Finite Group Databasebeta

← #27452132

Subgroups — #27452132

Order 13568.

832 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20279056C211yesmaximal, characteristic, complemented
2#20276032C211yesmaximal, characteristic, complemented
3#20275894C211yesmaximal, characteristic
4#20275861C211yesmaximal, characteristic, complemented
5#20275718C211yesmaximal, characteristic
6#20275672C211yesmaximal, characteristic, complemented
7#20275390C211yesmaximal, characteristic, complemented
8#20275356C211yesmaximal, characteristic, complemented
9#20275323C211yesmaximal, characteristic, complemented
10#20275283C211yesmaximal, characteristic, complemented
11#20275049C211yesmaximal, characteristic, complemented
12#20275022C211yesmaximal, characteristic, complemented
13#20274805C211yesmaximal, characteristic, complemented
14C4×#13133138C211yesmaximal, characteristic, complemented
15C53×#2361C211yesmaximal, characteristic
16C22×#11719923C2211yescharacteristic
17C2×#13133209C2211yescharacteristic
18C2×#13133196C2211yescharacteristic
19C2×#13133150C2211yescharacteristic
20C2×C4⋊#88322C2211yescharacteristic
21C2×#13133140C2211yescharacteristic
22C2×#13133140C2211yescharacteristic
23C2×#13133140C2211yescharacteristic
24C2×#13133138C2211yescharacteristic
25C2×#13133138C2211yescharacteristic
26C2×#13133138C2211yescharacteristic
27C2×C22⋊#88322C2211yescharacteristic
28C2×C22⋊#88322C2211yescharacteristic
29C2×C4×#88322C2211yescharacteristic
30#16810564—21no—
31#16810551—21no—
32#16810529C2211yescharacteristic
33#16810529C2211yescharacteristic
34#16810511—21no—
35#16810487—21no—
36#16810483C2211yescharacteristic
37C4×C4⋊Dic53C2211yescharacteristic
38#16810385C2211yescharacteristic
39#16810385C2211yescharacteristic
40#16810385C2211yescharacteristic
41#16810385C2211yescharacteristic
42#16810385C2211yescharacteristic
43#16810385C2211yescharacteristic
44#16810385C2211yescharacteristic
45#16810385C2211yescharacteristic
46#16810378C2211yescharacteristic
47#16810378C2211yescharacteristic
48C2×#121×C53C2211yescharacteristic
49C2×C4×D4×C53—21no—
50C2×C4×D4×C53C2211yescharacteristic
51C22×C4⋊C4×C53C2211yescharacteristic
52C53×#390C2211yescharacteristic
53C53×#389C2211yescharacteristic
54C53×#388—21no—
55C53×#386—21no—
56C53×#385—21no—
57C4×C4⋊C4×C53C2211yescharacteristic
58C4×C22⋊C4×C53C2211yescharacteristic
59C22×Dic106C2311yescharacteristic
60C2×#11719923—22no—
61C2×#11719923—41no—
62C2×#11719923—41no—
63C22×#88322C2311yescharacteristic
64C22×#88322C2311yescharacteristic
65C22×#88322C2311yescharacteristic
66C22×#88322C2311yescharacteristic
67#13133209—22no—
68#13133209—41no—
69#13133196—22no—
70#13133196—41no—
71C2×C4⋊Dic53C2311yescharacteristic
72C2×C4⋊Dic53C2311yescharacteristic
73C2×C4⋊Dic53C2311yescharacteristic
74C2×#11719919—21no—
75C2×#11719919—21no—
76C2×#11719919—21no—
77C2×C22⋊Dic53—21no—
78C2×C22⋊Dic53—21no—
79C2×C4×Dic53—21no—
80#13133150—22no—
81C4⋊#88322—41no—
82#13133140—22no—
83#13133140—22no—
84#13133140—41no—
85#13133138—22no—
86#13133138—22no—
87#13133138—41no—
88C22⋊#88322—41no—
89C22⋊#88322—41no—
90C4×#88322—41no—
