Finite Group Databasebeta

← #27465804

Subgroups — #27465804

Order 13568.

810 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810965C211yesmaximal, characteristic
2C2×#16810964C211yesmaximal, characteristic
3#20277243C211yesmaximal, characteristic
4#20277118C211yesmaximal, characteristic
5#20276195C211yesmaximal, characteristic, complemented
6#20276190C211yesmaximal, characteristic, complemented
7#20276127C211yesmaximal, characteristic, complemented
8#20276127C211yesmaximal, characteristic, complemented
9#20275486C211yesmaximal, characteristic, complemented
10#20275482C211yesmaximal, characteristic, complemented
11#20274807C211yesmaximal, characteristic
12#20274729C211yesmaximal, characteristic, complemented
13#20274729C211yesmaximal, characteristic, complemented
14#20274693C211yesmaximal, characteristic
15C53×#1825C211yesmaximal, characteristic
16C2×#13133196C2211yescharacteristic
17C2×#13133182C2211yescharacteristic
18C2×#13133176C2211yescharacteristic
19C22×C4×Dic53C2211yescharacteristic
20#16810965—24no—
21#16810964—24no—
22C2×#13133151C2211yescharacteristic
23C2×#13133150C2211yescharacteristic
24C2×#13133149C2211yescharacteristic
25C2×#13133148C2211yescharacteristic
26C2×#13133146C2211yescharacteristic
27C2×#13133146C2211yescharacteristic
28C2×#13133142C2211yescharacteristic
29C2×#13133142C2211yescharacteristic
30C2×#13133140C2211yescharacteristic
31C2×#13133140C2211yescharacteristic
32C2×C22⋊#88322C2211yescharacteristic
33#16810528C2211yescharacteristic
34#16810519C2211yescharacteristic
35#16810505—21no—
36#16810495—21no—
37#16810492C2211yescharacteristic
38#16810492C2211yescharacteristic
39#16810488C2211yescharacteristic
40#16810488C2211yescharacteristic
41#16810481—21no—
42#16810481—21no—
43C4⋊C4×Dic53C2211yescharacteristic
44C4×#11719919—21no—
45C4×#11719919—21no—
46#16810474—21no—
47C22⋊C4×Dic53—21no—
48#16810385C2211yescharacteristic
49#16810385C2211yescharacteristic
50#16810385C2211yescharacteristic
51#16810385C2211yescharacteristic
52#16810377C2211yescharacteristic
53#16810377C2211yescharacteristic
54C2×#117×C53C2211yescharacteristic
55C2×C22⋊C8×C53C2211yescharacteristic
56C53×#380C2211yescharacteristic
57C53×#340C2211yescharacteristic
58C53×#340C2211yescharacteristic
59C53×#340C2211yescharacteristic
60C53×#340C2211yescharacteristic
61C22×Dic106C2311yescharacteristic
62C23×Dic53D411yescharacteristic
63C23×Dic53D411yescharacteristic
64C22×#88322C2311yescharacteristic
65C22×#88322C2311yescharacteristic
66#13133196—22no—
67#13133196D414yes—
68#13133182—22no—
69#13133182—22no—
70#13133176—22no—
71#13133176—22no—
72C2×C4⋊Dic53C2311yescharacteristic
73C2×C4⋊Dic53C2311yescharacteristic
74C2×C4⋊Dic53C2311yescharacteristic
75C2×#11719919—21no—
76C2×#11719919—21no—
77C2×#11719919—21no—
78C2×#11719919—21no—
79C2×#11719919—21no—
80C2×#11719919—21no—
81C2×#11719919—21no—
82C2×C22⋊Dic53—21no—
83C2×C22⋊Dic53—21no—
84C2×C4×Dic53—22no—
85C2×C4×Dic53—22no—
86C2×C4×Dic53—21no—
87C2×C4×Dic53—21no—
88C2×C4×Dic53—21no—
89C2×C4×Dic53D411yescharacteristic
