Finite Group Databasebeta

← #27456789

Subgroups — #27456789

Order 13568.

784 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20279111C211yesmaximal, characteristic, complemented
2#20277735C211yesmaximal, characteristic
3#20276131C211yesmaximal, characteristic, complemented
4#20276101C211yesmaximal, characteristic, complemented
5#20275901C211yesmaximal, characteristic
6#20275480C211yesmaximal, characteristic, complemented
7#20275449C211yesmaximal, characteristic, complemented
8#20275324C211yesmaximal, characteristic, complemented
9#20275306C211yesmaximal, characteristic, complemented
10#20275296C211yesmaximal, characteristic, complemented
11#20274964C211yesmaximal, characteristic, complemented
12#20274905C211yesmaximal, characteristic, complemented
13#20274803C211yesmaximal, characteristic, complemented
14#102×Dic53C211yesmaximal, characteristic, complemented
15C53×#2033C211yesmaximal, characteristic
16C2×D4×Dic53—21no—
17C2×D4×Dic53C2211yescharacteristic
18C22×C4⋊Dic53C2211yescharacteristic
19C2×#13133151C2211yescharacteristic
20C2×#13133151C2211yescharacteristic
21C2×#13133148C2211yescharacteristic
22C2×#13133146C2211yescharacteristic
23C2×#13133138C2211yescharacteristic
24C2×#13133138C2211yescharacteristic
25C2×C22⋊#88322C2211yescharacteristic
26C2×C8×Dic53C2211yescharacteristic
27C53⋊#401—21no—
28#16810551—21no—
29#16810526C2211yescharacteristic
30#16810522C2211yescharacteristic
31#16810522C2211yescharacteristic
32#16810519C2211yescharacteristic
33#16810516—21no—
34#16810516—21no—
35#16810508—21no—
36#16810503—21no—
37#16810498—21no—
38#16810488C2211yescharacteristic
39#16810483C2211yescharacteristic
40C4⋊C4×Dic53C2211yescharacteristic
41C4⋊C4×Dic53C2211yescharacteristic
42C4×C22⋊Dic53C2211yescharacteristic
43#16810386C2211yescharacteristic
44#16810386C2211yescharacteristic
45#16810385C2211yescharacteristic
46#16810385C2211yescharacteristic
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—
60C23×Dic53—21no—
61C22×#88322C2311yescharacteristic
62C22×#88322C2311yescharacteristic
63D4×Dic53—22no—
64D4×Dic53—41no—
65D4×Dic53—44no—
66C2×C4⋊Dic53—21no—
67C2×C4⋊Dic53—21no—
68C2×C4⋊Dic53—21no—
69C2×C4⋊Dic53—22no—
70C2×C4⋊Dic53—41no—
71C2×C4⋊Dic53C2311yescharacteristic
72C2×C4⋊Dic53C2311yescharacteristic
73C2×C4⋊Dic53C2311yescharacteristic
74C2×C4⋊Dic53C2311yescharacteristic
75C2×#11719919—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×Dic53C2311yescharacteristic
85C2×C4×Dic53C2311yescharacteristic
86#13133151—22no—
87#13133151—41no—
88#13133148—22no—
89#13133146—22no—
90#13133138—22no—
91#13133138—41no—
92C22⋊#88322—41no—
93C8×Dic53—41no—
94#13133124—21no—
95#13133124—21no—
96#13133124—21no—
97#13133124—21no—
98#13133124—21no—
99#13133124—21no—
100#13133124—21no—
101#13133124—21no—
102#13133124D411yescharacteristic
103#13133124D411yescharacteristic
104C22×D4×C53—21no—
105C22×D4×C53C2311yescharacteristic
106C23×C4×C53C2311yescharacteristic
107C22×C8×C53C2311yescharacteristic
108C22×C8×C53C2311yescharacteristic
109#121×C53—22no—
110#121×C53—22no—
111C2×C4⋊C4×C53C2311yescharacteristic
112C2×C4⋊C4×C53C2311yescharacteristic
113C2×C4⋊C4×C53C2311yescharacteristic
114C2×C22⋊C4×C53—21no—
115C2×C22⋊C4×C53—21no—
116#107×C53—22no—
117#102×C53—22no—
118#102×C53—22no—
119C22⋊C8×C53—22no—
120#95×C53—21no—
121C24×C53—21no—
122C24×C53C2×D411yescharacteristic
123C2×D4×C53—24no—
