Finite Group Databasebeta

← #27440132

Subgroups — #27440132

Order 13568.

1046 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811022C211yesmaximal, characteristic, complemented
2C2×#16810923C211yesmaximal, characteristic
3#20280103C211yesmaximal, characteristic, complemented
4#20276050C211yesmaximal, characteristic, complemented
5#20276050C211yesmaximal, characteristic, complemented
6#20276045C211yesmaximal, characteristic, complemented
7#20276040C211yesmaximal, characteristic, complemented
8#20275673C211yesmaximal, characteristic, complemented
9#20275652C211yesmaximal, characteristic, complemented
10#20274976C211yesmaximal, characteristic
11#20274976C211yesmaximal, characteristic
12#20274962C211yesmaximal, characteristic
13#20274962C211yesmaximal, characteristic
14C4×#13133138C211yesmaximal, characteristic
15C2×C53×#554C211yesmaximal, characteristic
16C2×#13133209C2211yescharacteristic
17#16811022—24no—
18#16810923—24no—
19C2×#13133150C2211yescharacteristic
20C2×#13133150C2211yescharacteristic
21C2×#13133150C2211yescharacteristic
22C2×#13133148C2211yescharacteristic
23C2×#13133148C2211yescharacteristic
24C2×#13133148C2211yescharacteristic
25C2×C4⋊#88322C2211yescharacteristic
26C2×#13133138C2211yescharacteristic
27C2×#13133138C2211yescharacteristic
28C2×#13133138C2211yescharacteristic
29C2×#13133138C2211yescharacteristic
30C2×C4×#88322C2211yescharacteristic
31#16810554—21no—
32#16810542—21no—
33#16810521C2211yescharacteristic
34#16810521C2211yescharacteristic
35#16810521C2211yescharacteristic
36#16810521C2211yescharacteristic
37#16810518—21no—
38#16810518—21no—
39C4×C4⋊Dic53C2211yescharacteristic
40#16810385C2211yescharacteristic
41#16810385C2211yescharacteristic
42#16810385C2211yescharacteristic
43#16810385C2211yescharacteristic
44#16810385C2211yescharacteristic
45#16810385C2211yescharacteristic
46#16810378C2211yescharacteristic
47#16810378C2211yescharacteristic
48#16810378C2211yescharacteristic
49#16810378C2211yescharacteristic
50C53×#554—24no—
51C53×#554—24no—
52C2×C4⋊Q8×C53C2211yescharacteristic
53C2×#125×C53C2211yescharacteristic
54C2×#125×C53C2211yescharacteristic
55C2×C22⋊Q8×C53—21no—
56C2×C22⋊Q8×C53—21no—
57C2×C22⋊Q8×C53—21no—
58C2×C4×Q8×C53C2211yescharacteristic
59C2×C4×D4×C53C2211yescharacteristic
60C2×C4×D4×C53C2211yescharacteristic
61C2×C4×D4×C53C2211yescharacteristic
62C22×C4⋊C4×C53—21no—
63C22×#88322C2311yescharacteristic
64C22×#88322C2311yescharacteristic
65C22×#88322C2311yescharacteristic
66C22×#88322C2311yescharacteristic
67#13133209—22no—
68#13133209—22no—
