Finite Group Databasebeta

← #27457285

Subgroups — #27457285

Order 13568.

762 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20277771C211yesmaximal, characteristic, complemented
2#20276259C211yesmaximal, characteristic, complemented
3#20276032C211yesmaximal, characteristic
4#20275961C211yesmaximal, characteristic, complemented
5#20275797C211yesmaximal, characteristic
6#20275568C211yesmaximal, characteristic
7#20275349C211yesmaximal, characteristic, complemented
8#20275339C211yesmaximal, characteristic
9#20275290C211yesmaximal, characteristic, complemented
10#20275274C211yesmaximal, characteristic, complemented
11#20275073C211yesmaximal, characteristic
12#20275073C211yesmaximal, characteristic
13#20274717C211yesmaximal, characteristic, complemented
14C4×#13133140C211yesmaximal, characteristic, complemented
15C53×#2512C211yesmaximal, characteristic
16C22×#11719923C2211yescharacteristic
17C22×#11719923C2211yescharacteristic
18C2×#13133217C2211yescharacteristic
19C2×C4×Dic106C2211yescharacteristic
20C2×C4×Dic106C2211yescharacteristic
21C2×#13133148C2211yescharacteristic
22C2×#13133148C2211yescharacteristic
23C2×C4⋊#88322C2211yescharacteristic
24C2×#13133140C2211yescharacteristic
25C2×#13133140C2211yescharacteristic
26C2×#13133140C2211yescharacteristic
27C2×#13133140C2211yescharacteristic
28C2×#13133140C2211yescharacteristic
29C2×#13133140C2211yescharacteristic
30C2×#13133138C2211yescharacteristic
31C2×#13133138C2211yescharacteristic
32C2×C4×#88322C2211yescharacteristic
33#16810529C2211yescharacteristic
34#16810529C2211yescharacteristic
35#16810519—21no—
36#16810506—21no—
37#16810506—21no—
38C4⋊C4×Dic53—21no—
39C4×C4⋊Dic53C2211yescharacteristic
40#16810385C2211yescharacteristic
41#16810385C2211yescharacteristic
42#16810385C2211yescharacteristic
43#16810385C2211yescharacteristic
44#16810378C2211yescharacteristic
45#16810378C2211yescharacteristic
46#16810378C2211yescharacteristic
47#16810378C2211yescharacteristic
48C2×#126×C53—21no—
49C53×#397—21no—
50C53×#396—21no—
51C53×#390C2211yescharacteristic
52C53×#390C2211yescharacteristic
53C53×#390C2211yescharacteristic
54C53×#389C2211yescharacteristic
55C53×#388—21no—
56C53×#384C2211yescharacteristic
57C4×C4⋊C4×C53C2211yescharacteristic
58C4×C4⋊C4×C53C2211yescharacteristic
59C22×Dic106C2311yescharacteristic
60C22×Dic106C2311yescharacteristic
61C2×#11719923—22no—
62C2×#11719923—22no—
63C2×#11719923—22no—
64C2×#11719923—41no—
65C2×#11719923—41no—
66C2×#11719923—41no—
67C22×#88322C2311yescharacteristic
68C22×#88322C2311yescharacteristic
69C22×#88322C2311yescharacteristic
70C22×#88322C2311yescharacteristic
71#13133217—22no—
72#13133217—22no—
73C4×Dic106—22no—
74C4×Dic106—22no—
75C4×Dic106—41no—
76C4×Dic106—41no—
77C2×C4⋊Dic53—21no—
78C2×C4⋊Dic53—21no—
79C2×C4⋊Dic53C2311yescharacteristic
80C2×C4⋊Dic53C2311yescharacteristic
81C2×#11719919—21no—
