Finite Group Databasebeta

← #28323904

Subgroups — #28323904

Order 13568.

710 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20290837C211yesmaximal, characteristic
2#20290814C211yesmaximal, characteristic, complemented
3#20290791C211yesmaximal, characteristic, complemented
4#20290791C211yesmaximal, characteristic, complemented
5#20290787C211yesmaximal, characteristic, complemented
6#20290786C211yesmaximal, characteristic, complemented
7#20290769C211yesmaximal, characteristic, complemented
8#20290769C211yesmaximal, characteristic, complemented
9#20290695C211yesmaximal, characteristic
10#20290661C211yesmaximal, characteristic, complemented
11#20290574C211yesmaximal, characteristic, complemented
12#20290514C211yesmaximal, characteristic, complemented
13#20290456C211yesmaximal, characteristic, complemented
14#20290444C211yesmaximal, characteristic, complemented
15#20277981C211yesmaximal, characteristic
16C22×#11719936C2211yescharacteristic
17C2×#13133284C2211yescharacteristic
18C2×#13133284C2211yescharacteristic
19C2×C4⋊#88324C2211yescharacteristic
20C2×C4⋊#88324C2211yescharacteristic
21C2×C4⋊#88324C2211yescharacteristic
22C2×#13133274C2211yescharacteristic
23C2×#13133274C2211yescharacteristic
24C2×#13133274C2211yescharacteristic
25C2×#13133274C2211yescharacteristic
26C2×#13133274C2211yescharacteristic
27C2×#13133274C2211yescharacteristic
28C2×#13133274C2211yescharacteristic
29C2×#13133272C2211yescharacteristic
30C2×#13133272C2211yescharacteristic
31C2×C4×#88324C2211yescharacteristic
32C2×C4×#88324C2211yescharacteristic
33C2×C4×#88324C2211yescharacteristic
34#16811489C2211yescharacteristic
35#16811489C2211yescharacteristic
36#16811489C2211yescharacteristic
37#16811489C2211yescharacteristic
38#16811489C2211yescharacteristic
39#16811489C2211yescharacteristic
40#16811489C2211yescharacteristic
41#16811489C2211yescharacteristic
42#16811489C2211yescharacteristic
43#16811489C2211yescharacteristic
44C2×#13133179C2211yescharacteristic
45C2×#13133175C2211yescharacteristic
46#16810558C411yescharacteristic
47#16810553C411yescharacteristic
48#16810552C411yescharacteristic
49#16810552C411yescharacteristic
50#16810550C411yescharacteristic
51#16810541C411yescharacteristic
52#16810541C411yescharacteristic
53#16810474C2211yescharacteristic
54#16810474C2211yescharacteristic
55#16810470C2211yescharacteristic
56#16810470C2211yescharacteristic
57#16810470C2211yescharacteristic
58C53×#400C411yescharacteristic
59C2×#11719936—22no—
60C2×#11719936—41no—
61C2×#11719936—41no—
62C22×#88324C2311yescharacteristic
63C22×#88324C2311yescharacteristic
64C22×#88324C2311yescharacteristic
65C22×#88324C2311yescharacteristic
66C22×#88324C2311yescharacteristic
67C22×#88324C2311yescharacteristic
68C22×#88324C2311yescharacteristic
69C22×#88324C2311yescharacteristic
70#13133284—41no—
71#13133284—41no—
72C4⋊#88324—22no—
73C4⋊#88324—41no—
74C4⋊#88324—41no—
75#13133274—22no—
76#13133274—22no—
77#13133274—22no—
78#13133274—41no—
79#13133274—41no—
80#13133274—41no—
81#13133274—41no—
82#13133272—22no—
83#13133272—41no—
84C4×#88324—41no—
85C4×#88324—41no—
