#88322—22no—
82C4⋊#88322D414yes—
83#13133141—22no—
84#13133141—22no—
85#13133141—22no—
86#13133141—22no—
87C22⋊#88322—22no—
88C22⋊#88322—22no—
89#13133128—22no—
90C4×#88322—22no—
91#13133124—21no—
92#13133124—21no—
93#13133124—21no—
94#13133124—21no—
95#13133124—21no—
96#13133124—21no—
97C22×Q8×C53C2311yescharacteristic
98C23×C4×C53C2311yescharacteristic
99C22⋊Q8×C53—22no—
100C22⋊Q8×C53—22no—
101C4×Q8×C53—22no—
102C4×Q8×C53—22no—
103#117×C53—22no—
104#117×C53—22no—
105C2×C4⋊C4×C53—21no—
106C2×C4⋊C4×C53—21no—
107C2×C4⋊C4×C53—21no—
108C2×C4⋊C4×C53D411yescharacteristic
109C2×C4⋊C4×C53D411yescharacteristic
110C2×C4⋊C4×C53C2311yescharacteristic
111C2×C4⋊C4×C53C2311yescharacteristic
112C2×C4⋊C4×C53C2311yescharacteristic
113C2×C22⋊C4×C53—21no—
114C2×C22⋊C4×C53—21no—
115C2×C22⋊C4×C53D411yescharacteristic
116C2×C22⋊C4×C53D411yescharacteristic
117C2×C42×C53—21no—
118C2×C42×C53—21no—
119C2×C42×C53C2311yescharacteristic
120C2×C42×C53C2311yescharacteristic
121#95×C53—21no—
122#95×C53—21no—
123#95×C53—21no—
124#95×C53—21no—
125#95×C53—21no—
126C24×C53C2×D411yescharacteristic
127C2×Q8×C53—21no—
128C2×Q8×C53—21no—
129C2×Q8×C53—21no—
130C2×Q8×C53—21no—
131C2×Q8×C53—21no—
132C2×Q8×C53#4112yes—
133C22×C4×C53—21no—
134C22×C4×C53—21no—
135C22×C4×C53—21no—
136C22×C4×C53—21no—
137C22×C4×C53—21no—
138C22×C4×C53—21no—
139C22×C4×C53—21no—
140C22×C4×C53—21no—
141C22×C4×C53—21no—
142C22×C4×C53—21no—
143C22×C4×C53C2×D411yescharacteristic
144C22×C4×C53C2×D411yescharacteristic
145C22×C4×C53C2×D411yescharacteristic
146C22×C4×C53C2×D411yescharacteristic
147C22×C4×C53C2×D411yescharacteristic
148C22×C4×C53C2411yescharacteristic
149C22×Dic53—21no—
150C22×Dic53—21no—
151C22×Dic53—21no—
152C22×Dic53—21no—
153C2×Dic106—42no—
154C2×Dic106—81no—
155C2×Dic106—81no—
156C4⋊C4×C53—24no—
157C4⋊C4×C53—22no—
158C4⋊C4×C53—22no—
159C4⋊C4×C53—21no—
160C4⋊C4×C53—21no—
161C4⋊C4×C53—21no—
162C4⋊C4×C53—21no—
163C4⋊C4×C53—21no—
164C4⋊C4×C53—42no—
165C4⋊C4×C53—81no—
166C4⋊C4×C53#4112yes—
167C22⋊C4×C53—24no—
168C22⋊C4×C53—22no—
169C22⋊C4×C53—22no—
170C22⋊C4×C53—81no—
171C42×C53—21no—
172C42×C53—21no—
173C42×C53—42no—
174C42×C53—81no—
175C42×C53C2×D412yes—
176C42×C53#4112yes—
177C22⋊Dic53—81no—
178C22⋊Dic53—81no—
179C4⋊Dic53—21no—
180C4⋊Dic53—21no—
181C4⋊Dic53—21no—
182C4⋊Dic53—21no—
183C4⋊Dic53—21no—
184C4⋊Dic53—21no—
185C4⋊Dic53—21no—
186C4⋊Dic53—21no—
187C4⋊Dic53—42no—
188#11719919—81no—
189#11719919—81no—
190C2×#88322—21no—
191C2×#88322—21no—
192C2×#88322—21no—
193C2×#88322—21no—
194C2×#88322—21no—
195C2×#88322—21no—
196C2×#88322—21no—
197C2×#88322—21no—
198C2×#88322—21no—
199C2×#88322—21no—
200C2×#88322C2×D412yes—
201C2×#88322C2×D412yes—
202C23×C53—21no—
203C23×C53—21no—
204C23×C53—21no—
205C23×C53—21no—
206C23×C53—21no—
207C23×C53—21no—
208C23×C53—21no—
209C23×C53C22×D411yescharacteristic
210Q8×C53—22no—
211Q8×C53—22no—
212Q8×C53—41no—
213Q8×C53—41no—
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—22no—
223C2×C4×C53—22no—
224C2×C4×C53—21no—
225C2×C4×C53—21no—
226C2×C4×C53—21no—
227C2×C4×C53—21no—
228C2×C4×C53—21no—
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—41no—
237C2×C4×C53—41no—
238C2×C4×C53—41no—
239C2×C4×C53—41no—
240C2×C4×C53—41no—
241C2×C4×C53—41no—
242C2×C4×C53—41no—
243C2×C4×C53#12112yes—
244C2×C4×C53#12112yes—
245C2×C4×C53C22×D411yescharacteristic
246C2×C4×C53C2×#4111yescharacteristic
247C2×C4×C53C2×#4111yescharacteristic
248C2×C4×C53C2×#4111yescharacteristic
249C2×C4×C532−1+411yescharacteristic
