Finite Group Databasebeta

← #27425468

Subgroups — #27425468

Order 13568.

738 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810931C211yesmaximal, characteristic, complemented
2#20280545C211yesmaximal, characteristic
3#20277397C211yesmaximal, characteristic
4#20276148C211yesmaximal, characteristic
5#20276138C211yesmaximal, characteristic
6#20275919C211yesmaximal, characteristic
7#20275919C211yesmaximal, characteristic
8#20275670C211yesmaximal, characteristic, complemented
9#20275474C211yesmaximal, characteristic, complemented
10#20275453C211yesmaximal, characteristic, complemented
11#20275448C211yesmaximal, characteristic, complemented
12#20275405C211yesmaximal, characteristic, complemented
13#20275391C211yesmaximal, characteristic, complemented
14#20274721C211yesmaximal, characteristic, complemented
15C2×C53×#433C211yesmaximal, characteristic
16C2×#13133209C2211yescharacteristic
17C2×#13133196C2211yescharacteristic
18C2×C4×Dic106C2211yescharacteristic
19#16810931—24no—
20C2×#13133151C2211yescharacteristic
21C2×#13133151C2211yescharacteristic
22C2×#13133151C2211yescharacteristic
23C2×#13133149C2211yescharacteristic
24C2×#13133142C2211yescharacteristic
25C2×#13133142C2211yescharacteristic
26C2×#13133142C2211yescharacteristic
27C2×#13133142C2211yescharacteristic
28#16810556—21no—
29#16810556—21no—
30#16810546—21no—
31#16810546—21no—
32#16810529C2211yescharacteristic
33#16810521C2211yescharacteristic
34#16810521C2211yescharacteristic
35#16810521C2211yescharacteristic
36#16810518—21no—
37#16810517—21no—
38#16810511—21no—
39#16810504—21no—
40#16810493C2211yescharacteristic
41#16810483C2211yescharacteristic
42#16810483C2211yescharacteristic
43#16810483C2211yescharacteristic
44C4×C4⋊Dic53C2211yescharacteristic
45#16810386C2211yescharacteristic
46#16810386C2211yescharacteristic
47#16810386C2211yescharacteristic
48#16810386C2211yescharacteristic
49#16810386C2211yescharacteristic
50#16810386C2211yescharacteristic
51#16810386C2211yescharacteristic
52#16810386C2211yescharacteristic
53C2×#117×C53C2211yescharacteristic
54C53×#433—24no—
55C2×C4⋊C8×C53C2211yescharacteristic
56C2×C4⋊C8×C53C2211yescharacteristic
57C2×C22⋊C8×C53C2211yescharacteristic
58C2×C22⋊C8×C53C2211yescharacteristic
59C2×#97×C53C2211yescharacteristic
60C2×C4×C8×C53C2211yescharacteristic
61C22×Dic106C2311yescharacteristic
62#13133209—22no—
63#13133209—22no—
64#13133196—22no—
65#13133196—22no—
66C4×Dic106—22no—
67C4×Dic106—22no—
68C2×C4⋊Dic53—21no—
69C2×C4⋊Dic53—21no—
70C2×C4⋊Dic53C2311yescharacteristic
71C2×C4⋊Dic53C2311yescharacteristic
72C2×C4⋊Dic53C2311yescharacteristic
73C2×C4⋊Dic53C2311yescharacteristic
74C2×C4⋊Dic53C2311yescharacteristic
75C2×C4⋊Dic53C2311yescharacteristic
76C2×C4⋊Dic53C2311yescharacteristic
77C2×#11719919—21no—
78C2×#11719919—21no—