86C4×#88324—41no—
87C22×D212C2×C411yescharacteristic
88C22×C4×D53C2311yescharacteristic
89C22×C4×D53C2311yescharacteristic
90#13133179—41no—
91#13133179—41no—
92#13133175—41no—
93#13133175—41no—
94C2×C4⋊Dic53C2×C411yescharacteristic
95C2×#11719919C2×C411yescharacteristic
96C2×#11719919C2×C411yescharacteristic
97C2×#11719919C2×C411yescharacteristic
98C2×#11719919C2×C411yescharacteristic
99C2×#11719921C2×C411yescharacteristic
100C2×#11719921C2×C411yescharacteristic
101C2×#11719921C2×C411yescharacteristic
102C2×#11719921C2×C411yescharacteristic
103C2×#11719921C2×C411yescharacteristic
104C2×#11719921C2×C411yescharacteristic
105C2×#11719921C2×C411yescharacteristic
106C2×#11719921C2×C411yescharacteristic
107C2×#11719921C2×C411yescharacteristic
108C2×#11719921C2×C411yescharacteristic
109C2×C4×Dic53C2311yescharacteristic
110C2×C4×Dic53C2311yescharacteristic
111C2×C4×Dic53C2311yescharacteristic
112C2×C4×Dic53C2311yescharacteristic
113C2×C4×Dic53C2311yescharacteristic
114#13133124C2×C411yescharacteristic
115#13133124C2×C411yescharacteristic
116#13133124C2×C411yescharacteristic
117#13133124C2×C411yescharacteristic
118#13133124C2×C411yescharacteristic
119C2×C4⋊C4×C53C2×C411yescharacteristic
120C2×C4⋊C4×C53C2×C411yescharacteristic
121#95×C53C2×C411yescharacteristic
122#95×C53C2×C411yescharacteristic
123#95×C53C2×C411yescharacteristic
124#95×C53C2×C411yescharacteristic
125#95×C53C2×C411yescharacteristic
126C23×D53C22×C411yescharacteristic
127C23×D53C22×C411yescharacteristic
128C22×C4×C53C22×C411yescharacteristic
129C22×C4×C53C22×C411yescharacteristic
130C22×C4×C53C22×C411yescharacteristic
131C22×C4×C53C22×C411yescharacteristic
132C22×C4×C53C22×C411yescharacteristic
133C22×C4×C53C22×C411yescharacteristic
134C22×C4×C53C22×C411yescharacteristic
135C22×Dic53C22×C411yescharacteristic
136C22×Dic53C22×C411yescharacteristic
137C22×Dic53C22×C411yescharacteristic
138C22×Dic53C22×C411yescharacteristic
139C22×Dic53C22×C411yescharacteristic
140C22×Dic53C2411yescharacteristic
141C2×D212—22no—
142C2×D212—41no—
143C2×D212—41no—
144C2×C4×D53—21no—
145C2×C4×D53—21no—
146C2×C4×D53—21no—
147C2×C4×D53—21no—
148C2×C4×D53—21no—
149C2×C4×D53—21no—
150C2×C4×D53—21no—
151C2×C4×D53—21no—
152C2×C4×D53—21no—
153C2×C4×D53—21no—
154C2×C4×D53—21no—
155C2×C4×D53—21no—
156C2×#88324—21no—
157C2×#88324—21no—
158C2×#88324—21no—
159C2×#88324—21no—
160C2×#88324—21no—
161C2×#88324—21no—
162C2×#88324—21no—
163C2×#88324—21no—
164C2×#88324—21no—
165C2×#88324—21no—
166C2×#88324—21no—
167C2×#88324—21no—
168C2×#88324—21no—
169C2×#88324—21no—
170C2×#88324—21no—
171C2×#88324—21no—
172C2×#88324—21no—
173C2×#88324—21no—
174C2×#88324—21no—
175C2×#88324—21no—
176C2×#88324—21no—
177C2×#88324—21no—
178C2×#88324#4112yes—
179C2×#88324#4112yes—
180#11719936—44no—
181C4⋊C4×C53—41no—
182C4⋊C4×C53—41no—
183#11719921—22no—
184#11719921—22no—
185#11719921—41no—
186#11719921—41no—
187#11719921—41no—
188#11719921—41no—
189#11719921—41no—
190#11719921—41no—
191#11719921—41no—
192#11719921—41no—
193C4⋊Dic53—41no—
194#11719919—22no—
195#11719919—22no—
196#11719919—22no—
197#11719919—41no—