69C2×C4⋊Dic53C2311yescharacteristic
70C2×C4⋊Dic53C2311yescharacteristic
71C2×C4⋊Dic53C2311yescharacteristic
72C2×C4⋊Dic53C2311yescharacteristic
73C2×#11719919—21no—
74C2×#11719919—21no—
75C2×#11719919—21no—
76C2×#11719919—21no—
77C2×#11719919—21no—
78C2×#11719919—21no—
79C2×C4×Dic53—21no—
80#13133150—22no—
81#13133150—22no—
82#13133150—22no—
83#13133148—22no—
84#13133148—22no—
85#13133148—22no—
86C4⋊#88322—22no—
87#13133138—22no—
88#13133138—22no—
89#13133138Q814yescomplemented
90#13133138Q814yescomplemented
91C4×#88322—22no—
92#13133124—21no—
93#13133124—21no—
94#13133124—21no—
95#13133124—21no—
96#13133124—21no—
97C22×Q8×C53—21no—
98C22×Q8×C53D411yescharacteristic
99C22×D4×C53C2311yescharacteristic
100C23×C4×C53—21no—
101C23×C4×C53—21no—
102C23×C4×C53—21no—
103C4⋊Q8×C53—22no—
104C4⋊Q8×C53—22no—
105#125×C53—22no—
106#125×C53—22no—
107#125×C53—22no—
108#125×C53D414yes—
109C22⋊Q8×C53—28no—
110C22⋊Q8×C53—28no—
111C22⋊Q8×C53—28no—
112C4×Q8×C53—22no—
113C4×Q8×C53—22no—
114C4×D4×C53—22no—
115C4×D4×C53—22no—
116C4×D4×C53—22no—
117C4×D4×C53—22no—
118C4×D4×C53—22no—
119C4×D4×C53D414yes—
120C2×C4⋊C4×C53—28no—
121C2×C4⋊C4×C53—21no—
122C2×C4⋊C4×C53—21no—
123C2×C4⋊C4×C53—21no—
124C2×C4⋊C4×C53—21no—
125C2×C4⋊C4×C53—21no—
126C2×C4⋊C4×C53D411yescharacteristic
127C2×C4⋊C4×C53D411yescharacteristic
128C2×C4⋊C4×C53D411yescharacteristic
129C2×C4⋊C4×C53C2311yescharacteristic
130C2×C4⋊C4×C53C2311yescharacteristic
131C2×C4⋊C4×C53C2311yescharacteristic
132C2×C22⋊C4×C53—21no—
133C2×C22⋊C4×C53—21no—
134C2×C22⋊C4×C53—21no—
135C2×C42×C53C2311yescharacteristic
136C2×C42×C53C2311yescharacteristic
137C2×C42×C53C2311yescharacteristic
138C24×C53—21no—
139C2×Q8×C53—24no—
140C2×Q8×C53—22no—
141C2×Q8×C53—22no—
142C2×Q8×C53—41no—
143C2×Q8×C53—81no—
144C2×Q8×C53—81no—
145C2×D4×C53—21no—
146C2×D4×C53—21no—
147C2×D4×C53—21no—
148C2×D4×C53C2×D412yes—
149C2×D4×C53C2×Q812yes—
150C2×D4×C53C2×Q812yes—
151C22×C4×C53—24no—
152C22×C4×C53—21no—
153C22×C4×C53—21no—
154C22×C4×C53—21no—
155C22×C4×C53—24no—
156C22×C4×C53—24no—
157C22×C4×C53—42no—
158C22×C4×C53—42no—
159C22×C4×C53—81no—
160C22×C4×C53—81no—
161C22×C4×C53—81no—
162C22×C4×C53—81no—
163C22×C4×C53C2×D411yescharacteristic
164C22×C4×C53C2×D411yescharacteristic
165C22×C4×C53C2×D411yescharacteristic
166C22×C4×C53C2×D411yescharacteristic
167C22×C4×C53C2×D411yescharacteristic
168C22×C4×C53C2×D411yescharacteristic
169C22×C4×C53C2411yescharacteristic
170C22×Dic53—21no—
171C22×Dic53—21no—
172C22×Dic53—21no—
173C22×Dic53—21no—
174C4⋊C4×C53—24no—
175C4⋊C4×C53—24no—
176C4⋊C4×C53—24no—
177C4⋊C4×C53—22no—
