Finite Group Databasebeta

← #27395300

Subgroups — #27395300

Order 13568.

1296 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#121×Dic53C211yesmaximal, characteristic, complemented
2C2×#16810526C211yesmaximal, characteristic, complemented
3#20275592C212yesmaximal, complemented
4#20275414C212yesmaximal, complemented
5#20275344C212yesmaximal, complemented
6#20275065C212yesmaximal, complemented
7#20275033C211yesmaximal, characteristic, complemented
8#20274910C211yesmaximal, characteristic, complemented
9#20274646C211yesmaximal, characteristic, complemented
10#20274479C211yesmaximal, characteristic, complemented
11C2×C53×#447C211yesmaximal, characteristic
12C2×D4×Dic53—21no—
13C2×D4×Dic53C2211yescharacteristic, complemented
14C22×C4⋊Dic53C2211yescharacteristic
15C22×C22⋊Dic53—21no—
16C22×C22⋊Dic53—21no—
17C22×C4×Dic53C2211yescharacteristic
18C2×#13133149C2212yescomplemented
19C2×#13133138C2212yes—
20C2×#13133138C2212yes—
21C2×#13133138C2212yescomplemented
22C2×C22⋊#88322C2211yescharacteristic
23C2×#13133127C2212yescomplemented
24#16810526—21no—
25#16810526—21no—
26#16810526C2212yes—
27#16810526C2212yes—
28#16810526C2211yescharacteristic, complemented
29#16810522C2211yescharacteristic, complemented
30#16810516—21no—
31#16810503—21no—
32#16810483C2211yescharacteristic
33C4⋊C4×Dic53C2211yescharacteristic, complemented
34C22⋊C4×Dic53—21no—
35#16810386C2212yescomplemented
36#16810385C2212yescomplemented
37#16810377C2212yescomplemented
38C23×D4×C53C2211yescharacteristic
39C22×D8×C53C2212yes—
40C2×#121×C53C2211yescharacteristic
41C53×#447C418yescomplemented
42C2×#102×C53C2212yes—
43C2×C22⋊C8×C53C2211yescharacteristic
44C23×Dic53—21no—
45C23×Dic53—21no—
46C23×Dic53D411yescharacteristic
47C23×Dic53D411yescharacteristic
48C22×#88322C2312yes—
49D4×Dic53—44no—
50D4×Dic53—42no—
51C2×C4⋊Dic53—21no—
52C2×C4⋊Dic53—21no—
53C2×C4⋊Dic53—42no—
54C2×C4⋊Dic53C2311yescharacteristic
55C2×C4⋊Dic53C2312yes—
56C2×C4⋊Dic53C2311yescharacteristic
57C2×C22⋊Dic53—21no—
58C2×C22⋊Dic53—21no—
59C2×C22⋊Dic53—21no—
60C2×C22⋊Dic53—21no—
61C2×C22⋊Dic53—22no—
62C2×C22⋊Dic53—22no—
63C2×C22⋊Dic53—22no—
64C2×C22⋊Dic53—22no—
65C2×C22⋊Dic53—44no—
66C2×C22⋊Dic53—44no—
67C2×C4×Dic53—21no—
68C2×C4×Dic53—42no—
69C2×C4×Dic53D411yescharacteristic
70C2×C4×Dic53D411yescharacteristic
71C2×C4×Dic53C2311yescharacteristic
72C2×C4×Dic53C2311yescharacteristic
73#13133149—42no—
74#13133138—42no—
75#13133138—42no—
76#13133138—42no—
77C22⋊#88322—22no—
78#13133127—42no—
79#13133124—21no—
80#13133124—21no—
81#13133124—21no—
82C25×C53—21no—
83C22×D4×C53—21no—
84C22×D4×C53—21no—
85C22×D4×C53—22no—
86C22×D4×C53—42no—
87C22×D4×C53—42no—
88C22×D4×C53C2×C414yescomplemented
89C22×D4×C53C2312yes—
90C22×D4×C53C2311yescharacteristic
91C23×C4×C53C2311yescharacteristic
92C2×D8×C53—42no—
93C2×D8×C53—42no—
94C2×D8×C53C2×C418yescomplemented
95C22×C8×C53C2312yes—
96#121×C53—22no—
97#121×C53C2×C414yes—
98C2×C4⋊C4×C53C2311yescharacteristic
99C2×C22⋊C4×C53—21no—
100#102×C53C2×C418yescomplemented
101C22⋊C8×C53C2×C414yescomplemented
102C24×C53—24no—
103C24×C53—21no—
104C24×C53—22no—
105C24×C53—42no—
106C24×C53—42no—
107C24×C53—42no—
108C24×C53—42no—
109C24×C53—42no—
110C24×C53—42no—
111C24×C53C2×D411yescharacteristic
112C2×D4×C53—24no—
113C2×D4×C53—24no—
114C2×D4×C53—21no—
115C2×D4×C53—22no—
116C2×D4×C53—22no—
117C2×D4×C53—22no—
118C2×D4×C53—22no—