250C2×C4×C532−1+411yescharacteristic
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—
260C2×Dic53—41no—
261C2×Dic53—41no—
262C2×Dic53—41no—
263Dic106—84no—
264#88322—22no—
265#88322—22no—
266#88322—41no—
267#88322—41no—
268#24714—531nomaximal, Sylow
269C22×C53—41no—
270C22×C53—41no—
271C22×C53—41no—
272C22×C53—41no—
273C22×C53—41no—
274C22×C53—41no—
275C22×C53—41no—
276C22×C53C2×#12111yescharacteristic
277C22×C53#52511yescharacteristic
278C22×C53#53611yescharacteristic
279C22×C53#57611yescharacteristic
280C22×C53#57711yescharacteristic
281C22×C53#57811yescharacteristic
282C22×C53#57811yescharacteristic
283C4×C53—22no—
284C4×C53—22no—
285C4×C53—42no—
286C4×C53—41no—
287C4×C53—42no—
288C4×C53—41no—
289C4×C53—41no—
290C4×C53—81no—
291C4×C53—81no—
292C4×C53#54711yescharacteristic
293C4×C53#54811yescharacteristic
294C4×C53#55211yescharacteristic
295C4×C53#55611yescharacteristic
296Dic53—81no—
297Dic53—81no—
298Dic53—81no—
299Dic53—81no—
300C2×#477—531no—
301C2×#433—531no—
302#2628—531no—
303#2323D5311yescharacteristic
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—81no—
316C2×C53#241511yescharacteristic
317C2×C53#305911yescharacteristic
318C2×C53#306211yescharacteristic
319C2×C53#312111yescharacteristic
320C2×C53#312411yescharacteristic
321C2×C53#324211yescharacteristic
322C2×C53#324311yescharacteristic
323C2×C4⋊Q8—531no—
324C2×C22⋊Q8C2×D5311yescharacteristic
325C2×C4×Q8C2×D5311yescharacteristic
326C2×#117C2×D5311yescharacteristic
327#477—1064no—
328#433—1064no—
329C2×#107—531no—
330C2×#107—531no—
331C2×#107—531no—
332C2×#106—531no—
333C2×C4⋊C8—531no—
334C2×C4⋊C8—531no—
335C2×#103—531no—
336C2×#103—531no—
337C2×#103—531no—
338C2×#103—531no—
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#391C2×D5311yescharacteristic
348#389—21no—
349#389—531no—
350#389—531no—
351#389C2×D5311yescharacteristic
352#388—21no—
353#388—21no—
354#387—21no—
355#382—531no—
356#380—531no—
357#380—531no—
358#380C2×D5311yescharacteristic
359C4×C4⋊C4C2×D5311yescharacteristic
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—1061no—
370C22×Q8C22×D5311yescharacteristic
371C23×C4C22×D5311yescharacteristic
372C22×C8—531no—
373C22×C8—531no—
374C22×C8—531no—
375C22×C8—531no—
376C4⋊Q8—1062no—
377C4⋊Q8—1062no—
378C22⋊Q8—22no—
379C22⋊Q8—22no—
380C4×Q8—22no—
381C4×Q8—22no—
382#117—22no—
383#117—22no—
384C2×C4⋊C4—21no—
385C2×C4⋊C4—21no—
386C2×C4⋊C4—21no—
387C2×C4⋊C4—531no—
388C2×C4⋊C4—531no—
389C2×C4⋊C4—531no—
390C2×C4⋊C4—531no—
391C2×C4⋊C4—1061no—
392C2×C4⋊C4—1061no—
393C2×C4⋊C4—1061no—
394C2×C4⋊C4C22⋊D5311yescharacteristic
395C2×C4⋊C4C22⋊D5311yescharacteristic
396C2×C4⋊C4C22×D5311yescharacteristic
397C2×C4⋊C4C22×D5311yescharacteristic
398C2×C4⋊C4C22×D5311yescharacteristic
399C2×C22⋊C4—21no—
400C2×C22⋊C4—21no—
401C2×C22⋊C4—1061no—
402C2×C22⋊C4—1061no—
403C2×C22⋊C4C22⋊D5311yescharacteristic
404C2×C22⋊C4C22⋊D5311yescharacteristic
405C2×C42—21no—
406C2×C42—21no—
407C2×C42C22×D5311yescharacteristic
408C2×C42C22×D5311yescharacteristic
409#107—1062no—
410#107—1062no—
411#107—1062no—
412#106—1062no—
413C4⋊C8—534no—
414C4⋊C8—1062no—
415#103—1062no—
416#103—1062no—
417#103—1062no—
418#103—1062no—
419C22⋊C8—1062no—
420C22⋊C8—1062no—
421#97—1062no—
422C4×C8—1062no—
423#95—21no—
424#95—21no—
425#95—21no—
426#95—21no—
427#95—21no—
428#95—1061no—
429#95—1061no—
430#95—1061no—
431#95—1061no—
432#95—1061no—
433#95—1061no—
434C24C2×C22⋊D5311yescharacteristic
435C2×Q8—21no—
436C2×Q8—21no—
437C2×Q8—21no—
