Finite Group Databasebeta

← #27456907

Subgroups — #27456907

Order 13568.

742 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#20279150C211yesmaximal, characteristic, complemented
2#20277757C211yesmaximal, characteristic
3#20276121C211yesmaximal, characteristic, complemented
4#20276099C211yesmaximal, characteristic, complemented
5#20275894C211yesmaximal, characteristic
6#20275470C211yesmaximal, characteristic, complemented
7#20275447C211yesmaximal, characteristic, complemented
8#20275318C211yesmaximal, characteristic, complemented
9#20275309C211yesmaximal, characteristic, complemented
10#20275296C211yesmaximal, characteristic, complemented
11#20274974C211yesmaximal, characteristic, complemented
12#20274912C211yesmaximal, characteristic, complemented
13#20274811C211yesmaximal, characteristic, complemented
14#102×Dic53C211yesmaximal, characteristic, complemented
15C53×#2033C211yesmaximal, characteristic
16C2×D4×Dic53C2211yescharacteristic
17C2×#13133176C2211yescharacteristic
18C2×#13133150C2211yescharacteristic
19C2×#13133150C2211yescharacteristic
20C2×#13133148C2211yescharacteristic
21C2×#13133146C2211yescharacteristic
22C2×#13133138C2211yescharacteristic
23C2×#13133138C2211yescharacteristic
24C2×C22⋊#88322C2211yescharacteristic
25C2×C8×Dic53C2211yescharacteristic
26#16810564—21no—
27#16810546—21no—
28#16810526—21no—
29#16810526C2211yescharacteristic
30#16810522C2211yescharacteristic
31#16810522C2211yescharacteristic
32#16810519C2211yescharacteristic
33#16810517—21no—
34#16810516—21no—
35#16810508—21no—
36#16810503—21no—
37#16810492C2211yescharacteristic
38#16810483C2211yescharacteristic
39C4⋊C4×Dic53C2211yescharacteristic
40C4⋊C4×Dic53C2211yescharacteristic
41C22⋊C4×Dic53—21no—
42C4×C22⋊Dic53C2211yescharacteristic
43#16810386C2211yescharacteristic
44#16810386C2211yescharacteristic
45#16810386C2211yescharacteristic
46#16810386C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810385C2211yescharacteristic
49#16810385C2211yescharacteristic
50#16810377C2211yescharacteristic
51#16810377C2211yescharacteristic
52C2×#121×C53C2211yescharacteristic
53C2×#107×C53C2211yescharacteristic
54C2×#102×C53C2211yescharacteristic
55C2×#102×C53C2211yescharacteristic
56C2×C22⋊C8×C53C2211yescharacteristic
57C53×#380C2211yescharacteristic
58C53×#340C2211yescharacteristic
59C23×Dic53—21no—
60C22×#88322C2311yescharacteristic
61C22×#88322C2311yescharacteristic
62D4×Dic53—22no—
63D4×Dic53—41no—
64#13133176—22no—
65#13133176—41no—
66C2×C4⋊Dic53—21no—
67C2×C4⋊Dic53—21no—
68C2×C4⋊Dic53C2311yescharacteristic
69C2×C4⋊Dic53C2311yescharacteristic
70C2×C4⋊Dic53C2311yescharacteristic
71C2×C4⋊Dic53C2311yescharacteristic
72C2×#11719919—21no—
73C2×#11719919—21no—
74C2×#11719919—21no—
75C2×C22⋊Dic53—21no—
76C2×C22⋊Dic53—21no—
77C2×C22⋊Dic53—21no—
78C2×C22⋊Dic53—21no—