119C2×D4×C53—22no—
120C2×D4×C53—22no—
121C2×D4×C53—22no—
122C2×D4×C53—48no—
123C2×D4×C53—42no—
124C2×D4×C53—42no—
125C2×D4×C53—48no—
126C2×D4×C53—42no—
127C2×D4×C53—42no—
128C2×D4×C53C22×C414yes—
129C2×D4×C53C22×C412yes—
130C22×C4×C53—22no—
131C22×C4×C53—22no—
132C22×C4×C53—21no—
133C22×C4×C53C22×C412yes—
134C22×C4×C53C2×D411yescharacteristic
135C22×C4×C53#4112yes—
136C22×C4×C53C2411yescharacteristic
137C22×Dic53—22no—
138C22×Dic53—22no—
139C22×Dic53—22no—
140C22×Dic53—21no—
141C22×Dic53—21no—
142C22×Dic53—22no—
143C22×Dic53—41no—
144C22×Dic53—41no—
145C22×Dic53—41no—
146C22×Dic53—42no—
147C22×Dic53—42no—
148C22×Dic53—42no—
149C22×Dic53—42no—
150C22×Dic53—42no—
151C22×Dic53—42no—
152C22×Dic53C2×D411yescharacteristic
153C22×Dic53C2×D411yescharacteristic
154C22×Dic53C2×D411yescharacteristic
155C22×Dic53C2×D411yescharacteristic
156D8×C53—48no—
157C2×C8×C53—22no—
158C2×C8×C53—22no—
159C2×C8×C53C22×C414yes—
160C4⋊C4×C53—21no—
161C4⋊C4×C53C22×C412yes—
162C22⋊C4×C53—24no—
163C22⋊Dic53—42no—
164C22⋊Dic53—44no—
165C22⋊Dic53—44no—
166C22⋊Dic53—44no—
167C22⋊Dic53—42no—
168C22⋊Dic53—42no—
169C22⋊Dic53—42no—
170C22⋊Dic53—44no—
171C4⋊Dic53—22no—
172C4⋊Dic53—22no—
173C4⋊Dic53—22no—
174C4⋊Dic53—22no—
175C4⋊Dic53—42no—
176C4⋊Dic53—42no—
177C4×Dic53—22no—
178C4×Dic53—22no—
179C4×Dic53—41no—
180C4×Dic53—41no—
181C4×Dic53—42no—
182C2×#88322—22no—
183C2×#88322—22no—
184C2×#88322—22no—
185C23×C53—21no—
186C23×C53—21no—
187C23×C53—21no—
188C23×C53—21no—
189C23×C53—24no—
190C23×C53—22no—
191C23×C53—48no—
192C23×C53—48no—
193C23×C53—41no—
194C23×C53—42no—
195C23×C53—42no—
196C23×C53—41no—
197C23×C53—41no—
198C23×C53—41no—
199C23×C53—41no—
200C23×C53—42no—
201C23×C53—42no—
202C23×C53—42no—
203C23×C53—42no—
204C23×C53—41no—
205C23×C53—48no—
206C23×C53—84no—
207C23×C53—84no—
208C23×C53—84no—
209C23×C53—84no—
210C23×C53C4×D412yes—
211C23×C53AGL(1,ℤ/8)12yes—
212C23×C53AGL(1,ℤ/8)12yes—
213C23×C53C22×D411yescharacteristic
214D4×C53—24no—
215D4×C53—24no—
216D4×C53—24no—
217D4×C53—48no—
218D4×C53—48no—
219D4×C53—42no—
220D4×C53—42no—
221D4×C53—42no—
222C2×C4×C53—21no—
223C2×C4×C53—21no—
224C2×C4×C53—24no—
225C2×C4×C53—24no—
226C2×C4×C53—22no—
227C2×C4×C53—41no—
228C2×C4×C53—41no—
229C2×C4×C53C4×D412yes—
230C2×C4×C53C23×C411yescharacteristic
231C2×C4×C53C2×#4111yescharacteristic
232C2×C4×C53H(5,𝔽2)12yes—
233C2×C4×C53H(5,𝔽2)12yes—
234C2×Dic53—21no—
235C2×Dic53—21no—
236C2×Dic53—21no—
237C2×Dic53—21no—
238C2×Dic53—21no—
239C2×Dic53—21no—
240C2×Dic53—21no—
241C2×Dic53—21no—
242C2×Dic53—21no—
243C2×Dic53—21no—
244C2×Dic53—21no—
245C2×Dic53—44no—
246C2×Dic53—44no—
247C2×Dic53—41no—
248C2×Dic53—42no—
249C2×Dic53—41no—
250C2×Dic53—41no—
251C2×Dic53—41no—
252C2×Dic53—42no—
253C2×Dic53—42no—
254C2×Dic53—41no—
255C2×Dic53—41no—
256C2×Dic53—84no—
257C2×Dic53—84no—
258C2×Dic53AGL(1,ℤ/8)12yes—
259C8×C53—24no—
260#88322—42no—
261#22046—531nomaximal, Sylow
262C22×C53—22no—
263C22×C53—22no—
264C22×C53—22no—
265C22×C53—22no—
266C22×C53—22no—
267C22×C53—22no—
268C22×C53—44no—
269C22×C53—42no—
270C22×C53—42no—
271C22×C53—44no—
272C22×C53—42no—
273C22×C53—44no—
274C22×C53—816no—
275C22×C53—81no—
276C22×C53—82no—
277C22×C53—81no—
278C22×C53—82no—
279C22×C53—81no—
280C22×C53—82no—
281C22×C53—81no—
282C22×C53—82no—
283C22×C53#44214yes—
284C22×C53C2×C4×D411yescharacteristic
285C22×C53#53411yescharacteristic