91#13133124—21no—
92#13133124—21no—
93#13133124—21no—
94#13133124—21no—
95#13133124—21no—
96#13133124—21no—
97C22×D4×C53—21no—
98C22×D4×C53C2311yescharacteristic
99C23×C4×C53—21no—
100C23×C4×C53—21no—
101C23×C4×C53C2311yescharacteristic
102#121×C53—22no—
103#121×C53—22no—
104C4×D4×C53—22no—
105C4×D4×C53—41no—
106C4×D4×C53—44no—
107C2×C4⋊C4×C53—21no—
108C2×C4⋊C4×C53—21no—
109C2×C4⋊C4×C53—22no—
110C2×C4⋊C4×C53—41no—
111C2×C4⋊C4×C53C2311yescharacteristic
112C2×C4⋊C4×C53C2311yescharacteristic
113C2×C4⋊C4×C53C2311yescharacteristic
114C2×C22⋊C4×C53—21no—
115C2×C22⋊C4×C53—21no—
116C2×C22⋊C4×C53—21no—
117C2×C22⋊C4×C53—21no—
118C2×C22⋊C4×C53D411yescharacteristic
119C2×C22⋊C4×C53D411yescharacteristic
120C2×C42×C53—21no—
121C2×C42×C53C2311yescharacteristic
122C2×C42×C53C2311yescharacteristic
123#95×C53—21no—
124#95×C53—21no—
125#95×C53D411yescharacteristic
126#95×C53D411yescharacteristic
127C24×C53—21no—
128C24×C53C2×D411yescharacteristic
129C2×D4×C53—24no—
130C2×D4×C53—21no—
131C2×D4×C53—21no—
132C2×D4×C53—21no—
133C2×D4×C53—21no—
134C2×D4×C53—21no—
135C2×D4×C53—42no—
136C2×D4×C53—42no—
137C2×D4×C53#4112yes—
138C22×C4×C53—21no—
139C22×C4×C53—21no—
140C22×C4×C53—21no—
141C22×C4×C53—21no—
142C22×C4×C53—21no—
143C22×C4×C53—21no—
144C22×C4×C53—21no—
145C22×C4×C53—21no—
146C22×C4×C53—21no—
147C22×C4×C53—42no—
148C22×C4×C53—42no—
149C22×C4×C53—42no—
150C22×C4×C53—42no—
151C22×C4×C53—81no—
152C22×C4×C53—81no—
153C22×C4×C53C2×D411yescharacteristic
154C22×C4×C53C2×D411yescharacteristic
155C22×C4×C53C2×D411yescharacteristic
156C22×C4×C53C2×D411yescharacteristic
157C22×C4×C53C2×D411yescharacteristic
158C22×C4×C53C2411yescharacteristic
159C22×Dic53—21no—
160C22×Dic53—21no—
161C22×Dic53—21no—
162C22×Dic53—21no—
163C2×Dic106—21no—
164C2×Dic106—21no—
165C2×Dic106—21no—
166C2×Dic106—21no—
167C2×Dic106—21no—
168C2×Dic106—21no—
169C4⋊C4×C53—21no—
170C4⋊C4×C53—21no—
171C4⋊C4×C53—21no—
172C4⋊C4×C53—21no—
173C4⋊C4×C53—42no—
174C4⋊C4×C53—42no—
175C4⋊C4×C53#4112yes—
176C4⋊C4×C53#4112yes—
177C22⋊C4×C53—24no—
178C22⋊C4×C53—22no—
179C22⋊C4×C53—22no—
180C22⋊C4×C53—42no—
181C22⋊C4×C53—42no—
182C22⋊C4×C53—81no—
183C42×C53—21no—
184C42×C53—21no—
185C42×C53—21no—
186C42×C53—21no—
187C42×C53—42no—
188C22⋊Dic53—42no—
189C22⋊Dic53—42no—
190#11719923—44no—
191C4⋊Dic53—21no—
192C4⋊Dic53—21no—
193C4⋊Dic53—21no—
194C4⋊Dic53—21no—
195C4⋊Dic53—21no—
196C4⋊Dic53#4112yes—
197#11719919—42no—
198#11719919—42no—
199#11719919—42no—
200C4×Dic53—42no—
201C2×#88322—21no—
202C2×#88322—21no—
203C2×#88322—21no—
204C2×#88322—21no—
205C2×#88322—21no—
206C2×#88322—21no—
207C2×#88322—21no—
208C2×#88322—21no—
209C2×#88322—21no—
210C2×#88322—21no—
211C2×#88322—21no—
212C2×#88322—21no—
213C23×C53—21no—
214C23×C53—21no—
215C23×C53—21no—
216C23×C53—21no—
217C23×C53—21no—
218C23×C53—21no—
219C23×C53—22no—
220C23×C53—41no—
221C23×C53—41no—
222C23×C53—41no—
223C23×C53—41no—
224C23×C53—41no—
225C23×C53—41no—
226C23×C53#13512yes—
227C23×C53C22×D411yescharacteristic
228D4×C53—22no—
229D4×C53—22no—
230D4×C53—48no—
231D4×C53—41no—
232D4×C53—41no—
233C2×C4×C53—21no—
234C2×C4×C53—21no—