79C2×C22⋊Dic53—21no—
80C2×C22⋊Dic53D411yescharacteristic
81C2×C22⋊Dic53D411yescharacteristic
82C2×C4×Dic53—21no—
83C2×C4×Dic53—21no—
84C2×C4×Dic53—21no—
85C2×C4×Dic53C2311yescharacteristic
86C2×C4×Dic53C2311yescharacteristic
87#13133150—22no—
88#13133150—41no—
89#13133148—22no—
90#13133146—22no—
91#13133138—22no—
92#13133138—22no—
93C22⋊#88322—22no—
94C8×Dic53—41no—
95#13133124—21no—
96#13133124—21no—
97#13133124—21no—
98#13133124—21no—
99#13133124—21no—
100#13133124—21no—
101#13133124D411yescharacteristic
102#13133124D411yescharacteristic
103C22×D4×C53—21no—
104C22×D4×C53C2311yescharacteristic
105C23×C4×C53C2311yescharacteristic
106C22×C8×C53C2311yescharacteristic
107C22×C8×C53C2311yescharacteristic
108#121×C53—22no—
109#121×C53—22no—
110C2×C4⋊C4×C53C2311yescharacteristic
111C2×C4⋊C4×C53C2311yescharacteristic
112C2×C4⋊C4×C53C2311yescharacteristic
113C2×C22⋊C4×C53—21no—
114C2×C22⋊C4×C53—21no—
115#107×C53—22no—
116#102×C53—22no—
117#102×C53—41no—
118C22⋊C8×C53—41no—
119#95×C53—21no—
120C24×C53—21no—
121C24×C53C2×D411yescharacteristic
122C2×D4×C53—24no—
123C2×D4×C53—21no—
124C2×D4×C53—21no—
125C2×D4×C53—21no—
126C2×D4×C53—21no—
127C2×D4×C53—21no—
128C2×D4×C53—81no—
129C2×D4×C53—81no—
130C2×D4×C53#4112yes—
131C22×C4×C53—21no—
132C22×C4×C53—21no—
133C22×C4×C53—21no—
134C22×C4×C53—21no—
135C22×C4×C53—21no—
136C22×C4×C53—21no—
137C22×C4×C53—21no—
138C22×C4×C53—21no—
139C22×C4×C53—21no—
140C22×C4×C53C2×D411yescharacteristic
141C22×C4×C53C2411yescharacteristic
142C22×Dic53—21no—
143C22×Dic53—21no—
144C22×Dic53—21no—
145C22×Dic53—21no—
146C22×Dic53—42no—
147C22×Dic53—42no—
148C22×Dic53—81no—
149C22×Dic53C2×D411yescharacteristic
150C22×Dic53C2×D411yescharacteristic
151C22×Dic53C2×D411yescharacteristic
152C22×Dic53C2×D411yescharacteristic
153C2×C8×C53—21no—
154C2×C8×C53—21no—
155C2×C8×C53—21no—
156C2×C8×C53—21no—
157C2×C8×C53—21no—
158C2×C8×C53—21no—
159C4⋊C4×C53—21no—
160C4⋊C4×C53—21no—
161C4⋊C4×C53—21no—
162C4⋊C4×C53—21no—
163C4⋊C4×C53—21no—
164C4⋊C4×C53#4112yes—
165C22⋊C4×C53—24no—
166C22⋊C4×C53—81no—
167C22⋊Dic53—22no—
168C22⋊Dic53—22no—
169C22⋊Dic53—42no—
170C22⋊Dic53—42no—
171C22⋊Dic53—42no—
172C22⋊Dic53—81no—
173C22⋊Dic53—81no—
174C4⋊Dic53—21no—
175C4⋊Dic53—21no—
176C4⋊Dic53—21no—
177C4⋊Dic53—21no—
178C4⋊Dic53—21no—
179C4⋊Dic53—21no—
180C4⋊Dic53—21no—
181C4⋊Dic53—42no—
182C4⋊Dic53—81no—
183C4⋊Dic53#4112yes—
184#11719919—81no—
185#11719919—81no—
186#11719919—81no—
187C4×Dic53—21no—
188C4×Dic53—21no—
189C4×Dic53—21no—
190C4×Dic53—21no—
191C4×Dic53—42no—
192C4×Dic53—81no—
193C4×Dic53—81no—
194C2×#88322—21no—
195C2×#88322—21no—
196C2×#88322—21no—
197C2×#88322—21no—
198C2×#88322—21no—
199C2×#88322#4112yes—
200C23×C53—21no—
201C23×C53—22no—
202C23×C53—21no—
203C23×C53—21no—
204C23×C53—21no—
205C23×C53—21no—
206C23×C53—21no—
