Finite Group Databasebeta

← #27428666

Subgroups — #27428666

Order 13568.

800 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20279012C211yesmaximal, characteristic
2#20276277C211yesmaximal, characteristic, complemented
3#20276192C211yesmaximal, characteristic, complemented
4#20276151C211yesmaximal, characteristic, complemented
5#20276141C211yesmaximal, characteristic, complemented
6#20275307C211yesmaximal, characteristic, complemented
7#20275293C211yesmaximal, characteristic, complemented
8#20275013C211yesmaximal, characteristic
9#20274908C211yesmaximal, characteristic, complemented
10#20274908C211yesmaximal, characteristic, complemented
11#20274807C211yesmaximal, characteristic, complemented
12#20274801C211yesmaximal, characteristic, complemented
13#20274746C211yesmaximal, characteristic, complemented
14#20274434C211yesmaximal, characteristic, complemented
15D4×C4⋊C4×C53C211yesmaximal, characteristic
16C22×#11719917C2211yescharacteristic
17C2×#13133176C2211yescharacteristic
18C2×#13133150C2211yescharacteristic
19C2×#13133148C2211yescharacteristic
20C2×#13133148C2211yescharacteristic
21C2×#13133148C2211yescharacteristic
22C2×C22⋊#88322C2211yescharacteristic
23C2×C22⋊#88322C2211yescharacteristic
24C2×C22⋊#88322C2211yescharacteristic
25#16810551—21no—
26#16810537—21no—
27#16810521C2211yescharacteristic
28#16810521C2211yescharacteristic
29#16810511—21no—
30#16810504—21no—
31#16810483C2211yescharacteristic
32#16810483C2211yescharacteristic
33#16810483C2211yescharacteristic
34#16810385C2211yescharacteristic
35#16810385C2211yescharacteristic
36#16810385C2211yescharacteristic
37#16810385C2211yescharacteristic
38#16810385C2211yescharacteristic
39#16810385C2211yescharacteristic
40#16810385C2211yescharacteristic
41#16810385C2211yescharacteristic
42#16810385C2211yescharacteristic
43#16810385C2211yescharacteristic
44#16810385C2211yescharacteristic
45#16810385C2211yescharacteristic
46#16810378C2211yescharacteristic
47#16810378C2211yescharacteristic
48C2×C4×D4×C53—21no—
49C2×C4×D4×C53C2211yescharacteristic
50C22×C4⋊C4×C53C2211yescharacteristic
51C22×C4⋊C4×C53C2211yescharacteristic
52C53×#389—21no—
53C53×#385—21no—
54C53×#385—21no—
55C53×#384C2211yescharacteristic
56C53×#380C2211yescharacteristic
57C53×#380C2211yescharacteristic
58C4×C4⋊C4×C53C2211yescharacteristic
59C2×#11719917—22no—
60C2×#11719917—22no—
61C2×#11719917—41no—
62C22×#88322C2311yescharacteristic
63C22×#88322C2311yescharacteristic
64C22×#88322C2311yescharacteristic
65C22×#88322C2311yescharacteristic
66#13133176—22no—
67#13133176—22no—
68C2×C4⋊Dic53C2311yescharacteristic
69C2×C4⋊Dic53C2311yescharacteristic
70C2×C4⋊Dic53C2311yescharacteristic
71C2×C4⋊Dic53C2311yescharacteristic
72C2×#11719919—21no—
73C2×#11719919—21no—
74C2×C22⋊Dic53—21no—
75C2×C22⋊Dic53—21no—
76C2×C22⋊Dic53—21no—
77C2×C22⋊Dic53—21no—
78C2×C4×Dic53—21no—
79#13133150—22no—