286C22×C53#53511yescharacteristic
287C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
288C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
289C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
290C22×C53#57511yescharacteristic
291C4×C53—22no—
292C4×C53—42no—
293C4×C53#51812yes—
294C4×C53#53812yes—
295Dic53—22no—
296Dic53—41no—
297Dic53—41no—
298Dic53—41no—
299Dic53—81no—
300Dic53—81no—
301Dic53—82no—
302C2×#447D5311yescharacteristic
303C4×#121—531no—
304C2×#390—531no—
305#1972—532no—
306#1936—532no—
307#1926—532no—
308#1879—532no—
309#1874—531no—
310#1853—531no—
311#1786—531no—
312#1749—531no—
313C2×C53—24no—
314C2×C53—82no—
315C2×C53—84no—
316C2×C53#232111yescharacteristic
317C2×C53C2×#44211yescharacteristic
318C2×C53C4×AGL(1,ℤ/8)11yescharacteristic
319C2×C53#319211yescharacteristic
320C2×C53#319411yescharacteristic
321C2×C53#321311yescharacteristic
322C2×C53#321711yescharacteristic
323C23×D4C2×D5311yescharacteristic
324C22×D8C2×D5312yes—
325C2×#121C2×D5311yescharacteristic
326C2×C4×D4—531no—
327C2×C4×D4—1061no—
328C22×C4⋊C4—531no—
329C22×C22⋊C4—1061no—
330C22×C22⋊C4—1061no—
331C22×C42—531no—
332#447Dic5318yescomplemented
333C2×#106—532no—
334C2×#102—532no—
335C2×#102—532no—
336C2×#102—532no—
337C2×#102C2×D5312yes—
338C2×C22⋊C8—531no—
339C2×C22⋊C8C2×D5311yescharacteristic
340C2×#97—532no—
341#390—532no—
342#390—532no—
343#390—531no—
344#390—1061no—
345#390—1061no—
346#389—531no—
347#388—1061no—
348#386—1061no—
349#380—531no—
350C4×C4⋊C4—531no—
351C4×C22⋊C4—1061no—
352#340—532no—
353#340—532no—
354#336—532no—
355C53#2204611yescharacteristic, complemented, Sylow
356C25—21no—
357C22×D4—21no—
358C22×D4—21no—
359C22×D4—22no—
360C22×D4—42no—
361C22×D4—42no—
362C22×D4C2×Dic5314yescomplemented
363C22×D4C22×D5311yescharacteristic
364C22×D4C22×D5312yes—
365C23×C4—531no—
366C23×C4—531no—
367C23×C4—1061no—
368C23×C4—1061no—
369C23×C4C22×D5311yescharacteristic
370C2×D8—42no—
371C2×D8—42no—
372C2×D8C2×Dic5318yescomplemented
373C22×C8—532no—
374C22×C8C22×D5312yes—
375#121—22no—
376#121C2×Dic5314yes—
377C4×D4—2122no—
378C4×D4—2124no—
379C2×C4⋊C4—532no—
380C2×C4⋊C4—531no—
381C2×C4⋊C4—531no—
382C2×C4⋊C4—1061no—
383C2×C4⋊C4—1061no—
384C2×C4⋊C4—2122no—
385C2×C4⋊C4C22×D5311yescharacteristic
386C2×C22⋊C4—21no—
387C2×C22⋊C4—1062no—
388C2×C22⋊C4—1062no—
389C2×C22⋊C4—1062no—
390C2×C22⋊C4—1062no—
391C2×C22⋊C4—1061no—
392C2×C22⋊C4—1061no—
393C2×C22⋊C4—1061no—
394C2×C22⋊C4—1061no—
395C2×C22⋊C4—2124no—
396C2×C22⋊C4—2124no—
397C2×C42—531no—
398C2×C42—531no—
399C2×C42—531no—
400C2×C42—531no—
401C2×C42—1061no—
402C2×C42—2122no—
403#106—2122no—
404#102—2122no—
405#102—2122no—
406#102—2122no—
407#102C2×Dic5318yescomplemented
408C22⋊C8—1062no—
409C22⋊C8C2×Dic5314yescomplemented
410#97—2122no—
411#95—1061no—
412#95—1061no—
413#95—1061no—
414C24—24no—
415C24—21no—
416C24—22no—
417C24—42no—
418C24—42no—
419C24—42no—
420C24—42no—
421C24—42no—
422C24—42no—
423C24D4×D5311yescharacteristic
424C2×D4—24no—
425C2×D4—24no—
426C2×D4—21no—
427C2×D4—22no—
428C2×D4—22no—
429C2×D4—22no—
430C2×D4—22no—
431C2×D4—22no—
432C2×D4—22no—
433C2×D4—22no—
434C2×D4—48no—
435C2×D4—42no—
436C2×D4—42no—
437C2×D4—48no—
438C2×D4—42no—
439C2×D4—42no—
440C2×D4C22×Dic5314yes—
441C2×D4C22×Dic5312yes—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—21no—
445C22×C4—531no—
446C22×C4—531no—
447C22×C4—531no—
448C22×C4—531no—
