Finite Group Databasebeta

← #27275915

Subgroups — #27275915

Order 13568.

900 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20275033C211yesmaximal, characteristic
2#20275023C211yesmaximal, characteristic, complemented
3C2×#16810360C211yesmaximal, characteristic, complemented
4C2×#16810357C211yesmaximal, characteristic
5C2×#16810349C211yesmaximal, characteristic, complemented
6#20272968C211yesmaximal, characteristic, complemented
7C2×C53×#539C211yesmaximal, characteristic
8C22×C4⋊Dic53C2211yescharacteristic
9C2×#13133141—21no—
10C2×#13133138—21no—
11C2×C22⋊#88322C2211yescharacteristic
12C2×C22⋊#88322C2211yescharacteristic
13C2×#13133124—21no—
14C2×#13133124—21no—
15C2×#13133124C2211yescharacteristic
16#16810385—21no—
17#16810385—21no—
18#16810360—22no—
19#16810357—22no—
20#16810349—22no—
21C53×#539—22no—
22C53×#539—22no—
23C53×#539—22no—
24C53×#539C414yescomplemented
25C2×#126×C53—21no—
26C2×#125×C53—21no—
27C2×#123×C53—21no—
28C2×#123×C53—21no—
29C2×C22⋊Q8×C53—21no—
30C2×C22⋊Q8×C53C2211yescharacteristic
31C2×#121×C53C2211yescharacteristic
32C2×C4×D4×C53—21no—
33C22×C4⋊C4×C53C2211yescharacteristic
34C23×Dic53—21no—
35C23×Dic53—21no—
36C22×#88322—21no—
37C22×#88322—21no—
38C2×C4⋊Dic53—21no—
39C2×C4⋊Dic53—21no—
40C2×C4⋊Dic53—42no—
41#13133141—42no—
42#13133138—42no—
43C22⋊#88322—21no—
44C22⋊#88322—21no—
45C22⋊#88322—21no—
46C22⋊#88322—21no—
47#13133124—21no—
48#13133124—21no—
49#13133124—42no—
50#13133124—42no—
51C22×Q8×C53D411yescharacteristic
52C22×D4×C53—21no—
53C22×D4×C53D411yescharacteristic
54C23×C4×C53—21no—
55C23×C4×C53D411yescharacteristic
56C23×C4×C53D411yescharacteristic
57C23×C4×C53C2311yescharacteristic
58#126×C53—24no—
59#126×C53—24no—
60#125×C53—24no—
61#125×C53—24no—
62#123×C53—24no—
63#123×C53—24no—
64#123×C53—24no—
65#123×C53—24no—
66C22⋊Q8×C53—24no—
67C22⋊Q8×C53—24no—
68C22⋊Q8×C53—21no—
69C22⋊Q8×C53—21no—
70C22⋊Q8×C53C2×C412yes—
71C22⋊Q8×C53D412yes—
72#121×C53—21no—
73#121×C53—21no—
74#121×C53C2×C412yes—
75#121×C53D412yes—
76C4×D4×C53—24no—
77C4×D4×C53—24no—
78C2×C4⋊C4×C53—21no—
79C2×C4⋊C4×C53—21no—
80C2×C4⋊C4×C53—21no—
81C2×C4⋊C4×C53—21no—
82C2×C4⋊C4×C53—21no—
83C2×C4⋊C4×C53—21no—
84C2×C4⋊C4×C53—21no—
85C2×C4⋊C4×C53—21no—
86C2×C4⋊C4×C53C2×C412yes—
87C2×C4⋊C4×C53D411yescharacteristic
88C2×C4⋊C4×C53D411yescharacteristic
89C2×C4⋊C4×C53D412yes—
90C2×C22⋊C4×C53—21no—
91C2×C22⋊C4×C53—21no—
92C2×C22⋊C4×C53—21no—
93C2×C22⋊C4×C53—21no—
94C2×C22⋊C4×C53—21no—
95C2×C42×C53—21no—
96C24×C53—21no—
97C24×C53C2×D411yescharacteristic
98C2×Q8×C53—21no—
99C2×Q8×C53—41no—
100C2×Q8×C53—41no—
101C2×Q8×C53C22⋊C412yes—
102C2×D4×C53—22no—
103C2×D4×C53—22no—