79C2×#11719919—21no—
80C2×#11719919—21no—
81C2×#11719919—21no—
82C2×#11719919—21no—
83C2×#11719919—21no—
84C2×C22⋊Dic53—21no—
85C2×C22⋊Dic53—21no—
86C2×C22⋊Dic53—21no—
87C2×C22⋊Dic53—21no—
88C2×C4×Dic53—21no—
89C2×C4×Dic53—21no—
90#13133151—22no—
91#13133151—22no—
92#13133151—22no—
93#13133149—22no—
94#13133142—22no—
95#13133142—22no—
96#13133142—22no—
97#13133142—22no—
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#13133124—21no—
107C23×C4×C53C2311yescharacteristic
108C22×C8×C53C2311yescharacteristic
109C22×C8×C53C2311yescharacteristic
110C22×C8×C53C2311yescharacteristic
111C22×C8×C53C2311yescharacteristic
112#117×C53—22no—
113#117×C53—22no—
114C2×C4⋊C4×C53D411yescharacteristic
115C2×C4⋊C4×C53D411yescharacteristic
116C2×C22⋊C4×C53D411yescharacteristic
117C2×C22⋊C4×C53D411yescharacteristic
118C2×C42×C53C2311yescharacteristic
119C2×C42×C53C2311yescharacteristic
120C4⋊C8×C53—22no—
121C4⋊C8×C53D414yes—
122C22⋊C8×C53—22no—
123C22⋊C8×C53—22no—
124#97×C53—22no—
125C4×C8×C53—22no—
126C24×C53C2×D411yescharacteristic
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×C53C2×D411yescharacteristic
134C22×C4×C53C2×D411yescharacteristic
135C22×C4×C53C2×D411yescharacteristic
136C22×C4×C53C2×D411yescharacteristic
137C22×C4×C53C2×D411yescharacteristic
138C22×C4×C53C2411yescharacteristic
139C22×Dic53—21no—
140C22×Dic53—21no—
141C22×Dic53—21no—
142C22×Dic53—21no—
143C22×Dic53—21no—
144C22×Dic53—21no—
145C22×Dic53—21no—
146C22×Dic53—21no—
147C2×Dic106—21no—
148C2×Dic106—21no—
149C2×Dic106—21no—
150C2×Dic106—21no—
151C2×Dic106—21no—
152C2×Dic106#4112yes—
153C2×C8×C53—21no—
154C2×C8×C53—21no—
155C2×C8×C53—21no—
156C2×C8×C53—21no—
157C2×C8×C53—21no—
158C2×C8×C53—21no—
159C2×C8×C53—21no—
160C2×C8×C53—21no—
161C2×C8×C53—21no—
162C2×C8×C53—21no—
163C2×C8×C53C2×D412yes—
164C2×C8×C53C2×D412yes—
165C4⋊C4×C53—22no—
166C4⋊C4×C53—22no—
167C22⋊C4×C53—22no—
168C22⋊C4×C53—22no—
169C42×C53—21no—
170C42×C53—21no—
171C42×C53C2×D412yes—
172C42×C53#4112yes—
173C22⋊Dic53—24no—
174C22⋊Dic53—81no—
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—21no—
185C4⋊Dic53—21no—
186C4⋊Dic53—21no—
187C4⋊Dic53—21no—
188C4⋊Dic53—21no—
189C4⋊Dic53—21no—
190C4⋊Dic53—81no—
191C4⋊Dic53—81no—
192C4⋊Dic53#4112yes—
193#11719919—24no—
194#11719919—42no—
195#11719919—42no—
196#11719919—42no—
197#11719919—81no—
198#11719919—81no—
199#11719919—81no—
200C4×Dic53—42no—
201C4×Dic53—81no—
202C23×C53—21no—
203C23×C53—21no—
204C23×C53—21no—
205C23×C53—21no—
206C23×C53—21no—
207C23×C53—21no—
208C23×C53—21no—
209C23×C53C22×D411yescharacteristic
210C2×C4×C53—21no—
211C2×C4×C53—21no—
212C2×C4×C53—21no—
213C2×C4×C53—21no—
214C2×C4×C53—21no—
215C2×C4×C53—21no—
216C2×C4×C53—21no—
217C2×C4×C53—21no—
218C2×C4×C53—21no—
219C2×C4×C53—21no—
220C2×C4×C53—21no—
221C2×C4×C53—21no—
222C2×C4×C53—21no—