449C22×C4—1062no—
450C22×C4—1062no—
451C22×C4—1062no—
452C22×C4—1061no—
453C22×C4—1061no—
454C22×C4—1062no—
455C22×C4—2122no—
456C22×C4—2121no—
457C22×C4—2121no—
458C22×C4—2121no—
459C22×C4—2122no—
460C22×C4—2122no—
461C22×C4—2122no—
462C22×C4—2122no—
463C22×C4—2122no—
464C22×C4D4×D5311yescharacteristic
465C22×C4#1171994712yes—
466C22×C4C22×Dic5312yes—
467C22×C4C23×D5311yescharacteristic
468D8—48no—
469C2×C8—22no—
470C2×C8—22no—
471C2×C8—1062no—
472C2×C8—1062no—
473C2×C8—1062no—
474C2×C8C22×Dic5314yes—
475C4⋊C4—21no—
476C4⋊C4—1062no—
477C4⋊C4—1062no—
478C4⋊C4—1062no—
479C4⋊C4—1062no—
480C4⋊C4—2122no—
481C4⋊C4—2122no—
482C4⋊C4C22×Dic5312yes—
483C22⋊C4—24no—
484C22⋊C4—2124no—
485C22⋊C4—2124no—
486C22⋊C4—2124no—
487C22⋊C4—2124no—
488C22⋊C4—2122no—
489C22⋊C4—2122no—
490C22⋊C4—2122no—
491C22⋊C4—2122no—
492C42—1062no—
493C42—1062no—
494C42—2121no—
495C42—2121no—
496C42—2122no—
497C23—21no—
498C23—21no—
499C23—21no—
500C23—21no—
501C23—22no—
502C23—24no—
503C23—48no—
504C23—48no—
505C23—42no—
506C23—41no—
507C23—42no—
508C23—41no—
509C23—41no—
510C23—42no—
511C23—41no—
512C23—42no—
513C23—41no—
514C23—42no—
515C23—42no—
516C23—41no—
517C23—48no—
518C23—84no—
519C23—84no—
520C23—84no—
521C23—84no—
522C23D4×Dic5312yes—
523C23#1313324112yes—
524C23#1313324112yes—
525C23C2×D4×D5311yescharacteristic
526D4—24no—
527D4—24no—
528D4—24no—
529D4—48no—
530D4—48no—
531D4—42no—
532D4—42no—
533D4—42no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—22no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—532no—
542C2×C4—1061no—
543C2×C4—1061no—
544C2×C4—1061no—
545C2×C4—1061no—
546C2×C4—1061no—
547C2×C4—1061no—
548C2×C4—1061no—
549C2×C4—1061no—
550C2×C4—1061no—
551C2×C4—1061no—
552C2×C4—1061no—
553C2×C4—4244no—
554C2×C4—4244no—
555C2×C4—2124no—
556C2×C4—2122no—
557C2×C4—2121no—
558C2×C4—2121no—
559C2×C4—2122no—
560C2×C4—2121no—
561C2×C4—2121no—
562C2×C4—2121no—
563C2×C4—2122no—
564C2×C4—2121no—
565C2×C4—2124no—
566C2×C4D4×Dic5312yes—
567C2×C4C23×Dic5311yescharacteristic
568C2×C4C2×#1171994711yescharacteristic
569C2×C4#1313326212yes—
570C2×C4#1313326212yes—
571C8—24no—
572C8—2122no—
573C22—22no—
574C22—22no—
575C22—22no—
576C22—22no—
577C22—22no—
578C22—22no—
579C22—42no—
580C22—44no—
581C22—44no—
582C22—42no—
583C22—44no—
584C22—42no—
585C22—816no—
586C22—82no—
587C22—81no—
588C22—82no—
589C22—81no—
590C22—81no—
591C22—82no—
592C22—82no—
593C22—81no—
594C22#1681073514yes—
595C22C2×D4×Dic5311yescharacteristic
596C22#1681118311yescharacteristic
597C22#1681118811yescharacteristic
598C22AGL(1,ℤ/8)×D5311yescharacteristic
599C22C2×#1313324111yescharacteristic
600C22C2×#1313324111yescharacteristic
601C22#1681141911yescharacteristic
602C4—22no—
603C4—42no—
604C4—1062no—
605C4—4241no—
606C4—4242no—
607C4—4241no—
608C4—2121no—
609C4—2121no—
610C4—2121no—
611C4#1681109812yes—
612C4#1681120812yes—
613C2—24no—
614C2—84no—
615C2—82no—
616C2#2027737611yescharacteristic
617C2C2×#1681073511yescharacteristic
618C2AGL(1,ℤ/8)×Dic5311yescharacteristic
619C2#2028551311yescharacteristic
620C2#2028552611yescharacteristic
621C2#2028569911yescharacteristic
622C2#2028573111yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
Fitting, prime radicalC2×C53×#447C2
socleC23×C53C22×D4
cosocleC22×C4×C53C24
hypercenterC2×#447D53
hypocenterC53#22046
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#447

