Finite Group Databasebeta

← #27417979

Subgroups — #27417979

Order 13568.

1434 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810900C211yesmaximal, characteristic, complemented
2C2×C4×#11719922C211yesmaximal, characteristic, complemented
3#20279628C211yesmaximal, characteristic, complemented
4#20275894C212yesmaximal, complemented
5#20275711C212yesmaximal, complemented
6#20275677C211yesmaximal, characteristic, complemented
7#20275510C211yesmaximal, characteristic, complemented
8#20275032C212yesmaximal, complemented
9#20274903C212yesmaximal, complemented
10#20274715C211yesmaximal, characteristic, complemented
11C2×C53×#547C211yesmaximal, characteristic, complemented
12C22×#11719922C2211yescharacteristic
13C2×#13133191C2212yes—
14C2×#13133188—21no—
15C2×C4×D212C2211yescharacteristic
16#16810900—24no—
17C4×#11719922—24no—
18C2×#13133150C2212yes—
19C2×#13133150C2211yescharacteristic
20C2×C4⋊#88322C2211yescharacteristic
21C2×#13133138C2211yescharacteristic
22C2×#13133138C2212yes—
23C2×#13133137C2212yes—
24C2×#13133137C2211yescharacteristic
25C2×C22⋊#88322C2212yes—
26C2×C22⋊#88322C2212yes—
27C2×C4×#88322C2211yescharacteristic
28#16810539—22no—
29#16810493C2211yescharacteristic
30#16810483C2212yes—
31#16810480—21no—
32#16810385C2212yes—
33#16810385C2212yes—
34#16810385C2212yes—
35C22×#41×C53—21no—
36C53×#547—24no—
37C53×#547—24no—
38C2×C4⋊Q8×C53C2211yescharacteristic
39C2×#123×C53—22no—
40C2×C22⋊Q8×C53—21no—
41C2×#121×C53C2212yes—
42C2×C4×D4×C53C2211yescharacteristic
43C2×C4×D4×C53C2211yescharacteristic
44C22×C4⋊C4×C53C2212yes—
45C22×D212C2311yescharacteristic
46C22×C22⋊D53—22no—
47C22×C4×D53—21no—
48C2×#11719922—22no—
49C2×#11719922—22no—
50C2×#11719922—22no—
51C22×#88322C2311yescharacteristic
52C22×#88322C2312yes—
53C22×#88322C2311yescharacteristic
54#13133191—24no—
55#13133191—42no—
56#13133188—44no—
57C4×D212—22no—
58C4×D212—22no—
59C2×C4⋊Dic53—21no—
60C2×C4⋊Dic53—21no—
61C2×C4⋊Dic53C2311yescharacteristic
62C2×C4⋊Dic53C2312yes—
63C2×C22⋊Dic53—22no—
64C2×#11719921—21no—
65C2×#11719921—22no—
66#13133150—24no—
67#13133150—22no—
68C4⋊#88322—22no—
69#13133138—22no—
70#13133138—42no—
71#13133137—24no—
72#13133137—22no—
73C22⋊#88322—24no—
74C22⋊#88322—42no—
75C4×#88322—22no—
76#13133124—22no—
77C2×#41×C53—28no—
78C2×#41×C53—44no—
79C2×#41×C53—44no—
80C22×Q8×C53—21no—
81C22×D4×C53—22no—
82C22×D4×C53D411yescharacteristic
83C22×D4×C53C2311yescharacteristic
84C23×C4×C53—22no—
85C23×C4×C53—21no—
86C23×C4×C53C2312yes—
87C4⋊Q8×C53—22no—
88C4⋊Q8×C53D414yescomplemented
89#123×C53—216no—
90C22⋊Q8×C53—28no—
91#121×C53—24no—
92#121×C53—24no—
93C4×D4×C53—22no—
94C4×D4×C53—22no—
95C4×D4×C53—22no—
96C4×D4×C53D414yescomplemented
97C2×C4⋊C4×C53—24no—
98C2×C4⋊C4×C53—24no—
99C2×C4⋊C4×C53—22no—
100C2×C4⋊C4×C53—21no—
101C2×C4⋊C4×C53D411yescharacteristic
102C2×C4⋊C4×C53C2311yescharacteristic