104C2×D4×C53—21no—
105C2×D4×C53—41no—
106C2×D4×C53—41no—
107C2×D4×C53—41no—
108C2×D4×C53—41no—
109C2×D4×C53—41no—
110C2×D4×C53—41no—
111C2×D4×C53C22⋊C412yes—
112C22×C4×C53—21no—
113C22×C4×C53—21no—
114C22×C4×C53—21no—
115C22×C4×C53—21no—
116C22×C4×C53—21no—
117C22×C4×C53—21no—
118C22×C4×C53—21no—
119C22×C4×C53—21no—
120C22×C4×C53—21no—
121C22×C4×C53—21no—
122C22×C4×C53—24no—
123C22×C4×C53—21no—
124C22×C4×C53—41no—
125C22×C4×C53—41no—
126C22×C4×C53—41no—
127C22×C4×C53—41no—
128C22×C4×C53—81no—
129C22×C4×C53—81no—
130C22×C4×C53C22⋊C412yes—
131C22×C4×C53C22⋊C412yes—
132C22×C4×C53C22×C411yescharacteristic
133C22×C4×C53C2×D411yescharacteristic
134C22×C4×C53C2×D411yescharacteristic
135C22×C4×C53C2×D411yescharacteristic
136C22×C4×C53C2×D411yescharacteristic
137C22×C4×C53C2×D411yescharacteristic
138C22×Dic53—22no—
139C22×Dic53—22no—
140C22×Dic53—41no—
141C22×Dic53—41no—
142C22×Dic53—41no—
143C22×Dic53—41no—
144C22×Dic53—41no—
145C22×Dic53—41no—
146C22×Dic53—41no—
147C22×Dic53—41no—
148C22×Dic53—41no—
149C22×Dic53—41no—
150C22×Dic53—41no—
151C22×Dic53—41no—
152C4⋊C4×C53—24no—
153C4⋊C4×C53—24no—
154C4⋊C4×C53—24no—
155C4⋊C4×C53—22no—
156C4⋊C4×C53—22no—
157C4⋊C4×C53—22no—
158C4⋊C4×C53—22no—
159C4⋊C4×C53—22no—
160C4⋊C4×C53—22no—
161C4⋊C4×C53—21no—
162C4⋊C4×C53—21no—
163C4⋊C4×C53C22⋊C412yes—
164C4⋊C4×C53C22⋊C412yes—
165C22⋊C4×C53—24no—
166C22⋊C4×C53—24no—
167C22⋊C4×C53—24no—
168C22⋊C4×C53—22no—
169C22⋊C4×C53—22no—
170C22⋊C4×C53—22no—
171C22⋊C4×C53—22no—
172C42×C53—24no—
173C4⋊Dic53—22no—
174C4⋊Dic53—22no—
175C4⋊Dic53—41no—
176C4⋊Dic53—41no—
177C2×#88322—41no—
178C2×#88322—41no—
179C2×#88322—41no—
180C2×#88322—41no—
181C2×#88322—41no—
182C2×#88322—41no—
183C23×C53—21no—
184C23×C53—21no—
185C23×C53—21no—
186C23×C53—21no—
187C23×C53—22no—
188C23×C53—41no—
189C23×C53—41no—
190C23×C53—41no—
191C23×C53—41no—
192C23×C53—41no—
193C23×C53—41no—
194C23×C53C2×C22⋊C411yescharacteristic
195C23×C53#12011yescharacteristic
196C23×C53#12011yescharacteristic
197C23×C53#13511yescharacteristic
198C23×C53#13511yescharacteristic
199C23×C53AGL(1,ℤ/8)11yescharacteristic
200C23×C53#13711yescharacteristic
201Q8×C53—42no—
202Q8×C53—81no—
203D4×C53—42no—
204D4×C53—42no—
205D4×C53—42no—
206D4×C53—42no—
207D4×C53—42no—
208D4×C53—81no—
209C2×C4×C53—21no—
210C2×C4×C53—21no—
211C2×C4×C53—21no—
212C2×C4×C53—21no—
213C2×C4×C53—22no—
214C2×C4×C53—22no—
215C2×C4×C53—22no—
216C2×C4×C53—22no—
217C2×C4×C53—22no—
218C2×C4×C53—22no—
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—41no—
232C2×C4×C53—41no—
233C2×C4×C53—41no—
234C2×C4×C53—41no—
235C2×C4×C53—41no—
236C2×C4×C53—42no—
237C2×C4×C53—81no—
238C2×C4×C53—84no—