198C4×Dic53—21no—
199C4×Dic53—21no—
200C4×Dic53—21no—
201C4×Dic53—21no—
202C4×Dic53—21no—
203C4×Dic53—21no—
204C4×Dic53—21no—
205C4×Dic53—21no—
206C4×Dic53—21no—
207C4×Dic53—21no—
208C23×C53C23×C411yescharacteristic
209C22×D53—21no—
210C22×D53—21no—
211C22×D53—21no—
212C22×D53—21no—
213C22×D53—21no—
214C22×D53—21no—
215C22×D53—21no—
216C22×D53—21no—
217C22×D53—21no—
218C22×D53—21no—
219C22×D53—21no—
220C22×D53—21no—
221C22×D53—21no—
222C22×D53—21no—
223C2×C4×C53—21no—
224C2×C4×C53—21no—
225C2×C4×C53—21no—
226C2×C4×C53—21no—
227C2×C4×C53—21no—
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—21no—
239C2×C4×C53—21no—
240C2×C4×C53—21no—
241C2×C4×C53—21no—
242C2×C4×C53—21no—
243C2×C4×C53#13112yes—
244C2×Dic53—21no—
245C2×Dic53—21no—
246C2×Dic53—21no—
247C2×Dic53—21no—
248C2×Dic53—21no—
249C2×Dic53—21no—
250C2×Dic53—21no—
251C2×Dic53—21no—
252C2×Dic53—21no—
253C2×Dic53—21no—
254C2×Dic53—21no—
255C2×Dic53—21no—
256C2×Dic53—21no—
257C2×Dic53—21no—
258C2×Dic53#13112yes—
259C2×Dic53C2×#4111yescharacteristic
260C2×Dic53C2×#4111yescharacteristic
261C2×Dic53H(5,𝔽2)11yescharacteristic
262C2×Dic53H(5,𝔽2)11yescharacteristic
263C2×Dic53H(5,𝔽2)11yescharacteristic
264C2×Dic532−1+411yescharacteristic
265D212—44no—
266C4×D53—41no—
267C4×D53—41no—
268C4×D53—41no—
269C4×D53—41no—
270C4×D53—41no—
271C4×D53—41no—
272C4×D53—41no—
273C4×D53—41no—
274#88324—22no—
275#88324—22no—
276#88324—41no—
277#88324—41no—
278#88324—41no—
279#88324—41no—
280#88324—41no—
281#88324—41no—
282#22360—531nomaximal, Sylow
283C22×C53C4×#4111yescharacteristic
284C22×C53#51811yescharacteristic
285C22×C53#52011yescharacteristic
286C22×C53C2×#13111yescharacteristic
287C22×C53C2×#13111yescharacteristic
288C22×C53#56811yescharacteristic
289C22×C53#56811yescharacteristic
290C2×D53—41no—
291C2×D53—41no—
292C2×D53—41no—
293C2×D53—41no—
294C2×D53—41no—
295C2×D53—41no—
296C2×D53—41no—
297C2×D53—41no—
298C2×D53—41no—
299C2×D53—41no—
300C2×D53—41no—
301C2×D53—41no—
302C2×D53—41no—
303C2×D53—41no—
304C4×C53—22no—
305C4×C53—41no—
306C4×C53—41no—
307C4×C53—41no—
308C4×C53—41no—
309C4×C53—41no—
310C4×C53—41no—
311Dic53—22no—
312Dic53—41no—
313Dic53—41no—
314Dic53—41no—
315Dic53—41no—
316Dic53#53811yescharacteristic
317Dic53#53911yescharacteristic
318Dic53#54011yescharacteristic
319Dic53#54211yescharacteristic
320#2395—531no—
321#1954—531no—
322#1939—531no—
323#1924—531no—
324#1924—531no—
325#1923—531no—
326#1923—531no—
327#1916—531no—
328#1916—531no—
329#1870—531no—
330#1851—531no—
331#1798—531no—
332#1771—531no—
333#1753—531no—
334#1750—531no—
335C2×C53#233511yescharacteristic
336C2×C53#296911yescharacteristic
337C2×C53#296911yescharacteristic
338C2×C53#297611yescharacteristic
339C2×C53#298111yescharacteristic
340C2×C53#299411yescharacteristic
341C2×C53#299511yescharacteristic
342D53—81no—
343D53—81no—
344C22×M16—531no—
345C2×C4×D4—531no—
346C2×#117—531no—
347C2×C4⋊C8—531no—
348C2×C4⋊C8—531no—
349C2×C4⋊C8—531no—
350C2×C4⋊C8—531no—
351C2×C4⋊C8—531no—
352C2×C22⋊C8—531no—
353C2×C22⋊C8—531no—
354C2×C22⋊C8—531no—
355C2×C22⋊C8—531no—
356C2×C22⋊C8—531no—
357C2×C22⋊C8—531no—