80#13133148—22no—
81#13133148—41no—
82#13133148—41no—
83C22⋊#88322—22no—
84C22⋊#88322—22no—
85C22⋊#88322—41no—
86#13133124—21no—
87#13133124—21no—
88#13133124—21no—
89#13133124—21no—
90#13133124—21no—
91C22×D4×C53D411yescharacteristic
92C23×C4×C53—21no—
93C23×C4×C53—21no—
94C23×C4×C53C2311yescharacteristic
95C23×C4×C53C2311yescharacteristic
96C4×D4×C53—22no—
97C4×D4×C53—22no—
98C4×D4×C53—44no—
99C2×C4⋊C4×C53—21no—
100C2×C4⋊C4×C53—21no—
101C2×C4⋊C4×C53—21no—
102C2×C4⋊C4×C53—21no—
103C2×C4⋊C4×C53—22no—
104C2×C4⋊C4×C53—22no—
105C2×C4⋊C4×C53—41no—
106C2×C4⋊C4×C53—41no—
107C2×C4⋊C4×C53D411yescharacteristic
108C2×C4⋊C4×C53C2311yescharacteristic
109C2×C4⋊C4×C53C2311yescharacteristic
110C2×C4⋊C4×C53C2311yescharacteristic
111C2×C4⋊C4×C53C2311yescharacteristic
112C2×C22⋊C4×C53—21no—
113C2×C22⋊C4×C53—21no—
114C2×C22⋊C4×C53D411yescharacteristic
115C2×C22⋊C4×C53D411yescharacteristic
116C2×C42×C53—21no—
117C2×C42×C53C2311yescharacteristic
118#95×C53—21no—
119#95×C53—21no—
120#95×C53—21no—
121C24×C53C2×D411yescharacteristic
122C24×C53C2×D411yescharacteristic
123C2×D4×C53—22no—
124C2×D4×C53—22no—
125C2×D4×C53—22no—
126C22×C4×C53—21no—
127C22×C4×C53—21no—
128C22×C4×C53—21no—
129C22×C4×C53—21no—
130C22×C4×C53—21no—
131C22×C4×C53—21no—
132C22×C4×C53—21no—
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—42no—
141C22×C4×C53—42no—
142C22×C4×C53—42no—
143C22×C4×C53—42no—
144C22×C4×C53—81no—
145C22×C4×C53—81no—
146C22×C4×C53C2×D411yescharacteristic
147C22×C4×C53C2×D411yescharacteristic
148C22×C4×C53C2×D411yescharacteristic
149C22×C4×C53C2×D411yescharacteristic
150C22×C4×C53#4112yes—
151C22×C4×C53#4112yes—
152C22×C4×C53C2411yescharacteristic
153C22×Dic53—21no—
154C22×Dic53—21no—
155C22×Dic53—21no—
156C22×Dic53—21no—
157C4⋊C4×C53—22no—
158C4⋊C4×C53—21no—
159C4⋊C4×C53—21no—
160C4⋊C4×C53—21no—
161C4⋊C4×C53—21no—
162C4⋊C4×C53—21no—
163C4⋊C4×C53—21no—
164C4⋊C4×C53—21no—
165C4⋊C4×C53—42no—
166C4⋊C4×C53—42no—
167C4⋊C4×C53—42no—
168C4⋊C4×C53—81no—
169C4⋊C4×C53#4112yes—
170C22⋊C4×C53—22no—
171C22⋊C4×C53—22no—
172C22⋊C4×C53—42no—
173C22⋊C4×C53—42no—
174C42×C53—21no—
175C42×C53—21no—
176C42×C53—42no—
177C22⋊Dic53—42no—
178C22⋊Dic53—42no—
179C22⋊Dic53—81no—
180C22⋊Dic53—81no—
181C4⋊Dic53—21no—
182C4⋊Dic53—21no—
183C4⋊Dic53—21no—
184C4⋊Dic53—21no—
185C4⋊Dic53—21no—
186C4⋊Dic53—21no—
187C4⋊Dic53—21no—
188C4⋊Dic53—21no—
189#11719919—42no—
190#11719919—81no—
191C4×Dic53—42no—
192#11719917—44no—
193C2×#88322—21no—
194C2×#88322—21no—
195C2×#88322—21no—
196C2×#88322—21no—
197C2×#88322—21no—
198C2×#88322—21no—
199C2×#88322—21no—
200C2×#88322—21no—
201C2×#88322—21no—
202C2×#88322—21no—
203C2×#88322—21no—
204C2×#88322#4112yes—
205C23×C53—21no—
206C23×C53—21no—
207C23×C53—21no—
208C23×C53—21no—
209C23×C53—21no—
210C23×C53—21no—
211C23×C53—21no—
212C23×C53—21no—
213C23×C53—21no—
214C23×C53—21no—
215C23×C53—21no—
216C23×C53—21no—
217C23×C53#12312yes—