103C2×C4⋊C4×C53C2312yes—
104C2×C22⋊C4×C53—21no—
105C2×C22⋊C4×C53—22no—
106C2×C22⋊C4×C53D412yes—
107C2×C42×C53C2311yescharacteristic
108C24×C53—21no—
109C24×C53C2×D412yes—
110C23×D53—21no—
111#41×C53—432no—
112C2×Q8×C53—24no—
113C2×Q8×C53—42no—
114C2×Q8×C53—42no—
115C2×D4×C53—28no—
116C2×D4×C53—22no—
117C2×D4×C53—22no—
118C2×D4×C53—21no—
119C2×D4×C53—21no—
120C2×D4×C53—21no—
121C2×D4×C53—21no—
122C2×D4×C53—21no—
123C2×D4×C53—44no—
124C2×D4×C53—44no—
125C2×D4×C53D814yescomplemented
126C2×D4×C53C2×D412yes—
127C22×C4×C53—28no—
128C22×C4×C53—21no—
129C22×C4×C53—22no—
130C22×C4×C53—24no—
131C22×C4×C53—22no—
132C22×C4×C53—22no—
133C22×C4×C53—22no—
134C22×C4×C53—22no—
135C22×C4×C53—22no—
136C22×C4×C53—22no—
137C22×C4×C53—44no—
138C22×C4×C53—44no—
139C22×C4×C53—42no—
140C22×C4×C53—42no—
141C22×C4×C53C2×D412yes—
142C22×C4×C53C2×D411yescharacteristic
143C22×C4×C53C2×D411yescharacteristic
144C22×C4×C53C2411yescharacteristic
145C2×C22⋊D53—44no—
146C2×C22⋊D53—44no—
147C2×C22⋊D53—82no—
148C22×Dic53—21no—
149C22×Dic53—22no—
150C2×D212—21no—
151C2×D212—21no—
152C2×D212—21no—
153C2×D212—21no—
154C2×D212—21no—
155C2×D212#4112yescomplemented
156C2×C4×D53—42no—
157C2×C4×D53—42no—
158C2×C4×D53—42no—
159C4⋊C4×C53—28no—
160C4⋊C4×C53—24no—
161C4⋊C4×C53—22no—
162C4⋊C4×C53—21no—
163C4⋊C4×C53—22no—
164C4⋊C4×C53C2×D412yes—
165C4⋊C4×C53C2×D414yes—
166C22⋊C4×C53—28no—
167C22⋊C4×C53—24no—
168C22⋊C4×C53—24no—
169C42×C53C2×D412yes—
170C42×C53C2×D412yes—
171C22⋊Dic53—44no—
172#11719922—28no—
173#11719921—44no—
174#11719921—42no—
175C4⋊Dic53—21no—
176C4⋊Dic53—21no—
177C4⋊Dic53—22no—
178C4⋊Dic53—22no—
179C4⋊Dic53—42no—
180C4⋊Dic53—81no—
181C2×#88322—21no—
182C2×#88322—21no—
183C2×#88322—21no—
184C2×#88322—22no—
185C2×#88322—22no—
186C2×#88322—21no—
187C2×#88322—22no—
188C2×#88322—21no—
189C2×#88322#4112yescomplemented
190C23×C53—22no—
191C23×C53—22no—
192C23×C53—22no—
193C23×C53—22no—
194C23×C53—22no—
195C23×C53—22no—
196C23×C53—22no—
197C23×C53—41no—
198C23×C53—41no—
199C23×C53—41no—
200C23×C53—41no—
201C23×C53—41no—
202C23×C53—41no—
203C23×C53C2×D814yes—
204C23×C53C22×D411yescharacteristic
205C22×D53—22no—
206C22×D53—41no—
207C22×D53—41no—
208C22×D53—41no—
209C22×D53—41no—
210C22×D53—41no—
211C22×D53—41no—
212Q8×C53—48no—
213D4×C53—22no—
214D4×C53—22no—
215D4×C53—28no—
216D4×C53—416no—
217D4×C53—41no—
218D4×C53—41no—
219C2×C4×C53—21no—
220C2×C4×C53—21no—
221C2×C4×C53—22no—
222C2×C4×C53—24no—
223C2×C4×C53—22no—
224C2×C4×C53—22no—
225C2×C4×C53—22no—
226C2×C4×C53—416no—
227C2×C4×C53—41no—
228C2×C4×C53—42no—
229C2×C4×C53—41no—
230C2×C4×C53—42no—
231C2×C4×C53—42no—
232C2×C4×C53—42no—
233C2×C4×C53—42no—
234C2×C4×C53—42no—
235C2×C4×C53—48no—
236C2×C4×C53#12012yes—
237C2×C4×C53#12012yes—
238C2×C4×C53C2×D812yes—
239C2×C4×C53AGL(1,ℤ/8)12yes—
240C2×C4×C53C22×D411yescharacteristic
