Finite Group Databasebeta

← #27486406

Subgroups — #27486406

Order 13568.

778 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#20277192C211yesmaximal, characteristic, complemented
2#20276282C211yesmaximal, characteristic, complemented
3#20276266C211yesmaximal, characteristic, complemented
4#20275951C211yesmaximal, characteristic
5#20275809C211yesmaximal, characteristic
6#20275353C211yesmaximal, characteristic, complemented
7#20275339C211yesmaximal, characteristic
8#20275073C211yesmaximal, characteristic
9#20275059C211yesmaximal, characteristic, complemented
10#20275056C211yesmaximal, characteristic, complemented
11#20275049C211yesmaximal, characteristic
12#20274983C211yesmaximal, characteristic, complemented
13#20274962C211yesmaximal, characteristic
14#20274858C211yesmaximal, characteristic, complemented
15C4×#124×C53C211yesmaximal, characteristic
16C22×#11719925C2211yescharacteristic
17C22×#11719923C2211yescharacteristic
18C2×C4×Dic106C2211yescharacteristic
19C2×#13133150C2211yescharacteristic
20C2×#13133148C2211yescharacteristic
21C2×#13133141C2211yescharacteristic
22C2×#13133141C2211yescharacteristic
23C2×#13133141C2211yescharacteristic
24C2×#13133140C2211yescharacteristic
25C2×#13133140C2211yescharacteristic
26C2×#13133138C2211yescharacteristic
27C2×#13133138C2211yescharacteristic
28C2×#13133138C2211yescharacteristic
29#16810529C2211yescharacteristic
30#16810529C2211yescharacteristic
31#16810521C2211yescharacteristic
32#16810521C2211yescharacteristic
33#16810507—21no—
34#16810507—21no—
35C4×#11719919—21no—
36C4×C4⋊Dic53C2211yescharacteristic
37#16810469—21no—
38#16810385C2211yescharacteristic
39#16810385C2211yescharacteristic
40#16810385C2211yescharacteristic
41#16810385C2211yescharacteristic
42#16810378C2211yescharacteristic
43#16810378C2211yescharacteristic
44#16810378C2211yescharacteristic
45#16810378C2211yescharacteristic
46#16810378C2211yescharacteristic
47#16810378C2211yescharacteristic
48C2×#124×C53C2211yescharacteristic
49C2×C4×Q8×C53C2211yescharacteristic
50C2×C4×D4×C53C2211yescharacteristic
51C53×#391C2211yescharacteristic
52C53×#390C2211yescharacteristic
53C53×#388—21no—
54C53×#388—21no—
55C53×#382C2211yescharacteristic
56C4×C22⋊C4×C53—21no—
57C4×C22⋊C4×C53—21no—
58C43×C53C2211yescharacteristic
59C22×Dic106C2311yescharacteristic
60C2×#11719925—22no—
61C2×#11719925—22no—
62C2×#11719925—41no—
63C2×#11719923—22no—
64C2×#11719923—22no—
65C2×#11719923—41no—
66C22×#88322C2311yescharacteristic
67C22×#88322C2311yescharacteristic
68C22×#88322C2311yescharacteristic
69C22×#88322C2311yescharacteristic
70C4×Dic106—22no—
71C4×Dic106—22no—
72C2×C4⋊Dic53C2311yescharacteristic
73C2×C4⋊Dic53C2311yescharacteristic
74C2×C4⋊Dic53C2311yescharacteristic
75C2×#11719919—21no—
76C2×#11719919—21no—
77C2×#11719919—21no—
78C2×C4×Dic53—21no—
79C2×C4×Dic53—21no—