218C23×C53AGL(1,ℤ/8)12yes—
219C23×C53C22×D411yescharacteristic
220D4×C53—28no—
221C2×C4×C53—21no—
222C2×C4×C53—21no—
223C2×C4×C53—21no—
224C2×C4×C53—21no—
225C2×C4×C53—22no—
226C2×C4×C53—22no—
227C2×C4×C53—22no—
228C2×C4×C53—22no—
229C2×C4×C53—21no—
230C2×C4×C53—21no—
231C2×C4×C53—21no—
232C2×C4×C53—22no—
233C2×C4×C53—21no—
234C2×C4×C53—21no—
235C2×C4×C53—21no—
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—41no—
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—84no—
252C2×C4×C53—84no—
253C2×C4×C53#12312yes—
254C2×C4×C53#13712yes—
255C2×C4×C53C2×#4111yescharacteristic
256C2×C4×C53C2×#4111yescharacteristic
257C2×C4×C53C2×#4111yescharacteristic
258C2×C4×C53C2×#4111yescharacteristic
259C2×C4×C53H(5,𝔽2)11yescharacteristic
260C2×C4×C532−1+411yescharacteristic
261C2×Dic53—41no—
262C2×Dic53—41no—
263C2×Dic53—41no—
264C2×Dic53—41no—
265C2×Dic53—41no—
266C2×Dic53—41no—
267C2×Dic53—41no—
268C2×Dic53—41no—
269C2×Dic53—41no—
270C2×Dic53—41no—
271C2×Dic53—41no—
272C2×Dic53—41no—
273#88322—22no—
274#88322—41no—
275#88322—41no—
276#88322—41no—
277#24949—531nomaximal, Sylow
278C22×C53—22no—
279C22×C53—22no—
280C22×C53—22no—
281C22×C53—22no—
282C22×C53—22no—
283C22×C53—22no—
284C22×C53—41no—
285C22×C53—41no—
286C22×C53—41no—
287C22×C53—41no—
288C22×C53—41no—
289C22×C53—41no—
290C22×C53—41no—
291C22×C53—41no—
292C22×C53C2×#12311yescharacteristic
293C22×C53#54611yescharacteristic
294C22×C53#54711yescharacteristic
295C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
296C22×C53C2×#13711yescharacteristic
297C22×C53#57711yescharacteristic
298C22×C53#57711yescharacteristic
299C4×C53—22no—
300C4×C53—22no—
301C4×C53—41no—
302C4×C53—41no—
303C4×C53—42no—
304C4×C53—81no—
305C4×C53—81no—
306C4×C53—81no—
307C4×C53#52911yescharacteristic
308C4×C53#53911yescharacteristic
309C4×C53#55111yescharacteristic
310C4×C53#55211yescharacteristic
311Dic53—81no—
312Dic53—81no—
313Dic53—81no—
314Dic53—81no—
315#2485—531no—
316D4×C4⋊C4D5311yescharacteristic
317#2100—531no—
318#2084—531no—
319#2077—531no—
320#2075—531no—
321#1921—531no—
322#1919—531no—
323#1870—531no—
324#1853—531no—
325#1853—531no—
326#1825—531no—
327#1823—531no—
328#1809—531no—
329#1736—531no—
330C2×C53—42no—
331C2×C53—42no—
332C2×C53#241911yescharacteristic
333C2×C53#309211yescharacteristic
334C2×C53#309911yescharacteristic
335C2×C53#331011yescharacteristic
336C2×C53#332911yescharacteristic
337C2×C53#334511yescharacteristic
338C2×C53#335611yescharacteristic
339C22×M16—531no—
340C2×C4×D4—21no—
341C2×C4×D4C2×D5311yescharacteristic
342C2×#117—531no—
343C22×C4⋊C4C2×D5311yescharacteristic
344C22×C4⋊C4C2×D5311yescharacteristic
345C2×#107—531no—
346C2×#106—531no—
347C2×#106—531no—
348C2×#106—531no—
349C2×C22⋊C8—531no—
350C2×C22⋊C8—531no—
351C2×C22⋊C8—531no—
352#397—1061no—
353#394—1061no—
354#389—21no—
355#389—531no—
356#389—531no—
357#388—1061no—
