Finite Group Databasebeta

← #27456758

Subgroups — #27456758

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#20279126C211yesmaximal, characteristic, complemented
2#20276128C211yesmaximal, characteristic, complemented
3#20276089C211yesmaximal, characteristic, complemented
4#20275901C211yesmaximal, characteristic
5#20275894C211yesmaximal, characteristic
6#20275483C211yesmaximal, characteristic, complemented
7#20275440C211yesmaximal, characteristic, complemented
8#20275323C211yesmaximal, characteristic, complemented
9#20275293C211yesmaximal, characteristic, complemented
10#20275283C211yesmaximal, characteristic, complemented
11#20274976C211yesmaximal, characteristic, complemented
12#20274908C211yesmaximal, characteristic, complemented
13#20274805C211yesmaximal, characteristic, complemented
14C4×#13133138C211yesmaximal, characteristic, complemented
15C53×#2361C211yesmaximal, characteristic
16C2×#13133150C2211yescharacteristic
17C2×#13133148C2211yescharacteristic
18C2×#13133148C2211yescharacteristic
19C2×#13133148C2211yescharacteristic
20C2×C4⋊#88322C2211yescharacteristic
21C2×#13133138C2211yescharacteristic
22C2×#13133138C2211yescharacteristic
23C2×#13133138C2211yescharacteristic
24C2×#13133138C2211yescharacteristic
25C2×C22⋊#88322C2211yescharacteristic
26C2×C22⋊#88322C2211yescharacteristic
27C2×C4×#88322C2211yescharacteristic
28#16810564—21no—
29#16810551—21no—
30#16810521C2211yescharacteristic
31#16810521C2211yescharacteristic
32#16810511—21no—
33#16810490—21no—
34#16810489C2211yescharacteristic
35#16810483C2211yescharacteristic
36#16810483C2211yescharacteristic
37C4×C4⋊Dic53C2211yescharacteristic
38#16810385C2211yescharacteristic
39#16810385C2211yescharacteristic
40#16810385C2211yescharacteristic
41#16810385C2211yescharacteristic
42#16810385C2211yescharacteristic
43#16810385C2211yescharacteristic
44#16810385C2211yescharacteristic
45#16810385C2211yescharacteristic
46#16810378C2211yescharacteristic
47#16810378C2211yescharacteristic
48C2×#121×C53C2211yescharacteristic
49C2×C4×D4×C53—21no—
50C2×C4×D4×C53C2211yescharacteristic
51C22×C4⋊C4×C53C2211yescharacteristic
52C53×#390C2211yescharacteristic
53C53×#389C2211yescharacteristic
54C53×#388—21no—
55C53×#386—21no—
56C53×#385—21no—
57C4×C4⋊C4×C53C2211yescharacteristic
58C4×C22⋊C4×C53C2211yescharacteristic
59C22×#88322C2311yescharacteristic
60C22×#88322C2311yescharacteristic
61C22×#88322C2311yescharacteristic
62C22×#88322C2311yescharacteristic
63C2×C4⋊Dic53C2311yescharacteristic
64C2×C4⋊Dic53C2311yescharacteristic
65C2×C4⋊Dic53C2311yescharacteristic
66C2×C4⋊Dic53C2311yescharacteristic
67C2×#11719919—21no—
68C2×#11719919—21no—
69C2×C22⋊Dic53—21no—
70C2×C22⋊Dic53—21no—
71C2×C4×Dic53—21no—
72#13133150—22no—
73#13133148—22no—
74#13133148—22no—
75#13133148—41no—
76C4⋊#88322—22no—
77#13133138—22no—
78#13133138—22no—
79#13133138—22no—
80#13133138—41no—
81C22⋊#88322—22no—
82C22⋊#88322—41no—
83C4×#88322—41no—
84#13133124—21no—
85#13133124—21no—
86#13133124—21no—
87#13133124—21no—
88#13133124—21no—
89#13133124—21no—
90#13133124—21no—
91C22×D4×C53—21no—
92C22×D4×C53C2311yescharacteristic
93C23×C4×C53—21no—
94C23×C4×C53—21no—
