Finite Group Databasebeta

← #27484109

Subgroups — #27484109

Order 13568.

828 conjugacy classes of subgroups.

The subgroup lattice has 634 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20283065C211yesmaximal, characteristic
2#20277243C211yesmaximal, characteristic
3#20276280C211yesmaximal, characteristic, complemented
4#20276250C211yesmaximal, characteristic, complemented
5#20276041C211yesmaximal, characteristic, complemented
6#20275855C211yesmaximal, characteristic, complemented
7#20275309C211yesmaximal, characteristic, complemented
8#20275305C211yesmaximal, characteristic, complemented
9#20275138C211yesmaximal, characteristic
10#20275136C211yesmaximal, characteristic
11#20274970C211yesmaximal, characteristic, complemented
12#20274966C211yesmaximal, characteristic, complemented
13#20274742C211yesmaximal, characteristic
14C2×#16810377C211yesmaximal, characteristic
15C53×#1808C211yesmaximal, characteristic
16C23×#88322C2211yescharacteristic
17C2×#13133216C2211yescharacteristic
18C2×#13133196C2211yescharacteristic
19C2×#13133194—21no—
20C2×#13133182C2211yescharacteristic
21C22×C4×Dic53C2211yescharacteristic
22C2×#13133150C2211yescharacteristic
23C2×#13133150C2211yescharacteristic
24C2×#13133142C2211yescharacteristic
25C2×#13133142C2211yescharacteristic
26C2×#13133140C2211yescharacteristic
27C2×#13133140C2211yescharacteristic
28#16810528C2211yescharacteristic
29#16810527—21no—
30#16810522C2211yescharacteristic
31#16810520—21no—
32#16810519C2211yescharacteristic
33#16810519C2211yescharacteristic
34#16810505—21no—
35#16810495—21no—
36#16810492C2211yescharacteristic
37#16810483C2211yescharacteristic
38#16810481—21no—
39C4⋊C4×Dic53C2211yescharacteristic
40#16810474—21no—
41C22⋊C4×Dic53—21no—
42#16810386C2211yescharacteristic
43#16810386C2211yescharacteristic
44#16810385C2211yescharacteristic
45#16810385C2211yescharacteristic
46#16810385C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810377C2211yescharacteristic
49#16810377C2211yescharacteristic
50#16810377C2211yescharacteristic
51#16810377C2211yescharacteristic
52C23×C8×C53C2211yescharacteristic
53C2×#107×C53C2211yescharacteristic
54C2×#107×C53C2211yescharacteristic
55C53×#380C2211yescharacteristic
56C53×#380C2211yescharacteristic
57C53×#340C2211yescharacteristic
58C53×#340C2211yescharacteristic
59C22×Dic106—21no—
60C22×Dic106C2311yescharacteristic
61C23×Dic53D411yescharacteristic
62C23×Dic53D411yescharacteristic
63C22×#88322—22no—
64C22×#88322—22no—
65C22×#88322—41no—
66C22×#88322C2311yescharacteristic
67C22×#88322C2311yescharacteristic
68#13133216—41no—
69#13133216—41no—
70#13133196—22no—
71#13133196Q814yescomplemented
72#13133194—44no—
73#13133182—41no—
74#13133182—41no—
75C2×C4⋊Dic53—21no—
76C2×C4⋊Dic53C2311yescharacteristic
77C2×C4⋊Dic53C2311yescharacteristic
78C2×C4⋊Dic53C2311yescharacteristic
79C2×#11719919—21no—
80C2×#11719919—21no—