223C2×C4×C53—41no—
224C2×C4×C53—41no—
225C2×C4×C53—41no—
226C2×C4×C53—41no—
227C2×C4×C53#12112yes—
228C2×C4×C53#12112yes—
229C2×C4×C53C22×D411yescharacteristic
230C2×C4×C53C2×#4111yescharacteristic
231C2×C4×C53C2×#4111yescharacteristic
232C2×C4×C53C2×#4111yescharacteristic
233C2×C4×C532−1+411yescharacteristic
234C2×C4×C532−1+411yescharacteristic
235C2×Dic53—22no—
236C2×Dic53—22no—
237C2×Dic53—41no—
238C2×Dic53—41no—
239C2×Dic53—41no—
240C2×Dic53—41no—
241C2×Dic53—41no—
242C2×Dic53—41no—
243C2×Dic53—41no—
244C2×Dic53—41no—
245C2×Dic53—41no—
246C2×Dic53—41no—
247C2×Dic53—41no—
248C2×Dic53—41no—
249C2×Dic53—41no—
250C2×Dic53—41no—
251C2×Dic53—41no—
252C2×Dic53—41no—
253C2×Dic53—41no—
254C2×Dic53—41no—
255C2×Dic53—41no—
256C2×Dic53—41no—
257C2×Dic53—41no—
258C2×Dic53—41no—
259Dic106—22no—
260Dic106—22no—
261Dic106—41no—
262Dic106—41no—
263C8×C53—22no—
264C8×C53—22no—
265C8×C53—41no—
266C8×C53—41no—
267#24715—531nomaximal, Sylow
268C22×C53—41no—
269C22×C53—41no—
270C22×C53—41no—
271C22×C53—41no—
272C22×C53—41no—
273C22×C53—41no—
274C22×C53—41no—
275C22×C53C2×#12111yescharacteristic
276C22×C53#52511yescharacteristic
277C22×C53#53611yescharacteristic
278C22×C53#57611yescharacteristic
279C22×C53#57711yescharacteristic
280C22×C53#57811yescharacteristic
281C22×C53#57811yescharacteristic
282C4×C53—22no—
283C4×C53—22no—
284C4×C53—41no—
285C4×C53—41no—
286C4×C53—41no—
287C4×C53#54711yescharacteristic
288C4×C53#54811yescharacteristic
289C4×C53#55211yescharacteristic
290C4×C53#55611yescharacteristic
291Dic53—42no—
292Dic53—42no—
293Dic53—81no—
294Dic53—81no—
295Dic53—81no—
296Dic53—81no—
297Dic53—81no—
298Dic53—81no—
299C2×#479—531no—
300C2×#433D5311yescharacteristic
301#2629—531no—
302#2323—531no—
303#2077—531no—
304#2075—531no—
305#2036—531no—
306#2036—531no—
307#1989—531no—
308#1947—531no—
309#1943—531no—
310#1942—531no—
311#1935—531no—
312#1933—531no—
313#1803—531no—
314C2×C53—81no—
315C2×C53#241511yescharacteristic
316C2×C53#306111yescharacteristic
317C2×C53#306311yescharacteristic
318C2×C53#311911yescharacteristic
319C2×C53#312711yescharacteristic
320C2×C53#324211yescharacteristic
321C2×C53#324311yescharacteristic
322C2×#125—531no—
323C2×C22⋊Q8—531no—
324C2×C4×Q8—531no—
325C2×#117C2×D5311yescharacteristic
326#479—1064no—
327#433—24no—
328C2×#107—531no—
329C2×#107—531no—
330C2×#107—531no—
331C2×#106—531no—
332C2×C4⋊C8C2×D5311yescharacteristic
333C2×C4⋊C8C2×D5311yescharacteristic
334C2×#103—531no—
335C2×#103—531no—
336C2×#103—531no—
337C2×#103—531no—
338C2×C22⋊C8C2×D5311yescharacteristic
339C2×C22⋊C8C2×D5311yescharacteristic
340C2×#97C2×D5311yescharacteristic
341C2×C4×C8C2×D5311yescharacteristic
342#398—1061no—
343#398—1061no—
344#396—1061no—
345#396—1061no—
346#391—531no—
347#389—531no—
348#389—531no—
349#389—531no—
350#389—1061no—
351#388—1061no—