358#387—1061no—
359#385—21no—
360#385—21no—
361#384C2×D5311yescharacteristic
362#380—531no—
363#380—531no—
364#380—531no—
365#380C2×D5311yescharacteristic
366#380C2×D5311yescharacteristic
367C4×C4⋊C4C2×D5311yescharacteristic
368#340—531no—
369#340—531no—
370#340—531no—
371#340—531no—
372#340—531no—
373#340—531no—
374#340—531no—
375#340—531no—
376#340—531no—
377#340—531no—
378#340—531no—
379#340—531no—
380#336—531no—
381#336—531no—
382C53#2494911yescharacteristic, complemented, Sylow
383C22×D4C22⋊D5311yescharacteristic
384C23×C4—21no—
385C23×C4—21no—
386C23×C4C22×D5311yescharacteristic
387C23×C4C22×D5311yescharacteristic
388C2×M16—1062no—
389C2×M16—1062no—
390C2×M16—2121no—
391C22×C8—531no—
392C22×C8—531no—
393C22×C8—531no—
394C22×C8—531no—
395C4×D4—22no—
396C4×D4—22no—
397C4×D4—44no—
398#117—1062no—
399#117—1062no—
400C2×C4⋊C4—22no—
401C2×C4⋊C4—22no—
402C2×C4⋊C4—21no—
403C2×C4⋊C4—21no—
404C2×C4⋊C4—21no—
405C2×C4⋊C4—21no—
406C2×C4⋊C4—41no—
407C2×C4⋊C4—41no—
408C2×C4⋊C4—531no—
409C2×C4⋊C4—531no—
410C2×C4⋊C4—531no—
411C2×C4⋊C4—531no—
412C2×C4⋊C4—1061no—
413C2×C4⋊C4—1061no—
414C2×C4⋊C4C22⋊D5311yescharacteristic
415C2×C4⋊C4C22×D5311yescharacteristic
416C2×C4⋊C4C22×D5311yescharacteristic
417C2×C4⋊C4C22×D5311yescharacteristic
418C2×C4⋊C4C22×D5311yescharacteristic
419C2×C22⋊C4—21no—
420C2×C22⋊C4—21no—
421C2×C22⋊C4—1061no—
422C2×C22⋊C4—1061no—
423C2×C22⋊C4—1061no—
424C2×C22⋊C4—1061no—
425C2×C22⋊C4C22⋊D5311yescharacteristic
426C2×C22⋊C4C22⋊D5311yescharacteristic
427C2×C42—21no—
428C2×C42—1061no—
429C2×C42C22×D5311yescharacteristic
430#107—1062no—
431#106—1062no—
432#106—2121no—
433#106—2121no—
434C22⋊C8—1062no—
435C22⋊C8—1062no—
436C22⋊C8—2121no—
437#95—21no—
438#95—21no—
439#95—21no—
440#95—1061no—
441#95—1061no—
442#95—1061no—
443#95—1061no—
444#95—1061no—
445C24C2×C22⋊D5311yescharacteristic
446C24C2×C22⋊D5311yescharacteristic
447C2×D4—22no—
448C2×D4—22no—
449C2×D4—22no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4—42no—
468C22×C4—81no—
469C22×C4—81no—
470C22×C4—1061no—
471C22×C4—1061no—
472C22×C4—1061no—
473C22×C4—1061no—
474C22×C4#1171994512yes—
475C22×C4#1171994512yes—
476C22×C4C2×C22⋊D5311yescharacteristic
477C22×C4C2×C22⋊D5311yescharacteristic
478C22×C4C2×C22⋊D5311yescharacteristic
479C22×C4C2×C22⋊D5311yescharacteristic
480C22×C4C23×D5311yescharacteristic
481M16—2124no—
482C2×C8—532no—
483C2×C8—1061no—
484C2×C8—1061no—
485C2×C8—1061no—
486C2×C8—1061no—
487C2×C8—1061no—
488C2×C8—1061no—
489C2×C8—1061no—
490C2×C8—1061no—
491C2×C8—1061no—
492C2×C8—1061no—
493C2×C8—1061no—
494C4⋊C4—22no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—42no—
503C4⋊C4—42no—
504C4⋊C4—42no—
505C4⋊C4—81no—
506C4⋊C4—1061no—
507C4⋊C4—1061no—
508C4⋊C4—1061no—
509C4⋊C4—1061no—
510C4⋊C4—1061no—
511C4⋊C4—1061no—
512C4⋊C4—1061no—
513C4⋊C4—1061no—
514C4⋊C4—4241no—
515C4⋊C4—2122no—
516C4⋊C4#1171994712yes—
517C22⋊C4—22no—