82C2×#11719919—21no—
83C2×C4×Dic53—21no—
84C2×C4×Dic53—21no—
85C2×C4×Dic53—21no—
86#13133148—22no—
87#13133148—41no—
88C4⋊#88322—41no—
89#13133140—22no—
90#13133140—22no—
91#13133140—22no—
92#13133140—41no—
93#13133140—41no—
94#13133140—41no—
95#13133138—22no—
96#13133138—41no—
97C4×#88322—41no—
98#13133124—21no—
99#13133124—21no—
100#13133124—21no—
101#13133124—21no—
102#13133124—21no—
103C22×D4×C53C2311yescharacteristic
104#126×C53—44no—
105C2×C4⋊C4×C53—21no—
106C2×C4⋊C4×C53—21no—
107C2×C4⋊C4×C53—21no—
108C2×C4⋊C4×C53D411yescharacteristic
109C2×C4⋊C4×C53D411yescharacteristic
110C2×C4⋊C4×C53C2311yescharacteristic
111C2×C4⋊C4×C53C2311yescharacteristic
112C2×C4⋊C4×C53C2311yescharacteristic
113C2×C22⋊C4×C53—21no—
114C2×C22⋊C4×C53—21no—
115C2×C22⋊C4×C53—21no—
116C2×C22⋊C4×C53—21no—
117C2×C22⋊C4×C53—21no—
118C2×C22⋊C4×C53—21no—
119C2×C42×C53—21no—
120C2×C42×C53C2311yescharacteristic
121C2×C42×C53C2311yescharacteristic
122C2×C42×C53C2311yescharacteristic
123#95×C53—21no—
124#95×C53—21no—
125#95×C53D411yescharacteristic
126#95×C53D411yescharacteristic
127C24×C53—21no—
128C2×D4×C53—21no—
129C2×D4×C53—21no—
130C2×D4×C53—21no—
131C2×D4×C53—21no—
132C2×D4×C53—21no—
133C2×D4×C53—21no—
134C22×C4×C53—21no—
135C22×C4×C53—21no—
136C22×C4×C53—21no—
137C22×C4×C53C2×D411yescharacteristic
138C22×C4×C53C2×D411yescharacteristic
139C22×C4×C53C2×D411yescharacteristic
140C22×C4×C53C2×D411yescharacteristic
141C22×C4×C53C2×D411yescharacteristic
142C22×C4×C53C2×D411yescharacteristic
143C22×C4×C53C2411yescharacteristic
144C22×Dic53—21no—
145C22×Dic53—21no—
146C22×Dic53—21no—
147C22×Dic53—21no—
148C2×Dic106—21no—
149C2×Dic106—21no—
150C2×Dic106—21no—
151C2×Dic106—21no—
152C2×Dic106—21no—
153C2×Dic106—21no—
154C2×Dic106—21no—
155C2×Dic106—21no—
156C2×Dic106—21no—
157C2×Dic106—21no—
158C2×Dic106#4112yes—
159C2×Dic106#4112yes—
160C4⋊C4×C53—21no—
161C4⋊C4×C53—21no—
162C4⋊C4×C53—21no—
163C4⋊C4×C53—21no—
164C4⋊C4×C53—21no—
165C4⋊C4×C53—21no—
166C4⋊C4×C53—41no—
167C4⋊C4×C53—42no—
168C4⋊C4×C53—42no—
169C4⋊C4×C53—81no—
170C4⋊C4×C53SD1614yescomplemented
171C22⋊C4×C53—42no—
172C22⋊C4×C53—42no—
173C22⋊C4×C53—42no—
174C22⋊C4×C53—42no—
175C22⋊C4×C53—81no—
176C22⋊C4×C53—81no—
177C42×C53—21no—
178C42×C53—21no—
179C42×C53—21no—
180C42×C53—21no—
181C42×C53—21no—
182C42×C53—21no—
183C42×C53—42no—
184#11719923—44no—
185#11719923—44no—
186C4⋊Dic53—24no—
187C4⋊Dic53—24no—
188C4⋊Dic53—21no—
189C4⋊Dic53—21no—
190C4⋊Dic53—21no—
191C4⋊Dic53#4112yes—
192#11719919—42no—
193#11719919—42no—
194C4×Dic53—42no—
195C4×Dic53—42no—
196C4×Dic53—42no—
197C2×#88322—21no—
198C2×#88322—21no—
199C2×#88322—21no—
200C2×#88322—21no—
201C2×#88322—21no—
202C2×#88322—21no—
203C2×#88322—21no—
204C2×#88322—21no—
205C2×#88322—21no—
206C2×#88322—21no—
207C2×#88322—21no—
208C2×#88322#4112yes—
209C23×C53—41no—
210C23×C53—41no—
211C23×C53—41no—
212C23×C53—41no—
213C23×C53—41no—