358C2×C22⋊C8—531no—
359C2×#97—531no—
360C2×#97—531no—
361C2×C4×C8—531no—
362C2×C4×C8—531no—
363C2×C4×C8—531no—
364#400#615711yescharacteristic
365#399—531no—
366#398—531no—
367#397—531no—
368#397—531no—
369#397—531no—
370#394—531no—
371#394—531no—
372C4×C4⋊C4—531no—
373C4×C4⋊C4—531no—
374C4×C22⋊C4—531no—
375C4×C22⋊C4—531no—
376C4×C22⋊C4—531no—
377#336—531no—
378#336—531no—
379#336—531no—
380#336—531no—
381#336—531no—
382#336—531no—
383#336—531no—
384#336—531no—
385#336—531no—
386#336—531no—
387C53#2236011yescharacteristic, complemented, Sylow
388C22×D4—531no—
389C23×C4—531no—
390C23×C4—531no—
391C2×M16—1062no—
392C2×M16—2121no—
393C2×M16—2121no—
394C22×C8—531no—
395C22×C8—531no—
396C22×C8—531no—
397C22×C8—531no—
398C22×C8—531no—
399C22×C8—531no—
400C22×C8—531no—
401C22×C8—531no—
402C4×D4—2121no—
403C4×D4—2121no—
404#117—2121no—
405#117—2121no—
406C2×C4⋊C4—531no—
407C2×C4⋊C4—531no—
408C2×C4⋊C4—531no—
409C2×C4⋊C4—531no—
410C2×C4⋊C4—531no—
411C2×C4⋊C4C2×#615711yescharacteristic
412C2×C4⋊C4C2×#615711yescharacteristic
413C2×C22⋊C4—531no—
414C2×C22⋊C4—531no—
415C2×C22⋊C4—531no—
416C2×C22⋊C4—531no—
417C2×C22⋊C4—531no—
418C2×C22⋊C4—531no—
419C2×C22⋊C4—531no—
420C2×C22⋊C4—531no—
421C2×C22⋊C4—531no—
422C2×C22⋊C4—531no—
423C2×C42—531no—
424C2×C42—531no—
425C2×C42—531no—
426C2×C42—531no—
427C2×C42—531no—
428C4⋊C8—1062no—
429C4⋊C8—2121no—
430C4⋊C8—2121no—
431C4⋊C8—2121no—
432C4⋊C8—2121no—
433C22⋊C8—1062no—
434C22⋊C8—1062no—
435C22⋊C8—1062no—
436C22⋊C8—2121no—
437C22⋊C8—2121no—
438C22⋊C8—2121no—
439C22⋊C8—2121no—
440#97—1062no—
441#97—2121no—
442C4×C8—2121no—
443C4×C8—2121no—
444C4×C8—2121no—
445#95—531no—
446#95—531no—
447#95—531no—
448#95—531no—
449#95—531no—
450#95C2×#615711yescharacteristic
451#95C2×#615711yescharacteristic
452#95C2×#615711yescharacteristic
453#95C2×#615711yescharacteristic
454#95C2×#615711yescharacteristic
455C24—531no—
456C24—531no—
457C2×D4—1062no—
458C2×D4—2121no—
459C2×D4—2121no—
460C22×C4—531no—
461C22×C4—531no—
462C22×C4—531no—
463C22×C4—531no—
464C22×C4—531no—
465C22×C4—531no—
466C22×C4—1061no—
467C22×C4—1061no—
468C22×C4—1061no—
469C22×C4—1061no—
470C22×C4—1061no—
471C22×C4—1061no—
472C22×C4—1061no—
473C22×C4—1061no—
474C22×C4—1061no—
475C22×C4—1061no—
476C22×C4—1061no—
477C22×C4—1061no—
478C22×C4C22×#615711yescharacteristic
479C22×C4C22×#615711yescharacteristic
480C22×C4C22×#615711yescharacteristic
481C22×C4C22×#615711yescharacteristic
482C22×C4C22×#615711yescharacteristic
483C22×C4C22×#615711yescharacteristic
484C22×C4C22×#615711yescharacteristic
485M16—2124no—
486C2×C8—532no—
487C2×C8—532no—
488C2×C8—1061no—
489C2×C8—1061no—
490C2×C8—1061no—
491C2×C8—1061no—
492C2×C8—1061no—
493C2×C8—1061no—
494C2×C8—1061no—
495C2×C8—1061no—
496C2×C8—1061no—
497C2×C8—1061no—
498C2×C8—1061no—
499C2×C8—1061no—
500C2×C8—1061no—
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—
510C4⋊C4—41no—
511C4⋊C4—41no—
512C4⋊C4—1062no—
513C4⋊C4—1062no—
514C4⋊C4—1062no—
515C4⋊C4—2121no—
516C4⋊C4—2121no—
517C22⋊C4—1062no—
518C22⋊C4—1062no—
519C22⋊C4—2121no—
520C22⋊C4—2121no—
521C22⋊C4—2121no—
522C22⋊C4—2121no—