438C2×Q8—21no—
439C2×Q8—21no—
440C2×Q8—4241no—
441C2×Q8—4241no—
442C2×Q8—2122no—
443C2×Q8#1171994512yes—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—1061no—
455C22×C4—1061no—
456C22×C4—1061no—
457C22×C4—1061no—
458C22×C4C2×C22⋊D5311yescharacteristic
459C22×C4C2×C22⋊D5311yescharacteristic
460C22×C4C2×C22⋊D5311yescharacteristic
461C22×C4C2×C22⋊D5311yescharacteristic
462C22×C4C2×C22⋊D5311yescharacteristic
463C22×C4C23×D5311yescharacteristic
464C2×C8—532no—
465C2×C8—532no—
466C2×C8—1061no—
467C2×C8—1061no—
468C2×C8—1061no—
469C2×C8—1061no—
470C2×C8—1061no—
471C2×C8—1061no—
472C2×C8—1061no—
473C2×C8—1061no—
474C2×C8—1061no—
475C2×C8—1061no—
476C4⋊C4—24no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—42no—
485C4⋊C4—81no—
486C4⋊C4—1061no—
487C4⋊C4—1061no—
488C4⋊C4—1061no—
489C4⋊C4—1061no—
490C4⋊C4—1061no—
491C4⋊C4—1061no—
492C4⋊C4—1061no—
493C4⋊C4—1061no—
494C4⋊C4—4241no—
495C4⋊C4—4241no—
496C4⋊C4—2122no—
497C4⋊C4#1171994512yes—
498C22⋊C4—24no—
499C22⋊C4—22no—
500C22⋊C4—22no—
501C22⋊C4—81no—
502C22⋊C4—4241no—
503C22⋊C4—4241no—
504C42—21no—
505C42—21no—
506C42—42no—
507C42—81no—
508C42D4×D5312yes—
509C42#1171994712yes—
510C23—21no—
511C23—21no—
512C23—21no—
513C23—21no—
514C23—21no—
515C23—21no—
516C23—21no—
517C23C22×C22⋊D5311yescharacteristic
518Q8—22no—
519Q8—22no—
520Q8—41no—
521Q8—41no—
522Q8—4244no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
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#1313318912yes—
565C2×C4#1313318912yes—
566C2×C4C2×D4×D5311yescharacteristic
567C2×C4C2×#1171994711yescharacteristic
568C2×C4C2×#1171994511yescharacteristic
569C2×C4C2×#1171994511yescharacteristic
570C2×C4#1313326411yescharacteristic
571C2×C4#1313326411yescharacteristic
572C8—1062no—
573C8—1062no—
574C8—2121no—
575C8—2121no—
576C22—41no—
577C22—41no—
578C22—41no—
579C22—41no—
580C22—41no—
581C22—41no—
582C22—41no—
583C22C2×#1313318911yescharacteristic
584C22#1681113411yescharacteristic
585C22#1681119411yescharacteristic
586C22#1681142611yescharacteristic
587C22#1681143111yescharacteristic
588C22#1681143811yescharacteristic
589C22#1681143811yescharacteristic
590C4—22no—
591C4—22no—
592C4—42no—
593C4—41no—
594C4—42no—
595C4—41no—
596C4—41no—
597C4—81no—
598C4—81no—
599C4—4241no—
600C4—4241no—
601C4—4241no—
602C4—4241no—
603C4#1681127811yescharacteristic
604C4#1681128211yescharacteristic
605C4#1681131311yescharacteristic
606C4#1681134311yescharacteristic
607C2—81no—
608C2#2027814411yescharacteristic
609C2#2028436711yescharacteristic
610C2#2028439711yescharacteristic
611C2#2028494411yescharacteristic
612C2#2028497711yescharacteristic
613C2#2028597711yescharacteristic
614C2#2028599211yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2323C2
socleC23×C53C22×D4
prime radicalC24×C53C2×D4
hypercenter#2323D53
hypocenterC53#24714
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2323

Complements

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

NH ≅ G/Ntype
1. C2×#16810918C2semidirect
8. #20275667C2semidirect
9. #20275483C2semidirect
10. #20275455C2semidirect
11. #20275445C2semidirect
12. #20275408C2semidirect
13. #20275394C2semidirect
14. #20274730C2semidirect
368. C53#24714semidirect

Maximal subgroups

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

subgroupindex
C2×#168109182
C2×#168106842
#202805362
#202761462
#202761412
#202759152
#202759082
#202756672
#202754832
#202754552
#202754452
#202754082
#202753942
#202747302
C53×#23232
#2471453