207C23×C53—21no—
208C23×C53—41no—
209C23×C53—41no—
210C23×C53—41no—
211C23×C53—41no—
212C23×C53—41no—
213C23×C53—41no—
214C23×C53C22×D411yescharacteristic
215D4×C53—22no—
216D4×C53—22no—
217D4×C53—41no—
218D4×C53—41no—
219D4×C53—84no—
220C2×C4×C53—21no—
221C2×C4×C53—22no—
222C2×C4×C53—21no—
223C2×C4×C53—21no—
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×C53—41no—
236C2×C4×C53C2×#4111yescharacteristic
237C2×C4×C53C2×#4111yescharacteristic
238C2×C4×C53C2×#4111yescharacteristic
239C2×C4×C53C2×#4111yescharacteristic
240C2×C4×C53H(5,𝔽2)11yescharacteristic
241C2×C4×C532−1+411yescharacteristic
242C2×Dic53—21no—
243C2×Dic53—21no—
244C2×Dic53—21no—
245C2×Dic53—21no—
246C2×Dic53—22no—
247C2×Dic53—21no—
248C2×Dic53—21no—
249C2×Dic53—21no—
250C2×Dic53—21no—
251C2×Dic53—21no—
252C2×Dic53—21no—
253C2×Dic53—41no—
254C2×Dic53—41no—
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—84no—
265C2×Dic53#13512yes—
266C2×Dic53#13712yes—
267C8×C53—41no—
268C8×C53—41no—
269#88322—22no—
270#88322—41no—
271#27041—531nomaximal, Sylow
272C22×C53—41no—
273C22×C53—41no—
274C22×C53—41no—
275C22×C53—42no—
276C22×C53—41no—
277C22×C53—42no—
278C22×C53—41no—
279C22×C53—42no—
280C22×C53—41no—
281C22×C53—41no—
282C22×C53—81no—
283C22×C53—81no—
284C22×C53—81no—
285C22×C53—81no—
286C22×C53#52511yescharacteristic
287C22×C53#54611yescharacteristic
288C22×C53#54711yescharacteristic
289C22×C53C2×#13511yescharacteristic
290C22×C53C2×#13711yescharacteristic
291C22×C53#57611yescharacteristic
292C22×C53#57611yescharacteristic
293C4×C53—42no—
294C4×C53—41no—
295C4×C53—81no—
296C4×C53—81no—
297C4×C53#52911yescharacteristic
298C4×C53#53911yescharacteristic
299C4×C53#55311yescharacteristic
300C4×C53#55611yescharacteristic
301Dic53—22no—
302Dic53—22no—
303Dic53—41no—
304Dic53—41no—
305Dic53—42no—
306Dic53—81no—
307Dic53—81no—
308Dic53—81no—
309#2497—531no—
310#2363—531no—
311#2071—531no—
312#2067—531no—
313#2033—531no—
314#2033D5311yescharacteristic
315#1947—531no—
316#1942—531no—
317#1923—531no—
318#1921—531no—
319#1919—531no—
320#1864—531no—
321#1854—531no—
322#1825—531no—
323C4×#102—531no—
324C2×C53—81no—
325C2×C53—82no—
326C2×C53#246611yescharacteristic
327C2×C53#310711yescharacteristic
328C2×C53#312211yescharacteristic
329C2×C53#330511yescharacteristic
330C2×C53#331411yescharacteristic
331C2×C53#333911yescharacteristic
332C2×C53#335711yescharacteristic
333C2×#121C2×D5311yescharacteristic
334C2×C4×D4—531no—
335C2×#117—531no—
336C2×#107—531no—
337C2×#107—531no—
338C2×#107C2×D5311yescharacteristic
339C2×#106—531no—
340C2×C4⋊C8—531no—
341C2×#102—531no—
342C2×#102—531no—
343C2×#102C2×D5311yescharacteristic
344C2×#102C2×D5311yescharacteristic
345C2×C22⋊C8—531no—
346C2×C22⋊C8C2×D5311yescharacteristic
347C2×C4×C8—531no—
348#400—1061no—
349#396—1061no—
350#390—531no—
351#390—1061no—
