Finite Group Databasebeta

← #27446685

Subgroups — #27446685

Order 13568.

838 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810933C211yesmaximal, characteristic, complemented
2C22×#13133150C211yesmaximal, characteristic
3#20278440C211yesmaximal, characteristic
4#20277720C211yesmaximal, characteristic
5#20276264C211yesmaximal, characteristic, complemented
6#20276207C211yesmaximal, characteristic, complemented
7#20276111C211yesmaximal, characteristic, complemented
8#20276055C211yesmaximal, characteristic, complemented
9#20275484C211yesmaximal, characteristic, complemented
10#20275284C211yesmaximal, characteristic, complemented
11#20275136C211yesmaximal, characteristic
12#20275020C211yesmaximal, characteristic
13#20274970C211yesmaximal, characteristic, complemented
14#20274892C211yesmaximal, characteristic
15C53×#1853C211yesmaximal, characteristic
16C23×#88322C2211yescharacteristic
17C2×#13133208C2211yescharacteristic
18C2×#13133196C2211yescharacteristic
19C2×#13133182C2211yescharacteristic
20C22×C4⋊Dic53C2211yescharacteristic
21#16810933—24no—
22C2×#13133150—24no—
23C2×#13133150C2211yescharacteristic
24C2×#13133150C2211yescharacteristic
25C2×#13133150C2211yescharacteristic
26C2×#13133150C2211yescharacteristic
27C2×#13133146C2211yescharacteristic
28C2×#13133146C2211yescharacteristic
29C2×#13133142C2211yescharacteristic
30C2×#13133142C2211yescharacteristic
31C2×#13133140C2211yescharacteristic
32C2×#13133140C2211yescharacteristic
33#16810565—21no—
34#16810555—21no—
35#16810528C2211yescharacteristic
36#16810522C2211yescharacteristic
37#16810519C2211yescharacteristic
38#16810519C2211yescharacteristic
39#16810518—21no—
40#16810504—21no—
41#16810498—21no—
42#16810496—21no—
43#16810488C2211yescharacteristic
44#16810481—21no—
45C4⋊C4×Dic53C2211yescharacteristic
46C4×C22⋊Dic53—21no—
47C4×C22⋊Dic53C2211yescharacteristic
48#16810386C2211yescharacteristic
49#16810386C2211yescharacteristic
50#16810385C2211yescharacteristic
51#16810385C2211yescharacteristic
52#16810377C2211yescharacteristic
53#16810377C2211yescharacteristic
54C22×C4⋊C4×C53C2211yescharacteristic
55C2×#106×C53C2211yescharacteristic
56C2×#106×C53C2211yescharacteristic
57C2×C22⋊C8×C53C2211yescharacteristic
58C53×#380C2211yescharacteristic
59C53×#340C2211yescharacteristic
60C53×#340C2211yescharacteristic
61C22×Dic106C2311yescharacteristic
62C23×Dic53—21no—
63C22×#88322—22no—
64C22×#88322—22no—
65C22×#88322—22no—
66C22×#88322C2311yescharacteristic
67C22×#88322C2311yescharacteristic
68#13133208—22no—
69#13133208—41no—
70#13133196—22no—
71#13133196Q814yes—
72#13133182—22no—
73#13133182—41no—
74C2×C4⋊Dic53—21no—
75C2×C4⋊Dic53—21no—
76C2×C4⋊Dic53—21no—
77C2×C4⋊Dic53—22no—
78C2×C4⋊Dic53—41no—
79C2×C4⋊Dic53C2311yescharacteristic
