Finite Group Databasebeta

← #27413665

Subgroups — #27413665

Order 13568.

874 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810728C211yesmaximal, characteristic, complemented
2C2×#16810680C211yesmaximal, characteristic
3C4×#13133196C211yesmaximal, characteristic
4#20276216C211yesmaximal, characteristic, complemented
5#20276152C211yesmaximal, characteristic
6#20276110C211yesmaximal, characteristic, complemented
7#20276093C211yesmaximal, characteristic, complemented
8#20276037C211yesmaximal, characteristic, complemented
9#20275918C211yesmaximal, characteristic
10#20275908C211yesmaximal, characteristic
11#20275723C211yesmaximal, characteristic
12#20274613C211yesmaximal, characteristic, complemented
13#20274548C211yesmaximal, characteristic, complemented
14#20274545C211yesmaximal, characteristic, complemented
15C4×C22⋊Q8×C53C211yesmaximal, characteristic
16C22×#11719925C2211yescharacteristic
17C2×#13133196C2211yescharacteristic
18C2×C4×Dic106C2211yescharacteristic
19#16810728—24no—
20#16810680—24no—
21C2×#13133150C2211yescharacteristic
22C2×#13133148C2211yescharacteristic
23C2×#13133148C2211yescharacteristic
24C2×C4⋊#88322C2211yescharacteristic
25C2×C4⋊#88322C2211yescharacteristic
26C2×#13133141C2211yescharacteristic
27C2×#13133141C2211yescharacteristic
28C2×#13133141C2211yescharacteristic
29C2×#13133140C2211yescharacteristic
30C2×#13133140C2211yescharacteristic
31C2×#13133140C2211yescharacteristic
32C2×C22⋊#88322C2211yescharacteristic
33C2×C22⋊#88322C2211yescharacteristic
34C2×#13133128C2211yescharacteristic
35C2×#13133128C2211yescharacteristic
36#16810529C2211yescharacteristic
37#16810521C2211yescharacteristic
38#16810507—21no—
39#16810496—21no—
40#16810483C2211yescharacteristic
41C4×#11719919—21no—
42C4×C4⋊Dic53C2211yescharacteristic
43C4×C22⋊Dic53—21no—
44#16810385C2211yescharacteristic
45#16810385C2211yescharacteristic
46#16810385C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810385C2211yescharacteristic
49#16810385C2211yescharacteristic
50C2×C22⋊Q8×C53C2211yescharacteristic
51C2×C4×Q8×C53C2211yescharacteristic
52C22×C42×C53C2211yescharacteristic
53C53×#391C2211yescharacteristic
54C53×#389C2211yescharacteristic
55C53×#387—21no—
56C53×#385—21no—
57C53×#380C2211yescharacteristic
58C4×C4⋊C4×C53—21no—
59C4×C4⋊C4×C53C2211yescharacteristic
60C4×C22⋊C4×C53—21no—
61C22×Dic106C2311yescharacteristic
62C2×#11719925—22no—
63C2×#11719925—22no—
64C2×#11719925—22no—
65C22×#88322C2311yescharacteristic
66C22×#88322C2311yescharacteristic
67C22×#88322C2311yescharacteristic
68C22×#88322C2311yescharacteristic
69#13133196—22no—
70#13133196—22no—
71C4×Dic106—22no—
72C4×Dic106—22no—
73C2×C4⋊Dic53C2311yescharacteristic
74C2×C4⋊Dic53C2311yescharacteristic
75C2×C4⋊Dic53C2311yescharacteristic
76C2×#11719919—21no—
77C2×#11719919—21no—
78C2×#11719919—21no—
79C2×C22⋊Dic53—21no—
80C2×C22⋊Dic53—21no—
81C2×C4×Dic53—21no—
82C2×C4×Dic53—21no—
83#13133150—22no—
84#13133148—22no—
85#13133148—22no—