95C23×C4×C53C2311yescharacteristic
96#121×C53—22no—
97#121×C53—22no—
98C4×D4×C53—22no—
99C4×D4×C53—41no—
100C4×D4×C53—44no—
101C2×C4⋊C4×C53—21no—
102C2×C4⋊C4×C53—21no—
103C2×C4⋊C4×C53—22no—
104C2×C4⋊C4×C53—41no—
105C2×C4⋊C4×C53C2311yescharacteristic
106C2×C4⋊C4×C53C2311yescharacteristic
107C2×C4⋊C4×C53C2311yescharacteristic
108C2×C22⋊C4×C53—21no—
109C2×C22⋊C4×C53—21no—
110C2×C22⋊C4×C53—21no—
111C2×C22⋊C4×C53—21no—
112C2×C22⋊C4×C53D411yescharacteristic
113C2×C22⋊C4×C53D411yescharacteristic
114C2×C42×C53—21no—
115C2×C42×C53C2311yescharacteristic
116C2×C42×C53C2311yescharacteristic
117#95×C53—21no—
118#95×C53—21no—
119#95×C53D411yescharacteristic
120#95×C53D411yescharacteristic
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×C53—42no—
142C22×C4×C53—42no—
143C22×C4×C53—42no—
144C22×C4×C53—42no—
145C22×C4×C53—81no—
146C22×C4×C53—81no—
147C22×C4×C53C2×D411yescharacteristic
148C22×C4×C53C2×D411yescharacteristic
149C22×C4×C53C2×D411yescharacteristic
150C22×C4×C53C2×D411yescharacteristic
151C22×C4×C53C2×D411yescharacteristic
152C22×C4×C53C2411yescharacteristic
153C22×Dic53—21no—
154C22×Dic53—21no—
155C22×Dic53—21no—
156C22×Dic53—21no—
157C4⋊C4×C53—21no—
158C4⋊C4×C53—21no—
159C4⋊C4×C53—21no—
160C4⋊C4×C53—21no—
161C4⋊C4×C53—42no—
162C4⋊C4×C53—42no—
163C4⋊C4×C53#4112yes—
164C4⋊C4×C53#4112yes—
165C22⋊C4×C53—24no—
166C22⋊C4×C53—22no—
167C22⋊C4×C53—22no—
168C22⋊C4×C53—42no—
169C22⋊C4×C53—42no—
170C22⋊C4×C53—81no—
171C42×C53—21no—
172C42×C53—21no—
173C42×C53—21no—
174C42×C53—21no—
175C42×C53—42no—
176C22⋊Dic53—42no—
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—
186#11719919—81no—
187#11719919—81no—
188C4×Dic53—42no—
189C2×#88322—21no—
190C2×#88322—21no—
191C2×#88322—21no—
192C2×#88322—21no—
193C2×#88322—21no—
194C2×#88322—21no—
195C2×#88322—21no—
196C2×#88322—21no—
197C2×#88322—21no—
198C2×#88322—21no—
199C2×#88322—21no—
200C2×#88322#4112yes—
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—21no—
223C2×C4×C53—21no—
224C2×C4×C53—22no—
225C2×C4×C53—21no—
226C2×C4×C53—22no—
227C2×C4×C53—21no—
228C2×C4×C53—21no—
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—41no—
238C2×C4×C53—41no—
239C2×C4×C53—41no—
240C2×C4×C53—41no—
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—84no—
249C2×C4×C53—84no—
250C2×C4×C53AGL(1,ℤ/8)12yes—
251C2×C4×C53C2×#4111yescharacteristic
252C2×C4×C53C2×#4111yescharacteristic
253C2×C4×C53C2×#4111yescharacteristic
254C2×C4×C53C2×#4111yescharacteristic
255C2×C4×C53H(5,𝔽2)11yescharacteristic
256C2×C4×C532−1+411yescharacteristic
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—41no—
267C2×Dic53—41no—
268C2×Dic53—41no—
269#88322—22no—
270#88322—41no—
271#88322—41no—
272#88322—41no—
273#27031—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#57711yescharacteristic
294C22×C53#57711yescharacteristic
295C4×C53—22no—
296C4×C53—41no—
297C4×C53—41no—
298C4×C53—42no—
299C4×C53—41no—
300C4×C53—41no—
301C4×C53—42no—
302C4×C53—81no—
303C4×C53#52911yescharacteristic
304C4×C53#53911yescharacteristic
305C4×C53#55311yescharacteristic
306C4×C53#55611yescharacteristic
307Dic53—81no—
308Dic53—81no—
309Dic53—81no—
310Dic53—81no—
311#2495—531no—