352#389—531no—
353#389—531no—
354#389—531no—
355#388—1061no—
356#388—1061no—
357#387—1061no—
358#386—1061no—
359#384—531no—
360#380—531no—
361#380C2×D5311yescharacteristic
362C4×C4⋊C4—531no—
363C4×C4⋊C4—531no—
364C4×C22⋊C4—531no—
365C4×C22⋊C4—1061no—
366#340—531no—
367#340—531no—
368#340—531no—
369#340—531no—
370#340—531no—
371#340—531no—
372#340—531no—
373#340C2×D5311yescharacteristic
374#336—531no—
375#336—531no—
376C53#2704111yescharacteristic, complemented, Sylow
377C22×D4—21no—
378C22×D4C22×D5311yescharacteristic
379C23×C4—1061no—
380C23×C4C22×D5311yescharacteristic
381C22×C8—531no—
382C22×C8—531no—
383C22×C8C22×D5311yescharacteristic
384C22×C8C22×D5311yescharacteristic
385#121—22no—
386#121—22no—
387C4×D4—1062no—
388C4×D4—2121no—
389#117—1062no—
390#117—2121no—
391C2×C4⋊C4—531no—
392C2×C4⋊C4—531no—
393C2×C4⋊C4—531no—
394C2×C4⋊C4—531no—
395C2×C4⋊C4—1061no—
396C2×C4⋊C4—1061no—
397C2×C4⋊C4—1061no—
398C2×C4⋊C4—1061no—
399C2×C4⋊C4—1061no—
400C2×C4⋊C4C22×D5311yescharacteristic
401C2×C4⋊C4C22×D5311yescharacteristic
402C2×C4⋊C4C22×D5311yescharacteristic
403C2×C22⋊C4—21no—
404C2×C22⋊C4—21no—
405C2×C22⋊C4—531no—
406C2×C22⋊C4—531no—
407C2×C22⋊C4—1061no—
408C2×C22⋊C4—1061no—
409C2×C22⋊C4—1061no—
410C2×C22⋊C4—1061no—
411C2×C22⋊C4—1061no—
412C2×C42—531no—
413C2×C42—531no—
414C2×C42—1061no—
415C2×C42—1061no—
416C2×C42—1061no—
417#107—22no—
418#107—1062no—
419#107—2121no—
420#106—1062no—
421C4⋊C8—1062no—
422#102—22no—
423#102—41no—
424#102—1062no—
425#102—1062no—
426C22⋊C8—41no—
427C22⋊C8—1062no—
428C4×C8—2121no—
429#95—21no—
430#95—531no—
431#95—531no—
432#95—1061no—
433#95—1061no—
434#95—1061no—
435#95—1061no—
436#95—1061no—
437#95—1061no—
438C24—21no—
439C24D4×D5311yescharacteristic
440C2×D4—24no—
441C2×D4—21no—
442C2×D4—21no—
443C2×D4—21no—
444C2×D4—21no—
445C2×D4—21no—
446C2×D4—81no—
447C2×D4—81no—
448C2×D4#1171994712yes—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—531no—
459C22×C4—531no—
460C22×C4—531no—
461C22×C4—531no—
462C22×C4—1061no—
463C22×C4—1061no—
464C22×C4—1061no—
465C22×C4—1061no—
466C22×C4—4241no—
467C22×C4—2122no—
468C22×C4—2122no—
469C22×C4D4×D5311yescharacteristic
470C22×C4C23×D5311yescharacteristic
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—532no—
478C2×C8—1061no—
479C2×C8—1061no—
480C2×C8—1061no—
481C2×C8—1061no—
482C2×C8—1061no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—532no—
489C4⋊C4—1061no—
490C4⋊C4—1061no—
491C4⋊C4—1061no—
492C4⋊C4—1061no—
493C4⋊C4—1061no—
494C4⋊C4—1061no—
495C4⋊C4—1061no—
496C4⋊C4—4241no—
497C4⋊C4—4241no—
498C4⋊C4—4241no—
499C4⋊C4—4241no—
500C4⋊C4—2122no—
501C4⋊C4#1171994712yes—
502C22⋊C4—24no—
503C22⋊C4—81no—
504C22⋊C4—1062no—
505C22⋊C4—1062no—
506C22⋊C4—4241no—
507C22⋊C4—4241no—
508C22⋊C4—2122no—
509C22⋊C4—2122no—
510C22⋊C4—2122no—
511C42—1061no—
512C42—1061no—