86C4⋊#88322—22no—
87C4⋊#88322—22no—
88#13133141—22no—
89#13133141—22no—
90#13133141—22no—
91#13133140—22no—
92#13133140—22no—
93#13133140—22no—
94C22⋊#88322—22no—
95C22⋊#88322—22no—
96#13133128—22no—
97#13133128D414yes—
98#13133124—21no—
99#13133124—21no—
100#13133124—21no—
101#13133124—21no—
102#13133124—21no—
103C22×Q8×C53C2311yescharacteristic
104C23×C4×C53D411yescharacteristic
105C23×C4×C53D411yescharacteristic
106C23×C4×C53C2311yescharacteristic
107C22⋊Q8×C53—22no—
108C22⋊Q8×C53—22no—
109C4×Q8×C53—22no—
110C4×Q8×C53—22no—
111C2×C4⋊C4×C53—21no—
112C2×C4⋊C4×C53—21no—
113C2×C4⋊C4×C53—21no—
114C2×C4⋊C4×C53C2311yescharacteristic
115C2×C4⋊C4×C53C2311yescharacteristic
116C2×C4⋊C4×C53C2311yescharacteristic
117C2×C22⋊C4×C53—21no—
118C2×C22⋊C4×C53—21no—
119C2×C42×C53—22no—
120C2×C42×C53—22no—
121C2×C42×C53—21no—
122C2×C42×C53—21no—
123C2×C42×C53D411yescharacteristic
124C2×C42×C53D411yescharacteristic
125C2×C42×C53C2311yescharacteristic
126C2×C42×C53C2311yescharacteristic
127#95×C53—21no—
128#95×C53—21no—
129#95×C53—21no—
130#95×C53—21no—
131#95×C53—21no—
132C24×C53C2×D411yescharacteristic
133C2×Q8×C53—21no—
134C2×Q8×C53—21no—
135C2×Q8×C53—21no—
136C2×Q8×C53—21no—
137C2×Q8×C53—21no—
138C2×Q8×C53#4112yes—
139C22×C4×C53—22no—
140C22×C4×C53—21no—
141C22×C4×C53—21no—
142C22×C4×C53—22no—
143C22×C4×C53—22no—
144C22×C4×C53—22no—
145C22×C4×C53—21no—
146C22×C4×C53—21no—
147C22×C4×C53—21no—
148C22×C4×C53—21no—
149C22×C4×C53—21no—
150C22×C4×C53—21no—
151C22×C4×C53—21no—
152C22×C4×C53—21no—
153C22×C4×C53—41no—
154C22×C4×C53—41no—
155C22×C4×C53C2×D411yescharacteristic
156C22×C4×C53C2×D411yescharacteristic
157C22×C4×C53C2×D411yescharacteristic
158C22×C4×C53C2×D411yescharacteristic
159C22×C4×C53C2×D411yescharacteristic
160C22×C4×C53C2411yescharacteristic
161C22×Dic53—21no—
162C22×Dic53—21no—
163C22×Dic53—21no—
164C22×Dic53—21no—
165C2×Dic106—21no—
166C2×Dic106—21no—
167C2×Dic106—21no—
168C2×Dic106—21no—
169C2×Dic106—21no—
170C2×Dic106#4112yes—
171C4⋊C4×C53—24no—
172C4⋊C4×C53—21no—
173C4⋊C4×C53—21no—
174C4⋊C4×C53—21no—
175C4⋊C4×C53—21no—
176C4⋊C4×C53—21no—
177C4⋊C4×C53—42no—
178C4⋊C4×C53—42no—
179C4⋊C4×C53#4112yes—
180C22⋊C4×C53—24no—
181C22⋊C4×C53—42no—
182C42×C53—22no—
183C42×C53—22no—
184C42×C53—21no—
185C42×C53—21no—
186C42×C53—42no—
187C42×C53—42no—
188C42×C53C2×D412yes—
189C42×C53#4112yes—
190C22⋊Dic53—24no—
191C22⋊Dic53—42no—
192#11719925—28no—
193C4⋊Dic53—21no—
194C4⋊Dic53—21no—
195C4⋊Dic53—21no—
196C4⋊Dic53—21no—
197C4⋊Dic53—21no—
198C4⋊Dic53#4112yes—
199#11719919—24no—
200#11719919—42no—
201#11719919—42no—
202C4×Dic53—42no—
203C4×Dic53—42no—
204C2×#88322—21no—
205C2×#88322—21no—
206C2×#88322—21no—
207C2×#88322—21no—
208C2×#88322—21no—
209C2×#88322—21no—
210C2×#88322—21no—
211C2×#88322—21no—
212C2×#88322—21no—
213C2×#88322—21no—
214C2×#88322C2×D412yes—
215C2×#88322C2×D412yes—
216C23×C53—21no—
217C23×C53—21no—
218C23×C53—21no—
219C23×C53—21no—
220C23×C53—21no—
221C23×C53—21no—