214C23×C53—41no—
215C23×C53—41no—
216C23×C53C22×D411yescharacteristic
217D4×C53—41no—
218D4×C53—41no—
219D4×C53—41no—
220D4×C53—41no—
221C2×C4×C53—21no—
222C2×C4×C53—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—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×C53C2×SD1612yes—
246C2×C4×C53C2×SD1612yes—
247C2×C4×C53C2×SD1612yes—
248C2×C4×C53C2×#4111yescharacteristic
249C2×C4×C53C2×#4111yescharacteristic
250C2×C4×C53C2×#4111yescharacteristic
251C2×C4×C53C2×#4111yescharacteristic
252C2×C4×C53H(5,𝔽2)11yescharacteristic
253C2×C4×C53H(5,𝔽2)11yescharacteristic
254C2×Dic53—22no—
255C2×Dic53—22no—
256C2×Dic53—22no—
257C2×Dic53—41no—
258C2×Dic53—41no—
259C2×Dic53—41no—
260C2×Dic53—41no—
261C2×Dic53—41no—
262C2×Dic53—41no—
263C2×Dic53—41no—
264C2×Dic53—41no—
265C2×Dic53—41no—
266Dic106—22no—
267Dic106—22no—
268Dic106—22no—
269Dic106—22no—
270Dic106—41no—
271Dic106—41no—
272Dic106—41no—
273Dic106—41no—
274#88322—22no—
275#88322—41no—
276#88322—41no—
277#88322—41no—
278#27066—531nomaximal, Sylow
279C22×C53—81no—
280C22×C53—81no—
281C22×C53—81no—
282C22×C53—81no—
283C22×C53—81no—
284C22×C53—81no—
285C22×C53—81no—
286C22×C53#52511yescharacteristic
287C22×C53#54611yescharacteristic
288C22×C53#55011yescharacteristic
289C22×C53C22×SD1611yescharacteristic
290C22×C53#57511yescharacteristic
291C22×C53#57711yescharacteristic
292C22×C53#57811yescharacteristic
293C4×C53—22no—
294C4×C53—22no—
295C4×C53—22no—
296C4×C53—41no—
297C4×C53—41no—
298C4×C53—41no—
299C4×C53—81no—
300C4×C53—81no—
301C4×C53—81no—
302C4×C53#52911yescharacteristic
303C4×C53#53811yescharacteristic
304C4×C53#55111yescharacteristic
305C4×C53#55311yescharacteristic
306Dic53—42no—
307Dic53—42no—
308Dic53—42no—
309Dic53—81no—
310#2512D5311yescharacteristic
311#2365—531no—
312#2096—531no—
313#2055—531no—
314#2043—531no—
315#2013—531no—
316#1968—531no—
317#1927—531no—
318#1926—531no—
319#1919—531no—
320#1916—531no—
321#1880—531no—
322#1880—531no—
323#1803—531no—
324C4×#103—531no—
325C2×C53—161no—
326C2×C53#246711yescharacteristic
327C2×C53#310311yescharacteristic
328C2×C53#312111yescharacteristic
329C2×C53#329811yescharacteristic
330C2×C53#330711yescharacteristic
331C2×C53#339211yescharacteristic
332C2×C53#340211yescharacteristic
333C22×SD16—531no—
334C22×SD16—531no—
335C2×C4⋊Q8—531no—
336C2×#126—21no—
337C2×C4×Q8—531no—
338C2×C4×Q8—531no—
339C2×#106—531no—
340C2×#106—531no—
341C2×C4⋊C8—531no—
342C2×#103—531no—
343C2×#103—531no—
344C2×#103—531no—
345C2×#103—531no—
346C2×#103—531no—
347C2×#103—531no—
348C2×#102—531no—
349C2×#102—531no—
350C2×C4×C8—531no—
351#397—21no—
352#396—21no—
353#391—531no—
354#391—531no—
355#390C2×D5311yescharacteristic
356#390C2×D5311yescharacteristic
357#390C2×D5311yescharacteristic
358#389—1061no—
359#389C2×D5311yescharacteristic
360#388—21no—
361#387—1061no—
362#387—1061no—
363#384C2×D5311yescharacteristic
364C4×C4⋊C4—531no—
