Finite Group Databasebeta

← #27487071

Subgroups — #27487071

Order 13568.

724 conjugacy classes of subgroups.

The subgroup lattice has 630 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20277399C211yesmaximal, characteristic
2#20277399C211yesmaximal, characteristic
3#20276148C211yesmaximal, characteristic
4#20275915C211yesmaximal, characteristic
5#20275908C211yesmaximal, characteristic
6#20275359C211yesmaximal, characteristic, complemented
7#20275352C211yesmaximal, characteristic, complemented
8#20275310C211yesmaximal, characteristic, complemented
9#20275303C211yesmaximal, characteristic, complemented
10#20275058C211yesmaximal, characteristic, complemented
11#20275058C211yesmaximal, characteristic, complemented
12#20274985C211yesmaximal, characteristic, complemented
13#20274985C211yesmaximal, characteristic, complemented
14#20274784C211yesmaximal, characteristic
15C53×#2001C211yesmaximal, characteristic
16C2×Q8×Dic53C2211yescharacteristic
17C2×Q8×Dic53C2211yescharacteristic
18C2×#13133176C2211yescharacteristic
19C2×#13133150C2211yescharacteristic
20C2×#13133149C2211yescharacteristic
21C2×#13133148C2211yescharacteristic
22C2×#13133141C2211yescharacteristic
23C2×#13133141C2211yescharacteristic
24C2×#13133141C2211yescharacteristic
25C2×#13133141C2211yescharacteristic
26C2×C22⋊#88322C2211yescharacteristic
27#16810530C2211yescharacteristic
28#16810530C2211yescharacteristic
29#16810522—21no—
30#16810522—21no—
31#16810522C2211yescharacteristic
32#16810522C2211yescharacteristic
33#16810516—21no—
34#16810516—21no—
35#16810516—21no—
36#16810516—21no—
37#16810509—21no—
38#16810509—21no—
39#16810483C2211yescharacteristic
40#16810483C2211yescharacteristic
41C4⋊C4×Dic53C2211yescharacteristic
42C4⋊C4×Dic53C2211yescharacteristic
43#16810386C2211yescharacteristic
44#16810386C2211yescharacteristic
45#16810386C2211yescharacteristic
46#16810385C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810377C2211yescharacteristic
49#16810377C2211yescharacteristic
50#16810377C2211yescharacteristic
51#16810377C2211yescharacteristic
52C22×Q16×C53C2211yescharacteristic
53C2×C22⋊Q8×C53C2211yescharacteristic
54C2×C22⋊Q8×C53C2211yescharacteristic
55C2×#103×C53C2211yescharacteristic
56C2×#103×C53C2211yescharacteristic
57C2×C22⋊C8×C53C2211yescharacteristic
58C53×#340C2211yescharacteristic
59C22×#88322C2311yescharacteristic
60C22×#88322C2311yescharacteristic
61Q8×Dic53—41no—
62Q8×Dic53—41no—
63Q8×Dic53—41no—
64Q8×Dic53—41no—
65#13133176—41no—
66#13133176—41no—
67C2×C4⋊Dic53—21no—
68C2×C4⋊Dic53—21no—
69C2×C4⋊Dic53—21no—
70C2×C4⋊Dic53—21no—
71C2×C4⋊Dic53D411yescharacteristic
72C2×C4⋊Dic53D411yescharacteristic
73C2×C4⋊Dic53C2311yescharacteristic
74C2×C4⋊Dic53C2311yescharacteristic
75C2×C4⋊Dic53C2311yescharacteristic
76C2×C4⋊Dic53C2311yescharacteristic