518C22⋊C4—22no—
519C22⋊C4—42no—
520C22⋊C4—42no—
521C22⋊C4—4241no—
522C22⋊C4—4241no—
523C22⋊C4—2122no—
524C22⋊C4—2122no—
525C42—21no—
526C42—21no—
527C42—42no—
528C42—2122no—
529C23—21no—
530C23—21no—
531C23—21no—
532C23—21no—
533C23—21no—
534C23—21no—
535C23—21no—
536C23—21no—
537C23—21no—
538C23—21no—
539C23—21no—
540C23—21no—
541C23#1313320012yes—
542C23#1313324312yes—
543C23C22×C22⋊D5311yescharacteristic
544D4—28no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—84no—
576C2×C4—84no—
577C2×C4—2121no—
578C2×C4—2121no—
579C2×C4—2121no—
580C2×C4—2121no—
581C2×C4—2121no—
582C2×C4—2121no—
583C2×C4—2121no—
584C2×C4—2121no—
585C2×C4—2121no—
586C2×C4—2121no—
587C2×C4—2121no—
588C2×C4—2121no—
589C2×C4#1313320012yes—
590C2×C4#1313324812yes—
591C2×C4#41×D5311yescharacteristic
592C2×C4C2×#1171994711yescharacteristic
593C2×C4C2×#1171994511yescharacteristic
594C2×C4C2×#1171994511yescharacteristic
595C2×C4#1313326211yescharacteristic
596C2×C4#1313326411yescharacteristic
597C8—1062no—
598C8—2121no—
599C8—2121no—
600C8—2121no—
601C22—22no—
602C22—22no—
603C22—22no—
604C22—22no—
605C22—22no—
606C22—22no—
607C22—41no—
608C22—41no—
609C22—41no—
610C22—41no—
611C22—41no—
612C22—41no—
613C22—41no—
614C22—41no—
615C22C2×#1313320011yescharacteristic
616C22#1681126711yescharacteristic
617C22#1681127211yescharacteristic
618C22C2×#1313324311yescharacteristic
619C22C2×#1313324811yescharacteristic
620C22#1681143111yescharacteristic
621C22#1681143411yescharacteristic
622C4—22no—
623C4—22no—
624C4—42no—
625C4—41no—
626C4—41no—
627C4—81no—
628C4—81no—
629C4—81no—
630C4—4241no—
631C4—4241no—
632C4—4241no—
633C4—4241no—
634C4#1681115311yescharacteristic
635C4#1681121511yescharacteristic
636C4#1681130211yescharacteristic
637C4#1681131311yescharacteristic
638C2—42no—
639C2—42no—
640C2#2027818611yescharacteristic
641C2#2028463811yescharacteristic
642C2#2028470411yescharacteristic
643C2#2028673611yescharacteristic
644C2#2028700911yescharacteristic
645C2#2028718511yescharacteristic
646C2#2028733011yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingD4×C4⋊C4×C53C2
socleC23×C53C22×D4
prime radicalC22×D4×C53D4
hypercenterD4×C4⋊C4D53
hypocenterC53#24949
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ D4×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. #20276277C2semidirect
3. #20276192C2semidirect
4. #20276151C2semidirect
5. #20276141C2semidirect
6. #20275307C2semidirect
7. #20275293C2semidirect
9. #20274908C2semidirect
10. #20274908C2semidirect
11. #20274807C2semidirect
12. #20274801C2semidirect
13. #20274746C2semidirect
14. #20274434C2semidirect
382. C53#24949semidirect

Maximal subgroups

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

subgroupindex
#202790122
#202762772
#202761922
#202761512
#202761412
#202753072
#202752932
#202750132
#202749082
#202748072
#202748012
#202747462
#202744342
D4×C4⋊C4×C532
#2494953