239C2×C4×C53C2×C22⋊C411yescharacteristic
240C2×C4×C53C2×C22⋊C411yescharacteristic
241C2×C4×C53#12011yescharacteristic
242C2×C4×C53#12011yescharacteristic
243C2×Dic53—42no—
244C2×Dic53—42no—
245C2×Dic53—42no—
246C2×Dic53—42no—
247C2×Dic53—42no—
248C2×Dic53—42no—
249C2×Dic53—81no—
250C2×Dic53—81no—
251C2×Dic53—81no—
252C2×Dic53—81no—
253C2×Dic53—81no—
254C2×Dic53—81no—
255C2×Dic53—81no—
256C2×Dic53—81no—
257#88322—81no—
258#88322—81no—
259#8407—531nomaximal, Sylow
260C22×C53—21no—
261C22×C53—21no—
262C22×C53—21no—
263C22×C53—21no—
264C22×C53—21no—
265C22×C53—21no—
266C22×C53—21no—
267C22×C53—21no—
268C22×C53—21no—
269C22×C53—21no—
270C22×C53—21no—
271C22×C53—21no—
272C22×C53—42no—
273C22×C53—42no—
274C22×C53—42no—
275C22×C53—81no—
276C22×C53—81no—
277C22×C53—81no—
278C22×C53—81no—
279C22×C53#37911yescharacteristic
280C22×C53#41611yescharacteristic
281C22×C53#41711yescharacteristic
282C22×C53#44911yescharacteristic
283C22×C53#44911yescharacteristic
284C22×C53#45211yescharacteristic
285C22×C53#45211yescharacteristic
286C22×C53#45411yescharacteristic
287C22×C53#45611yescharacteristic
288C22×C53B(4,𝔽2)11yescharacteristic
289C22×C53#45811yescharacteristic
290C4×C53—21no—
291C4×C53—21no—
292C4×C53—21no—
293C4×C53—21no—
294C4×C53—42no—
295C4×C53—42no—
296C4×C53—42no—
297C4×C53—42no—
298C4×C53—42no—
299Dic53—82no—
300Dic53—82no—
301C2×#539D5311yescharacteristic
302#1874—531no—
303#1872—531no—
304C2×#328—531no—
305C2×#327—531no—
306C2×#324—531no—
307#1441—531no—
308C2×C53—21no—
309C2×C53—21no—
310C2×C53—21no—
311C2×C53—21no—
312C2×C53—82no—
313C2×C53#159711yescharacteristic
314C2×C53#160311yescharacteristic
315C2×C53#160911yescharacteristic
316C2×C53#161511yescharacteristic
317C2×C53#178811yescharacteristic
318C2×C53#179011yescharacteristic
319C2×C53#179311yescharacteristic
320#539—22no—
321#539—22no—
322#539—22no—
323#539Dic5314yescomplemented
324C2×#126—21no—
325C2×#125—21no—
326C2×#123—21no—
327C2×#123—21no—
328C2×C22⋊Q8—21no—
329C2×C22⋊Q8C2×D5311yescharacteristic
330C2×#121C2×D5311yescharacteristic
331C2×C4×D4—21no—
332C22×C4⋊C4—531no—
333C22×C4⋊C4C2×D5311yescharacteristic
334C2×#103—1061no—
335C2×#102—1061no—
336C2×C22⋊C8—531no—
337C2×C22⋊C8—531no—
338C2×#95—531no—
339C2×#95—1061no—
340C2×#95—1061no—
341#340—1061no—
342#340—1061no—
343#328—1062no—
344#327—1062no—
345#324—1062no—
346C53#840711yescharacteristic, complemented, Sylow
347C22×Q8C22⋊D5311yescharacteristic
348C22×D4—21no—
349C22×D4C22⋊D5311yescharacteristic
350C23×C4—21no—
351C23×C4—1061no—
352C23×C4—1061no—
353C23×C4C22⋊D5311yescharacteristic
354C23×C4C22⋊D5311yescharacteristic
355C23×C4C22×D5311yescharacteristic