80C2×C4⋊Dic53C2311yescharacteristic
81C2×C4⋊Dic53C2311yescharacteristic
82C2×#11719919—21no—
83C2×#11719919—21no—
84C2×#11719919—21no—
85C2×#11719919—21no—
86C2×#11719919—21no—
87C2×C22⋊Dic53—21no—
88C2×C22⋊Dic53D411yescharacteristic
89C2×C22⋊Dic53D411yescharacteristic
90C2×C4×Dic53—21no—
91C2×C4×Dic53—21no—
92C2×C4×Dic53C2311yescharacteristic
93C2×C4×Dic53C2311yescharacteristic
94#13133150—22no—
95#13133150—22no—
96#13133150—22no—
97#13133150D414yes—
98#13133146—22no—
99#13133146—22no—
100#13133142—22no—
101#13133142—41no—
102#13133140—22no—
103#13133140—22no—
104#13133124—21no—
105#13133124—21no—
106#13133124—21no—
107#13133124—21no—
108#13133124—21no—
109#13133124—21no—
110#13133124—21no—
111#13133124—21no—
112#13133124D411yescharacteristic
113#13133124D411yescharacteristic
114C23×C4×C53—21no—
115C23×C4×C53C2311yescharacteristic
116C22×C8×C53C2311yescharacteristic
117C22×C8×C53C2311yescharacteristic
118C2×C4⋊C4×C53—21no—
119C2×C4⋊C4×C53—22no—
120C2×C4⋊C4×C53—41no—
121C2×C4⋊C4×C53C2311yescharacteristic
122C2×C4⋊C4×C53C2311yescharacteristic
123C2×C4⋊C4×C53C2311yescharacteristic
124C2×C4⋊C4×C53C2311yescharacteristic
125C2×C22⋊C4×C53—21no—
126#106×C53—22no—
127#106×C53—41no—
128C22⋊C8×C53—22no—
129#95×C53—21no—
130C24×C53C2×D411yescharacteristic
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—42no—
140C22×C4×C53—42no—
141C22×C4×C53—81no—
142C22×C4×C53C2×D411yescharacteristic
143C22×C4×C53C2×Q812yes—
144C22×C4×C53#4112yes—
145C22×C4×C53C2411yescharacteristic
146C22×Dic53—21no—
147C22×Dic53—21no—
148C22×Dic53—21no—
149C22×Dic53—21no—
150C22×Dic53—42no—
151C22×Dic53—42no—
152C22×Dic53—81no—
153C22×Dic53C2×D411yescharacteristic
154C22×Dic53C2×D411yescharacteristic
155C22×Dic53C2×D411yescharacteristic
156C22×Dic53C2×D411yescharacteristic
157C2×Dic106—21no—
158C2×Dic106—21no—
159C2×Dic106—21no—
160C2×Dic106—21no—
161C2×Dic106—21no—
162C2×Dic106C2×Q812yes—
163C2×C8×C53—21no—
164C2×C8×C53—21no—
165C2×C8×C53—21no—
166C2×C8×C53—21no—
167C2×C8×C53—21no—
168C2×C8×C53—21no—
169C4⋊C4×C53—21no—
170C4⋊C4×C53—21no—
171C4⋊C4×C53—21no—
172C4⋊C4×C53—21no—
173C4⋊C4×C53—21no—
174C4⋊C4×C53—21no—
175C4⋊C4×C53—21no—
176C4⋊C4×C53—42no—
177C4⋊C4×C53C2×D412yes—
178C22⋊C4×C53—42no—
179C22⋊Dic53—24no—
180C22⋊Dic53—22no—
181C22⋊Dic53—22no—
182C4⋊Dic53—21no—
183C4⋊Dic53—21no—
184C4⋊Dic53—21no—
185C4⋊Dic53—21no—
186C4⋊Dic53—42no—
187C4⋊Dic53—42no—
188C4⋊Dic53—81no—
189C4⋊Dic53C2×D412yes—
190C4⋊Dic53C2×Q812yes—
191#11719919—24no—
192#11719919—42no—
193#11719919—42no—
194#11719919—81no—
195#11719919—81no—
196C4×Dic53—21no—
197C4×Dic53—21no—
198C4×Dic53—21no—
199C4×Dic53—21no—
200C4×Dic53—42no—
201C4×Dic53—81no—
202C2×#88322—28no—
203C2×#88322—21no—
204C2×#88322—21no—
205C2×#88322—21no—
206C2×#88322—21no—