124C2×D4×C53—21no—
125C2×D4×C53—21no—
126C2×D4×C53—21no—
127C2×D4×C53—21no—
128C2×D4×C53—21no—
129C2×D4×C53—42no—
130C2×D4×C53—42no—
131C2×D4×C53#4112yes—
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×C53—21no—
141C22×C4×C53C2×D411yescharacteristic
142C22×C4×C53C2411yescharacteristic
143C22×Dic53—21no—
144C22×Dic53—21no—
145C22×Dic53—21no—
146C22×Dic53—21no—
147C22×Dic53—42no—
148C22×Dic53—42no—
149C22×Dic53—42no—
150C22×Dic53—42no—
151C22×Dic53—81no—
152C22×Dic53—81no—
153C22×Dic53C2×D411yescharacteristic
154C22×Dic53C2×D411yescharacteristic
155C22×Dic53C2×D411yescharacteristic
156C22×Dic53C2×D411yescharacteristic
157C2×C8×C53—21no—
158C2×C8×C53—21no—
159C2×C8×C53—21no—
160C2×C8×C53—21no—
161C2×C8×C53—21no—
162C2×C8×C53#4112yes—
163C4⋊C4×C53—21no—
164C4⋊C4×C53—21no—
165C4⋊C4×C53—21no—
166C4⋊C4×C53—21no—
167C4⋊C4×C53—21no—
168C4⋊C4×C53#4112yes—
169C22⋊C4×C53—24no—
170C22⋊C4×C53—42no—
171C22⋊Dic53—22no—
172C22⋊Dic53—22no—
173C22⋊Dic53—42no—
174C22⋊Dic53—42no—
175C22⋊Dic53—81no—
176C22⋊Dic53—81no—
177C4⋊Dic53—21no—
178C4⋊Dic53—21no—
179C4⋊Dic53—21no—
180C4⋊Dic53—21no—
181C4⋊Dic53—21no—
182C4⋊Dic53—21no—
183C4⋊Dic53—21no—
184C4⋊Dic53—42no—
185C4⋊Dic53—42no—
186C4⋊Dic53—42no—
187C4⋊Dic53#4112yes—
188#11719919—81no—
189C4×Dic53—21no—
190C4×Dic53—21no—
191C4×Dic53—21no—
192C4×Dic53—21no—
193C4×Dic53—42no—
194C4×Dic53—81no—
195C2×#88322—21no—
196C2×#88322—21no—
197C2×#88322—21no—
198C2×#88322—21no—
199C2×#88322—21no—
200C2×#88322—21no—
201C23×C53—21no—
202C23×C53—21no—
203C23×C53—21no—
204C23×C53—21no—
205C23×C53—21no—
206C23×C53—21no—
207C23×C53—22no—
208C23×C53—41no—
209C23×C53—41no—
210C23×C53—41no—
211C23×C53—41no—
212C23×C53—41no—
213C23×C53—41no—
214C23×C53#13512yes—
215C23×C53C22×D411yescharacteristic
216D4×C53—22no—
217D4×C53—22no—
218D4×C53—48no—
219D4×C53—41no—
220D4×C53—41no—
221C2×C4×C53—21no—
222C2×C4×C53—22no—
223C2×C4×C53—21no—
224C2×C4×C53—21no—
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×C53—41no—
237C2×C4×C53C2×#4111yescharacteristic
238C2×C4×C53C2×#4111yescharacteristic
239C2×C4×C53C2×#4111yescharacteristic
240C2×C4×C53C2×#4111yescharacteristic
241C2×C4×C53H(5,𝔽2)11yescharacteristic
242C2×C4×C532−1+411yescharacteristic
243C2×Dic53—21no—
244C2×Dic53—21no—
245C2×Dic53—22no—
246C2×Dic53—21no—
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—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×Dic53—41no—
266C2×Dic53—84no—
267C2×Dic53—84no—
268C2×Dic53AGL(1,ℤ/8)12yes—
269C8×C53—22no—
270C8×C53—41no—
271#88322—41no—
272#88322—41no—
273#27033—531nomaximal, Sylow
274C22×C53—22no—
275C22×C53—22no—
276C22×C53—22no—
277C22×C53—41no—
278C22×C53—41no—
279C22×C53—41no—
280C22×C53—41no—
281C22×C53—42no—
282C22×C53—42no—
283C22×C53—42no—
284C22×C53—81no—
285C22×C53—81no—
286C22×C53—81no—
287C22×C53—81no—
288C22×C53#52511yescharacteristic
289C22×C53#54611yescharacteristic
290C22×C53#54711yescharacteristic
291C22×C53C2×#13511yescharacteristic
292C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
293C22×C53#57611yescharacteristic
294C22×C53#57811yescharacteristic
295C4×C53—42no—
296C4×C53—41no—
297C4×C53—81no—
298C4×C53—81no—
299C4×C53#52911yescharacteristic
300C4×C53#53911yescharacteristic
301C4×C53#55311yescharacteristic
302C4×C53#55611yescharacteristic
303Dic53—22no—
304Dic53—41no—
305Dic53—41no—
306Dic53—41no—
307Dic53—42no—
308Dic53—81no—
309Dic53—81no—