235C2×C4×C53—21no—
236C2×C4×C53—22no—
237C2×C4×C53—21no—
238C2×C4×C53—21no—
239C2×C4×C53—22no—
240C2×C4×C53—21no—
241C2×C4×C53—21no—
242C2×C4×C53—21no—
243C2×C4×C53—21no—
244C2×C4×C53—21no—
245C2×C4×C53—21no—
246C2×C4×C53—21no—
247C2×C4×C53—21no—
248C2×C4×C53—21no—
249C2×C4×C53—41no—
250C2×C4×C53—41no—
251C2×C4×C53—41no—
252C2×C4×C53—41no—
253C2×C4×C53—41no—
254C2×C4×C53—41no—
255C2×C4×C53—41no—
256C2×C4×C53—41no—
257C2×C4×C53—41no—
258C2×C4×C53—41no—
259C2×C4×C53—41no—
260C2×C4×C53—84no—
261C2×C4×C53—84no—
262C2×C4×C53AGL(1,ℤ/8)12yes—
263C2×C4×C53C2×#4111yescharacteristic
264C2×C4×C53C2×#4111yescharacteristic
265C2×C4×C53C2×#4111yescharacteristic
266C2×C4×C53C2×#4111yescharacteristic
267C2×C4×C53H(5,𝔽2)11yescharacteristic
268C2×C4×C53H(5,𝔽2)11yescharacteristic
269C2×Dic53—22no—
270C2×Dic53—41no—
271C2×Dic53—41no—
272C2×Dic53—41no—
273C2×Dic53—41no—
274C2×Dic53—41no—
275C2×Dic53—41no—
276C2×Dic53—41no—
277C2×Dic53—41no—
278C2×Dic53—41no—
279C2×Dic53—41no—
280C2×Dic53—41no—
281Dic106—41no—
282Dic106—41no—
283Dic106—41no—
284Dic106—41no—
285#88322—41no—
286#88322—41no—
287#88322—41no—
288#88322—41no—
289#26721—531nomaximal, Sylow
290C22×C53—22no—
291C22×C53—22no—
292C22×C53—22no—
293C22×C53—41no—
294C22×C53—41no—
295C22×C53—41no—
296C22×C53—41no—
297C22×C53—42no—
298C22×C53—42no—
299C22×C53—42no—
300C22×C53—81no—
301C22×C53—81no—
302C22×C53—81no—
303C22×C53—81no—
304C22×C53#52511yescharacteristic
305C22×C53#54611yescharacteristic
306C22×C53#54611yescharacteristic
307C22×C53C2×#13511yescharacteristic
308C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
309C22×C53#57711yescharacteristic
310C22×C53#57811yescharacteristic
311C4×C53—22no—
312C4×C53—41no—
313C4×C53—42no—
314C4×C53—41no—
315C4×C53—41no—
316C4×C53—41no—
317C4×C53—42no—
318C4×C53—81no—
319C4×C53#52911yescharacteristic
320C4×C53#53811yescharacteristic
321C4×C53#55511yescharacteristic
322C4×C53#55911yescharacteristic
323Dic53—42no—
324Dic53—81no—
325Dic53—81no—
326Dic53—81no—
327#2489—531no—
328#2361D5311yescharacteristic
329#2055—531no—
330#2033—531no—
331#2025—531no—
332#2000—531no—
333#1989—531no—
334#1933—531no—
335#1928—531no—
336#1923—531no—
337#1918—531no—
338#1877—531no—
339#1872—531no—
340#1824—531no—
341C4×#102—531no—
342C2×C53—42no—
343C2×C53—82no—
344C2×C53#245911yescharacteristic
345C2×C53#310511yescharacteristic
346C2×C53#312011yescharacteristic
347C2×C53#330011yescharacteristic
348C2×C53#330111yescharacteristic
349C2×C53#331011yescharacteristic
350C2×C53#333011yescharacteristic
351C22×SD16—531no—
352C2×#125—531no—
353C2×C22⋊Q8—531no—
354C2×#121C2×D5311yescharacteristic
355C2×C4×D4—21no—
356C2×C4×D4C2×D5311yescharacteristic
357C22×C4⋊C4C2×D5311yescharacteristic
358C2×#107—531no—
359C2×C4⋊C8—531no—
360C2×#103—531no—
361C2×#103—531no—
362C2×#103—531no—
363C2×#102—531no—
364C2×#102—531no—
365C2×#102—531no—
366C2×C22⋊C8—531no—
367C2×C22⋊C8—531no—
368C2×C4×C8—531no—
369#400—1061no—
370#397—1061no—
371#391—531no—
372#391—531no—
373#390C2×D5311yescharacteristic
374#389C2×D5311yescharacteristic
375#388—21no—
376#388—1061no—
377#386—21no—
