Finite Group Databasebeta

← #27447975

Subgroups — #27447975

Order 13568.

756 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20278407C211yesmaximal, characteristic
2#20277720C211yesmaximal, characteristic
3#20276285C211yesmaximal, characteristic, complemented
4#20276214C211yesmaximal, characteristic, complemented
5#20276091C211yesmaximal, characteristic, complemented
6#20275855C211yesmaximal, characteristic, complemented
7#20275661C211yesmaximal, characteristic, complemented
8#20275295C211yesmaximal, characteristic, complemented
9#20275146C211yesmaximal, characteristic
10#20275025C211yesmaximal, characteristic
11#20274980C211yesmaximal, characteristic, complemented
12#20274897C211yesmaximal, characteristic
13#20274729C211yesmaximal, characteristic, complemented
14#20274434C211yesmaximal, characteristic
15C53×#1853C211yesmaximal, characteristic
16C22×#11719917C2211yescharacteristic
17C2×#13133216C2211yescharacteristic
18C2×#13133196C2211yescharacteristic
19C2×#13133182C2211yescharacteristic
20C2×#13133176C2211yescharacteristic
21C2×#13133146C2211yescharacteristic
22C2×#13133146C2211yescharacteristic
23C2×#13133142C2211yescharacteristic
24C2×#13133142C2211yescharacteristic
25C2×#13133140C2211yescharacteristic
26C2×#13133140C2211yescharacteristic
27#16810546—21no—
28#16810535—21no—
29#16810528C2211yescharacteristic
30#16810527—21no—
31#16810522C2211yescharacteristic
32#16810519C2211yescharacteristic
33#16810519C2211yescharacteristic
34#16810518—21no—
35#16810506—21no—
36#16810504—21no—
37#16810496—21no—
38#16810488C2211yescharacteristic
39C4⋊C4×Dic53C2211yescharacteristic
40C4×C22⋊Dic53—21no—
41C4×C22⋊Dic53C2211yescharacteristic
42#16810386C2211yescharacteristic
43#16810386C2211yescharacteristic
44#16810385C2211yescharacteristic
45#16810385C2211yescharacteristic
46#16810385C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810385C2211yescharacteristic
49#16810385C2211yescharacteristic
50#16810377C2211yescharacteristic
51#16810377C2211yescharacteristic
52C22×C4⋊C4×C53C2211yescharacteristic
53C2×#106×C53C2211yescharacteristic
54C2×#106×C53C2211yescharacteristic
55C2×C22⋊C8×C53C2211yescharacteristic
56C53×#380C2211yescharacteristic
57C53×#340C2211yescharacteristic
58C53×#340C2211yescharacteristic
59C22×Dic106—21no—
60C22×Dic106C2311yescharacteristic
61C2×#11719917—22no—
62C2×#11719917—41no—
63C2×#11719917—41no—
64C22×#88322C2311yescharacteristic
65C22×#88322C2311yescharacteristic
66#13133216—22no—
67#13133216—41no—
68#13133196—22no—
69#13133196Q814yescomplemented
70#13133182—22no—
71#13133182—41no—
72#13133176—22no—
73#13133176—41no—
74C2×C4⋊Dic53—21no—
75C2×C4⋊Dic53C2311yescharacteristic
76C2×C4⋊Dic53C2311yescharacteristic
77C2×C4⋊Dic53C2311yescharacteristic
78C2×#11719919—21no—