77C2×C22⋊Dic53—21no—
78C2×C22⋊Dic53—21no—
79C2×C22⋊Dic53D411yescharacteristic
80C2×C22⋊Dic53D411yescharacteristic
81C2×C4×Dic53—21no—
82C2×C4×Dic53—21no—
83C2×C4×Dic53—21no—
84C2×C4×Dic53—21no—
85C2×C4×Dic53C2311yescharacteristic
86C2×C4×Dic53C2311yescharacteristic
87#13133150—22no—
88#13133149—41no—
89#13133148—22no—
90#13133141—22no—
91#13133141—22no—
92#13133141—41no—
93#13133141—41no—
94C22⋊#88322—41no—
95#13133124—21no—
96#13133124—21no—
97#13133124—21no—
98#13133124—21no—
99#13133124—21no—
100#13133124—21no—
101#13133124—21no—
102#13133124—21no—
103#13133124—21no—
104#13133124—21no—
105C22×Q8×C53C2311yescharacteristic
106C22×Q8×C53C2311yescharacteristic
107C23×C4×C53C2311yescharacteristic
108C2×Q16×C53—22no—
109C2×Q16×C53—22no—
110C2×Q16×C53—41no—
111C22×C8×C53C2311yescharacteristic
112C22×C8×C53C2311yescharacteristic
113C22⋊Q8×C53—22no—
114C22⋊Q8×C53—22no—
115C22⋊Q8×C53—22no—
116C22⋊Q8×C53—22no—
117C2×C4⋊C4×C53—21no—
118C2×C4⋊C4×C53—21no—
119C2×C4⋊C4×C53C2311yescharacteristic
120C2×C4⋊C4×C53C2311yescharacteristic
121C2×C22⋊C4×C53—21no—
122C2×C22⋊C4×C53—21no—
123#103×C53—22no—
124#103×C53—22no—
125C22⋊C8×C53—41no—
126C24×C53C2×D411yescharacteristic
127C2×Q8×C53—21no—
128C2×Q8×C53—21no—
129C2×Q8×C53—21no—
130C2×Q8×C53—21no—
131C2×Q8×C53—21no—
132C2×Q8×C53—21no—
133C2×Q8×C53—21no—
134C2×Q8×C53—21no—
135C2×Q8×C53—21no—
136C2×Q8×C53—21no—
137C2×Q8×C53#4112yes—
138C2×Q8×C53#4112yes—
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—21no—
148C22×C4×C53—21no—
149C22×C4×C53C2×D411yescharacteristic
150C22×C4×C53C2411yescharacteristic
151C22×Dic53—21no—
152C22×Dic53—21no—
153C22×Dic53—21no—
154C22×Dic53—21no—
155C22×Dic53C2×D411yescharacteristic
156C22×Dic53C2×D411yescharacteristic
157C22×Dic53C2×D411yescharacteristic
158C22×Dic53C2×D411yescharacteristic
159Q16×C53—44no—
160C2×C8×C53—21no—
161C2×C8×C53—21no—
162C2×C8×C53—21no—
163C2×C8×C53—21no—
164C2×C8×C53—21no—
165C2×C8×C53—21no—
166C4⋊C4×C53—24no—
167C4⋊C4×C53—24no—
168C4⋊C4×C53—21no—
169C4⋊C4×C53—21no—
170C4⋊C4×C53#4112yes—
171C4⋊C4×C53#4112yes—
172C22⋊C4×C53—24no—
173C22⋊C4×C53—24no—
174C22⋊Dic53—41no—
175C22⋊Dic53—41no—
176C22⋊Dic53—81no—
177C22⋊Dic53—81no—
178C4⋊Dic53—21no—
179C4⋊Dic53—21no—
180C4⋊Dic53—21no—
181C4⋊Dic53—21no—
182C4⋊Dic53—21no—
183C4⋊Dic53—21no—
184C4⋊Dic53—21no—
185C4⋊Dic53—21no—
186C4⋊Dic53—41no—
187C4⋊Dic53—41no—
188C4⋊Dic53—42no—
189C4⋊Dic53—42no—
190C4⋊Dic53—81no—
191C4⋊Dic53—81no—
192C4×Dic53—21no—
193C4×Dic53—21no—
194C4×Dic53—21no—
195C4×Dic53—21no—
196C4×Dic53—42no—
197C4×Dic53—42no—
198C4×Dic53—81no—
199C4×Dic53—81no—
200C2×#88322—21no—
201C2×#88322—21no—
202C2×#88322—21no—
203C2×#88322—21no—
204C2×#88322—21no—
205C2×#88322—21no—
206C23×C53—21no—
207C23×C53—21no—
208C23×C53—21no—
209C23×C53—21no—
210C23×C53—21no—