513C42—1061no—
514C42—1061no—
515C42—4241no—
516C42—4241no—
517C42—2122no—
518C23—21no—
519C23—22no—
520C23—21no—
521C23—21no—
522C23—21no—
523C23—21no—
524C23—21no—
525C23—21no—
526C23—41no—
527C23—41no—
528C23—41no—
529C23—41no—
530C23—41no—
531C23—41no—
532C23C2×D4×D5311yescharacteristic
533D4—22no—
534D4—22no—
535D4—41no—
536D4—41no—
537D4—84no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—22no—
541C2×C4—21no—
542C2×C4—41no—
543C2×C4—41no—
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—532no—
555C2×C4—532no—
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—1062no—
567C2×C4—4244no—
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#41×D5311yescharacteristic
580C2×C4#41×D5311yescharacteristic
581C2×C4C2×#1171994711yescharacteristic
582C2×C4C2×#1171994711yescharacteristic
583C2×C4#1313326211yescharacteristic
584C2×C4#1313326411yescharacteristic
585C8—41no—
586C8—41no—
587C8—1062no—
588C8—2121no—
589C22—41no—
590C22—42no—
591C22—41no—
592C22—41no—
593C22—41no—
594C22—42no—
595C22—41no—
596C22—41no—
597C22—41no—
598C22—42no—
599C22—81no—
600C22—81no—
601C22—81no—
602C22—81no—
603C22#1681112911yescharacteristic
604C22#1681126211yescharacteristic
605C22#1681127111yescharacteristic
606C22#135×D5311yescharacteristic
607C22#137×D5311yescharacteristic
608C22#1681142411yescharacteristic
609C22#1681142711yescharacteristic
610C4—42no—
611C4—41no—
612C4—81no—
613C4—81no—
614C4—1062no—
615C4—1062no—
616C4—4241no—
617C4—4241no—
618C4—4241no—
619C4—2122no—
620C4—2121no—
621C4—2121no—
622C4#1681115011yescharacteristic
623C4#1681121811yescharacteristic
624C4#1681132011yescharacteristic
625C4#1681134211yescharacteristic
626C2—81no—
627C2—82no—
628C2#2027874411yescharacteristic
629C2#2028477911yescharacteristic
630C2#2028495211yescharacteristic
631C2#2028666011yescharacteristic
632C2#2028679111yescharacteristic
633C2#2028711011yescharacteristic
634C2#2028733411yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2033C2
socleC23×C53C22×D4
prime radicalC2×#121×C53C22
hypercenter#2033D53
hypocenterC53#27041
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2033

Complements

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

NH ≅ G/Ntype
1. #20279150C2semidirect
3. #20276121C2semidirect
4. #20276099C2semidirect
6. #20275470C2semidirect
7. #20275447C2semidirect
8. #20275318C2semidirect
9. #20275309C2semidirect
10. #20275296C2semidirect
11. #20274974C2semidirect
12. #20274912C2semidirect
13. #20274811C2semidirect
14. #102×Dic53C2semidirect
376. C53#27041semidirect

Maximal subgroups

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

subgroupindex
#202791502
#202777572
#202761212
#202760992
#202758942
#202754702
#202754472
#202753182
#202753092
#202752962
#202749742
#202749122
#202748112
#102×Dic532
C53×#20332
#2704153