81C2×#11719919—21no—
82C2×#11719919—21no—
83C2×#11719919—21no—
84C2×#11719919—21no—
85C2×#11719919—21no—
86C2×C22⋊Dic53—21no—
87C2×C22⋊Dic53—21no—
88C2×C4×Dic53—21no—
89C2×C4×Dic53—21no—
90C2×C4×Dic53—41no—
91C2×C4×Dic53—41no—
92C2×C4×Dic53D411yescharacteristic
93C2×C4×Dic53D411yescharacteristic
94C2×C4×Dic53C2311yescharacteristic
95C2×C4×Dic53C2311yescharacteristic
96#13133150—22no—
97#13133150—41no—
98#13133142—22no—
99#13133142—41no—
100#13133140—22no—
101#13133140—41no—
102#13133124—21no—
103#13133124—21no—
104#13133124—21no—
105#13133124—21no—
106#13133124—21no—
107#13133124—21no—
108#13133124—21no—
109C23×C4×C53C2311yescharacteristic
110C22×C8×C53—22no—
111C22×C8×C53—22no—
112C22×C8×C53—41no—
113C22×C8×C53C2311yescharacteristic
114C22×C8×C53C2311yescharacteristic
115C2×C4⋊C4×C53C2311yescharacteristic
116C2×C4⋊C4×C53C2311yescharacteristic
117C2×C4⋊C4×C53C2311yescharacteristic
118C2×C4⋊C4×C53C2311yescharacteristic
119C2×C22⋊C4×C53—21no—
120C2×C22⋊C4×C53—21no—
121#107×C53—22no—
122#107×C53—41no—
123#95×C53—21no—
124#95×C53—21no—
125C24×C53C2×D411yescharacteristic
126C22×C4×C53—21no—
127C22×C4×C53—21no—
128C22×C4×C53—21no—
129C22×C4×C53—21no—
130C22×C4×C53—21no—
131C22×C4×C53—21no—
132C22×C4×C53—21no—
133C22×C4×C53—21no—
134C22×C4×C53C2×D411yescharacteristic
135C22×C4×C53C2×Q812yes—
136C22×C4×C53#4112yes—
137C22×C4×C53C2411yescharacteristic
138C22×Dic53—22no—
139C22×Dic53—21no—
140C22×Dic53—21no—
141C22×Dic53—22no—
142C22×Dic53—21no—
143C22×Dic53—21no—
144C22×Dic53—41no—
145C22×Dic53—41no—
146C22×Dic53—41no—
147C22×Dic53Q1614yes—
148C22×Dic53C2×D411yescharacteristic
149C22×Dic53C2×D411yescharacteristic
150C22×Dic53C2×D411yescharacteristic
151C22×Dic53C2×D411yescharacteristic
152C2×Dic106—21no—
153C2×Dic106—21no—
154C2×Dic106—21no—
155C2×Dic106—21no—
156C2×Dic106—21no—
157C2×Dic106—42no—
158C2×Dic106—42no—
159C2×Dic106—42no—
160C2×Dic106C2×Q812yes—
161C2×C8×C53—21no—
162C2×C8×C53—21no—
163C2×C8×C53—21no—
164C2×C8×C53—21no—
165C2×C8×C53—21no—
166C2×C8×C53—44no—
167C2×C8×C53#4112yes—
168C4⋊C4×C53—21no—
169C4⋊C4×C53—21no—
170C4⋊C4×C53—21no—
171C4⋊C4×C53—21no—
172C4⋊C4×C53—21no—
173C4⋊C4×C53—21no—
174C4⋊C4×C53—21no—
175C4⋊C4×C53—21no—
176C22⋊C4×C53—42no—
177C22⋊C4×C53—42no—
178C22⋊Dic53—24no—
179C22⋊Dic53—42no—
180C4⋊Dic53—21no—
181C4⋊Dic53—21no—
182C4⋊Dic53—21no—
183C4⋊Dic53—21no—
184C4⋊Dic53—21no—
185C4⋊Dic53—42no—
186C4⋊Dic53C2×Q812yes—
187#11719919—24no—
188#11719919—42no—
189#11719919—42no—
190#11719919—42no—
191#11719919—42no—
192#11719919—42no—
193#11719919—42no—
194C4×Dic53—21no—
195C4×Dic53—21no—
196C4×Dic53—21no—
197C4×Dic53—21no—
198C4×Dic53—41no—
199C4×Dic53—41no—
200C4×Dic53—42no—
201C4×Dic53—42no—
202C2×#88322—21no—
203C2×#88322—21no—
204C2×#88322—21no—
205C2×#88322—21no—
206C2×#88322—21no—
207C2×#88322—44no—
208C2×#88322#4112yes—
209C23×C53—21no—
210C23×C53—21no—
211C23×C53—21no—
212C23×C53—21no—