222C23×C53#13712yes—
223C23×C53C22×D411yescharacteristic
224Q8×C53—22no—
225Q8×C53—22no—
226Q8×C53—41no—
227Q8×C53—41no—
228C2×C4×C53—21no—
229C2×C4×C53—21no—
230C2×C4×C53—21no—
231C2×C4×C53—21no—
232C2×C4×C53—21no—
233C2×C4×C53—21no—
234C2×C4×C53—21no—
235C2×C4×C53—21no—
236C2×C4×C53—21no—
237C2×C4×C53—21no—
238C2×C4×C53—22no—
239C2×C4×C53—22no—
240C2×C4×C53—21no—
241C2×C4×C53—21no—
242C2×C4×C53—44no—
243C2×C4×C53—44no—
244C2×C4×C53—41no—
245C2×C4×C53—41no—
246C2×C4×C53—41no—
247C2×C4×C53—41no—
248C2×C4×C53—41no—
249C2×C4×C53—41no—
250C2×C4×C53—41no—
251C2×C4×C53—41no—
252C2×C4×C53—41no—
253C2×C4×C53—41no—
254C2×C4×C53—41no—
255C2×C4×C53—41no—
256C2×C4×C53—41no—
257C2×C4×C53—41no—
258C2×C4×C53#12112yes—
259C2×C4×C53#12112yes—
260C2×C4×C53#13712yes—
261C2×C4×C53C22×D411yescharacteristic
262C2×C4×C53C2×#4111yescharacteristic
263C2×C4×C53C2×#4111yescharacteristic
264C2×C4×C53C2×#4111yescharacteristic
265C2×C4×C53C2×#4111yescharacteristic
266C2×C4×C53C2×#4111yescharacteristic
267C2×Dic53—22no—
268C2×Dic53—22no—
269C2×Dic53—41no—
270C2×Dic53—41no—
271C2×Dic53—41no—
272C2×Dic53—41no—
273C2×Dic53—41no—
274C2×Dic53—41no—
275C2×Dic53—41no—
276C2×Dic53—41no—
277C2×Dic53—41no—
278C2×Dic53—41no—
279Dic106—22no—
280Dic106—22no—
281Dic106—41no—
282Dic106—41no—
283#88322—22no—
284#88322—22no—
285#88322—41no—
286#88322—41no—
287#23709—531nomaximal, Sylow
288C22×C53—22no—
289C22×C53—22no—
290C22×C53—22no—
291C22×C53—41no—
292C22×C53—41no—
293C22×C53—41no—
294C22×C53—41no—
295C22×C53C2×#12111yescharacteristic
296C22×C53#52511yescharacteristic
297C22×C53#52511yescharacteristic
298C22×C53C2×#13711yescharacteristic
299C22×C53C2×#13711yescharacteristic
300C22×C53#57511yescharacteristic
301C22×C53#57511yescharacteristic
302C4×C53—22no—
303C4×C53—22no—
304C4×C53—22no—
305C4×C53—42no—
306C4×C53—41no—
307C4×C53—42no—
308C4×C53—41no—
309C4×C53—81no—
310C4×C53—81no—
311C4×C53#52511yescharacteristic
312C4×C53#52911yescharacteristic
313C4×C53#52911yescharacteristic
314C4×C53#53211yescharacteristic
315Dic53—42no—
316Dic53—42no—
317Dic53—81no—
318Dic53—81no—
319C2×#441—531no—
320C2×#432—531no—
321C4×C22⋊Q8—531no—
322C4×C22⋊Q8D5311yescharacteristic
323#2088—531no—
324#2077—531no—
325#2068—531no—
326#2066—531no—
327#2056—531no—
328#2036—531no—
329#2035—531no—
330#2001—531no—
331#1776—531no—
332#1763—531no—
333#1763—531no—
334C2×C53—42no—
335C2×C53#239211yescharacteristic
336C2×C53#305611yescharacteristic
337C2×C53#305611yescharacteristic
338C2×C53#311111yescharacteristic
339C2×C53#311311yescharacteristic
340C2×C53#311611yescharacteristic
341C2×C53#311711yescharacteristic
342C22×Q16—531no—
343C2×C22⋊Q8—531no—
344C2×C22⋊Q8C2×D5311yescharacteristic
345C2×C4×Q8—531no—
346C2×C4×Q8C2×D5311yescharacteristic
347C22×C42C2×D5311yescharacteristic
348#441—1064no—
349#432—1064no—
350C2×#107—531no—
351C2×#106—531no—
352C2×#106—531no—
353C2×C4⋊C8—531no—
354C2×C4⋊C8—531no—
355C2×#103—531no—
356C2×#103—531no—
357C2×#103—531no—