356C22×C8—1061no—
357C22×C8—1061no—
358#126—24no—
359#126—24no—
360#125—24no—
361#125—24no—
362#123—24no—
363#123—24no—
364#123—24no—
365#123—24no—
366C22⋊Q8—24no—
367C22⋊Q8—24no—
368C22⋊Q8—21no—
369C22⋊Q8—21no—
370C22⋊Q8C2×Dic5312yes—
371C22⋊Q8C22⋊D5312yes—
372#121—21no—
373#121—21no—
374#121C2×Dic5312yes—
375#121C22⋊D5312yes—
376C4×D4—24no—
377C4×D4—24no—
378C2×C4⋊C4—21no—
379C2×C4⋊C4—21no—
380C2×C4⋊C4—21no—
381C2×C4⋊C4—21no—
382C2×C4⋊C4—21no—
383C2×C4⋊C4—21no—
384C2×C4⋊C4—21no—
385C2×C4⋊C4—21no—
386C2×C4⋊C4—1061no—
387C2×C4⋊C4—1061no—
388C2×C4⋊C4—2122no—
389C2×C4⋊C4C2×Dic5312yes—
390C2×C4⋊C4C22⋊D5312yes—
391C2×C4⋊C4C22⋊D5311yescharacteristic
392C2×C4⋊C4C22⋊D5311yescharacteristic
393C2×C22⋊C4—21no—
394C2×C22⋊C4—21no—
395C2×C22⋊C4—21no—
396C2×C22⋊C4—21no—
397C2×C22⋊C4—21no—
398C2×C42—21no—
399#103—2122no—
400#102—2122no—
401C22⋊C8—1061no—
402C22⋊C8—1061no—
403C22⋊C8—1061no—
404C22⋊C8—1061no—
405#95—1061no—
406#95—1061no—
407#95—2122no—
408#95—2122no—
409C24—21no—
410C24C2×C22⋊D5311yescharacteristic
411C2×Q8—21no—
412C2×Q8—41no—
413C2×Q8—41no—
414C2×Q8C22⋊Dic5312yes—
415C2×D4—22no—
416C2×D4—22no—
417C2×D4—21no—
418C2×D4—41no—
419C2×D4—41no—
420C2×D4—41no—
421C2×D4—41no—
422C2×D4—41no—
423C2×D4—41no—
424C2×D4C22⋊Dic5312yes—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—24no—
436C22×C4—21no—
437C22×C4—41no—
438C22×C4—41no—
439C22×C4—41no—
440C22×C4—41no—
441C22×C4—81no—
442C22×C4—81no—
443C22×C4—1062no—
444C22×C4—1062no—
445C22×C4—2121no—
446C22×C4—2121no—
447C22×C4—2121no—
448C22×C4—2121no—
449C22×C4—2121no—
450C22×C4—2121no—
451C22×C4—2121no—
452C22×C4—2121no—
453C22×C4—2121no—
454C22×C4—2121no—
455C22×C4—2121no—
456C22×C4—2121no—
457C22×C4C22⋊Dic5312yes—
458C22×C4C22⋊Dic5312yes—
459C22×C4C22×Dic5311yescharacteristic
460C22×C4C2×C22⋊D5311yescharacteristic
461C22×C4C2×C22⋊D5311yescharacteristic
462C22×C4C2×C22⋊D5311yescharacteristic
463C22×C4C2×C22⋊D5311yescharacteristic
464C22×C4C2×C22⋊D5311yescharacteristic
465C2×C8—2121no—
466C2×C8—2121no—
467C2×C8—2121no—
468C2×C8—2121no—
469C2×C8—2121no—
470C2×C8—2121no—
471C4⋊C4—24no—
472C4⋊C4—24no—
473C4⋊C4—24no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—1062no—
483C4⋊C4—1062no—
484C4⋊C4—2121no—
485C4⋊C4—2121no—
486C4⋊C4C22⋊Dic5312yes—
487C4⋊C4C22⋊Dic5312yes—
488C22⋊C4—24no—
489C22⋊C4—24no—
490C22⋊C4—24no—
491C22⋊C4—22no—
492C22⋊C4—22no—
493C22⋊C4—22no—
494C22⋊C4—22no—
495C42—24no—
496C23—21no—
497C23—21no—
498C23—21no—
499C23—21no—
500C23—22no—
501C23—41no—
502C23—41no—
503C23—41no—
504C23—41no—
505C23—41no—
506C23—41no—
507C23C2×C22⋊Dic5311yescharacteristic
508C23#1313318611yescharacteristic