79C2×#11719919—21no—
80C2×#11719919—21no—
81C2×#11719919—21no—
82C2×#11719919—21no—
83C2×C22⋊Dic53—21no—
84C2×C22⋊Dic53—21no—
85C2×C22⋊Dic53D411yescharacteristic
86C2×C22⋊Dic53D411yescharacteristic
87C2×C4×Dic53—21no—
88C2×C4×Dic53—21no—
89C2×C4×Dic53—21no—
90C2×C4×Dic53C2311yescharacteristic
91C2×C4×Dic53C2311yescharacteristic
92#13133146—41no—
93#13133146—41no—
94#13133142—22no—
95#13133142—41no—
96#13133140—41no—
97#13133140—41no—
98#13133124—21no—
99#13133124—21no—
100#13133124—21no—
101#13133124—21no—
102#13133124—21no—
103#13133124—21no—
104#13133124—21no—
105#13133124—21no—
106#13133124D411yescharacteristic
107#13133124D411yescharacteristic
108C23×C4×C53—21no—
109C23×C4×C53C2311yescharacteristic
110C22×C8×C53C2311yescharacteristic
111C22×C8×C53C2311yescharacteristic
112C2×C4⋊C4×C53—21no—
113C2×C4⋊C4×C53—22no—
114C2×C4⋊C4×C53—41no—
115C2×C4⋊C4×C53C2311yescharacteristic
116C2×C4⋊C4×C53C2311yescharacteristic
117C2×C4⋊C4×C53C2311yescharacteristic
118C2×C4⋊C4×C53C2311yescharacteristic
119C2×C22⋊C4×C53—21no—
120#106×C53—22no—
121#106×C53—41no—
122C22⋊C8×C53—22no—
123#95×C53—21no—
124C24×C53C2×D411yescharacteristic
125C22×C4×C53—21no—
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—42no—
134C22×C4×C53—42no—
135C22×C4×C53—81no—
136C22×C4×C53C2×D411yescharacteristic
137C22×C4×C53C2×Q812yes—
138C22×C4×C53#4112yes—
139C22×C4×C53C2411yescharacteristic
140C22×Dic53—21no—
141C22×Dic53—21no—
142C22×Dic53—21no—
143C22×Dic53—21no—
144C22×Dic53C2×D411yescharacteristic
145C22×Dic53C2×D411yescharacteristic
146C22×Dic53C2×D411yescharacteristic
147C22×Dic53C2×D411yescharacteristic
148C2×Dic106—21no—
149C2×Dic106—21no—
150C2×Dic106—21no—
151C2×Dic106—21no—
152C2×Dic106—21no—
153C2×Dic106—42no—
154C2×Dic106—42no—
155C2×Dic106—81no—
156C2×Dic106C2×Q812yes—
157C2×C8×C53—21no—
158C2×C8×C53—21no—
159C2×C8×C53—21no—
160C2×C8×C53—21no—
161C2×C8×C53—21no—
162C2×C8×C53—21no—
163C4⋊C4×C53—21no—
164C4⋊C4×C53—21no—
165C4⋊C4×C53—21no—
166C4⋊C4×C53—21no—
167C4⋊C4×C53—21no—
168C4⋊C4×C53—21no—
169C4⋊C4×C53—21no—
170C4⋊C4×C53—42no—
171C4⋊C4×C53#4112yes—
172C22⋊C4×C53—81no—
173C22⋊Dic53—24no—
174C22⋊Dic53—22no—
175C22⋊Dic53—41no—
176C22⋊Dic53—42no—
177C4⋊Dic53—21no—
178C4⋊Dic53—21no—
179C4⋊Dic53—21no—
180C4⋊Dic53—21no—
181C4⋊Dic53—42no—
182C4⋊Dic53C2×Q812yes—
183C4⋊Dic53#4112yes—
184#11719919—24no—
185#11719919—42no—
186#11719919—42no—
187#11719919—81no—
188#11719919—81no—
189C4×Dic53—21no—
190C4×Dic53—21no—
191C4×Dic53—21no—
192C4×Dic53—21no—
193C4×Dic53—42no—
194C4×Dic53—42no—
195C4×Dic53—81no—
196#11719917—44no—
197C2×#88322—21no—
198C2×#88322—21no—
199C2×#88322—21no—
200C2×#88322—21no—
201C2×#88322—21no—
202C2×#88322—21no—
203C23×C53—21no—
204C23×C53—21no—
205C23×C53—21no—
206C23×C53—21no—
207C23×C53—21no—