358C2×#103—531no—
359C2×#103—531no—
360C2×#103—531no—
361C2×C22⋊C8—531no—
362C2×C22⋊C8—531no—
363C2×#97—531no—
364C2×#97—531no—
365#391—531no—
366#391C2×D5311yescharacteristic
367#389—531no—
368#389C2×D5311yescharacteristic
369#387—21no—
370#387—1061no—
371#385—21no—
372#385—1061no—
373#380—531no—
374#380C2×D5311yescharacteristic
375C4×C4⋊C4—21no—
376C4×C4⋊C4—531no—
377C4×C4⋊C4—1061no—
378C4×C4⋊C4C2×D5311yescharacteristic
379C4×C22⋊C4—21no—
380C4×C22⋊C4—1061no—
381#340—531no—
382#340—531no—
383#340—531no—
384#340—531no—
385#340—531no—
386#340—531no—
387C53#2370911yescharacteristic, complemented, Sylow
388C22×Q8—531no—
389C22×Q8C22×D5311yescharacteristic
390C23×C4C22⋊D5311yescharacteristic
391C23×C4C22⋊D5311yescharacteristic
392C23×C4C22×D5311yescharacteristic
393C2×Q16—1062no—
394C2×Q16—1062no—
395C2×Q16—1062no—
396C22×C8—531no—
397C22×C8—531no—
398C22×C8—531no—
399C22×C8—531no—
400C22⋊Q8—22no—
401C22⋊Q8—22no—
402C22⋊Q8—1062no—
403C22⋊Q8—1062no—
404C4×Q8—22no—
405C4×Q8—22no—
406C4×Q8—1062no—
407C4×Q8—1062no—
408C2×C4⋊C4—21no—
409C2×C4⋊C4—21no—
410C2×C4⋊C4—21no—
411C2×C4⋊C4—531no—
412C2×C4⋊C4—531no—
413C2×C4⋊C4—531no—
414C2×C4⋊C4—1061no—
415C2×C4⋊C4—1061no—
416C2×C4⋊C4—1061no—
417C2×C4⋊C4C22×D5311yescharacteristic
418C2×C4⋊C4C22×D5311yescharacteristic
419C2×C4⋊C4C22×D5311yescharacteristic
420C2×C22⋊C4—21no—
421C2×C22⋊C4—21no—
422C2×C22⋊C4—1061no—
423C2×C22⋊C4—1061no—
424C2×C42—22no—
425C2×C42—22no—
426C2×C42—21no—
427C2×C42—21no—
428C2×C42—1061no—
429C2×C42—1061no—
430C2×C42C22⋊D5311yescharacteristic
431C2×C42C22⋊D5311yescharacteristic
432C2×C42C22×D5311yescharacteristic
433C2×C42C22×D5311yescharacteristic
434#107—1062no—
435#106—1062no—
436#106—1062no—
437C4⋊C8—1062no—
438C4⋊C8—1062no—
439#103—1062no—
440#103—1062no—
441#103—1062no—
442#103—1062no—
443#103—1062no—
444#103—1062no—
445C22⋊C8—1062no—
446C22⋊C8—1062no—
447#97—534no—
448#97—1062no—
449#95—21no—
450#95—21no—
451#95—21no—
452#95—21no—
453#95—21no—
454#95—1061no—
455#95—1061no—
456#95—1061no—
457#95—1061no—
458#95—1061no—
459C24C2×C22⋊D5311yescharacteristic
460C2×Q8—21no—
461C2×Q8—21no—
462C2×Q8—21no—
463C2×Q8—21no—
464C2×Q8—21no—
465C2×Q8—532no—
466C2×Q8—1061no—
467C2×Q8—1061no—
468C2×Q8—1061no—
469C2×Q8—1061no—
470C2×Q8—1061no—
471C2×Q8#1171994512yes—
472C22×C4—22no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—41no—
487C22×C4—41no—
488C22×C4—1061no—
489C22×C4—1061no—
490C22×C4—1061no—
491C22×C4—1061no—
492C22×C4C2×C22⋊D5311yescharacteristic
493C22×C4C2×C22⋊D5311yescharacteristic
494C22×C4C2×C22⋊D5311yescharacteristic
495C22×C4C2×C22⋊D5311yescharacteristic
496C22×C4C2×C22⋊D5311yescharacteristic
497C22×C4C23×D5311yescharacteristic
498Q16—1068no—
499C2×C8—532no—
500C2×C8—532no—
501C2×C8—1061no—
502C2×C8—1061no—
503C2×C8—1061no—
504C2×C8—1061no—
505C2×C8—1061no—
506C2×C8—1061no—
507C2×C8—1061no—
508C2×C8—1061no—
509C2×C8—1061no—
510C2×C8—1061no—
511C4⋊C4—24no—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—42no—
