Finite Group Databasebeta

← #27447703

Subgroups — #27447703

Order 13568.

818 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
1#20278411C211yesmaximal, characteristic
2#20277720C211yesmaximal, characteristic
3#20276275C211yesmaximal, characteristic, complemented
4#20276219C211yesmaximal, characteristic, complemented
5#20276097C211yesmaximal, characteristic, complemented
6#20275855C211yesmaximal, characteristic, complemented
7#20275665C211yesmaximal, characteristic, complemented
8#20275291C211yesmaximal, characteristic, complemented
9#20275138C211yesmaximal, characteristic
10#20275022C211yesmaximal, characteristic
11#20274984C211yesmaximal, characteristic, complemented
12#20274908C211yesmaximal, characteristic
13#20274893C211yesmaximal, characteristic
14#20274719C211yesmaximal, characteristic, complemented
15C2×C53×#340C211yesmaximal, characteristic
16C2×#13133216C2211yescharacteristic
17C2×#13133196—21no—
18C2×#13133196C2211yescharacteristic
19C2×#13133182C2211yescharacteristic
20C2×#13133148C2211yescharacteristic
21C2×#13133148C2211yescharacteristic
22C2×#13133146C2211yescharacteristic
23C2×#13133146C2211yescharacteristic
24C2×#13133142C2211yescharacteristic
25C2×#13133142C2211yescharacteristic
26C2×#13133140C2211yescharacteristic
27C2×#13133140C2211yescharacteristic
28C2×C22⋊#88322C2211yescharacteristic
29#16810556—21no—
30#16810543—21no—
31#16810528C2211yescharacteristic
32#16810522C2211yescharacteristic
33#16810519C2211yescharacteristic
34#16810519C2211yescharacteristic
35#16810518—21no—
36#16810504—21no—
37#16810496—21no—
38#16810496—21no—
39#16810492C2211yescharacteristic
40#16810483C2211yescharacteristic
41C4⋊C4×Dic53C2211yescharacteristic
42C4×C22⋊Dic53—21no—
43C4×C22⋊Dic53C2211yescharacteristic
44#16810386C2211yescharacteristic
45#16810386C2211yescharacteristic
46#16810385C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810385C2211yescharacteristic
49#16810385C2211yescharacteristic
50#16810377C2211yescharacteristic
51#16810377C2211yescharacteristic
52C23×C8×C53C2211yescharacteristic
53C22×C4⋊C4×C53C2211yescharacteristic
54C22×C4⋊C4×C53C2211yescharacteristic
55C53×#340C2211yescharacteristic
56C53×#340C2211yescharacteristic
57C53×#340C2211yescharacteristic
58C53×#340C2211yescharacteristic
59C22×Dic106—21no—
60C22×Dic106C2311yescharacteristic
61C22×#88322C2311yescharacteristic
62C22×#88322C2311yescharacteristic
63#13133216—22no—
64#13133216—41no—
65#13133196—22no—
66#13133196—44no—
67#13133196Q814yescomplemented
68#13133182—22no—
69#13133182—41no—
70C2×C4⋊Dic53—21no—
71C2×C4⋊Dic53C2311yescharacteristic
72C2×C4⋊Dic53C2311yescharacteristic
73C2×C4⋊Dic53C2311yescharacteristic
74C2×#11719919—21no—
75C2×#11719919—21no—
76C2×#11719919—21no—
77C2×#11719919—21no—
78C2×#11719919—21no—
79C2×#11719919—21no—
80C2×#11719919—21no—
81C2×C22⋊Dic53—21no—
82C2×C22⋊Dic53—21no—
83C2×C22⋊Dic53D411yescharacteristic
84C2×C22⋊Dic53D411yescharacteristic
85C2×C4×Dic53—21no—
86C2×C4×Dic53—21no—
87C2×C4×Dic53C2311yescharacteristic
88C2×C4×Dic53C2311yescharacteristic
89#13133148—22no—
90#13133148—41no—
91#13133146—41no—
92#13133146—41no—
93#13133142—41no—
94#13133142—41no—
95#13133140—22no—
96#13133140—41no—
97C22⋊#88322—22no—
98#13133124—21no—
99#13133124—21no—
100#13133124—21no—
101#13133124—21no—
102#13133124—21no—
103#13133124—21no—
104#13133124—21no—
105#13133124D411yescharacteristic