211C23×C53—21no—
212C23×C53—21no—
213C23×C53C22×D411yescharacteristic
214Q8×C53—22no—
215Q8×C53—22no—
216Q8×C53—22no—
217Q8×C53—22no—
218Q8×C53—41no—
219Q8×C53—41no—
220Q8×C53—41no—
221Q8×C53—41no—
222C2×C4×C53—21no—
223C2×C4×C53—21no—
224C2×C4×C53—22no—
225C2×C4×C53—22no—
226C2×C4×C53—22no—
227C2×C4×C53—22no—
228C2×C4×C53—21no—
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×C53—41no—
236C2×C4×C53—41no—
237C2×C4×C53—41no—
238C2×C4×C53—41no—
239C2×C4×C53—41no—
240C2×C4×C53—41no—
241C2×C4×C53C2×#4111yescharacteristic
242C2×C4×C53C2×#4111yescharacteristic
243C2×C4×C53C2×#4111yescharacteristic
244C2×C4×C53C2×#4111yescharacteristic
245C2×C4×C53H(5,𝔽2)11yescharacteristic
246C2×C4×C53H(5,𝔽2)11yescharacteristic
247C2×Dic53—21no—
248C2×Dic53—21no—
249C2×Dic53—21no—
250C2×Dic53—21no—
251C2×Dic53—21no—
252C2×Dic53—21no—
253C2×Dic53—21no—
254C2×Dic53—21no—
255C2×Dic53—21no—
256C2×Dic53—21no—
257C2×Dic53—21no—
258C2×Dic53—21no—
259C2×Dic53—41no—
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—
270C2×Dic53—41no—
271C8×C53—41no—
272C8×C53—41no—
273#88322—41no—
274#88322—41no—
275#29425—531nomaximal, Sylow
276C22×C53—41no—
277C22×C53—41no—
278C22×C53—41no—
279C22×C53—41no—
280C22×C53—41no—
281C22×C53—41no—
282C22×C53—41no—
283C22×C53#52511yescharacteristic
284C22×C53#52511yescharacteristic
285C22×C53#53411yescharacteristic
286C22×C53#57711yescharacteristic
287C22×C53#57711yescharacteristic
288C22×C53#57811yescharacteristic
289C22×C53#57811yescharacteristic
290C4×C53—41no—
291C4×C53—42no—
292C4×C53—42no—
293C4×C53—42no—
294C4×C53—42no—
295C4×C53#55111yescharacteristic
296C4×C53#55311yescharacteristic
297C4×C53#55311yescharacteristic
298C4×C53#55911yescharacteristic
299Dic53—41no—
300Dic53—41no—
301Dic53—41no—
302Dic53—41no—
303Dic53—81no—
304Dic53—81no—
305Dic53—81no—
306Dic53—81no—
307#2323—531no—
308#2323—531no—
309#2077—531no—
310#2036—531no—
311#2035—531no—
312#2001D5311yescharacteristic
313#1928—531no—
314#1927—531no—
315#1921—531no—
316#1920—531no—
317#1878—531no—
318#1878—531no—
319#1865—531no—
320#1865—531no—
321#1818—531no—
322C2×C53—81no—
323C2×C53#260011yescharacteristic
324C2×C53#312011yescharacteristic
325C2×C53#312111yescharacteristic
326C2×C53#312511yescharacteristic
327C2×C53#312711yescharacteristic
328C2×C53#320111yescharacteristic
329C2×C53#320111yescharacteristic
330C22×Q16C2×D5311yescharacteristic
331C2×C22⋊Q8C2×D5311yescharacteristic
332C2×C22⋊Q8C2×D5311yescharacteristic
333C2×C4×Q8—531no—
334C2×C4×Q8—531no—
335C2×#117—531no—
336C2×#107—531no—
337C2×#106—531no—
338C2×#106—531no—
339C2×#103—531no—
340C2×#103—531no—
341C2×#103—531no—
342C2×#103—531no—
343C2×#103C2×D5311yescharacteristic
344C2×#103C2×D5311yescharacteristic
345C2×C22⋊C8—531no—
346C2×C22⋊C8C2×D5311yescharacteristic
347#391—531no—