241C2×C4×C53C22×D411yescharacteristic
242C2×C4×C53C2×#4111yescharacteristic
243C2×C4×C53C2×#4111yescharacteristic
244C2×C4×C53H(5,𝔽2)11yescharacteristic
245C2×C4×C532−1+411yescharacteristic
246C22⋊D53—88no—
247C2×Dic53—41no—
248C2×Dic53—42no—
249C2×Dic53—41no—
250C2×Dic53—42no—
251C2×Dic53—41no—
252C2×Dic53—42no—
253D212—22no—
254D212—22no—
255D212—41no—
256D212—41no—
257C4×D53—84no—
258#88322—22no—
259#88322—41no—
260#88322—42no—
261#24116—531nomaximal, Sylow
262C22×C53—24no—
263C22×C53—24no—
264C22×C53—24no—
265C22×C53—42no—
266C22×C53—42no—
267C22×C53—42no—
268C22×C53—42no—
269C22×C53—42no—
270C22×C53—42no—
271C22×C53—42no—
272C22×C53—81no—
273C22×C53—81no—
274C22×C53—81no—
275C22×C53—81no—
276C22×C53C2×#12011yescharacteristic
277C22×C53#54611yescharacteristic
278C22×C53#54711yescharacteristic
279C22×C53C22×D811yescharacteristic
280C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
281C22×C53#57711yescharacteristic
282C22×C53#57811yescharacteristic
283C2×D53—42no—
284C2×D53—42no—
285C2×D53—42no—
286C2×D53—81no—
287C2×D53—81no—
288C2×D53—81no—
289C2×D53—81no—
290C4×C53—22no—
291C4×C53—42no—
292C4×C53—42no—
293C4×C53—44no—
294C4×C53#44714yes—
295C4×C53#52511yescharacteristic
296C4×C53#53711yescharacteristic
297C4×C53#54811yescharacteristic
298C4×C53#55011yescharacteristic
299Dic53—81no—
300Dic53—82no—
301C2×#547D5311yescharacteristic, complemented
302C2×#474—531no—
303C2×C4×D8—531no—
304#2537—531no—
305#2033—532no—
306#1999—532no—
307#1990—531no—
308#1954—531no—
309#1874—532no—
310#1852—532no—
311#1802—531no—
312C2×C53—44no—
313C2×C53—82no—
314C2×C53#240711yescharacteristic
315C2×C53C2×#44711yescharacteristic
316C2×C53#301311yescharacteristic
317C2×C53#329411yescharacteristic
318C2×C53#333011yescharacteristic
319C2×C53#333411yescharacteristic
320C2×C53#335411yescharacteristic
321D53—82no—
322C22×#41—21no—
323C22×D8—531no—
324#547—24no—
325#547—24no—
326C2×C4⋊Q8C2×D5311yescharacteristic
327C2×#123—22no—
328C2×C22⋊Q8—21no—
329C2×#121—532no—
330C2×#121—1061no—
331C2×#121C2×D5312yes—
332C2×C4×D4—531no—
333C2×C4×D4C2×D5311yescharacteristic
334C2×C4×D4C2×D5311yescharacteristic
335C22×C4⋊C4C2×D5312yes—
336#474—1064no—
337C4×D8—1064no—
338C2×#107—531no—
339C2×#107—532no—
340C2×C4⋊C8—531no—
341C2×#102—531no—
342C2×#102—531no—
343C2×#102—532no—
344C2×#102—532no—
345C2×C22⋊C8—532no—
346C2×C22⋊C8—532no—
347C2×C4×C8—531no—
348#394—1062no—
349#384—531no—
350#380—532no—
351#380—1061no—
352#340—532no—
353#340—532no—
354#340—532no—
355C53#2411611yescharacteristic, complemented, Sylow
356C2×#41—28no—
357C2×#41—44no—
358C2×#41—44no—
359C22×Q8—21no—
360C22×D4—22no—
361C22×D4—531no—
362C22×D4—1062no—
363C22×D4C22⋊D5311yescharacteristic
364C22×D4C22×D5311yescharacteristic
365C23×C4—22no—
366C23×C4—21no—
367C23×C4—1061no—
368C23×C4C22×D5312yes—
369C2×D8—1062no—
370C2×D8—1062no—
371C2×D8—1062no—