207C2×#88322C2×D412yes—
208C2×#88322#4112yes—
209C23×C53—21no—
210C23×C53—21no—
211C23×C53—21no—
212C23×C53—21no—
213C23×C53C22⋊Q812yes—
214C23×C53#13512yes—
215C23×C53#13712yes—
216C23×C53C22×D411yescharacteristic
217C2×C4×C53—21no—
218C2×C4×C53—21no—
219C2×C4×C53—22no—
220C2×C4×C53—22no—
221C2×C4×C53—22no—
222C2×C4×C53—22no—
223C2×C4×C53—22no—
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—84no—
236C2×C4×C53C22⋊Q812yes—
237C2×C4×C53C22×D411yescharacteristic
238C2×C4×C53C22×Q811yescharacteristic
239C2×C4×C53C2×#4111yescharacteristic
240C2×C4×C53C2×#4111yescharacteristic
241C2×C4×C53H(5,𝔽2)11yescharacteristic
242C2×C4×C532−1+411yescharacteristic
243C2×Dic53—22no—
244C2×Dic53—22no—
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—21no—
256C2×Dic53—21no—
257C2×Dic53—21no—
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—84no—
268Dic106—22no—
269Dic106—22no—
270Dic106—41no—
271Dic106—41no—
272C8×C53—41no—
273C8×C53—41no—
274#88322—22no—
275#88322—22no—
276#26344—531nomaximal, Sylow
277C22×C53—22no—
278C22×C53—22no—
279C22×C53—22no—
280C22×C53—22no—
281C22×C53—22no—
282C22×C53—22no—
283C22×C53#47914yescomplemented
284C22×C53C2×C22⋊Q811yescharacteristic
285C22×C53D4211yescharacteristic
286C22×C53#54711yescharacteristic
287C22×C53C2×#13511yescharacteristic
288C22×C53C2×#13711yescharacteristic
289C22×C53#57611yescharacteristic
290C22×C53#57811yescharacteristic
291C4×C53—22no—
292C4×C53—42no—
293C4×C53—81no—
294C4×C53—81no—
295C4×C53—81no—
296C4×C53#52511yescharacteristic
297C4×C53#53911yescharacteristic
298C4×C53D4×Q811yescharacteristic
299C4×C53#55411yescharacteristic
300Dic53—41no—
301Dic53—42no—
302Dic53—41no—
303Dic53—42no—
304Dic53—41no—
305Dic53—41no—
306Dic53—42no—
307Dic53—81no—
308C2×#479—531no—
309C22×#107—531no—
310#2442—531no—
311#2360—531no—
312#2097—531no—
313#2086—531no—
314#2068—531no—
315#2059—531no—
316#1949—531no—
317#1918—531no—
318#1892—531no—
319#1872—531no—
320#1863—531no—
321#1853D5311yescharacteristic
322#1850—531no—
323C2×C53—24no—
324C2×C53#244911yescharacteristic
325C2×C53C2×#47911yescharacteristic
326C2×C53#308311yescharacteristic
327C2×C53#327011yescharacteristic
328C2×C53#328011yescharacteristic
329C2×C53#333611yescharacteristic
330C2×C53#334611yescharacteristic
331C23×C8—531no—
332C2×#125—531no—
333C2×C22⋊Q8—531no—
334C2×C4×Q8—531no—
335C22×C4⋊C4—531no—
336C22×C4⋊C4C2×D5311yescharacteristic
337#479—1064no—
338C2×#107—531no—
339C2×#107—531no—
340C2×#107—531no—
341C2×#107—531no—
342C2×#107—1064no—
343C2×#106C2×D5311yescharacteristic
344C2×#106C2×D5311yescharacteristic
345C2×C4⋊C8—531no—
346C2×C4⋊C8—531no—
347C2×#103—531no—
348C2×#103—531no—
349C2×#103—531no—
350C2×#103—531no—
351C2×C22⋊C8C2×D5311yescharacteristic
352#400—1061no—