312#2361D5311yescharacteristic
313#2073—531no—
314#2065—531no—
315#2034—531no—
316#2033—531no—
317#1949—531no—
318#1941—531no—
319#1923—531no—
320#1919—531no—
321#1918—531no—
322#1864—531no—
323#1853—531no—
324#1824—531no—
325C4×#102—531no—
326C2×C53—42no—
327C2×C53—82no—
328C2×C53#246611yescharacteristic
329C2×C53#310511yescharacteristic
330C2×C53#312511yescharacteristic
331C2×C53#330111yescharacteristic
332C2×C53#331011yescharacteristic
333C2×C53#333711yescharacteristic
334C2×C53#335411yescharacteristic
335C2×#121C2×D5311yescharacteristic
336C2×C4×D4—21no—
337C2×C4×D4C2×D5311yescharacteristic
338C22×C4⋊C4C2×D5311yescharacteristic
339C2×#107—531no—
340C2×#106—531no—
341C2×#106—531no—
342C2×#106—531no—
343C2×C4⋊C8—531no—
344C2×#102—531no—
345C2×#102—531no—
346C2×#102—531no—
347C2×#102—531no—
348C2×C22⋊C8—531no—
349C2×C22⋊C8—531no—
350C2×C4×C8—531no—
351#400—1061no—
352#397—1061no—
353#390C2×D5311yescharacteristic
354#389—531no—
355#389—531no—
356#389C2×D5311yescharacteristic
357#388—21no—
358#388—1061no—
359#386—21no—
360#385—21no—
361#383—1061no—
362#382—531no—
363#380—531no—
364#380—531no—
365C4×C4⋊C4—531no—
366C4×C4⋊C4C2×D5311yescharacteristic
367C4×C22⋊C4C2×D5311yescharacteristic
368#340—531no—
369#340—531no—
370#340—531no—
371#340—531no—
372#340—531no—
373#340—531no—
374#340—531no—
375#340—531no—
376#336—531no—
377#336—531no—
378C53#2703111yescharacteristic, complemented, Sylow
379C22×D4—21no—
380C22×D4C22×D5311yescharacteristic
381C23×C4—21no—
382C23×C4—21no—
383C23×C4C22×D5311yescharacteristic
384C22×C8—531no—
385C22×C8—531no—
386C22×C8—531no—
387C22×C8—531no—
388#121—22no—
389#121—22no—
390C4×D4—22no—
391C4×D4—41no—
392C4×D4—44no—
393C2×C4⋊C4—22no—
394C2×C4⋊C4—21no—
395C2×C4⋊C4—21no—
396C2×C4⋊C4—41no—
397C2×C4⋊C4—531no—
398C2×C4⋊C4—531no—
399C2×C4⋊C4—531no—
400C2×C4⋊C4—531no—
401C2×C4⋊C4—1061no—
402C2×C4⋊C4—1061no—
403C2×C4⋊C4C22×D5311yescharacteristic
404C2×C4⋊C4C22×D5311yescharacteristic
405C2×C4⋊C4C22×D5311yescharacteristic
406C2×C22⋊C4—21no—
407C2×C22⋊C4—21no—
408C2×C22⋊C4—21no—
409C2×C22⋊C4—21no—
410C2×C22⋊C4—1061no—
411C2×C22⋊C4—1061no—
412C2×C22⋊C4C22⋊D5311yescharacteristic
413C2×C22⋊C4C22⋊D5311yescharacteristic
414C2×C42—21no—
415C2×C42—1061no—
416C2×C42C22×D5311yescharacteristic
417C2×C42C22×D5311yescharacteristic
418#107—1062no—
419#106—1062no—
420#106—1062no—
421#106—2121no—
422C4⋊C8—1062no—
423#102—1062no—
424#102—1062no—
425#102—1062no—
426#102—2121no—
427C22⋊C8—1062no—
428C22⋊C8—2121no—
429C4×C8—2121no—
430#95—21no—
431#95—21no—
432#95—1061no—
433#95—1061no—
434#95—1061no—
435#95—1061no—
436#95—1061no—
437#95—1061no—
438#95—1061no—
439#95C22⋊D5311yescharacteristic
440#95C22⋊D5311yescharacteristic
441C24—21no—
442C24C2×C22⋊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#1171994512yes—
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—42no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4—42no—
465C22×C4—81no—
466C22×C4—81no—
467C22×C4—1061no—
468C22×C4—1061no—
469C22×C4—1061no—
470C22×C4—1061no—
471C22×C4C2×C22⋊D5311yescharacteristic
472C22×C4C2×C22⋊D5311yescharacteristic
473C22×C4C2×C22⋊D5311yescharacteristic
474C22×C4C2×C22⋊D5311yescharacteristic