178C4⋊C4×C53—24no—
179C4⋊C4×C53—24no—
180C4⋊C4×C53—22no—
181C4⋊C4×C53—22no—
182C4⋊C4×C53—21no—
183C4⋊C4×C53—21no—
184C4⋊C4×C53—21no—
185C4⋊C4×C53C2×D412yes—
186C4⋊C4×C53C2×D412yes—
187C4⋊C4×C53C2×D412yes—
188C22⋊C4×C53—24no—
189C22⋊C4×C53—24no—
190C22⋊C4×C53—24no—
191C42×C53—21no—
192C42×C53—21no—
193C42×C53—21no—
194C42×C53—21no—
195C42×C53C2×D412yes—
196C42×C53C2×D412yes—
197C4⋊Dic53—21no—
198C4⋊Dic53—21no—
199C4⋊Dic53—21no—
200C4⋊Dic53—21no—
201C4⋊Dic53—21no—
202C4⋊Dic53—21no—
203C4⋊Dic53C2×Q812yes—
204C4⋊Dic53C2×Q812yes—
205#11719919—42no—
206#11719919—42no—
207#11719919—42no—
208#11719919—42no—
209#11719919—81no—
210#11719919—81no—
211C4×Dic53—81no—
212C2×#88322—21no—
213C2×#88322—21no—
214C2×#88322—21no—
215C2×#88322—21no—
216C2×#88322—21no—
217C2×#88322—21no—
218C2×#88322—21no—
219C2×#88322—21no—
220C2×#88322—21no—
221C2×#88322—21no—
222C2×#88322C2×Q812yes—
223C2×#88322C2×Q812yes—
224C23×C53—22no—
225C23×C53—22no—
226C23×C53—22no—
227C23×C53—41no—
228C23×C53—41no—
229C23×C53—41no—
230C23×C53—41no—
231C23×C53C22×D411yescharacteristic
232Q8×C53—44no—
233Q8×C53—84no—
234D4×C53—22no—
235D4×C53—22no—
236D4×C53—22no—
237D4×C53C4⋊Q814yes—
238C2×C4×C53—21no—
239C2×C4×C53—21no—
240C2×C4×C53—21no—
241C2×C4×C53—21no—
242C2×C4×C53—21no—
243C2×C4×C53—21no—
244C2×C4×C53—21no—
245C2×C4×C53—22no—
246C2×C4×C53—22no—
247C2×C4×C53—22no—
248C2×C4×C53—21no—
249C2×C4×C53—21no—
250C2×C4×C53—21no—
251C2×C4×C53—21no—
252C2×C4×C53—21no—
253C2×C4×C53—21no—
254C2×C4×C53—21no—
255C2×C4×C53—21no—
256C2×C4×C53—48no—
257C2×C4×C53—41no—
258C2×C4×C53—41no—
259C2×C4×C53—41no—
260C2×C4×C53—41no—
261C2×C4×C53—41no—
262C2×C4×C53—41no—
263C2×C4×C53—84no—
264C2×C4×C53—84no—
265C2×C4×C53#12012yes—
266C2×C4×C53#12012yes—
267C2×C4×C53#13512yes—
268C2×C4×C53C22×D411yescharacteristic
269C2×C4×C53C22×D411yescharacteristic
270C2×C4×C53C22×Q811yescharacteristic
271C2×C4×C53C22×Q811yescharacteristic
272C2×C4×C532−1+411yescharacteristic
273C2×C4×C532−1+411yescharacteristic
274C2×Dic53—22no—
275C2×Dic53—22no—
276C2×Dic53—41no—
277C2×Dic53—41no—
278C2×Dic53—41no—
279C2×Dic53—41no—
280C2×Dic53—41no—
281C2×Dic53—41no—
282C2×Dic53—41no—
283C2×Dic53—41no—
284C2×Dic53—41no—
285C2×Dic53—41no—
286#88322—22no—
287#88322—22no—
288#88322—41no—
289#88322—41no—
290#25843—531nomaximal, Sylow
291C22×C53—24no—
292C22×C53—42no—
293C22×C53—42no—
294C22×C53—42no—
295C22×C53—42no—
296C22×C53—42no—
297C22×C53—42no—
298C22×C53C2×#12011yescharacteristic
299C22×C53D4×Q811yescharacteristic
300C22×C53D4×Q811yescharacteristic
301C22×C53C2×#13511yescharacteristic