518C4⋊C4—42no—
519C4⋊C4—532no—
520C4⋊C4—1064no—
521C4⋊C4—1061no—
522C4⋊C4—1061no—
523C4⋊C4—1061no—
524C4⋊C4—1061no—
525C4⋊C4—1061no—
526C4⋊C4—2122no—
527C4⋊C4—2122no—
528C4⋊C4#1171994512yes—
529C22⋊C4—24no—
530C22⋊C4—42no—
531C22⋊C4—1064no—
532C22⋊C4—2122no—
533C42—22no—
534C42—22no—
535C42—21no—
536C42—21no—
537C42—42no—
538C42—42no—
539C42—2122no—
540C42—2122no—
541C42D4×D5312yes—
542C42#1171994712yes—
543C23—21no—
544C23—21no—
545C23—21no—
546C23—21no—
547C23—21no—
548C23—21no—
549C23#1313324812yes—
550C23C22×C22⋊D5311yescharacteristic
551Q8—22no—
552Q8—22no—
553Q8—41no—
554Q8—41no—
555Q8—1062no—
556Q8—1062no—
557Q8—2121no—
558Q8—2121no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—1062no—
590C2×C4—1062no—
591C2×C4—2121no—
592C2×C4—2121no—
593C2×C4—2121no—
594C2×C4—2121no—
595C2×C4—2121no—
596C2×C4—2121no—
597C2×C4—2121no—
598C2×C4—2121no—
599C2×C4—2121no—
600C2×C4—2121no—
601C2×C4#1313318912yes—
602C2×C4#1313318912yes—
603C2×C4#1313324712yes—
604C2×C4C2×D4×D5311yescharacteristic
605C2×C4#41×D5311yescharacteristic
606C2×C4#41×D5311yescharacteristic
607C2×C4C2×#1171994711yescharacteristic
608C2×C4C2×#1171994511yescharacteristic
609C2×C4C2×#1171994511yescharacteristic
610C8—1062no—
611C8—1062no—
612C8—2121no—
613C8—2121no—
614C22—22no—
615C22—22no—
616C22—22no—
617C22—41no—
618C22—41no—
619C22—41no—
620C22—41no—
621C22C2×#1313318911yescharacteristic
622C22#1681113011yescharacteristic
623C22#1681113411yescharacteristic
624C22C2×#1313324711yescharacteristic
625C22C2×#1313324811yescharacteristic
626C22#1681142111yescharacteristic
627C22#1681142111yescharacteristic
628C4—22no—
629C4—22no—
630C4—22no—
631C4—41no—
632C4—41no—
633C4—42no—
634C4—42no—
635C4—81no—
636C4—81no—
637C4—4241no—
638C4—4241no—
639C4—2122no—
640C4—2122no—
641C4#1681113211yescharacteristic
642C4#1681115311yescharacteristic
643C4#1681115511yescharacteristic
644C4#1681117011yescharacteristic
645C2—42no—
646C2#2027795711yescharacteristic
647C2#2028434111yescharacteristic
648C2#2028434711yescharacteristic
649C2#2028481811yescharacteristic
650C2#2028484111yescharacteristic
651C2#2028487711yescharacteristic
652C2#2028488611yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC4×C22⋊Q8×C53C2
socleC23×C53C22×D4
prime radicalC24×C53C2×D4
hypercenterC4×C22⋊Q8D53
hypocenterC53#23709
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C22×C42 ◁ C4×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×#16810728C2semidirect
4. #20276216C2semidirect
6. #20276110C2semidirect
7. #20276093C2semidirect
8. #20276037C2semidirect
12. #20274613C2semidirect
13. #20274548C2semidirect
14. #20274545C2semidirect
387. C53#23709semidirect

Maximal subgroups

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

subgroupindex
C2×#168107282
C2×#168106802
C4×#131331962
#202762162
#202761522
#202761102
#202760932
#202760372
#202759182
#202759082
#202757232
#202746132
#202745482
#202745452
C4×C22⋊Q8×C532
#2370953