378#385—21no—
379#382—1061no—
380#380—531no—
381C4×C4⋊C4—531no—
382C4×C4⋊C4C2×D5311yescharacteristic
383C4×C22⋊C4C2×D5311yescharacteristic
384#340—531no—
385#340—531no—
386#340—531no—
387#340—531no—
388#340—531no—
389#340—531no—
390#340—531no—
391#340—531no—
392#336—531no—
393#336—531no—
394C53#2672111yescharacteristic, complemented, Sylow
395C22×Q8—531no—
396C22×D4—21no—
397C22×D4C22×D5311yescharacteristic
398C23×C4—21no—
399C23×C4—21no—
400C23×C4C22×D5311yescharacteristic
401C2×SD16—1062no—
402C2×SD16—2121no—
403C2×SD16—2121no—
404C22×C8—531no—
405C22×C8—531no—
406C22×C8—531no—
407C22×C8—531no—
408#125—1062no—
409#125—2121no—
410C22⋊Q8—1062no—
411C22⋊Q8—2121no—
412#121—22no—
413#121—22no—
414C4×D4—22no—
415C4×D4—41no—
416C4×D4—44no—
417C2×C4⋊C4—22no—
418C2×C4⋊C4—21no—
419C2×C4⋊C4—21no—
420C2×C4⋊C4—41no—
421C2×C4⋊C4—531no—
422C2×C4⋊C4—531no—
423C2×C4⋊C4—531no—
424C2×C4⋊C4—1061no—
425C2×C4⋊C4—1061no—
426C2×C4⋊C4—1061no—
427C2×C4⋊C4C22×D5311yescharacteristic
428C2×C4⋊C4C22×D5311yescharacteristic
429C2×C4⋊C4C22×D5311yescharacteristic
430C2×C22⋊C4—21no—
431C2×C22⋊C4—21no—
432C2×C22⋊C4—21no—
433C2×C22⋊C4—21no—
434C2×C22⋊C4—1061no—
435C2×C22⋊C4—1061no—
436C2×C22⋊C4C22⋊D5311yescharacteristic
437C2×C22⋊C4C22⋊D5311yescharacteristic
438C2×C42—21no—
439C2×C42—1061no—
440C2×C42C22×D5311yescharacteristic
441C2×C42C22×D5311yescharacteristic
442#107—1062no—
443C4⋊C8—2121no—
444#103—1062no—
445#103—1062no—
446#103—2121no—
447#102—1062no—
448#102—1062no—
449#102—2121no—
450C22⋊C8—2121no—
451C22⋊C8—2121no—
452C4×C8—2121no—
453#95—21no—
454#95—21no—
455#95—1061no—
456#95—1061no—
457#95—1061no—
458#95—1061no—
459#95—1061no—
460#95—1061no—
461#95C22⋊D5311yescharacteristic
462#95C22⋊D5311yescharacteristic
463C24—21no—
464C24C2×C22⋊D5311yescharacteristic
465C2×Q8—1061no—
466C2×Q8—1061no—
467C2×Q8—1061no—
468C2×Q8—1061no—
469C2×Q8—1061no—
470C2×Q8—1061no—
471C2×D4—24no—
472C2×D4—21no—
473C2×D4—21no—
474C2×D4—21no—
475C2×D4—21no—
476C2×D4—21no—
477C2×D4—42no—
478C2×D4—42no—
479C2×D4#1171994512yes—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—21no—
487C22×C4—21no—
488C22×C4—21no—
489C22×C4—42no—
490C22×C4—42no—
491C22×C4—42no—
492C22×C4—42no—
493C22×C4—81no—
494C22×C4—81no—
495C22×C4—1061no—
496C22×C4—1061no—
497C22×C4—1061no—
498C22×C4—1061no—
499C22×C4C2×C22⋊D5311yescharacteristic
500C22×C4C2×C22⋊D5311yescharacteristic
501C22×C4C2×C22⋊D5311yescharacteristic
502C22×C4C2×C22⋊D5311yescharacteristic
503C22×C4C2×C22⋊D5311yescharacteristic
504C22×C4C23×D5311yescharacteristic
505SD16—2124no—
506C2×C8—1061no—
507C2×C8—1061no—
508C2×C8—1061no—
509C2×C8—1061no—
510C2×C8—1061no—
511C2×C8—1061no—
512C2×C8—1061no—
513C2×C8—1061no—
514C2×C8—1061no—
515C2×C8—1061no—
516C2×C8—1061no—
517C2×C8—1061no—
518C4⋊C4—21no—
519C4⋊C4—21no—
520C4⋊C4—21no—
521C4⋊C4—21no—
522C4⋊C4—42no—
523C4⋊C4—42no—
524C4⋊C4—532no—
525C4⋊C4—1061no—
526C4⋊C4—1061no—
527C4⋊C4—1061no—
528C4⋊C4—1061no—
529C4⋊C4—1061no—
530C4⋊C4—2122no—
531C4⋊C4—2122no—
532C4⋊C4—2122no—
533C4⋊C4#1171994512yes—
534C4⋊C4#1171994712yes—
535C22⋊C4—24no—
536C22⋊C4—22no—
537C22⋊C4—22no—