80#13133150—41no—
81#13133148—41no—
82#13133141—22no—
83#13133141—22no—
84#13133141—41no—
85#13133140—41no—
86#13133140—41no—
87#13133138—22no—
88#13133138—22no—
89#13133138—41no—
90#13133124—21no—
91#13133124—21no—
92#13133124—21no—
93#13133124—21no—
94#13133124—21no—
95#13133124—21no—
96#13133124—21no—
97C22×Q8×C53C2311yescharacteristic
98C22×D4×C53C2311yescharacteristic
99C23×C4×C53—21no—
100#124×C53—22no—
101#124×C53—22no—
102C4×Q8×C53—41no—
103C4×Q8×C53—41no—
104C4×D4×C53—41no—
105C4×D4×C53—41no—
106C2×C4⋊C4×C53—21no—
107C2×C4⋊C4×C53C2311yescharacteristic
108C2×C4⋊C4×C53C2311yescharacteristic
109C2×C22⋊C4×C53—21no—
110C2×C22⋊C4×C53—21no—
111C2×C22⋊C4×C53—21no—
112C2×C22⋊C4×C53—21no—
113C2×C22⋊C4×C53—21no—
114C2×C42×C53—21no—
115C2×C42×C53D411yescharacteristic
116C2×C42×C53D411yescharacteristic
117C2×C42×C53D411yescharacteristic
118C2×C42×C53D411yescharacteristic
119C2×C42×C53C2311yescharacteristic
120C2×C42×C53C2311yescharacteristic
121C2×C42×C53C2311yescharacteristic
122#95×C53—21no—
123#95×C53—21no—
124#95×C53—21no—
125#95×C53—21no—
126C24×C53—21no—
127C2×Q8×C53—21no—
128C2×Q8×C53—21no—
129C2×Q8×C53—21no—
130C2×Q8×C53—21no—
131C2×Q8×C53—21no—
132C2×Q8×C53#4112yes—
133C2×D4×C53—21no—
134C2×D4×C53—21no—
135C2×D4×C53—21no—
136C2×D4×C53—21no—
137C2×D4×C53—21no—
138C2×D4×C53#4112yes—
139C22×C4×C53—21no—
140C22×C4×C53—21no—
141C22×C4×C53—21no—
142C22×C4×C53—42no—
143C22×C4×C53—42no—
144C22×C4×C53—42no—
145C22×C4×C53C2×D411yescharacteristic
146C22×C4×C53C2×D411yescharacteristic
147C22×C4×C53C2×D411yescharacteristic
148C22×C4×C53C2×D411yescharacteristic
149C22×C4×C53C2×D411yescharacteristic
150C22×C4×C53C2×D411yescharacteristic
151C22×C4×C53C2411yescharacteristic
152C22×Dic53—21no—
153C22×Dic53—21no—
154C22×Dic53—21no—
155C22×Dic53—21no—
156C2×Dic106—21no—
157C2×Dic106—21no—
158C2×Dic106—21no—
159C2×Dic106—21no—
160C2×Dic106—21no—
161C2×Dic106#4112yes—
162C4⋊C4×C53—21no—
163C4⋊C4×C53—21no—
164C4⋊C4×C53—21no—
165C4⋊C4×C53—21no—
166C4⋊C4×C53—42no—
167C22⋊C4×C53—24no—
168C22⋊C4×C53—24no—
169C22⋊C4×C53—42no—
170C22⋊C4×C53—42no—
171C22⋊C4×C53—42no—
172C42×C53—22no—
173C42×C53—22no—
174C42×C53—21no—
175C42×C53—21no—
176C42×C53—21no—
177C42×C53—21no—
178C42×C53—21no—
179C42×C53—21no—
180C42×C53—41no—
181C42×C53—41no—
182C42×C53—42no—
183#11719925—44no—
184#11719923—44no—
185C4⋊Dic53—21no—
186C4⋊Dic53—21no—
187C4⋊Dic53—21no—
188C4⋊Dic53—21no—
189C4⋊Dic53—21no—
190C4⋊Dic53#4112yes—
191#11719919—24no—
192#11719919—42no—
193#11719919—42no—
194C4×Dic53—24no—
195C4×Dic53—42no—
196C2×#88322—21no—
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—
208C23×C53—22no—
209C23×C53—41no—
210C23×C53—41no—
211C23×C53—41no—
212C23×C53—41no—
213C23×C53—41no—
214C23×C53—41no—
215C23×C53C22×D411yescharacteristic
216Q8×C53—22no—
217Q8×C53—22no—
218Q8×C53—41no—
219Q8×C53—41no—