365C4×C4⋊C4—1061no—
366C4×C4⋊C4C2×D5311yescharacteristic
367C4×C4⋊C4C2×D5311yescharacteristic
368#340—531no—
369#340—531no—
370#340—531no—
371#340—531no—
372#336—531no—
373#336—531no—
374#336—531no—
375#336—531no—
376C53#2706611yescharacteristic, complemented, Sylow
377C22×Q8—531no—
378C22×Q8—531no—
379C22×D4C22×D5311yescharacteristic
380C2×SD16—1062no—
381C2×SD16—1062no—
382C2×SD16—1062no—
383C2×SD16—2121no—
384C2×SD16—2121no—
385C2×SD16—2121no—
386C22×C8—531no—
387C22×C8—531no—
388C22×C8—531no—
389C22×C8—531no—
390C4⋊Q8—1062no—
391C4⋊Q8—1062no—
392#126—44no—
393C4×Q8—1062no—
394C4×Q8—1062no—
395C4×Q8—2121no—
396C4×Q8—2121no—
397C2×C4⋊C4—21no—
398C2×C4⋊C4—21no—
399C2×C4⋊C4—21no—
400C2×C4⋊C4—531no—
401C2×C4⋊C4—531no—
402C2×C4⋊C4—1061no—
403C2×C4⋊C4—1061no—
404C2×C4⋊C4—1061no—
405C2×C4⋊C4—1061no—
406C2×C4⋊C4C22⋊D5311yescharacteristic
407C2×C4⋊C4C22⋊D5311yescharacteristic
408C2×C4⋊C4C22×D5311yescharacteristic
409C2×C4⋊C4C22×D5311yescharacteristic
410C2×C4⋊C4C22×D5311yescharacteristic
411C2×C22⋊C4—21no—
412C2×C22⋊C4—21no—
413C2×C22⋊C4—21no—
414C2×C22⋊C4—21no—
415C2×C22⋊C4—21no—
416C2×C22⋊C4—21no—
417C2×C42—21no—
418C2×C42—1061no—
419C2×C42—1061no—
420C2×C42—1061no—
421C2×C42C22×D5311yescharacteristic
422C2×C42C22×D5311yescharacteristic
423C2×C42C22×D5311yescharacteristic
424#106—1062no—
425#106—2121no—
426C4⋊C8—2121no—
427#103—1062no—
428#103—1062no—
429#103—1062no—
430#103—2121no—
431#103—2121no—
432#103—2121no—
433#102—1062no—
434#102—2121no—
435C4×C8—2121no—
436#95—21no—
437#95—21no—
438#95—1061no—
439#95—1061no—
440#95—1061no—
441#95—1061no—
442#95—1061no—
443#95C22⋊D5311yescharacteristic
444#95C22⋊D5311yescharacteristic
445C24—21no—
446C2×Q8—532no—
447C2×Q8—532no—
448C2×Q8—1061no—
449C2×Q8—1061no—
450C2×Q8—1061no—
451C2×Q8—1061no—
452C2×Q8—1061no—
453C2×Q8—1061no—
454C2×Q8—1061no—
455C2×Q8—1061no—
456C2×Q8—1061no—
457C2×Q8—1061no—
458C2×D4—21no—
459C2×D4—21no—
460C2×D4—21no—
461C2×D4—21no—
462C2×D4—21no—
463C2×D4—21no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—1061no—
468C22×C4—1061no—
469C22×C4—1061no—
470C22×C4—1061no—
471C22×C4C2×C22⋊D5311yescharacteristic
472C22×C4C2×C22⋊D5311yescharacteristic
473C22×C4C2×C22⋊D5311yescharacteristic
474C22×C4C2×C22⋊D5311yescharacteristic
475C22×C4C2×C22⋊D5311yescharacteristic
476C22×C4C2×C22⋊D5311yescharacteristic
477C22×C4C23×D5311yescharacteristic
478SD16—2124no—
479SD16—2124no—
480C2×C8—532no—
481C2×C8—1061no—
482C2×C8—1061no—
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—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—41no—
499C4⋊C4—42no—
500C4⋊C4—42no—
501C4⋊C4—81no—
502C4⋊C4—532no—
503C4⋊C4—1064no—
504C4⋊C4—1064no—
505C4⋊C4—1061no—
506C4⋊C4—1061no—
507C4⋊C4—1061no—
508C4⋊C4—2122no—
509C4⋊C4—2122no—
510C4⋊C4#1171992314yescomplemented
511C22⋊C4—42no—
512C22⋊C4—42no—
513C22⋊C4—42no—
514C22⋊C4—42no—
515C22⋊C4—81no—