352#388—1061no—
353#387—1061no—
354#384—531no—
355#380—531no—
356#380—531no—
357#380—531no—
358C4×C4⋊C4—531no—
359#340—531no—
360#340—531no—
361#340—531no—
362#340—531no—
363#340—531no—
364#340—531no—
365#340—531no—
366#340—531no—
367C53#2471511yescharacteristic, complemented, Sylow
368C22×Q8—531no—
369C23×C4C22×D5311yescharacteristic
370C22×C8C22×D5311yescharacteristic
371C22×C8C22×D5311yescharacteristic
372C22×C8C22×D5311yescharacteristic
373C22×C8C22×D5311yescharacteristic
374#125—1062no—
375#125—1062no—
376C22⋊Q8—1062no—
377C22⋊Q8—1062no—
378C4×Q8—1062no—
379C4×Q8—1062no—
380#117—22no—
381#117—22no—
382C2×C4⋊C4—531no—
383C2×C4⋊C4—531no—
384C2×C4⋊C4—531no—
385C2×C4⋊C4—531no—
386C2×C4⋊C4—531no—
387C2×C4⋊C4—531no—
388C2×C4⋊C4—531no—
389C2×C4⋊C4—1061no—
390C2×C4⋊C4—1061no—
391C2×C4⋊C4—1061no—
392C2×C4⋊C4—1061no—
393C2×C4⋊C4—1061no—
394C2×C4⋊C4—1061no—
395C2×C4⋊C4—1061no—
396C2×C4⋊C4—1061no—
397C2×C4⋊C4—1061no—
398C2×C4⋊C4D21211yescharacteristic
399C2×C4⋊C4D21211yescharacteristic
400C2×C22⋊C4—1061no—
401C2×C22⋊C4—1061no—
402C2×C22⋊C4—1061no—
403C2×C22⋊C4—1061no—
404C2×C22⋊C4D21211yescharacteristic
405C2×C22⋊C4D21211yescharacteristic
406C2×C42—1061no—
407C2×C42—1061no—
408C2×C42C22×D5311yescharacteristic
409C2×C42C22×D5311yescharacteristic
410#107—1062no—
411#107—1062no—
412#107—1062no—
413#106—1062no—
414C4⋊C8—22no—
415C4⋊C8C22⋊D5314yes—
416#103—1062no—
417#103—1062no—
418#103—1062no—
419#103—1062no—
420C22⋊C8—22no—
421C22⋊C8—22no—
422#97—22no—
423C4×C8—22no—
424#95—1061no—
425#95—1061no—
426#95—1061no—
427#95—1061no—
428#95—1061no—
429#95—1061no—
430#95—1061no—
431#95—1061no—
432#95—1061no—
433C24C2×D21211yescharacteristic
434C2×Q8—532no—
435C2×Q8—1061no—
436C2×Q8—1061no—
437C2×Q8—1061no—
438C2×Q8—1061no—
439C2×Q8—1061no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—1061no—
447C22×C4—1061no—
448C22×C4—1061no—
449C22×C4—1061no—
450C22×C4—1061no—
451C22×C4—1061no—
452C22×C4—1061no—
453C22×C4—1061no—
454C22×C4C2×D21211yescharacteristic
455C22×C4C2×D21211yescharacteristic
456C22×C4C2×D21211yescharacteristic
457C22×C4C2×D21211yescharacteristic
458C22×C4C2×D21211yescharacteristic
459C22×C4C23×D5311yescharacteristic
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8C2×C22⋊D5312yes—
471C2×C8C2×C22⋊D5312yes—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—532no—
475C4⋊C4—1064no—
476C4⋊C4—1061no—
477C4⋊C4—1061no—
478C4⋊C4—1061no—
479C4⋊C4—1061no—
480C4⋊C4—1061no—
481C4⋊C4—1061no—
482C4⋊C4—1061no—
483C4⋊C4—1061no—
484C4⋊C4—1061no—
485C4⋊C4—1061no—
486C4⋊C4—1061no—
487C4⋊C4—1061no—
488C4⋊C4—1061no—
489C4⋊C4—4241no—
490C4⋊C4—4241no—
491C4⋊C4—4241no—
492C4⋊C4—4241no—
493C4⋊C4—4241no—
494C4⋊C4—2122no—
495C4⋊C4—2122no—
496C4⋊C4—2122no—
497C22⋊C4—22no—
498C22⋊C4—22no—
499C22⋊C4—1064no—
500C22⋊C4—4241no—
501C22⋊C4—4241no—