509C23#1313318611yescharacteristic
510C23#1313323711yescharacteristic
511C23#1313323711yescharacteristic
512C23#1313324211yescharacteristic
513C23#1313324711yescharacteristic
514Q8—42no—
515Q8—81no—
516D4—42no—
517D4—42no—
518D4—42no—
519D4—42no—
520D4—42no—
521D4—81no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
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—42no—
550C2×C4—81no—
551C2×C4—84no—
552C2×C4—4241no—
553C2×C4—4241no—
554C2×C4—4241no—
555C2×C4—4241no—
556C2×C4—4241no—
557C2×C4—4241no—
558C2×C4—4241no—
559C2×C4—4241no—
560C2×C4—2122no—
561C2×C4—2122no—
562C2×C4—2122no—
563C2×C4—2122no—
564C2×C4—2122no—
565C2×C4—2122no—
566C2×C4C2×C22⋊Dic5311yescharacteristic
567C2×C4C2×C22⋊Dic5311yescharacteristic
568C2×C4#1313318611yescharacteristic
569C2×C4#1313318611yescharacteristic
570C8—4241no—
571C8—4241no—
572C22—21no—
573C22—21no—
574C22—21no—
575C22—21no—
576C22—21no—
577C22—21no—
578C22—21no—
579C22—21no—
580C22—21no—
581C22—21no—
582C22—21no—
583C22—21no—
584C22—42no—
585C22—42no—
586C22—42no—
587C22—81no—
588C22—81no—
589C22—81no—
590C22—81no—
591C22#1681047911yescharacteristic
592C22#1681061911yescharacteristic
593C22#1681062411yescharacteristic
594C22#1681076811yescharacteristic
595C22#1681076811yescharacteristic
596C22#1681078511yescharacteristic
597C22#1681078511yescharacteristic
598C22#1681079611yescharacteristic
599C22#1681080611yescharacteristic
600C22#1681081011yescharacteristic
601C22#1681081311yescharacteristic
602C4—21no—
603C4—21no—
604C4—21no—
605C4—21no—
606C4—42no—
607C4—42no—
608C4—42no—
609C4—42no—
610C4—42no—
611C4—4242no—
612C4—4242no—
613C2—21no—
614C2—21no—
615C2—21no—
616C2—21no—
617C2—82no—
618C2#2027364411yescharacteristic
619C2#2027367811yescharacteristic
620C2#2027371211yescharacteristic
621C2#2027374611yescharacteristic
622C2#2027465511yescharacteristic
623C2#2027466311yescharacteristic
624C2#2027467411yescharacteristic
subgroupG/H
centerC23#13133186
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC2×C53×#539C2
socleC23×C53#120
prime radicalC2×#121×C53C22
cosocleC23×C4×C53C23
hypercenterC2×#539D53
hypocenterC53#8407
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#539

Complements

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

NH ≅ G/Ntype
2. #20275023C2semidirect
3. C2×#16810360C2semidirect
5. C2×#16810349C2semidirect
6. #20272968C2semidirect
24. C53×#539C4semidirect
323. #539Dic53semidirect
346. C53#8407semidirect

Maximal subgroups

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

subgroupindex
#202750332
#202750232
C2×#168103602
C2×#168103572
C2×#168103492
#202729682
C2×C53×#5392
#840753