348#391—531no—
349#389—531no—
350#389—531no—
351#389—1061no—
352#389—1061no—
353#388—1061no—
354#388—1061no—
355#388—1061no—
356#388—1061no—
357#387—1061no—
358#387—1061no—
359#380—531no—
360#380—531no—
361C4×C4⋊C4—531no—
362C4×C4⋊C4—531no—
363#340—531no—
364#340—531no—
365#340—531no—
366#340—531no—
367#340—531no—
368#340C2×D5311yescharacteristic
369#336—531no—
370#336—531no—
371#336—531no—
372#336—531no—
373C53#2942511yescharacteristic, complemented, Sylow
374C22×Q8C22×D5311yescharacteristic
375C22×Q8C22×D5311yescharacteristic
376C23×C4C22×D5311yescharacteristic
377C2×Q16—22no—
378C2×Q16—22no—
379C2×Q16—41no—
380C22×C8—531no—
381C22×C8—531no—
382C22×C8C22×D5311yescharacteristic
383C22×C8C22×D5311yescharacteristic
384C22⋊Q8—22no—
385C22⋊Q8—22no—
386C22⋊Q8—22no—
387C22⋊Q8—22no—
388C4×Q8—2121no—
389C4×Q8—2121no—
390C4×Q8—2121no—
391C4×Q8—2121no—
392#117—2121no—
393#117—2121no—
394C2×C4⋊C4—21no—
395C2×C4⋊C4—21no—
396C2×C4⋊C4—531no—
397C2×C4⋊C4—531no—
398C2×C4⋊C4—531no—
399C2×C4⋊C4—531no—
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×C22⋊C4—21no—
409C2×C22⋊C4—21no—
410C2×C22⋊C4—531no—
411C2×C22⋊C4—531no—
412C2×C22⋊C4—1061no—
413C2×C22⋊C4—1061no—
414C2×C42—531no—
415C2×C42—531no—
416C2×C42—1061no—
417C2×C42—1061no—
418C2×C42—1061no—
419C2×C42—1061no—
420#107—1062no—
421#106—1062no—
422#106—2121no—
423#103—22no—
424#103—22no—
425#103—1062no—
426#103—1062no—
427#103—2121no—
428#103—2121no—
429C22⋊C8—41no—
430C22⋊C8—2121no—
431#95—1061no—
432#95—1061no—
433#95—1061no—
434#95—1061no—
435#95—1061no—
436#95—1061no—
437#95—1061no—
438#95—1061no—
439#95—1061no—
440#95—1061no—
441C24D4×D5311yescharacteristic
442C2×Q8—21no—
443C2×Q8—21no—
444C2×Q8—21no—
445C2×Q8—21no—
446C2×Q8—21no—
447C2×Q8—21no—
448C2×Q8—21no—
449C2×Q8—21no—
450C2×Q8—21no—
451C2×Q8—21no—
452C2×Q8#1171994712yes—
453C2×Q8#1171994712yes—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—531no—
465C22×C4—531no—
466C22×C4—531no—
467C22×C4—531no—
468C22×C4—1061no—
469C22×C4—1061no—
470C22×C4—1061no—
471C22×C4—1061no—
472C22×C4D4×D5311yescharacteristic
473C22×C4C23×D5311yescharacteristic
474Q16—44no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—21no—
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—1061no—
482C2×C8—1061no—
483C2×C8—1061no—
484C2×C8—1061no—
485C2×C8—1061no—
486C2×C8—1061no—
487C4⋊C4—24no—
488C4⋊C4—24no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—1061no—
492C4⋊C4—1061no—
493C4⋊C4—1061no—
494C4⋊C4—1061no—
495C4⋊C4—1061no—
496C4⋊C4—1061no—
497C4⋊C4—1061no—
498C4⋊C4—1061no—
499C4⋊C4—4241no—
500C4⋊C4—4241no—
501C4⋊C4—2121no—
502C4⋊C4—2121no—
503C4⋊C4—2122no—
504C4⋊C4—2122no—
505C4⋊C4#1171994712yes—
506C4⋊C4#1171994712yes—
507C22⋊C4—24no—
508C22⋊C4—24no—
509C22⋊C4—4241no—
510C22⋊C4—4241no—
511C22⋊C4—2121no—
512C22⋊C4—2121no—