106#13133124D411yescharacteristic
107C23×C4×C53—21no—
108C23×C4×C53—21no—
109C23×C4×C53C2311yescharacteristic
110C22×C8×C53—22no—
111C22×C8×C53—41no—
112C22×C8×C53—41no—
113C22×C8×C53C2311yescharacteristic
114C22×C8×C53C2311yescharacteristic
115C2×C4⋊C4×C53—21no—
116C2×C4⋊C4×C53—21no—
117C2×C4⋊C4×C53—22no—
118C2×C4⋊C4×C53—22no—
119C2×C4⋊C4×C53—41no—
120C2×C4⋊C4×C53—41no—
121C2×C4⋊C4×C53C2311yescharacteristic
122C2×C4⋊C4×C53C2311yescharacteristic
123C2×C4⋊C4×C53C2311yescharacteristic
124C2×C4⋊C4×C53C2311yescharacteristic
125C24×C53C2×D411yescharacteristic
126C22×C4×C53—21no—
127C22×C4×C53—21no—
128C22×C4×C53—21no—
129C22×C4×C53—21no—
130C22×C4×C53—21no—
131C22×C4×C53—21no—
132C22×C4×C53—21no—
133C22×C4×C53—21no—
134C22×C4×C53—42no—
135C22×C4×C53—42no—
136C22×C4×C53—42no—
137C22×C4×C53—42no—
138C22×C4×C53—81no—
139C22×C4×C53—81no—
140C22×C4×C53C2×D411yescharacteristic
141C22×C4×C53C2×Q812yes—
142C22×C4×C53#4112yes—
143C22×C4×C53C2411yescharacteristic
144C22×Dic53—21no—
145C22×Dic53—21no—
146C22×Dic53—21no—
147C22×Dic53—21no—
148C22×Dic53C2×D411yescharacteristic
149C22×Dic53C2×D411yescharacteristic
150C22×Dic53C2×D411yescharacteristic
151C22×Dic53C2×D411yescharacteristic
152C2×Dic106—21no—
153C2×Dic106—21no—
154C2×Dic106—21no—
155C2×Dic106—21no—
156C2×Dic106—21no—
157C2×Dic106—42no—
158C2×Dic106—42no—
159C2×Dic106—81no—
160C2×Dic106C2×Q812yes—
161C2×C8×C53—21no—
162C2×C8×C53—21no—
163C2×C8×C53—21no—
164C2×C8×C53—21no—
165C2×C8×C53—21no—
166C2×C8×C53—21no—
167C2×C8×C53—44no—
168C4⋊C4×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—42no—
175C4⋊C4×C53—42no—
176C4⋊C4×C53#4112yes—
177C4⋊C4×C53#4112yes—
178C22⋊Dic53—24no—
179C22⋊Dic53—22no—
180C22⋊Dic53—42no—
181C22⋊Dic53SD1614yes—
182C4⋊Dic53—21no—
183C4⋊Dic53—21no—
184C4⋊Dic53—21no—
185C4⋊Dic53—21no—
186C4⋊Dic53—21no—
187C4⋊Dic53—42no—
188C4⋊Dic53C2×Q812yes—
189#11719919—24no—
190#11719919—42no—
191#11719919—42no—
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—21no—
203C2×#88322—21no—
204C2×#88322—21no—
205C2×#88322—21no—
206C2×#88322—21no—
207C2×#88322—21no—
208C23×C53—21no—
209C23×C53—21no—
210C23×C53—21no—
211C23×C53—21no—
212C23×C53C22⋊Q812yes—
213C23×C53C2×SD1612yes—
214C23×C53#13712yes—
215C23×C53C22×D411yescharacteristic
216C2×C4×C53—21no—
217C2×C4×C53—21no—
218C2×C4×C53—22no—
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—84no—
235C2×C4×C53—84no—
236C2×C4×C53C22⋊Q812yes—
237C2×C4×C53C22×Q811yescharacteristic
238C2×C4×C53C2×#4111yescharacteristic
239C2×C4×C53C2×#4111yescharacteristic
240C2×C4×C53C2×#4111yescharacteristic
241C2×C4×C53H(5,𝔽2)11yescharacteristic
242C2×C4×C53H(5,𝔽2)11yescharacteristic
243C2×Dic53—21no—
244C2×Dic53—21no—
245C2×Dic53—21no—
246C2×Dic53—21no—
247C2×Dic53—22no—
248C2×Dic53—22no—
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×Dic53C2×SD1612yes—
266C2×Dic53C2×SD1612yes—
267Dic106—22no—
268Dic106—22no—
269Dic106—41no—
270Dic106—41no—
271Dic106—84no—
272C8×C53—41no—
273C8×C53—41no—
274#88322—41no—
275#88322—41no—
276#26412—531nomaximal, Sylow
277C22×C53—22no—
278C22×C53—22no—
279C22×C53—22no—
280C22×C53—22no—
281C22×C53—22no—
282C22×C53—22no—
283C22×C53#47514yes—
284C22×C53C2×C22⋊Q811yescharacteristic
285C22×C53#54611yescharacteristic
286C22×C53#54611yescharacteristic
287C22×C53C22×SD1611yescharacteristic
288C22×C53C2×#13711yescharacteristic
289C22×C53#57611yescharacteristic
290C22×C53#57811yescharacteristic