310Dic53—81no—
311#2494—531no—
312#2361—531no—
313#2073—531no—
314#2067—531no—
315#2034—531no—
316#2033D5311yescharacteristic
317#1949—531no—
318#1942—531no—
319#1923—531no—
320#1921—531no—
321#1919—531no—
322#1862—531no—
323#1852—531no—
324#1824—531no—
325C4×#102—531no—
326C2×C53—42no—
327C2×C53—82no—
328C2×C53#246611yescharacteristic
329C2×C53#310511yescharacteristic
330C2×C53#312411yescharacteristic
331C2×C53#330011yescharacteristic
332C2×C53#331811yescharacteristic
333C2×C53#333911yescharacteristic
334C2×C53#335311yescharacteristic
335C2×#121C2×D5311yescharacteristic
336C2×C4×D4—531no—
337C2×C4×D4—1061no—
338C22×C4⋊C4—531no—
339C2×#107—531no—
340C2×#107—531no—
341C2×#107C2×D5311yescharacteristic
342C2×#106—531no—
343C2×C4⋊C8—531no—
344C2×#102—531no—
345C2×#102—531no—
346C2×#102C2×D5311yescharacteristic
347C2×#102C2×D5311yescharacteristic
348C2×C22⋊C8—531no—
349C2×C22⋊C8C2×D5311yescharacteristic
350C2×C4×C8—531no—
351#401—1061no—
352#397—1061no—
353#390—531no—
354#389—531no—
355#389—531no—
356#389—531no—
357#388—1061no—
358#388—1061no—
359#387—1061no—
360#386—1061no—
361#385—1061no—
362#382—531no—
363#380—531no—
364#380C2×D5311yescharacteristic
365C4×C4⋊C4—531no—
366C4×C4⋊C4—531no—
367C4×C22⋊C4—531no—
368#340—531no—
369#340—531no—
370#340—531no—
371#340—531no—
372#340—531no—
373#340—531no—
374#340—531no—
375#340C2×D5311yescharacteristic
376#336—531no—
377#336—531no—
378C53#2703311yescharacteristic, complemented, Sylow
379C22×D4—21no—
380C22×D4C22×D5311yescharacteristic
381C23×C4—1061no—
382C23×C4—1061no—
383C23×C4C22×D5311yescharacteristic
384C22×C8—531no—
385C22×C8—531no—
386C22×C8C22×D5311yescharacteristic
387C22×C8C22×D5311yescharacteristic
388#121—22no—
389#121—22no—
390C4×D4—1062no—
391C4×D4—2121no—
392C4×D4—2124no—
393C2×C4⋊C4—531no—
394C2×C4⋊C4—531no—
395C2×C4⋊C4—531no—
396C2×C4⋊C4—531no—
397C2×C4⋊C4—1062no—
398C2×C4⋊C4—1061no—
399C2×C4⋊C4—1061no—
400C2×C4⋊C4—1061no—
401C2×C4⋊C4—1061no—
402C2×C4⋊C4—2121no—
403C2×C4⋊C4C22×D5311yescharacteristic
404C2×C4⋊C4C22×D5311yescharacteristic
405C2×C4⋊C4C22×D5311yescharacteristic
406C2×C22⋊C4—21no—
407C2×C22⋊C4—21no—
408C2×C22⋊C4—531no—
409C2×C22⋊C4—531no—
410C2×C22⋊C4—1061no—
411C2×C22⋊C4—1061no—
412C2×C22⋊C4—1061no—
413C2×C22⋊C4—1061no—
414C2×C42—531no—
415C2×C42—531no—
416C2×C42—1061no—
417C2×C42—1061no—
418#107—22no—
419#107—1062no—
420#107—2121no—
421#106—1062no—
422C4⋊C8—1062no—
423#102—22no—
424#102—22no—
425#102—1062no—
426#102—2121no—
427C22⋊C8—22no—
428C22⋊C8—2121no—
429C4×C8—2121no—
430#95—21no—
431#95—531no—
432#95—531no—
433#95—1061no—
434#95—1061no—
435#95—1061no—
436#95—1061no—
437#95—1061no—
438#95—1061no—
439#95—1061no—
440#95—1061no—
441C24—21no—
442C24D4×D5311yescharacteristic
443C2×D4—24no—
444C2×D4—21no—
445C2×D4—21no—
446C2×D4—21no—
447C2×D4—21no—
448C2×D4—21no—
449C2×D4—42no—
450C2×D4—42no—
451C2×D4#1171994712yes—
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—21no—
461C22×C4—531no—
462C22×C4—531no—
463C22×C4—531no—
464C22×C4—531no—
465C22×C4—1061no—
466C22×C4—1061no—
467C22×C4—1061no—
468C22×C4—1061no—
469C22×C4—4241no—
470C22×C4—4241no—
471C22×C4—2122no—
472C22×C4—2122no—
473C22×C4—2122no—
474C22×C4—2122no—
475C22×C4D4×D5311yescharacteristic
476C22×C4C23×D5311yescharacteristic
477C2×C8—21no—
478C2×C8—21no—
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—1061no—
483C2×C8—1061no—
484C2×C8—1061no—