523C22⋊C4—2121no—
524C22⋊C4—2121no—
525C22⋊C4—2121no—
526C22⋊C4—2121no—
527C42—1061no—
528C42—1061no—
529C42—1061no—
530C42—1061no—
531C42—1061no—
532C42—1061no—
533C42—1061no—
534C42—1061no—
535C42—1061no—
536C42—1061no—
537C23—1061no—
538C23—1061no—
539C23—1061no—
540C23—1061no—
541C23—1061no—
542C23—1061no—
543C23—1061no—
544C23—1061no—
545C23—1061no—
546C23—1061no—
547C23—1061no—
548C23—1061no—
549C23—1061no—
550C23—1061no—
551C23C23×#615711yescharacteristic
552D4—2124no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
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—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—531no—
574C2×C4—531no—
575C2×C4—531no—
576C2×C4—531no—
577C2×C4—531no—
578C2×C4—531no—
579C2×C4—532no—
580C2×C4—1061no—
581C2×C4—1061no—
582C2×C4—1061no—
583C2×C4—1061no—
584C2×C4—1061no—
585C2×C4—1061no—
586C2×C4—1061no—
587C2×C4—1061no—
588C2×C4—1061no—
589C2×C4—1061no—
590C2×C4—1061no—
591C2×C4—1061no—
592C2×C4—1061no—
593C2×C4—1061no—
594C2×C4—2121no—
595C2×C4—2121no—
596C2×C4—2121no—
597C2×C4—2121no—
598C2×C4—2121no—
599C2×C4—2121no—
600C2×C4—2121no—
601C2×C4—2121no—
602C2×C4#1313330512yes—
603C8—1062no—
604C8—1062no—
605C8—2121no—
606C8—2121no—
607C8—2121no—
608C8—2121no—
609C8—2121no—
610C8—2121no—
611C22—2121no—
612C22—2121no—
613C22—2121no—
614C22—2121no—
615C22—2121no—
616C22—2121no—
617C22—2121no—
618C22—2121no—
619C22—2121no—
620C22—2121no—
621C22—2121no—
622C22—2121no—
623C22—2121no—
624C22—2121no—
625C22#41×#615711yescharacteristic
626C22#1681169011yescharacteristic
627C22#1681169211yescharacteristic
628C22D53⋊#13111yescharacteristic
629C22C2×#1313330511yescharacteristic
630C22#1681170211yescharacteristic
631C22#1681170211yescharacteristic
632C4—22no—
633C4—41no—
634C4—41no—
635C4—41no—
636C4—41no—
637C4—41no—
638C4—41no—
639C4—531no—
640C4—531no—
641C4—531no—
642C4—531no—
643C4—1062no—
644C4—2121no—
645C4—2121no—
646C4—2121no—
647C4—2121no—
648C2—4241no—
649C2—4241no—
650C2#2029121411yescharacteristic
651C2#2029155011yescharacteristic
652C2#2029155111yescharacteristic
653C2#2029158511yescharacteristic
654C2#2029160611yescharacteristic
655C2#2029166711yescharacteristic
656C2#2029167011yescharacteristic
subgroupG/H
center, FrattiniC23C23×#6157
commutator, socleC23×C53C23×C4
FittingC53×#400C4
prime radicalC22×D212C2×C4
cosocleC22×Dic53C24
hypercenter#400#6157
hypocenterC53#22360
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #400

Complements

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

NH ≅ G/Ntype
2. #20290814C2semidirect
3. #20290791C2semidirect
4. #20290791C2semidirect
5. #20290787C2semidirect
6. #20290786C2semidirect
7. #20290769C2semidirect
8. #20290769C2semidirect
10. #20290661C2semidirect
11. #20290574C2semidirect
12. #20290514C2semidirect
13. #20290456C2semidirect
14. #20290444C2semidirect
387. C53#22360semidirect

Maximal subgroups

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

subgroupindex
#202908372
#202908142
#202907912
#202907872
#202907862
#202907692
#202906952
#202906612
#202905742
#202905142
#202904562
#202904442
#202779812
#2236053