513C42—1061no—
514C42—1061no—
515C42—1061no—
516C42—1061no—
517C42—4241no—
518C42—4241no—
519C42—2122no—
520C42—2122no—
521C23—21no—
522C23—21no—
523C23—21no—
524C23—21no—
525C23—21no—
526C23—21no—
527C23—21no—
528C23C2×D4×D5311yescharacteristic
529Q8—22no—
530Q8—22no—
531Q8—22no—
532Q8—22no—
533Q8—41no—
534Q8—41no—
535Q8—41no—
536Q8—41no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—41no—
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—1061no—
557C2×C4—1061no—
558C2×C4—1061no—
559C2×C4—1061no—
560C2×C4—1061no—
561C2×C4—1061no—
562C2×C4—1061no—
563C2×C4—1061no—
564C2×C4—1061no—
565C2×C4—1061no—
566C2×C4—1061no—
567C2×C4—1061no—
568C2×C4—2121no—
569C2×C4—2121no—
570C2×C4—2121no—
571C2×C4—2121no—
572C2×C4—2121no—
573C2×C4—2121no—
574C2×C4—2121no—
575C2×C4—2121no—
576C2×C4—2121no—
577C2×C4—2121no—
578C2×C4—2121no—
579C2×C4—2121no—
580C2×C4C2×#1171994711yescharacteristic
581C2×C4C2×#1171994711yescharacteristic
582C2×C4C2×#1171994711yescharacteristic
583C2×C4C2×#1171994711yescharacteristic
584C2×C4#1313326211yescharacteristic
585C2×C4#1313326311yescharacteristic
586C8—41no—
587C8—41no—
588C8—2121no—
589C8—2121no—
590C22—41no—
591C22—41no—
592C22—41no—
593C22—41no—
594C22—41no—
595C22—41no—
596C22—41no—
597C22#1681112911yescharacteristic
598C22#1681112911yescharacteristic
599C22#1681118111yescharacteristic
600C22#1681143011yescharacteristic
601C22#1681143211yescharacteristic
602C22#1681143611yescharacteristic
603C22#1681143911yescharacteristic
604C4—42no—
605C4—41no—
606C4—42no—
607C4—42no—
608C4—42no—
609C4—4241no—
610C4—4241no—
611C4—4241no—
612C4—4241no—
613C4—2121no—
614C4—2121no—
615C4—2121no—
616C4—2121no—
617C4#1681130411yescharacteristic
618C4#1681132511yescharacteristic
619C4#1681132511yescharacteristic
620C4#1681135911yescharacteristic
621C2—81no—
622C2#2028029611yescharacteristic
623C2#2028492211yescharacteristic
624C2#2028494511yescharacteristic
625C2#2028498511yescharacteristic
626C2#2028500911yescharacteristic
627C2#2028559811yescharacteristic
628C2#2028560011yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2001C2
socleC23×C53C22×D4
prime radicalC24×C53C2×D4
hypercenter#2001D53
hypocenterC53#29425
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2001

Complements

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

NH ≅ G/Ntype
6. #20275359C2semidirect
7. #20275352C2semidirect
8. #20275310C2semidirect
9. #20275303C2semidirect
10. #20275058C2semidirect
11. #20275058C2semidirect
12. #20274985C2semidirect
13. #20274985C2semidirect
373. C53#29425semidirect

Maximal subgroups

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

subgroupindex
#202773992
#202761482
#202759152
#202759082
#202753592
#202753522
#202753102
#202753032
#202750582
#202749852
#202747842
C53×#20012
#2942553