485C2×C8—1061no—
486C2×C8—1061no—
487C2×C8—1061no—
488C2×C8#1171994712yes—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—532no—
495C4⋊C4—1061no—
496C4⋊C4—1061no—
497C4⋊C4—1061no—
498C4⋊C4—1061no—
499C4⋊C4—1061no—
500C4⋊C4—1061no—
501C4⋊C4—1061no—
502C4⋊C4—4241no—
503C4⋊C4—2122no—
504C4⋊C4—2122no—
505C4⋊C4—2122no—
506C4⋊C4#1171994712yes—
507C22⋊C4—24no—
508C22⋊C4—42no—
509C22⋊C4—1062no—
510C22⋊C4—1062no—
511C22⋊C4—4241no—
512C22⋊C4—4241no—
513C22⋊C4—2122no—
514C22⋊C4—2122no—
515C42—1061no—
516C42—1061no—
517C42—1061no—
518C42—1061no—
519C42—4241no—
520C42—2122no—
521C23—21no—
522C23—21no—
523C23—21no—
524C23—21no—
525C23—21no—
526C23—21no—
527C23—22no—
528C23—41no—
529C23—41no—
530C23—41no—
531C23—41no—
532C23—41no—
533C23—41no—
534C23#1313323612yes—
535C23C2×D4×D5311yescharacteristic
536D4—22no—
537D4—22no—
538D4—48no—
539D4—41no—
540D4—41no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
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—1061no—
559C2×C4—1061no—
560C2×C4—1061no—
561C2×C4—1062no—
562C2×C4—1061no—
563C2×C4—1061no—
564C2×C4—1061no—
565C2×C4—1061no—
566C2×C4—1061no—
567C2×C4—1061no—
568C2×C4—1061no—
569C2×C4—1061no—
570C2×C4—4244no—
571C2×C4—4244no—
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×C4—2121no—
581C2×C4—2121no—
582C2×C4—2121no—
583C2×C4#41×D5311yescharacteristic
584C2×C4C2×#1171994711yescharacteristic
585C2×C4C2×#1171994711yescharacteristic
586C2×C4C2×#1171994711yescharacteristic
587C2×C4#1313326311yescharacteristic
588C2×C4#1313326411yescharacteristic
589C8—22no—
590C8—41no—
591C8—2121no—
592C8—2121no—
593C22—22no—
594C22—22no—
595C22—22no—
596C22—41no—
597C22—41no—
598C22—41no—
599C22—41no—
600C22—42no—
601C22—42no—
602C22—42no—
603C22—81no—
604C22—81no—
605C22—81no—
606C22—81no—
607C22#1681112911yescharacteristic
608C22#1681126511yescharacteristic
609C22#1681127311yescharacteristic
610C22C2×#1313323611yescharacteristic
611C22AGL(1,ℤ/8)×D5311yescharacteristic
612C22#1681142511yescharacteristic
613C22#1681143911yescharacteristic
614C4—42no—
615C4—41no—
616C4—81no—
617C4—81no—
618C4—1062no—
619C4—4241no—
620C4—4241no—
621C4—4241no—
622C4—2121no—
623C4—2121no—
624C4—2122no—
625C4—2121no—
626C4#1681115411yescharacteristic
627C4#1681121711yescharacteristic
628C4#1681132511yescharacteristic
629C4#1681134211yescharacteristic
630C2—42no—
631C2—82no—
632C2#2027874611yescharacteristic
633C2#2028475711yescharacteristic
634C2#2028497811yescharacteristic
635C2#2028660011yescharacteristic
636C2#2028685911yescharacteristic
637C2#2028712011yescharacteristic
638C2#2028729411yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2033C2
socleC23×C53C22×D4
prime radicalC2×#121×C53C22
hypercenter#2033D53
hypocenterC53#27033
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. #20279111C2semidirect
3. #20276131C2semidirect
4. #20276101C2semidirect
6. #20275480C2semidirect
7. #20275449C2semidirect
8. #20275324C2semidirect
9. #20275306C2semidirect
10. #20275296C2semidirect
11. #20274964C2semidirect
12. #20274905C2semidirect
13. #20274803C2semidirect
14. #102×Dic53C2semidirect
378. C53#27033semidirect

Maximal subgroups

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

subgroupindex
#202791112
#202777352
#202761312
#202761012
#202759012
#202754802
#202754492
#202753242
#202753062
#202752962
#202749642
#202749052
#202748032
#102×Dic532
C53×#20332
#2703353