220D4×C53—22no—
221D4×C53—22no—
222D4×C53—41no—
223D4×C53—41no—
224C2×C4×C53—21no—
225C2×C4×C53—21no—
226C2×C4×C53—21no—
227C2×C4×C53—21no—
228C2×C4×C53—22no—
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—21no—
237C2×C4×C53—21no—
238C2×C4×C53—21no—
239C2×C4×C53—21no—
240C2×C4×C53—21no—
241C2×C4×C53—41no—
242C2×C4×C53—41no—
243C2×C4×C53—41no—
244C2×C4×C53—41no—
245C2×C4×C53—41no—
246C2×C4×C53—41no—
247C2×C4×C53—41no—
248C2×C4×C53—41no—
249C2×C4×C53—84no—
250C2×C4×C53#13512yes—
251C2×C4×C53#13712yes—
252C2×C4×C53C2×#4111yescharacteristic
253C2×C4×C53C2×#4111yescharacteristic
254C2×C4×C53C2×#4111yescharacteristic
255C2×C4×C53C2×#4111yescharacteristic
256C2×C4×C53H(5,𝔽2)11yescharacteristic
257C2×C4×C53H(5,𝔽2)11yescharacteristic
258C2×Dic53—22no—
259C2×Dic53—22no—
260C2×Dic53—41no—
261C2×Dic53—41no—
262C2×Dic53—41no—
263C2×Dic53—41no—
264C2×Dic53—41no—
265C2×Dic53—41no—
266C2×Dic53—41no—
267C2×Dic53—41no—
268C2×Dic53—41no—
269C2×Dic53—41no—
270Dic106—22no—
271Dic106—22no—
272Dic106—41no—
273Dic106—41no—
274#88322—41no—
275#88322—41no—
276#88322—41no—
277#88322—41no—
278#29375—531nomaximal, Sylow
279C22×C53—42no—
280C22×C53—42no—
281C22×C53—42no—
282C22×C53—81no—
283C22×C53—81no—
284C22×C53—81no—
285C22×C53—81no—
286C22×C53#52511yescharacteristic
287C22×C53#52511yescharacteristic
288C22×C53#53511yescharacteristic
289C22×C53C2×#13511yescharacteristic
290C22×C53C2×#13711yescharacteristic
291C22×C53#57511yescharacteristic
292C22×C53#57511yescharacteristic
293C4×C53—22no—
294C4×C53—22no—
295C4×C53—41no—
296C4×C53—41no—
297C4×C53—42no—
298C4×C53—41no—
299C4×C53—41no—
300C4×C53—81no—
301C4×C53—81no—
302C4×C53#55111yescharacteristic
303C4×C53#55111yescharacteristic
304C4×C53#55311yescharacteristic
305C4×C53#55311yescharacteristic
306Dic53—42no—
307Dic53—42no—
308Dic53—81no—
309Dic53—81no—
310C4×#124D5311yescharacteristic
311#2295—531no—
312#2101—531no—
313#2097—531no—
314#2042—531no—
315#2015—531no—
316#1928—531no—
317#1926—531no—
318#1880—531no—
319#1878—531no—
320#1878—531no—
321#1877—531no—
322#1865—531no—
323#1862—531no—
324#1839—531no—
325C2×C53—82no—
326C2×C53#259811yescharacteristic
327C2×C53#310211yescharacteristic
328C2×C53#310711yescharacteristic
329C2×C53#311111yescharacteristic
330C2×C53#311711yescharacteristic
331C2×C53#321111yescharacteristic
332C2×C53#321611yescharacteristic
333C22×Q16—531no—
334C22×SD16—531no—
335C2×#124C2×D5311yescharacteristic
336C2×C4×Q8—531no—
337C2×C4×Q8C2×D5311yescharacteristic
338C2×C4×D4C2×D5311yescharacteristic
339C2×#107—531no—
340C2×#106—531no—
341C2×#103—531no—
342C2×#103—531no—
343C2×#103—531no—
344C2×#103—531no—
345C2×#103—531no—
346C2×#102—531no—
347C2×#102—531no—
348C2×#102—531no—
349#391—531no—
350#391—531no—
351#391C2×D5311yescharacteristic
352#390C2×D5311yescharacteristic
353#389—531no—
354#389—531no—