302C22×C53#57511yescharacteristic
303C22×C53#57611yescharacteristic
304C22×C53#57711yescharacteristic
305C4×C53—22no—
306C4×C53—22no—
307C4×C53—22no—
308C4×C53—41no—
309C4×C53—42no—
310C4×C53—41no—
311C4×C53—41no—
312C4×C53—42no—
313C4×C53—42no—
314C4×C53C2×C4⋊Q811yescharacteristic
315C4×C53#53611yescharacteristic
316C4×C53D4×Q811yescharacteristic
317C4×C53D4×Q811yescharacteristic
318Dic53—42no—
319Dic53—42no—
320Dic53—81no—
321Dic53—81no—
322C2×#554D5311yescharacteristic
323C2×#499—531no—
324C2×#478—531no—
325#2584—531no—
326#2059—531no—
327#2059—531no—
328#2058—531no—
329#2057—531no—
330#1989—531no—
331#1984—531no—
332#1864—531no—
333#1864—531no—
334#1862—531no—
335#1862—531no—
336C4×#102—531no—
337C2×C53—44no—
338C2×C53#243411yescharacteristic
339C2×C53#300211yescharacteristic
340C2×C53#301211yescharacteristic
341C2×C53#338011yescharacteristic
342C2×C53#338111yescharacteristic
343C2×C53#338811yescharacteristic
344C2×C53#338911yescharacteristic
345#554—24no—
346#554—24no—
347C2×C4⋊Q8C2×D5311yescharacteristic
348C2×#125—531no—
349C2×#125C2×D5311yescharacteristic
350C2×#125C2×D5311yescharacteristic
351C2×C22⋊Q8—21no—
352C2×C22⋊Q8—21no—
353C2×C22⋊Q8—21no—
354C2×C4×Q8C2×D5311yescharacteristic
355C2×C4×D4C2×D5311yescharacteristic
356C2×C4×D4C2×D5311yescharacteristic
357C2×C4×D4C2×D5311yescharacteristic
358C22×C4⋊C4—21no—
359#499—1064no—
360#478—1064no—
361C2×#107—531no—
362C2×#107—531no—
363C2×#107—531no—
364C2×#106—531no—
365C2×#106—531no—
366C2×#106—531no—
367C2×C4⋊C8—531no—
368C2×#102—531no—
369C2×#102—531no—
370C2×#102—531no—
371C2×#102—531no—
372C2×C4×C8—531no—
373#398—1061no—
374#395—1061no—
375#389—531no—
376#389—531no—
377#389—531no—
378#389—531no—
379#389—1061no—
380#389—1061no—
381C4×C4⋊C4—531no—
382#340—531no—
383#340—531no—
384#340—531no—
385#340—531no—
386#340—531no—
387#340—531no—
388#336—531no—
389#336—531no—
390#336—531no—
391#336—531no—
392C53#2584311yescharacteristic, complemented, Sylow
393C22×Q8—21no—
394C22×Q8C22⋊D5311yescharacteristic
395C22×D4C22×D5311yescharacteristic
396C23×C4—21no—
397C23×C4—21no—
398C23×C4—21no—
399C22×C8—531no—
400C22×C8—531no—
401C22×C8—531no—
402C22×C8—531no—
403C4⋊Q8—22no—
404C4⋊Q8—22no—
405#125—22no—
406#125—22no—
407#125—22no—
408#125—1062no—
409#125—1062no—
410#125C22⋊D5314yes—
411C22⋊Q8—28no—
412C22⋊Q8—28no—
413C22⋊Q8—28no—
414C4×Q8—22no—
415C4×Q8—22no—
416C4×D4—22no—
417C4×D4—22no—
418C4×D4—22no—
419C4×D4—22no—
420C4×D4—22no—
421C4×D4C22⋊D5314yes—
422C2×C4⋊C4—28no—
423C2×C4⋊C4—21no—
424C2×C4⋊C4—21no—
425C2×C4⋊C4—21no—
426C2×C4⋊C4—21no—