213C23×C53C22⋊Q812yes—
214C23×C53C2×Q1612yes—
215C23×C53AGL(1,ℤ/8)12yes—
216C23×C53C22×D411yescharacteristic
217C2×C4×C53—21no—
218C2×C4×C53—21no—
219C2×C4×C53—22no—
220C2×C4×C53—22no—
221C2×C4×C53—22no—
222C2×C4×C53—22no—
223C2×C4×C53—41no—
224C2×C4×C53—41no—
225C2×C4×C53—41no—
226C2×C4×C53—41no—
227C2×C4×C53—41no—
228C2×C4×C53—41no—
229C2×C4×C53—41no—
230C2×C4×C53—41no—
231C2×C4×C53—41no—
232C2×C4×C53—41no—
233C2×C4×C53—41no—
234C2×C4×C53—41no—
235C2×C4×C53C22⋊Q812yes—
236C2×C4×C53C22×Q811yescharacteristic
237C2×C4×C53C2×#4111yescharacteristic
238C2×C4×C53C2×#4111yescharacteristic
239C2×C4×C53C2×#4111yescharacteristic
240C2×C4×C53H(5,𝔽2)11yescharacteristic
241C2×C4×C53H(5,𝔽2)11yescharacteristic
242C2×Dic53—21no—
243C2×Dic53—21no—
244C2×Dic53—21no—
245C2×Dic53—21no—
246C2×Dic53—21no—
247C2×Dic53—21no—
248C2×Dic53—22no—
249C2×Dic53—22no—
250C2×Dic53—21no—
251C2×Dic53—21no—
252C2×Dic53—21no—
253C2×Dic53—44no—
254C2×Dic53—44no—
255C2×Dic53—41no—
256C2×Dic53—41no—
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×Dic53C2×Q1612yes—
266C2×Dic53C2×Q1612yes—
267C2×Dic53#13512yes—
268Dic106—22no—
269Dic106—22no—
270Dic106—41no—
271Dic106—41no—
272Dic106—84no—
273C8×C53—22no—
274C8×C53—41no—
275#88322—22no—
276#88322—41no—
277#29160—531nomaximal, Sylow
278C22×C53—22no—
279C22×C53—22no—
280C22×C53—22no—
281C22×C53—22no—
282C22×C53—22no—
283C22×C53—22no—
284C22×C53#47714yes—
285C22×C53C2×C22⋊Q811yescharacteristic
286C22×C53#53211yescharacteristic
287C22×C53#53411yescharacteristic
288C22×C53C22×Q1611yescharacteristic
289C22×C53C2×#13511yescharacteristic
290C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
291C22×C53#57511yescharacteristic
292C4×C53—22no—
293C4×C53—81no—
294C4×C53—81no—
295C4×C53—81no—
296C4×C53—81no—
297C4×C53#55111yescharacteristic
298C4×C53#55111yescharacteristic
299C4×C53#55411yescharacteristic
300C4×C53#55411yescharacteristic
301Dic53—22no—
302Dic53—22no—
303Dic53—22no—
304Dic53—42no—
305Dic53—42no—
306Dic53—41no—
307Dic53—81no—
308Dic53—81no—
309#2857—531no—
310C4×C22⋊Q8—531no—
311#2100—531no—
312#2095—531no—
313#2057—531no—
314#2024—531no—
315#1921—531no—
316#1921—531no—
317#1892—531no—
318#1892—531no—
319#1863—531no—
320#1863—531no—
321#1808—531no—
322#1808D5311yescharacteristic
323C2×#336—531no—
324C2×C53—24no—
325C2×C53#256411yescharacteristic
326C2×C53C2×#47711yescharacteristic
327C2×C53#307611yescharacteristic
328C2×C53#317011yescharacteristic
329C2×C53#318011yescharacteristic
330C2×C53#318811yescharacteristic
331C2×C53#318911yescharacteristic
332C23×C8—531no—
333C23×C8C2×D5311yescharacteristic
334C2×C4⋊Q8—531no—
335C2×C22⋊Q8—531no—
336C2×C22⋊Q8—1061no—
337C2×C4×Q8—531no—
338C22×C42—531no—
339C2×#107—531no—
340C2×#107—531no—
341C2×#107C2×D5311yescharacteristic
342C2×#107C2×D5311yescharacteristic
343C2×#103—531no—
344C2×#103—531no—
345C2×#103—531no—
346C2×#103—531no—
347#391—531no—