208C23×C53—21no—
209C23×C53C22⋊Q812yes—
210C23×C53C22×D411yescharacteristic
211C2×C4×C53—21no—
212C2×C4×C53—21no—
213C2×C4×C53—22no—
214C2×C4×C53—22no—
215C2×C4×C53—22no—
216C2×C4×C53—22no—
217C2×C4×C53—22no—
218C2×C4×C53—41no—
219C2×C4×C53—41no—
220C2×C4×C53—41no—
221C2×C4×C53—41no—
222C2×C4×C53—41no—
223C2×C4×C53—41no—
224C2×C4×C53—41no—
225C2×C4×C53—41no—
226C2×C4×C53—41no—
227C2×C4×C53—41no—
228C2×C4×C53—41no—
229C2×C4×C53—84no—
230C2×C4×C53C22⋊Q812yes—
231C2×C4×C53C22×Q811yescharacteristic
232C2×C4×C53C2×#4111yescharacteristic
233C2×C4×C53C2×#4111yescharacteristic
234C2×C4×C53C2×#4111yescharacteristic
235C2×C4×C53H(5,𝔽2)11yescharacteristic
236C2×C4×C53H(5,𝔽2)11yescharacteristic
237C2×Dic53—21no—
238C2×Dic53—21no—
239C2×Dic53—22no—
240C2×Dic53—22no—
241C2×Dic53—21no—
242C2×Dic53—22no—
243C2×Dic53—21no—
244C2×Dic53—21no—
245C2×Dic53—21no—
246C2×Dic53—21no—
247C2×Dic53—21no—
248C2×Dic53—21no—
249C2×Dic53—21no—
250C2×Dic53—21no—
251C2×Dic53—41no—
252C2×Dic53—41no—
253C2×Dic53—41no—
254C2×Dic53—41no—
255C2×Dic53—41no—
256C2×Dic53—41no—
257C2×Dic53—41no—
258C2×Dic53—41no—
259C2×Dic53—41no—
260C2×Dic53#13712yes—
261Dic106—22no—
262Dic106—22no—
263Dic106—41no—
264Dic106—41no—
265Dic106—84no—
266C8×C53—41no—
267C8×C53—41no—
268#88322—41no—
269#88322—41no—
270#26430—531nomaximal, Sylow
271C22×C53—22no—
272C22×C53—22no—
273C22×C53—22no—
274C22×C53—41no—
275C22×C53—41no—
276C22×C53—41no—
277C22×C53—41no—
278C22×C53C2×C22⋊Q811yescharacteristic
279C22×C53#54611yescharacteristic
280C22×C53#54611yescharacteristic
281C22×C53C2×#13711yescharacteristic
282C22×C53#57511yescharacteristic
283C22×C53#57711yescharacteristic
284C22×C53#57711yescharacteristic
285C4×C53—22no—
286C4×C53—42no—
287C4×C53—81no—
288C4×C53—81no—
289C4×C53—81no—
290C4×C53#52911yescharacteristic
291C4×C53#53811yescharacteristic
292C4×C53#55411yescharacteristic
293C4×C53#55411yescharacteristic
294Dic53—22no—
295Dic53—41no—
296Dic53—42no—
297Dic53—42no—
298Dic53—41no—
299Dic53—41no—
300Dic53—42no—
301Dic53—81no—
302#2439—531no—
303#2360—531no—
304#2101—531no—
305#2088—531no—
306#2066—531no—
307#2024—531no—
308#1987—531no—
309#1919—531no—
310#1894—531no—
311#1873—531no—
312#1865—531no—
313#1853D5311yescharacteristic
314#1851—531no—
315#1805—531no—
316#1736—531no—
317C2×C53—42no—
318C2×C53#245011yescharacteristic
319C2×C53#307811yescharacteristic
320C2×C53#308411yescharacteristic
321C2×C53#331111yescharacteristic
322C2×C53#331611yescharacteristic
323C2×C53#332311yescharacteristic
324C2×C53#332911yescharacteristic
325C22×M16—531no—
326C2×C4⋊Q8—531no—
327C2×C22⋊Q8—531no—
328C2×C4×Q8—531no—
329C2×#117—531no—
330C22×C4⋊C4C2×D5311yescharacteristic
331C2×#106C2×D5311yescharacteristic
332C2×#106C2×D5311yescharacteristic
333C2×C4⋊C8—531no—
334C2×C4⋊C8—531no—