291C4×C53—22no—
292C4×C53—42no—
293C4×C53—42no—
294C4×C53—81no—
295C4×C53—81no—
296C4×C53#52911yescharacteristic
297C4×C53#53811yescharacteristic
298C4×C53#55411yescharacteristic
299C4×C53#55411yescharacteristic
300Dic53—22no—
301Dic53—22no—
302Dic53—42no—
303Dic53—42no—
304Dic53—41no—
305Dic53—41no—
306Dic53—81no—
307Dic53—81no—
308#2440—531no—
309#2360—531no—
310#2099—531no—
311#2088—531no—
312#2066—531no—
313#2024—531no—
314#1988—531no—
315#1919—531no—
316#1892—531no—
317#1872—531no—
318#1865—531no—
319#1853—531no—
320#1850—531no—
321#1803—531no—
322C2×#340D5311yescharacteristic
323C2×C53—24no—
324C2×C53#245011yescharacteristic
325C2×C53C2×#47511yescharacteristic
326C2×C53#308311yescharacteristic
327C2×C53#329511yescharacteristic
328C2×C53#329811yescharacteristic
329C2×C53#331211yescharacteristic
330C2×C53#331411yescharacteristic
331C23×C8C2×D5311yescharacteristic
332C2×C4⋊Q8—531no—
333C2×C22⋊Q8—531no—
334C2×C22⋊Q8—1061no—
335C2×C4×Q8—531no—
336C22×C4⋊C4C2×D5311yescharacteristic
337C22×C4⋊C4C2×D5311yescharacteristic
338C2×#106—531no—
339C2×#106—531no—
340C2×C4⋊C8—531no—
341C2×C4⋊C8—531no—
342C2×#103—531no—
343C2×#103—531no—
344C2×#103—531no—
345C2×#103—531no—
346C2×C22⋊C8—531no—
347#398—1061no—
348#395—1061no—
349#391—531no—
350#389—531no—
351#389—531no—
352#389—531no—
353#389—1061no—
354#387—1061no—
355#385—1061no—
356#385—1061no—
357#384—531no—
358#380—531no—
359C4×C4⋊C4—531no—
360C4×C22⋊C4—531no—
361C4×C22⋊C4—1061no—
362#340—531no—
363#340—531no—
364#340—531no—
365#340—531no—
366#340—531no—
367#340—531no—
368#340C2×D5311yescharacteristic
369#340C2×D5311yescharacteristic
370#340C2×D5311yescharacteristic
371#340C2×D5311yescharacteristic
372#336—531no—
373#336—531no—
374C53#2641211yescharacteristic, complemented, Sylow
375C22×Q8—531no—
376C22×Q8—1061no—
377C23×C4—21no—
378C23×C4—21no—
379C23×C4C22×D5311yescharacteristic
380C22×C8—22no—
381C22×C8—41no—
382C22×C8—41no—
383C22×C8—531no—
384C22×C8—531no—
385C22×C8C22×D5311yescharacteristic
386C22×C8C22×D5311yescharacteristic
387C4⋊Q8—1062no—
388C4⋊Q8—2121no—
389C22⋊Q8—534no—
390C22⋊Q8—1062no—
391C22⋊Q8—2124no—
392C4×Q8—1062no—
393C4×Q8—2121no—
394C2×C4⋊C4—22no—
395C2×C4⋊C4—22no—
396C2×C4⋊C4—21no—
397C2×C4⋊C4—21no—
398C2×C4⋊C4—41no—
399C2×C4⋊C4—41no—
400C2×C4⋊C4—531no—
401C2×C4⋊C4—531no—
402C2×C4⋊C4—531no—
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—1061no—
410C2×C4⋊C4—1061no—
411C2×C4⋊C4C22×D5311yescharacteristic
412C2×C4⋊C4C22×D5311yescharacteristic
413C2×C4⋊C4C22×D5311yescharacteristic
414C2×C4⋊C4C22×D5311yescharacteristic
415C2×C22⋊C4—531no—
416C2×C22⋊C4—531no—
417C2×C22⋊C4—1061no—
418C2×C22⋊C4—1061no—
419C2×C42—531no—
420C2×C42—531no—
421C2×C42—1061no—
422C2×C42—1061no—
423#106—1062no—
424#106—2121no—
425C4⋊C8—2121no—
426C4⋊C8—2121no—
427#103—1062no—
428#103—2121no—
429#103—2121no—
430#103—2121no—
431C22⋊C8—1062no—
432#95—531no—
433#95—531no—
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—532no—
443C2×Q8—1061no—
444C2×Q8—1061no—
445C2×Q8—1061no—
446C2×Q8—1061no—
447C2×Q8—1061no—
448C2×Q8—4241no—
449C2×Q8—2122no—
450C2×Q8—2122no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—81no—
464C22×C4—81no—
465C22×C4—531no—
466C22×C4—531no—
467C22×C4—531no—
468C22×C4—531no—
469C22×C4—1061no—
470C22×C4—1061no—
471C22×C4—1061no—
472C22×C4—1061no—
473C22×C4#1171994512yes—
474C22×C4D4×D5311yescharacteristic
475C22×C4Q8×D5312yes—