Complements

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

NH ≅ G/Ntype
1. #121×Dic53C2semidirect
2. C2×#16810526C2semidirect
3. #20275592C2semidirect
4. #20275414C2semidirect
5. #20275344C2semidirect
6. #20275065C2semidirect
7. #20275033C2semidirect
8. #20274910C2semidirect
9. #20274646C2semidirect
10. #20274479C2semidirect
13. C2×D4×Dic53C22semidirect
18. C2×#13133149C22semidirect
21. C2×#13133138C22semidirect
23. C2×#13133127C22semidirect
28. #16810526C22semidirect
29. #16810522C22semidirect
33. C4⋊C4×Dic53C22semidirect
35. #16810386C22semidirect
36. #16810385C22semidirect
37. #16810377C22semidirect
41. C53×#447C4semidirect
88. C22×D4×C53C2×C4semidirect
94. C2×D8×C53C2×C4semidirect
100. #102×C53C2×C4semidirect
101. C22⋊C8×C53C2×C4semidirect
332. #447Dic53semidirect
355. C53#22046semidirect
362. C22×D4C2×Dic53semidirect
372. C2×D8C2×Dic53semidirect
407. #102C2×Dic53semidirect
409. C22⋊C8C2×Dic53semidirect

Maximal subgroups

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

subgroupindex
#121×Dic532
C2×#168105262
#202755922
#202754142
#202753442
#202750652
#202750332
#202749102
#202746462
#202744792
C2×C53×#4472
#2204653