538C22⋊C4—42no—
539C22⋊C4—42no—
540C22⋊C4—81no—
541C22⋊C4—2122no—
542C22⋊C4—2122no—
543C42—21no—
544C42—21no—
545C42—21no—
546C42—21no—
547C42—42no—
548C42—2122no—
549C23—21no—
550C23—21no—
551C23—21no—
552C23—21no—
553C23—21no—
554C23—21no—
555C23—22no—
556C23—41no—
557C23—41no—
558C23—41no—
559C23—41no—
560C23—41no—
561C23—41no—
562C23#1313323712yes—
563C23C22×C22⋊D5311yescharacteristic
564Q8—2121no—
565Q8—2121no—
566Q8—2121no—
567Q8—2121no—
568D4—22no—
569D4—22no—
570D4—48no—
571D4—41no—
572D4—41no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—22no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—22no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—84no—
601C2×C4—84no—
602C2×C4—1062no—
603C2×C4—2121no—
604C2×C4—2121no—
605C2×C4—2121no—
606C2×C4—2121no—
607C2×C4—2121no—
608C2×C4—2121no—
609C2×C4—2121no—
610C2×C4—2121no—
611C2×C4—2121no—
612C2×C4—2121no—
613C2×C4—2121no—
614C2×C4#1313324312yes—
615C2×C4#41×D5311yescharacteristic
616C2×C4C2×#1171994711yescharacteristic
617C2×C4C2×#1171994511yescharacteristic
618C2×C4C2×#1171994511yescharacteristic
619C2×C4#1313326211yescharacteristic
620C2×C4#1313326311yescharacteristic
621C8—2121no—
622C8—2121no—
623C8—2121no—
624C8—2121no—
625C22—22no—
626C22—22no—
627C22—22no—
628C22—41no—
629C22—41no—
630C22—41no—
631C22—41no—
632C22—42no—
633C22—42no—
634C22—42no—
635C22—81no—
636C22—81no—
637C22—81no—
638C22—81no—
639C22#1681113411yescharacteristic
640C22#1681126111yescharacteristic
641C22#1681126711yescharacteristic
642C22C2×#1313323711yescharacteristic
643C22C2×#1313324311yescharacteristic
644C22#1681143411yescharacteristic
645C22#1681143811yescharacteristic
646C4—22no—
647C4—42no—
648C4—41no—
649C4—42no—
650C4—41no—
651C4—41no—
652C4—41no—
653C4—81no—
654C4—4241no—
655C4—4241no—
656C4—4241no—
657C4—2122no—
658C4#1681115311yescharacteristic
659C4#1681120611yescharacteristic
660C4#1681133811yescharacteristic
661C4#1681135611yescharacteristic
662C2—42no—
663C2—82no—
664C2#2027864611yescharacteristic
665C2#2028476211yescharacteristic
666C2#2028492911yescharacteristic
667C2#2028660211yescharacteristic
668C2#2028661111yescharacteristic
669C2#2028673611yescharacteristic
670C2#2028701411yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2361C2
socleC23×C53C22×D4
prime radicalC2×#121×C53C22
hypercenter#2361D53
hypocenterC53#26721
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2361

Complements

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

NH ≅ G/Ntype
1. #20279056C2semidirect
2. #20276032C2semidirect
4. #20275861C2semidirect
6. #20275672C2semidirect
7. #20275390C2semidirect
8. #20275356C2semidirect
9. #20275323C2semidirect
10. #20275283C2semidirect
11. #20275049C2semidirect
12. #20275022C2semidirect
13. #20274805C2semidirect
14. C4×#13133138C2semidirect
394. C53#26721semidirect

Maximal subgroups

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

subgroupindex
#202790562
#202760322
#202758942
#202758612
#202757182
#202756722
#202753902
#202753562
#202753232
#202752832
#202750492
#202750222
#202748052
C4×#131331382
C53×#23612
#2672153