427C2×C4⋊C4—21no—
428C2×C4⋊C4—531no—
429C2×C4⋊C4—531no—
430C2×C4⋊C4—531no—
431C2×C4⋊C4—531no—
432C2×C4⋊C4—1061no—
433C2×C4⋊C4—1061no—
434C2×C4⋊C4—1061no—
435C2×C4⋊C4—1061no—
436C2×C4⋊C4—1061no—
437C2×C4⋊C4—1061no—
438C2×C4⋊C4C22⋊D5311yescharacteristic
439C2×C4⋊C4C22⋊D5311yescharacteristic
440C2×C4⋊C4C22⋊D5311yescharacteristic
441C2×C4⋊C4C22×D5311yescharacteristic
442C2×C4⋊C4C22×D5311yescharacteristic
443C2×C4⋊C4C22×D5311yescharacteristic
444C2×C22⋊C4—21no—
445C2×C22⋊C4—21no—
446C2×C22⋊C4—21no—
447C2×C42—1061no—
448C2×C42C22×D5311yescharacteristic
449C2×C42C22×D5311yescharacteristic
450C2×C42C22×D5311yescharacteristic
451#107—1062no—
452#107—1062no—
453#107—1062no—
454#106—1062no—
455#106—1062no—
456#106—1062no—
457C4⋊C8—1062no—
458#102—534no—
459#102—534no—
460#102—1062no—
461#102—1062no—
462C4×C8—1062no—
463#95—1061no—
464#95—1061no—
465#95—1061no—
466#95—1061no—
467#95—1061no—
468C24—21no—
469C2×Q8—24no—
470C2×Q8—22no—
471C2×Q8—22no—
472C2×Q8—41no—
473C2×Q8—81no—
474C2×Q8—81no—
475C2×D4—21no—
476C2×D4—21no—
477C2×D4—21no—
478C2×D4Q8×D5312yes—
479C2×D4Q8×D5312yes—
480C2×D4C2×C22⋊D5312yes—
481C22×C4—24no—
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—24no—
486C22×C4—24no—
487C22×C4—42no—
488C22×C4—42no—
489C22×C4—81no—
490C22×C4—81no—
491C22×C4—81no—
492C22×C4—81no—
493C22×C4—1061no—
494C22×C4—1061no—
495C22×C4—1061no—
496C22×C4—1061no—
497C22×C4C2×C22⋊D5311yescharacteristic
498C22×C4C2×C22⋊D5311yescharacteristic
499C22×C4C2×C22⋊D5311yescharacteristic
500C22×C4C2×C22⋊D5311yescharacteristic
501C22×C4C2×C22⋊D5311yescharacteristic
502C22×C4C2×C22⋊D5311yescharacteristic
503C22×C4C23×D5311yescharacteristic
504C2×C8—532no—
505C2×C8—532no—
506C2×C8—1061no—
507C2×C8—1061no—
508C2×C8—1061no—
509C2×C8—1061no—
510C2×C8—1061no—
511C2×C8—1061no—
512C2×C8—1061no—
513C2×C8—1061no—
514C2×C8—1061no—
515C2×C8—1061no—
516C4⋊C4—24no—
517C4⋊C4—24no—
518C4⋊C4—24no—
519C4⋊C4—22no—
520C4⋊C4—24no—
521C4⋊C4—24no—
522C4⋊C4—22no—
523C4⋊C4—22no—
524C4⋊C4—21no—
525C4⋊C4—21no—
526C4⋊C4—21no—
527C4⋊C4—532no—
528C4⋊C4—532no—
529C4⋊C4—1061no—
530C4⋊C4—1061no—
531C4⋊C4—1061no—
532C4⋊C4—1061no—
533C4⋊C4—1061no—
534C4⋊C4—1061no—
535C4⋊C4—4241no—
536C4⋊C4—4241no—
537C4⋊C4—2122no—
538C4⋊C4—2122no—
539C4⋊C4—2122no—
540C4⋊C4—2122no—
541C4⋊C4C2×C22⋊D5312yes—
542C4⋊C4C2×C22⋊D5312yes—
543C4⋊C4C2×C22⋊D5312yes—
544C22⋊C4—24no—
545C22⋊C4—24no—
546C22⋊C4—24no—
547C42—21no—
548C42—21no—
549C42—21no—
550C42—21no—
551C42—4241no—
552C42C2×C22⋊D5312yes—
553C42C2×C22⋊D5312yes—
554C23—22no—
555C23—22no—
556C23—22no—
557C23—41no—
558C23—41no—