348#391—1061no—
349#389—531no—
350#389—531no—
351#389—531no—
352#389—1061no—
353#387—1061no—
354#385—1061no—
355#384—531no—
356#380—531no—
357#380—1061no—
358#380C2×D5311yescharacteristic
359#380C2×D5311yescharacteristic
360C4×C4⋊C4—531no—
361C4×C4⋊C4—1061no—
362C4×C22⋊C4—1061no—
363#340—531no—
364#340—531no—
365#340—531no—
366#340—531no—
367#340—531no—
368#340—531no—
369#340C2×D5311yescharacteristic
370#340C2×D5311yescharacteristic
371#336—531no—
372#336—531no—
373#336—531no—
374#336—531no—
375C53#2916011yescharacteristic, complemented, Sylow
376C22×Q8—531no—
377C22×Q8—1061no—
378C23×C4—531no—
379C23×C4—531no—
380C23×C4C22×D5311yescharacteristic
381C22×C8—22no—
382C22×C8—22no—
383C22×C8—41no—
384C22×C8—531no—
385C22×C8—531no—
386C22×C8—1062no—
387C22×C8—1062no—
388C22×C8—2121no—
389C22×C8C22×D5311yescharacteristic
390C22×C8C22×D5311yescharacteristic
391C4⋊Q8—2121no—
392C4⋊Q8—2121no—
393C22⋊Q8—534no—
394C22⋊Q8—1062no—
395C22⋊Q8—2124no—
396C4×Q8—2121no—
397C4×Q8—2121no—
398C2×C4⋊C4—531no—
399C2×C4⋊C4—531no—
400C2×C4⋊C4—531no—
401C2×C4⋊C4—1061no—
402C2×C4⋊C4—1061no—
403C2×C4⋊C4—1061no—
404C2×C4⋊C4—1061no—
405C2×C4⋊C4—1061no—
406C2×C4⋊C4—1061no—
407C2×C4⋊C4—1061no—
408C2×C4⋊C4—1061no—
409C2×C4⋊C4C22×D5311yescharacteristic
410C2×C4⋊C4C22×D5311yescharacteristic
411C2×C4⋊C4C22×D5311yescharacteristic
412C2×C4⋊C4C22×D5311yescharacteristic
413C2×C22⋊C4—21no—
414C2×C22⋊C4—21no—
415C2×C22⋊C4—1061no—
416C2×C22⋊C4—1061no—
417C2×C42—531no—
418C2×C42—531no—
419C2×C42—531no—
420C2×C42—531no—
421C2×C42—1061no—
422C2×C42—1061no—
423C2×C42—2121no—
424C2×C42—2121no—
425#107—22no—
426#107—41no—
427#107—1062no—
428#107—2121no—
429#103—1062no—
430#103—1062no—
431#103—2121no—
432#103—2121no—
433#95—21no—
434#95—21no—
435#95—1061no—
436#95—1061no—
437#95—1061no—
438#95—1061no—
439#95—1061no—
440#95—1061no—
441#95—1061no—
442C24D4×D5311yescharacteristic
443C2×Q8—532no—
444C2×Q8—1061no—
445C2×Q8—1061no—
446C2×Q8—1061no—
447C2×Q8—1061no—
448C2×Q8—1061no—
449C2×Q8—2122no—
450C2×Q8—2122no—
451C2×Q8—2122no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—531no—
461C22×C4—531no—
462C22×C4—531no—
463C22×C4—534no—
464C22×C4—531no—
465C22×C4—1062no—
466C22×C4—1062no—
467C22×C4—1061no—
468C22×C4—1061no—
469C22×C4—1061no—
470C22×C4—1061no—
471C22×C4—2121no—
472C22×C4—2121no—
473C22×C4—2121no—
474C22×C4#1171994512yes—
475C22×C4D4×D5311yescharacteristic
476C22×C4Q8×D5312yes—
477C22×C4C23×D5311yescharacteristic
478C2×C8—21no—
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—44no—
484C2×C8—532no—
485C2×C8—1061no—
486C2×C8—1061no—
487C2×C8—1061no—
488C2×C8—1061no—
489C2×C8—1061no—
490C2×C8—2124no—
491C2×C8#1171994712yes—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—532no—
501C4⋊C4—1064no—
502C4⋊C4—1061no—
503C4⋊C4—1061no—
504C4⋊C4—1061no—
505C4⋊C4—1061no—
506C4⋊C4—1061no—
507C4⋊C4—2122no—
508C4⋊C4—2122no—
509C4⋊C4—2122no—