502C22⋊C4—4241no—
503C42—21no—
504C42—21no—
505C42—4241no—
506C42—2122no—
507C42#1171994512yes—
508C42C2×C22⋊D5312yes—
509C23—21no—
510C23—21no—
511C23—21no—
512C23—21no—
513C23—21no—
514C23—21no—
515C23—21no—
516C23C22×D21211yescharacteristic
517Q8—1062no—
518Q8—1062no—
519Q8—2121no—
520Q8—2121no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—1062no—
539C2×C4—1062no—
540C2×C4—2121no—
541C2×C4—2121no—
542C2×C4—2121no—
543C2×C4—2121no—
544C2×C4—2121no—
545C2×C4—2121no—
546C2×C4—2121no—
547C2×C4—2121no—
548C2×C4—2121no—
549C2×C4—2121no—
550C2×C4—2121no—
551C2×C4—2121no—
552C2×C4—2121no—
553C2×C4—2121no—
554C2×C4—2121no—
555C2×C4—2121no—
556C2×C4—2121no—
557C2×C4—2121no—
558C2×C4—2121no—
559C2×C4—2121no—
560C2×C4—2121no—
561C2×C4—2121no—
562C2×C4#1313319112yes—
563C2×C4#1313319112yes—
564C2×C4C22×C22⋊D5311yescharacteristic
565C2×C4#41×D5311yescharacteristic
566C2×C4#41×D5311yescharacteristic
567C2×C4C2×#1171994511yescharacteristic
568C2×C4#1313326411yescharacteristic
569C2×C4#1313326411yescharacteristic
570C8—22no—
571C8—22no—
572C8—41no—
573C8—41no—
574C22—41no—
575C22—41no—
576C22—41no—
577C22—41no—
578C22—41no—
579C22—41no—
580C22—41no—
581C22C2×#1313319111yescharacteristic
582C22#1681113511yescharacteristic
583C22#1681119911yescharacteristic
584C22#1681142811yescharacteristic
585C22#1681143511yescharacteristic
586C22#1681144011yescharacteristic
587C22#1681144011yescharacteristic
588C4—22no—
589C4—22no—
590C4—41no—
591C4—41no—
592C4—41no—
593C4—4241no—
594C4—4241no—
595C4—4241no—
596C4—4241no—
597C4—4241no—
598C4—4241no—
599C4—2122no—
600C4—2122no—
601C4#1681127211yescharacteristic
602C4#1681128011yescharacteristic
603C4#1681131211yescharacteristic
604C4#1681134111yescharacteristic
605C2—81no—
606C2#2027813811yescharacteristic
607C2#2028438511yescharacteristic
608C2#2028440611yescharacteristic
609C2#2028491611yescharacteristic
610C2#2028501611yescharacteristic
611C2#2028598511yescharacteristic
612C2#2028600011yescharacteristic
subgroupG/H
centerC23C22×D212
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC2×C53×#433C2
socleC23×C53C22×D4
prime radicalC24×C53C2×D4
hypercenterC2×#433D53
hypocenterC53#24715
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#433

Complements

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

NH ≅ G/Ntype
1. C2×#16810931C2semidirect
8. #20275670C2semidirect
9. #20275474C2semidirect
10. #20275453C2semidirect
11. #20275448C2semidirect
12. #20275405C2semidirect
13. #20275391C2semidirect
14. #20274721C2semidirect
367. C53#24715semidirect

Maximal subgroups

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

subgroupindex
C2×#168109312
#202805452
#202773972
#202761482
#202761382
#202759192
#202756702
#202754742
#202754532
#202754482
#202754052
#202753912
#202747212
C2×C53×#4332
#2471553