353#398—1061no—
354#391—531no—
355#389—531no—
356#389—531no—
357#389—531no—
358#389—1061no—
359#387—1061no—
360#385—1061no—
361#385—1061no—
362#382—531no—
363#380—1061no—
364#380C2×D5311yescharacteristic
365C4×C4⋊C4—531no—
366C4×C22⋊C4—531no—
367C4×C22⋊C4—1061no—
368#340—531no—
369#340—531no—
370#340—531no—
371#340—531no—
372#340C2×D5311yescharacteristic
373#340C2×D5311yescharacteristic
374#336—531no—
375#336—531no—
376C53#2634411yescharacteristic, complemented, Sylow
377C22×Q8—531no—
378C23×C4—21no—
379C23×C4—1061no—
380C23×C4C22×D5311yescharacteristic
381C22×C8—531no—
382C22×C8—531no—
383C22×C8—1062no—
384C22×C8—1062no—
385C22×C8—1062no—
386C22×C8C22×D5311yescharacteristic
387C22×C8C22×D5311yescharacteristic
388#125—1062no—
389#125—2121no—
390C22⋊Q8—534no—
391C22⋊Q8—1062no—
392C4×Q8—1062no—
393C4×Q8—2121no—
394C2×C4⋊C4—22no—
395C2×C4⋊C4—21no—
396C2×C4⋊C4—41no—
397C2×C4⋊C4—531no—
398C2×C4⋊C4—531no—
399C2×C4⋊C4—531no—
400C2×C4⋊C4—1062no—
401C2×C4⋊C4—1061no—
402C2×C4⋊C4—1061no—
403C2×C4⋊C4—1061no—
404C2×C4⋊C4—1061no—
405C2×C4⋊C4—1061no—
406C2×C4⋊C4—1061no—
407C2×C4⋊C4—1061no—
408C2×C4⋊C4—1061no—
409C2×C4⋊C4—2121no—
410C2×C4⋊C4C22×D5311yescharacteristic
411C2×C4⋊C4C22×D5311yescharacteristic
412C2×C4⋊C4C22×D5311yescharacteristic
413C2×C4⋊C4C22×D5311yescharacteristic
414C2×C22⋊C4—21no—
415C2×C22⋊C4—531no—
416C2×C22⋊C4—531no—
417C2×C22⋊C4—1061no—
418C2×C42—531no—
419C2×C42—531no—
420C2×C42—1061no—
421C2×C42—1061no—
422#107—534no—
423#107—1062no—
424#107—1062no—
425#107—1062no—
426#106—22no—
427#106—41no—
428C4⋊C8—1062no—
429C4⋊C8—1062no—
430#103—1062no—
431#103—1062no—
432#103—1062no—
433#103—2121no—
434C22⋊C8—22no—
435#95—21no—
436#95—531no—
437#95—531no—
438#95—1061no—
439#95—1061no—
440#95—1061no—
441#95—1061no—
442#95—1061no—
443#95—1061no—
444#95—1061no—
445#95—1061no—
446C24D4×D5311yescharacteristic
447C2×Q8—532no—
448C2×Q8—1061no—
449C2×Q8—1061no—
450C2×Q8—1061no—
451C2×Q8—1061no—
452C2×Q8—1061no—
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—81no—
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×C4—4241no—
473C22×C4—2122no—
474C22×C4—2122no—
475C22×C4#1171994512yes—
476C22×C4D4×D5311yescharacteristic
477C22×C4Q8×D5312yes—
478C22×C4C23×D5311yescharacteristic
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—21no—
484C2×C8—21no—
485C2×C8—532no—
486C2×C8—532no—
487C2×C8—1068no—
488C2×C8—1061no—
489C2×C8—1061no—
490C2×C8—1061no—
491C2×C8—1061no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—42no—
500C4⋊C4—532no—
501C4⋊C4—532no—
502C4⋊C4—1064no—
503C4⋊C4—1061no—
504C4⋊C4—1061no—
505C4⋊C4—1061no—
506C4⋊C4—1061no—
507C4⋊C4—4241no—
508C4⋊C4—4241no—
509C4⋊C4—4241no—
510C4⋊C4—2122no—
511C4⋊C4—2122no—
512C4⋊C4—2122no—
513C4⋊C4—2122no—
514C4⋊C4D4×D5312yes—
515C22⋊C4—42no—