335C2×#103—531no—
336C2×#103—531no—
337C2×#103—531no—
338C2×#103—531no—
339C2×C22⋊C8C2×D5311yescharacteristic
340#396—1061no—
341#393—1061no—
342#391—531no—
343#391—1061no—
344#389—531no—
345#389—531no—
346#389—531no—
347#389—1061no—
348#387—1061no—
349#387—1061no—
350#385—1061no—
351#382—531no—
352#380C2×D5311yescharacteristic
353C4×C4⋊C4—531no—
354C4×C22⋊C4—531no—
355C4×C22⋊C4—1061no—
356#340—531no—
357#340—531no—
358#340—531no—
359#340—531no—
360#340—531no—
361#340—531no—
362#340—531no—
363#340—531no—
364#340C2×D5311yescharacteristic
365#340C2×D5311yescharacteristic
366#336—531no—
367#336—531no—
368C53#2643011yescharacteristic, complemented, Sylow
369C22×Q8—531no—
370C22×Q8—1061no—
371C23×C4—21no—
372C23×C4C22×D5311yescharacteristic
373C2×M16—1062no—
374C2×M16—2121no—
375C2×M16—2121no—
376C22×C8—531no—
377C22×C8—531no—
378C22×C8C22×D5311yescharacteristic
379C22×C8C22×D5311yescharacteristic
380C4⋊Q8—1062no—
381C4⋊Q8—2121no—
382C22⋊Q8—534no—
383C22⋊Q8—1062no—
384C4×Q8—1062no—
385C4×Q8—2121no—
386#117—1062no—
387#117—2121no—
388C2×C4⋊C4—22no—
389C2×C4⋊C4—21no—
390C2×C4⋊C4—41no—
391C2×C4⋊C4—531no—
392C2×C4⋊C4—531no—
393C2×C4⋊C4—531no—
394C2×C4⋊C4—1061no—
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×C4⋊C4C22×D5311yescharacteristic
404C2×C22⋊C4—21no—
405C2×C22⋊C4—531no—
406C2×C22⋊C4—531no—
407C2×C22⋊C4—1061no—
408C2×C22⋊C4—1061no—
409C2×C42—531no—
410C2×C42—531no—
411C2×C42—1061no—
412C2×C42—1061no—
413C2×C42—1061no—
414#106—22no—
415#106—41no—
416C4⋊C8—2121no—
417C4⋊C8—2121no—
418#103—1062no—
419#103—2121no—
420#103—2121no—
421#103—2121no—
422C22⋊C8—22no—
423#95—21no—
424#95—531no—
425#95—531no—
426#95—1061no—
427#95—1061no—
428#95—1061no—
429#95—1061no—
430#95—1061no—
431#95—1061no—
432#95—1061no—
433#95—1061no—
434C24D4×D5311yescharacteristic
435C2×Q8—532no—
436C2×Q8—1061no—
437C2×Q8—1061no—
438C2×Q8—1061no—
439C2×Q8—1061no—
440C2×Q8—1061no—
441C2×Q8—4241no—
442C2×Q8—2122no—
443C2×Q8—2122no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—81no—
455C22×C4—531no—
456C22×C4—531no—
457C22×C4—531no—
458C22×C4—531no—
459C22×C4—1061no—
460C22×C4—1061no—
461C22×C4—1061no—
462C22×C4—1061no—
463C22×C4#1171994512yes—
464C22×C4D4×D5311yescharacteristic
465C22×C4Q8×D5312yes—
466C22×C4C23×D5311yescharacteristic
467M16—2124no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—21no—
474C2×C8—1061no—
475C2×C8—1061no—
476C2×C8—1061no—
477C2×C8—1061no—
478C2×C8—1061no—
479C2×C8—1061no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—42no—
488C4⋊C4—532no—
489C4⋊C4—532no—
490C4⋊C4—1064no—
491C4⋊C4—1061no—
492C4⋊C4—1061no—
493C4⋊C4—1061no—
494C4⋊C4—1061no—
495C4⋊C4—4241no—
496C4⋊C4—4241no—
497C4⋊C4—2122no—
498C4⋊C4—2122no—
499C4⋊C4—2122no—
500C4⋊C4#1171994712yes—
501C22⋊C4—81no—