476C22×C4C23×D5311yescharacteristic
477C2×C8—21no—
478C2×C8—21no—
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—44no—
484C2×C8—1061no—
485C2×C8—1061no—
486C2×C8—1061no—
487C2×C8—1061no—
488C2×C8—1061no—
489C2×C8—1061no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—42no—
497C4⋊C4—42no—
498C4⋊C4—532no—
499C4⋊C4—1064no—
500C4⋊C4—1061no—
501C4⋊C4—1061no—
502C4⋊C4—1061no—
503C4⋊C4—1061no—
504C4⋊C4—1061no—
505C4⋊C4—4241no—
506C4⋊C4—4241no—
507C4⋊C4—2122no—
508C4⋊C4—2122no—
509C4⋊C4—2122no—
510C4⋊C4—2122no—
511C4⋊C4—2122no—
512C4⋊C4#1171994512yes—
513C4⋊C4#1171994512yes—
514C22⋊C4—534no—
515C22⋊C4—1064no—
516C22⋊C4—1062no—
517C22⋊C4—2122no—
518C42—1061no—
519C42—1061no—
520C42—1061no—
521C42—1061no—
522C42—4241no—
523C42—2122no—
524C23—21no—
525C23—21no—
526C23—21no—
527C23—21no—
528C23#1313319512yes—
529C23SD16×D5312yes—
530C23#1313324612yes—
531C23C2×D4×D5311yescharacteristic
532Q8—1062no—
533Q8—1062no—
534Q8—4244no—
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—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—84no—
556C2×C4—84no—
557C2×C4—532no—
558C2×C4—532no—
559C2×C4—1061no—
560C2×C4—1061no—
561C2×C4—1061no—
562C2×C4—1061no—
563C2×C4—1062no—
564C2×C4—1062no—
565C2×C4—1061no—
566C2×C4—1061no—
567C2×C4—1061no—
568C2×C4—1061no—
569C2×C4—1061no—
570C2×C4—1061no—
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#1313319512yes—
582C2×C4C2×Q8×D5311yescharacteristic
583C2×C4C2×#1171994511yescharacteristic
584C2×C4C2×#1171994511yescharacteristic
585C2×C4C2×#1171994511yescharacteristic
586C2×C4#1313326211yescharacteristic
587C2×C4#1313326211yescharacteristic
588C8—41no—
589C8—41no—
590C8—2121no—
591C8—2121no—
592C22—22no—
593C22—22no—
594C22—22no—
595C22—22no—
596C22—22no—
597C22—22no—
598C22#1681090714yes—
599C22C2×#1313319511yescharacteristic
600C22#1681126311yescharacteristic
601C22#1681126311yescharacteristic
602C22C2×SD16×D5311yescharacteristic
603C22C2×#1313324611yescharacteristic
604C22#1681142411yescharacteristic
605C22#1681143611yescharacteristic
606C4—22no—
607C4—42no—
608C4—42no—
609C4—81no—
610C4—81no—
611C4—1062no—
612C4—1062no—
613C4—4241no—
614C4—4241no—
615C4—2122no—
616C4—2121no—
617C4—2121no—
618C4—2122no—
619C4#1681115711yescharacteristic
620C4#1681121111yescharacteristic
621C4#1681132811yescharacteristic
622C4#1681132811yescharacteristic
623C2—24no—
624C2#2027853911yescharacteristic
625C2C2×#1681090711yescharacteristic
626C2#2028457111yescharacteristic
627C2#2028653511yescharacteristic
628C2#2028657611yescharacteristic
629C2#2028676211yescharacteristic
630C2#2028679211yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC2×C53×#340C2
socleC23×C53C22×D4
prime radicalC24×C53C2×D4
hypercenterC2×#340D53
hypocenterC53#26412
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#340

Complements

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

NH ≅ G/Ntype
3. #20276275C2semidirect
4. #20276219C2semidirect
5. #20276097C2semidirect
6. #20275855C2semidirect
7. #20275665C2semidirect
8. #20275291C2semidirect
11. #20274984C2semidirect
14. #20274719C2semidirect
67. #13133196Q8semidirect
374. C53#26412semidirect

Maximal subgroups

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

subgroupindex
#202784112
#202777202
#202762752
#202762192
#202760972
#202758552
#202756652
#202752912
#202751382
#202750222
#202749842
#202749082
#202748932
#202747192
C2×C53×#3402
#2641253