372C22×C8—531no—
373C22×C8—531no—
374C22×C8—532no—
375C4⋊Q8—22no—
376C4⋊Q8C22⋊D5314yescomplemented
377#123—216no—
378C22⋊Q8—28no—
379#121—24no—
380#121—24no—
381#121—1064no—
382#121—2122no—
383#121—2124no—
384C4×D4—22no—
385C4×D4—22no—
386C4×D4—22no—
387C4×D4—1062no—
388C4×D4—1062no—
389C4×D4C22⋊D5314yescomplemented
390C2×C4⋊C4—22no—
391C2×C4⋊C4—21no—
392C2×C4⋊C4—24no—
393C2×C4⋊C4—24no—
394C2×C4⋊C4—531no—
395C2×C4⋊C4—532no—
396C2×C4⋊C4—1061no—
397C2×C4⋊C4—1061no—
398C2×C4⋊C4C22⋊D5311yescharacteristic
399C2×C4⋊C4C22×D5311yescharacteristic
400C2×C4⋊C4C22×D5312yes—
401C2×C22⋊C4—21no—
402C2×C22⋊C4—22no—
403C2×C22⋊C4—1062no—
404C2×C22⋊C4—1061no—
405C2×C22⋊C4—1062no—
406C2×C22⋊C4C22⋊D5312yes—
407C2×C42C22×D5311yescharacteristic
408#107—1062no—
409#107—1064no—
410C4⋊C8—1062no—
411#102—1062no—
412#102—1062no—
413#102—1064no—
414#102—2122no—
415C22⋊C8—1064no—
416C22⋊C8—2122no—
417C4×C8—1062no—
418#95—1062no—
419C24—21no—
420C24—1061no—
421C24C2×C22⋊D5312yes—
422#41—432no—
423C2×Q8—24no—
424C2×Q8—42no—
425C2×Q8—42no—
426C2×D4—28no—
427C2×D4—22no—
428C2×D4—22no—
429C2×D4—21no—
430C2×D4—21no—
431C2×D4—21no—
432C2×D4—21no—
433C2×D4—21no—
434C2×D4—44no—
435C2×D4—44no—
436C2×D4—532no—
437C2×D4—1061no—
438C2×D4—1061no—
439C2×D4—1061no—
440C2×D4—1061no—
441C2×D4—1061no—
442C2×D4—4242no—
443C2×D4—2124no—
444C2×D4—2124no—
445C2×D4#1171992214yescomplemented
446C2×D4C2×C22⋊D5312yes—
447C22×C4—28no—
448C22×C4—21no—
449C22×C4—22no—
450C22×C4—24no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—44no—
458C22×C4—44no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4—1061no—
462C22×C4—1062no—
463C22×C4—2122no—
464C22×C4—2122no—
465C22×C4—2122no—
466C22×C4C2×C22⋊D5312yes—
467C22×C4C2×C22⋊D5311yescharacteristic
468C22×C4C2×C22⋊D5311yescharacteristic
469C22×C4C23×D5311yescharacteristic
470D8—1068no—
471C2×C8—532no—
472C2×C8—1061no—
473C2×C8—1061no—
474C2×C8—1061no—
475C2×C8—1062no—
476C2×C8—1062no—
477C2×C8—1061no—
478C2×C8—1062no—
479C2×C8—1061no—
480C4⋊C4—28no—
481C4⋊C4—24no—
482C4⋊C4—22no—
483C4⋊C4—21no—
484C4⋊C4—22no—
485C4⋊C4—1061no—
486C4⋊C4—1061no—
487C4⋊C4—1062no—
488C4⋊C4—1062no—
489C4⋊C4—4241no—
490C4⋊C4—2122no—
491C4⋊C4C2×C22⋊D5312yes—
492C4⋊C4C2×C22⋊D5314yes—
493C22⋊C4—28no—
494C22⋊C4—24no—
495C22⋊C4—24no—
496C22⋊C4—2124no—
497C22⋊C4—2124no—
498C22⋊C4—2122no—
499C42C2×C22⋊D5312yes—
500C42C2×C22⋊D5312yes—
501C23—22no—
502C23—22no—
503C23—22no—
504C23—22no—
505C23—22no—
506C23—22no—
507C23—22no—
508C23—41no—
509C23—41no—
510C23—41no—
511C23—41no—
512C23—41no—
513C23—41no—
514C23—1062no—
515C23—2121no—
516C23—2121no—
517C23—2121no—
518C23—2121no—
519C23—2121no—
520C23—2121no—
521C23C2×#1171992214yes—
522C23C22×C22⋊D5311yescharacteristic
523Q8—48no—
524D4—28no—
525D4—22no—
526D4—22no—
527D4—416no—
528D4—41no—
529D4—41no—
530D4—1062no—
531D4—1062no—
532D4—4248no—