502C22⋊C4—1062no—
503C22⋊C4—1064no—
504C22⋊C4—2121no—
505C22⋊C4—2122no—
506C42—1061no—
507C42—1061no—
508C42—1061no—
509C42—1061no—
510C42—4241no—
511C42—2122no—
512C42—2122no—
513C23—21no—
514C23—21no—
515C23—21no—
516C23—21no—
517C23—21no—
518C23—21no—
519C23#1313319512yes—
520C23C2×D4×D5311yescharacteristic
521Q8—1062no—
522Q8—1062no—
523Q8—4244no—
524Q8—2121no—
525Q8—2121no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—84no—
545C2×C4—532no—
546C2×C4—1061no—
547C2×C4—1061no—
548C2×C4—1061no—
549C2×C4—1062no—
550C2×C4—1062no—
551C2×C4—1062no—
552C2×C4—1061no—
553C2×C4—1061no—
554C2×C4—1061no—
555C2×C4—1061no—
556C2×C4—1061no—
557C2×C4—1061no—
558C2×C4—1061no—
559C2×C4—1061no—
560C2×C4—2121no—
561C2×C4—2121no—
562C2×C4—2121no—
563C2×C4—2121no—
564C2×C4—2121no—
565C2×C4—2121no—
566C2×C4—2121no—
567C2×C4—2121no—
568C2×C4—2121no—
569C2×C4#1313319512yes—
570C2×C4C2×Q8×D5311yescharacteristic
571C2×C4#41×D5311yescharacteristic
572C2×C4C2×#1171994711yescharacteristic
573C2×C4C2×#1171994511yescharacteristic
574C2×C4#1313326211yescharacteristic
575C2×C4#1313326311yescharacteristic
576C8—41no—
577C8—41no—
578C8—2121no—
579C8—2121no—
580C22—22no—
581C22—22no—
582C22—22no—
583C22—41no—
584C22—41no—
585C22—41no—
586C22—41no—
587C22C2×#1313319511yescharacteristic
588C22#1681126011yescharacteristic
589C22#1681126511yescharacteristic
590C22#137×D5311yescharacteristic
591C22#1681141911yescharacteristic
592C22#1681142911yescharacteristic
593C22#1681142911yescharacteristic
594C4—22no—
595C4—42no—
596C4—81no—
597C4—81no—
598C4—81no—
599C4—1062no—
600C4—4241no—
601C4—2122no—
602C4—2121no—
603C4—2121no—
604C4—2122no—
605C4—2121no—
606C4—2122no—
607C4#1681115311yescharacteristic
608C4#1681120611yescharacteristic
609C4#1681132711yescharacteristic
610C4#1681133111yescharacteristic
611C2—42no—
612C2#2027854111yescharacteristic
613C2#2028451911yescharacteristic
614C2#2028458211yescharacteristic
615C2#2028674411yescharacteristic
616C2#2028682911yescharacteristic
617C2#2028693411yescharacteristic
618C2#2028700211yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#1853C2
socleC23×C53C22×D4
prime radicalC24×C53C2×D4
hypercenter#1853D53
hypocenterC53#26430
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
3. #20276285C2semidirect
4. #20276214C2semidirect
5. #20276091C2semidirect
6. #20275855C2semidirect
7. #20275661C2semidirect
8. #20275295C2semidirect
11. #20274980C2semidirect
13. #20274729C2semidirect
69. #13133196Q8semidirect
368. C53#26430semidirect

Maximal subgroups

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

subgroupindex
#202784072
#202777202
#202762852
#202762142
#202760912
#202758552
#202756612
#202752952
#202751462
#202750252
#202749802
#202748972
#202747292
#202744342
C53×#18532
#2643053