355#388—21no—
356#388—21no—
357#387—1061no—
358#387—1061no—
359#382C2×D5311yescharacteristic
360C4×C4⋊C4—531no—
361C4×C4⋊C4—1061no—
362C4×C22⋊C4—21no—
363C4×C22⋊C4—21no—
364#376—1061no—
365C43C2×D5311yescharacteristic
366#340—531no—
367#340—531no—
368#340—531no—
369#340—531no—
370#336—531no—
371#336—531no—
372#336—531no—
373#336—531no—
374#336—531no—
375#336—531no—
376C53#2937511yescharacteristic, complemented, Sylow
377C22×Q8—531no—
378C22×Q8C22×D5311yescharacteristic
379C22×D4C22×D5311yescharacteristic
380C23×C4—21no—
381C2×Q16—1062no—
382C2×Q16—1062no—
383C2×Q16—2121no—
384C2×SD16—1062no—
385C2×SD16—1062no—
386C2×SD16—2121no—
387C22×C8—531no—
388C22×C8—531no—
389C22×C8—531no—
390C22×C8—531no—
391#124—22no—
392#124—22no—
393C4×Q8—41no—
394C4×Q8—41no—
395C4×Q8—1062no—
396C4×Q8—1062no—
397C4×D4—41no—
398C4×D4—41no—
399C2×C4⋊C4—21no—
400C2×C4⋊C4—531no—
401C2×C4⋊C4—531no—
402C2×C4⋊C4—531no—
403C2×C4⋊C4—1061no—
404C2×C4⋊C4—1061no—
405C2×C4⋊C4—1061no—
406C2×C4⋊C4C22×D5311yescharacteristic
407C2×C4⋊C4C22×D5311yescharacteristic
408C2×C22⋊C4—21no—
409C2×C22⋊C4—21no—
410C2×C22⋊C4—21no—
411C2×C22⋊C4—21no—
412C2×C22⋊C4—21no—
413C2×C42—21no—
414C2×C42—1061no—
415C2×C42—1061no—
416C2×C42C22⋊D5311yescharacteristic
417C2×C42C22⋊D5311yescharacteristic
418C2×C42C22⋊D5311yescharacteristic
419C2×C42C22⋊D5311yescharacteristic
420C2×C42C22×D5311yescharacteristic
421C2×C42C22×D5311yescharacteristic
422C2×C42C22×D5311yescharacteristic
423#107—2121no—
424#106—2121no—
425#103—1062no—
426#103—1062no—
427#103—2121no—
428#103—2121no—
429#103—2121no—
430#102—1062no—
431#102—1062no—
432#102—2121no—
433#95—21no—
434#95—21no—
435#95—21no—
436#95—21no—
437#95—1061no—
438#95—1061no—
439#95—1061no—
440#95—1061no—
441#95—1061no—
442#95—1061no—
443#95—1061no—
444C24—21no—
445C2×Q8—21no—
446C2×Q8—21no—
447C2×Q8—21no—
448C2×Q8—21no—
449C2×Q8—21no—
450C2×Q8—532no—
451C2×Q8—1061no—
452C2×Q8—1061no—
453C2×Q8—1061no—
454C2×Q8—1061no—
455C2×Q8—1061no—
456C2×Q8#1171994512yes—
457C2×D4—21no—
458C2×D4—21no—
459C2×D4—21no—
460C2×D4—21no—
461C2×D4—21no—
462C2×D4#1171994512yes—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—42no—
467C22×C4—42no—
468C22×C4—42no—
469C22×C4—1061no—
470C22×C4—1061no—
471C22×C4—1061no—
472C22×C4—1061no—
473C22×C4C2×C22⋊D5311yescharacteristic
474C22×C4C2×C22⋊D5311yescharacteristic
475C22×C4C2×C22⋊D5311yescharacteristic
476C22×C4C2×C22⋊D5311yescharacteristic
477C22×C4C2×C22⋊D5311yescharacteristic
478C22×C4C2×C22⋊D5311yescharacteristic
479C22×C4C23×D5311yescharacteristic
480Q16—2124no—
481SD16—2124no—
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—
492C2×C8—1061no—
493C2×C8—1061no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—42no—
499C4⋊C4—532no—
500C4⋊C4—1064no—
501C4⋊C4—1061no—
502C4⋊C4—1061no—
503C4⋊C4—1061no—
504C4⋊C4—1061no—
505C4⋊C4—1061no—