475C22×C4C2×C22⋊D5311yescharacteristic
476C22×C4C23×D5311yescharacteristic
477C2×C8—532no—
478C2×C8—1061no—
479C2×C8—1061no—
480C2×C8—1061no—
481C2×C8—1061no—
482C2×C8—1061no—
483C2×C8—1061no—
484C2×C8—1061no—
485C2×C8—1061no—
486C2×C8—1061no—
487C2×C8—1061no—
488C2×C8—1061no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—42no—
494C4⋊C4—42no—
495C4⋊C4—1061no—
496C4⋊C4—1061no—
497C4⋊C4—1061no—
498C4⋊C4—1061no—
499C4⋊C4—1061no—
500C4⋊C4—1061no—
501C4⋊C4—1061no—
502C4⋊C4—1061no—
503C4⋊C4—4241no—
504C4⋊C4—4241no—
505C4⋊C4#1171994512yes—
506C4⋊C4#1171994712yes—
507C22⋊C4—24no—
508C22⋊C4—22no—
509C22⋊C4—22no—
510C22⋊C4—42no—
511C22⋊C4—42no—
512C22⋊C4—81no—
513C22⋊C4—4241no—
514C22⋊C4—2122no—
515C42—21no—
516C42—21no—
517C42—21no—
518C42—21no—
519C42—42no—
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#1313323712yes—
535C23C22×C22⋊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—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—22no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—84no—
569C2×C4—84no—
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×C4—2121no—
581C2×C4—2121no—
582C2×C4#1313324312yes—
583C2×C4#41×D5311yescharacteristic
584C2×C4C2×#1171994711yescharacteristic
585C2×C4C2×#1171994511yescharacteristic
586C2×C4C2×#1171994511yescharacteristic
587C2×C4#1313326211yescharacteristic
588C2×C4#1313326411yescharacteristic
589C8—1062no—
590C8—2121no—
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#1681113411yescharacteristic
608C22#1681126711yescharacteristic
609C22#1681127211yescharacteristic
610C22C2×#1313323711yescharacteristic
611C22C2×#1313324311yescharacteristic
612C22#1681143111yescharacteristic
613C22#1681143411yescharacteristic
614C4—22no—
615C4—42no—
616C4—41no—
617C4—42no—
618C4—41no—
619C4—41no—
620C4—41no—
621C4—81no—
622C4—4241no—
623C4—4241no—
624C4—4241no—
625C4—4241no—
626C4#1681115311yescharacteristic
627C4#1681121511yescharacteristic
628C4#1681132311yescharacteristic
629C4#1681134311yescharacteristic
630C2—42no—
631C2—82no—
632C2#2027874511yescharacteristic
633C2#2028476211yescharacteristic
634C2#2028499211yescharacteristic
635C2#2028661711yescharacteristic
636C2#2028673611yescharacteristic
637C2#2028708811yescharacteristic
638C2#2028731011yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2361C2
socleC23×C53C22×D4
prime radicalC2×#121×C53C22
hypercenter#2361D53
hypocenterC53#27031
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2361

Complements

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

NH ≅ G/Ntype
1. #20279126C2semidirect
2. #20276128C2semidirect
3. #20276089C2semidirect
6. #20275483C2semidirect
7. #20275440C2semidirect
8. #20275323C2semidirect
9. #20275293C2semidirect
10. #20275283C2semidirect
11. #20274976C2semidirect
12. #20274908C2semidirect
13. #20274805C2semidirect
14. C4×#13133138C2semidirect
378. C53#27031semidirect

Maximal subgroups

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

subgroupindex
#202791262
#202761282
#202760892
#202759012
#202758942
#202754832
#202754402
#202753232
#202752932
#202752832
#202749762
#202749082
#202748052
C4×#131331382
C53×#23612
#2703153