516C22⋊C4—1064no—
517C22⋊C4—1062no—
518C22⋊C4—1062no—
519C42—1061no—
520C42—1061no—
521C42—1061no—
522C42—1061no—
523C42—4241no—
524C42—2122no—
525C23—21no—
526C23—21no—
527C23—21no—
528C23—21no—
529C23#1313319512yes—
530C23#1313323812yes—
531C23#1313324612yes—
532C23C2×D4×D5311yescharacteristic
533Q8—1062no—
534Q8—1062no—
535Q8—2121no—
536Q8—2121no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
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—84no—
556C2×C4—1061no—
557C2×C4—1061no—
558C2×C4—1062no—
559C2×C4—1062no—
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—1061no—
569C2×C4—1061no—
570C2×C4—1062no—
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#1313319512yes—
582C2×C4C2×D4×D5311yescharacteristic
583C2×C4C2×Q8×D5311yescharacteristic
584C2×C4#41×D5311yescharacteristic
585C2×C4C2×#1171994511yescharacteristic
586C2×C4#1313326211yescharacteristic
587C2×C4#1313326411yescharacteristic
588C8—41no—
589C8—41no—
590C8—1062no—
591C8—1062no—
592C22—22no—
593C22—22no—
594C22—22no—
595C22—22no—
596C22—22no—
597C22—22no—
598C22#1681093314yescomplemented
599C22C2×#1313319511yescharacteristic
600C22#1681125511yescharacteristic
601C22#1681127111yescharacteristic
602C22C2×#1313323811yescharacteristic
603C22C2×#1313324611yescharacteristic
604C22#1681142711yescharacteristic
605C22#1681143911yescharacteristic
606C4—22no—
607C4—42no—
608C4—81no—
609C4—81no—
610C4—81no—
611C4—4241no—
612C4—2121no—
613C4—2121no—
614C4—2122no—
615C4—2121no—
616C4—2122no—
617C4—2121no—
618C4—2122no—
619C4#1681113211yescharacteristic
620C4#1681121511yescharacteristic
621C4#1681128611yescharacteristic
622C4#1681132711yescharacteristic
623C2—24no—
624C2#2027852611yescharacteristic
625C2C2×#1681093311yescharacteristic
626C2#2028456511yescharacteristic
627C2#2028627811yescharacteristic
628C2#2028639311yescharacteristic
629C2#2028708111yescharacteristic
630C2#2028719811yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#1853C2
socleC23×C53C22×D4
prime radicalC24×C53C2×D4
hypercenter#1853D53
hypocenterC53#26344
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1853

Complements

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

NH ≅ G/Ntype
1. C2×#16810933C2semidirect
5. #20276264C2semidirect
6. #20276207C2semidirect
7. #20276111C2semidirect
8. #20276055C2semidirect
9. #20275484C2semidirect
10. #20275284C2semidirect
13. #20274970C2semidirect
283. C22×C53#479semidirect
376. C53#26344semidirect
598. C22#16810933semidirect

Maximal subgroups

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

subgroupindex
C2×#168109332
C22×#131331502
#202784402
#202777202
#202762642
#202762072
#202761112
#202760552
#202754842
#202752842
#202751362
#202750202
#202749702
#202748922
C53×#18532
#2634453