506C4⋊C4—2122no—
507C4⋊C4—2122no—
508C22⋊C4—24no—
509C22⋊C4—24no—
510C22⋊C4—42no—
511C22⋊C4—42no—
512C22⋊C4—42no—
513C42—22no—
514C42—22no—
515C42—21no—
516C42—21no—
517C42—21no—
518C42—21no—
519C42—21no—
520C42—21no—
521C42—41no—
522C42—41no—
523C42—42no—
524C42—1064no—
525C42—2122no—
526C23—22no—
527C23—41no—
528C23—41no—
529C23—41no—
530C23—41no—
531C23—41no—
532C23—41no—
533C23C22×C22⋊D5311yescharacteristic
534Q8—22no—
535Q8—22no—
536Q8—41no—
537Q8—41no—
538Q8—1062no—
539Q8—1062no—
540Q8—2121no—
541Q8—2121no—
542D4—22no—
543D4—22no—
544D4—41no—
545D4—41no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—22no—
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—21no—
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—84no—
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×C4—2121no—
584C2×C4#1313323712yes—
585C2×C4#1313324712yes—
586C2×C4#41×D5311yescharacteristic
587C2×C4#41×D5311yescharacteristic
588C2×C4C2×#1171994511yescharacteristic
589C2×C4C2×#1171994511yescharacteristic
590C2×C4#1313326211yescharacteristic
591C2×C4#1313326211yescharacteristic
592C8—2121no—
593C8—2121no—
594C8—2121no—
595C8—2121no—
596C22—42no—
597C22—42no—
598C22—42no—
599C22—81no—
600C22—81no—
601C22—81no—
602C22—81no—
603C22#1681113011yescharacteristic
604C22#1681113411yescharacteristic
605C22#1681119011yescharacteristic
606C22C2×#1313323711yescharacteristic
607C22C2×#1313324711yescharacteristic
608C22#1681142111yescharacteristic
609C22#1681142111yescharacteristic
610C4—22no—
611C4—22no—
612C4—41no—
613C4—41no—
614C4—41no—
615C4—41no—
616C4—42no—
617C4—81no—
618C4—81no—
619C4—4241no—
620C4—4241no—
621C4—2122no—
622C4—2122no—
623C4#1681130211yescharacteristic
624C4#1681130211yescharacteristic
625C4#1681132211yescharacteristic
626C4#1681132211yescharacteristic
627C2—82no—
628C2#2028027511yescharacteristic
629C2#2028472711yescharacteristic
630C2#2028478611yescharacteristic
631C2#2028482411yescharacteristic
632C2#2028488211yescharacteristic
633C2#2028568011yescharacteristic
634C2#2028572611yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC4×#124×C53C2
socleC23×C53C22×D4
prime radicalC22×D4×C53C23
hypercenterC4×#124D53
hypocenterC53#29375
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C43 ◁ C4×#124

Complements

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

NH ≅ G/Ntype
1. #20277192C2semidirect
2. #20276282C2semidirect
3. #20276266C2semidirect
6. #20275353C2semidirect
9. #20275059C2semidirect
10. #20275056C2semidirect
12. #20274983C2semidirect
14. #20274858C2semidirect
376. C53#29375semidirect

Maximal subgroups

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

subgroupindex
#202771922
#202762822
#202762662
#202759512
#202758092
#202753532
#202753392
#202750732
#202750592
#202750562
#202750492
#202749832
#202749622
#202748582
C4×#124×C532
#2937553