533D4—2121no—
534D4—2121no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—24no—
541C2×C4—22no—
542C2×C4—416no—
543C2×C4—41no—
544C2×C4—42no—
545C2×C4—41no—
546C2×C4—42no—
547C2×C4—48no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—4244no—
553C2×C4—2121no—
554C2×C4—2121no—
555C2×C4—2122no—
556C2×C4—2121no—
557C2×C4—2122no—
558C2×C4—2122no—
559C2×C4#1313318612yes—
560C2×C4#1313318612yes—
561C2×C4C2×#1171992212yes—
562C2×C4#1313324312yes—
563C2×C4C22×C22⋊D5311yescharacteristic
564C2×C4C22×C22⋊D5311yescharacteristic
565C2×C4#41×D5311yescharacteristic
566C2×C4#41×D5311yescharacteristic
567C2×C4#1313326311yescharacteristic
568C2×C4#1313326411yescharacteristic
569C8—1062no—
570C8—2122no—
571C8—2121no—
572C22—24no—
573C22—24no—
574C22—24no—
575C22—42no—
576C22—42no—
577C22—42no—
578C22—42no—
579C22—42no—
580C22—42no—
581C22—42no—
582C22—81no—
583C22—81no—
584C22—81no—
585C22—81no—
586C22—4241no—
587C22—4241no—
588C22—4241no—
589C22—4241no—
590C22—2122no—
591C22—2122no—
592C22—2122no—
593C22C2×#1313318611yescharacteristic
594C22#1681126111yescharacteristic
595C22#1681127211yescharacteristic
596C22C22×#1171992211yescharacteristic
597C22C2×#1313324311yescharacteristic
598C22#1681143111yescharacteristic
599C22#1681143811yescharacteristic
600C4—22no—
601C4—42no—
602C4—42no—
603C4—44no—
604C4—4242no—
605C4—4241no—
606C4#1681075714yes—
607C4#1681113011yescharacteristic
608C4#1681120111yescharacteristic
609C4#1681128011yescharacteristic
610C4#1681129211yescharacteristic
611C2—44no—
612C2—82no—
613C2—4242no—
614C2#2027805411yescharacteristic
615C2C2×#1681075711yescharacteristic
616C2#2028403011yescharacteristic
617C2#2028652211yescharacteristic
618C2#2028701411yescharacteristic
619C2#2028705811yescharacteristic
620C2#2028731011yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC2×C53×#547C2
socleC23×C53C22×D4
hypercenterC2×#547D53
hypocenterC53#24116
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#547

Complements

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

NH ≅ G/Ntype
1. C2×#16810900C2semidirect
2. C2×C4×#11719922C2semidirect
3. #20279628C2semidirect
4. #20275894C2semidirect
5. #20275711C2semidirect
6. #20275677C2semidirect
7. #20275510C2semidirect
8. #20275032C2semidirect
9. #20274903C2semidirect
10. #20274715C2semidirect
11. C2×C53×#547C2semidirect
88. C4⋊Q8×C53D4semidirect
96. C4×D4×C53D4semidirect
125. C2×D4×C53D8semidirect
155. C2×D212#41semidirect
189. C2×#88322#41semidirect
301. C2×#547D53semidirect
355. C53#24116semidirect
376. C4⋊Q8C22⋊D53semidirect
389. C4×D4C22⋊D53semidirect
445. C2×D4#11719922semidirect

Maximal subgroups

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

subgroupindex
C2×#168109002
C2×C4×#117199222
#202796282
#202758942
#202757112
#202756772
#202755102
#202750322
#202749032
#202747152
C2×C53×#5472
#2411653