559C23—41no—
560C23—41no—
561C23C22×C22⋊D5311yescharacteristic
562Q8—44no—
563Q8—84no—
564D4—22no—
565D4—22no—
566D4—22no—
567D4#1313321514yes—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—48no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—84no—
594C2×C4—84no—
595C2×C4—1062no—
596C2×C4—1062no—
597C2×C4—2121no—
598C2×C4—2121no—
599C2×C4—2121no—
600C2×C4—2121no—
601C2×C4—2121no—
602C2×C4—2121no—
603C2×C4—2121no—
604C2×C4—2121no—
605C2×C4—2121no—
606C2×C4—2121no—
607C2×C4#1313318612yes—
608C2×C4#1313318612yes—
609C2×C4#1313323712yes—
610C2×C4C22×C22⋊D5311yescharacteristic
611C2×C4C22×C22⋊D5311yescharacteristic
612C2×C4C2×Q8×D5311yescharacteristic
613C2×C4C2×Q8×D5311yescharacteristic
614C2×C4#1313326511yescharacteristic
615C2×C4#1313326511yescharacteristic
616C8—1062no—
617C8—1062no—
618C8—2121no—
619C8—2121no—
620C22—24no—
621C22—42no—
622C22—42no—
623C22—42no—
624C22—42no—
625C22—42no—
626C22—42no—
627C22C2×#1313318611yescharacteristic
628C22Q8×C22⋊D5311yescharacteristic
629C22Q8×C22⋊D5311yescharacteristic
630C22C2×#1313323711yescharacteristic
631C22#1681142111yescharacteristic
632C22#1681142611yescharacteristic
633C22#1681143411yescharacteristic
634C4—22no—
635C4—22no—
636C4—22no—
637C4—41no—
638C4—42no—
639C4—41no—
640C4—41no—
641C4—42no—
642C4—42no—
643C4—4241no—
644C4—4241no—
645C4—2122no—
646C4—2122no—
647C4C2×#1313321511yescharacteristic
648C4#1681119711yescharacteristic
649C4Q8×C22⋊D5311yescharacteristic
650C4Q8×C22⋊D5311yescharacteristic
651C2—44no—
652C2#2027835511yescharacteristic
653C2#2028393311yescharacteristic
654C2#2028401911yescharacteristic
655C2#2028757811yescharacteristic
656C2#2028759011yescharacteristic
657C2#2028766011yescharacteristic
658C2#2028766911yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC2×C53×#554C2
socleC23×C53C22×D4
prime radicalC22×D4×C53C23
hypercenterC2×#554D53
hypocenterC53#25843
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×#554

Complements

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

NH ≅ G/Ntype
1. C2×#16811022C2semidirect
3. #20280103C2semidirect
4. #20276050C2semidirect
5. #20276050C2semidirect
6. #20276045C2semidirect
7. #20276040C2semidirect
8. #20275673C2semidirect
9. #20275652C2semidirect
89. #13133138Q8semidirect
90. #13133138Q8semidirect
392. C53#25843semidirect

Maximal subgroups

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

subgroupindex
C2×#168110222
C2×#168109232
#202801032
#202760502
#202760452
#202760402
#202756732
#202756522
#202749762
#202749622
C4×#131331382
C2×C53×#5542
#2584353