516C22⋊C4—81no—
517C42—21no—
518C42—21no—
519C42—21no—
520C42—21no—
521C42—21no—
522C42—21no—
523C42—42no—
524C42—2122no—
525C42—2122no—
526C42—2122no—
527C23—41no—
528C23—41no—
529C23—41no—
530C23—41no—
531C23—41no—
532C23—41no—
533C23—41no—
534C23C22×C22⋊D5311yescharacteristic
535Q8—1062no—
536Q8—1062no—
537Q8—1062no—
538Q8—1062no—
539Q8—2121no—
540Q8—2121no—
541Q8—2121no—
542Q8—2121no—
543D4—41no—
544D4—41no—
545D4—41no—
546D4—41no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
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—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—1062no—
572C2×C4—1062no—
573C2×C4—1062no—
574C2×C4—2121no—
575C2×C4—2121no—
576C2×C4—2121no—
577C2×C4—2121no—
578C2×C4—2121no—
579C2×C4—2121no—
580C2×C4—2121no—
581C2×C4—2121no—
582C2×C4—2121no—
583C2×C4C2×#1171992312yes—
584C2×C4C2×#1171992312yes—
585C2×C4C2×#1171992312yes—
586C2×C4#41×D5311yescharacteristic
587C2×C4#41×D5311yescharacteristic
588C2×C4#41×D5311yescharacteristic
589C2×C4#41×D5311yescharacteristic
590C2×C4#1313326311yescharacteristic
591C2×C4#1313326311yescharacteristic
592C8—1062no—
593C8—2121no—
594C8—2121no—
595C8—2121no—
596C22—81no—
597C22—81no—
598C22—81no—
599C22—81no—
600C22—81no—
601C22—81no—
602C22—81no—
603C22#1681113011yescharacteristic
604C22#1681126111yescharacteristic
605C22#1681129211yescharacteristic
606C22C22×#1171992311yescharacteristic
607C22#1681142111yescharacteristic
608C22#1681143411yescharacteristic
609C22#1681143811yescharacteristic
610C4—22no—
611C4—22no—
612C4—22no—
613C4—41no—
614C4—41no—
615C4—41no—
616C4—81no—
617C4—81no—
618C4—81no—
619C4—4241no—
620C4—2122no—
621C4—2122no—
622C4—2122no—
623C4#1681115611yescharacteristic
624C4#1681121211yescharacteristic
625C4#1681129811yescharacteristic
626C4#1681132411yescharacteristic
627C2—161no—
628C2#2027876211yescharacteristic
629C2#2028474211yescharacteristic
630C2#2028493811yescharacteristic
631C2#2028657411yescharacteristic
632C2#2028668511yescharacteristic
633C2#2028769611yescharacteristic
634C2#2028779011yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2512C2
socleC23×C53C22×D4
prime radicalC22×D4×C53C23
hypercenter#2512D53
hypocenterC53#27066
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2512

Complements

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

NH ≅ G/Ntype
1. #20277771C2semidirect
2. #20276259C2semidirect
4. #20275961C2semidirect
7. #20275349C2semidirect
9. #20275290C2semidirect
10. #20275274C2semidirect
13. #20274717C2semidirect
14. C4×#13133140C2semidirect
170. C4⋊C4×C53SD16semidirect
376. C53#27066semidirect
510. C4⋊C4#11719923semidirect

Maximal subgroups

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

subgroupindex
#202777712
#202762592
#202760322
#202759612
#202757972
#202755682
#202753492
#202753392
#202752902
#202752742
#202750732
#202747172
C4×#131331402
C53×#25122
#2706653