510C4⋊C4—2122no—
511C4⋊C4—2122no—
512C4⋊C4—2122no—
513C4⋊C4—2122no—
514C22⋊C4—42no—
515C22⋊C4—42no—
516C22⋊C4—1064no—
517C22⋊C4—2122no—
518C42—1061no—
519C42—1061no—
520C42—1061no—
521C42—1061no—
522C42—2121no—
523C42—2121no—
524C42—2122no—
525C42—2122no—
526C23—21no—
527C23—21no—
528C23—21no—
529C23—21no—
530C23#1313319512yes—
531C23Q16×D5312yes—
532C23#1313324112yes—
533C23C2×D4×D5311yescharacteristic
534Q8—1062no—
535Q8—1062no—
536Q8—4244no—
537Q8—2121no—
538Q8—2121no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—532no—
558C2×C4—532no—
559C2×C4—532no—
560C2×C4—1061no—
561C2×C4—1061no—
562C2×C4—1061no—
563C2×C4—1061no—
564C2×C4—1061no—
565C2×C4—1061no—
566C2×C4—1062no—
567C2×C4—1062no—
568C2×C4—1061no—
569C2×C4—1061no—
570C2×C4—1061no—
571C2×C4—2124no—
572C2×C4—2124no—
573C2×C4—2121no—
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×C4#1313319512yes—
584C2×C4C2×Q8×D5311yescharacteristic
585C2×C4#41×D5311yescharacteristic
586C2×C4C2×#1171994711yescharacteristic
587C2×C4C2×#1171994511yescharacteristic
588C2×C4#1313326211yescharacteristic
589C2×C4#1313326311yescharacteristic
590C8—22no—
591C8—41no—
592C8—1062no—
593C8—2121no—
594C22—22no—
595C22—22no—
596C22—22no—
597C22—22no—
598C22—22no—
599C22—22no—
600C22#1681091714yes—
601C22C2×#1313319511yescharacteristic
602C22#1681116811yescharacteristic
603C22#1681118111yescharacteristic
604C22C2×Q16×D5311yescharacteristic
605C22#135×D5311yescharacteristic
606C22C2×#1313324111yescharacteristic
607C22#1681142011yescharacteristic
608C4—22no—
609C4—81no—
610C4—81no—
611C4—81no—
612C4—81no—
613C4—1062no—
614C4—1062no—
615C4—1062no—
616C4—4241no—
617C4—4241no—
618C4—2121no—
619C4—2122no—
620C4—2122no—
621C4#1681129711yescharacteristic
622C4#1681129911yescharacteristic
623C4#1681132711yescharacteristic
624C4#1681133111yescharacteristic
625C2—24no—
626C2#2027989311yescharacteristic
627C2C2×#1681091711yescharacteristic
628C2#2028450111yescharacteristic
629C2#2028529611yescharacteristic
630C2#2028540811yescharacteristic
631C2#2028547511yescharacteristic
632C2#2028548611yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#1808C2
socleC23×C53C22×D4
prime radicalC24×C53C2×D4
hypercenter#1808D53
hypocenterC53#29160
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1808

Complements

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

NH ≅ G/Ntype
3. #20276280C2semidirect
4. #20276250C2semidirect
5. #20276041C2semidirect
6. #20275855C2semidirect
7. #20275309C2semidirect
8. #20275305C2semidirect
11. #20274970C2semidirect
12. #20274966C2semidirect
71. #13133196Q8semidirect
375. C53#29160semidirect

Maximal subgroups

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

subgroupindex
#202830652
#202772432
#202762802
#202762502
#202760412
#202758552
#202753092
#202753052
#202751382
#202751362
#202749702
#202749662
#202747422
C2×#168103772
C53×#18082
#2916053