90C2×C4×Dic53D411yescharacteristic
91C2×C4×Dic53C2311yescharacteristic
92C2×C4×Dic53C2311yescharacteristic
93#13133151—22no—
94#13133150—22no—
95#13133149—22no—
96#13133148—22no—
97#13133146—22no—
98#13133146—22no—
99#13133142—22no—
100#13133142—22no—
101#13133140—22no—
102#13133140—22no—
103C22⋊#88322—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—
112C23×C4×C53C2311yescharacteristic
113C22×C8×C53C2311yescharacteristic
114C22×C8×C53C2311yescharacteristic
115#117×C53—22no—
116#117×C53—22no—
117C2×C4⋊C4×C53C2311yescharacteristic
118C2×C4⋊C4×C53C2311yescharacteristic
119C2×C4⋊C4×C53C2311yescharacteristic
120C2×C4⋊C4×C53C2311yescharacteristic
121C2×C22⋊C4×C53—21no—
122C2×C22⋊C4×C53—21no—
123C2×C42×C53—21no—
124C22⋊C8×C53—22no—
125#95×C53—21no—
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×C53—21no—
134C22×C4×C53—21no—
135C22×C4×C53C2×D411yescharacteristic
136C22×C4×C53C2×D412yes—
137C22×C4×C53#4112yes—
138C22×C4×C53C2411yescharacteristic
139C22×Dic53—21no—
140C22×Dic53—21no—
141C22×Dic53—22no—
142C22×Dic53—22no—
143C22×Dic53—22no—
144C22×Dic53—22no—
145C22×Dic53—21no—
146C22×Dic53—21no—
147C22×Dic53—41no—
148C22×Dic53—41no—
149C22×Dic53C2×D411yescharacteristic
150C22×Dic53C2×D411yescharacteristic
151C22×Dic53C2×D411yescharacteristic
152C22×Dic53C2×D411yescharacteristic
153C2×Dic106—21no—
154C2×Dic106—21no—
155C2×Dic106—21no—
156C2×Dic106—21no—
157C2×Dic106—21no—
158C2×Dic106C2×D412yes—
159C2×C8×C53—21no—
160C2×C8×C53—21no—
161C2×C8×C53—21no—
162C2×C8×C53—21no—
163C2×C8×C53—21no—
164C2×C8×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—21no—
171C4⋊C4×C53—21no—
172C4⋊C4×C53—21no—
173C22⋊C4×C53—42no—
174C22⋊C4×C53—42no—
175C42×C53—42no—
176C22⋊Dic53—24no—
177C22⋊Dic53—42no—
178C4⋊Dic53—21no—
179C4⋊Dic53—21no—
180C4⋊Dic53—21no—
181C4⋊Dic53—21no—
182C4⋊Dic53—21no—
183C4⋊Dic53C2×D412yes—
184#11719919—24no—
185#11719919—42no—
186#11719919—42no—
187#11719919—42no—
188#11719919—42no—
189#11719919—42no—
190#11719919—42no—
191C4×Dic53—21no—
192C4×Dic53—21no—
193C4×Dic53—22no—
194C4×Dic53—22no—
195C4×Dic53—42no—
196C4×Dic53—42no—
197C4×Dic53—42no—
198C4×Dic53#4112yes—
199C4×Dic53#4112yes—
200C2×#88322—21no—
201C2×#88322—21no—
202C2×#88322—21no—
203C2×#88322—21no—
204C2×#88322—21no—
205C2×#88322—21no—
206C23×C53—21no—
207C23×C53—21no—
208C23×C53—21no—
209C23×C53—21no—
210C23×C53—21no—
211C23×C53—21no—
212C23×C53#12112yes—
213C23×C53C22×D411yescharacteristic
214C2×C4×C53—21no—
215C2×C4×C53—21no—
216C2×C4×C53—22no—
217C2×C4×C53—22no—
218C2×C4×C53—22no—
219C2×C4×C53—22no—
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—41no—
230C2×C4×C53—41no—
231C2×C4×C53—41no—
232C2×C4×C53#12112yes—
233C2×C4×C53C22×D411yescharacteristic
234C2×C4×C53C2×#4111yescharacteristic
235C2×C4×C53C2×#4111yescharacteristic
236C2×C4×C53C2×#4111yescharacteristic
237C2×C4×C532−1+411yescharacteristic
238C2×C4×C532−1+411yescharacteristic
239C2×Dic53—21no—
240C2×Dic53—21no—
241C2×Dic53—21no—
242C2×Dic53—21no—
243C2×Dic53—21no—
244C2×Dic53—21no—
245C2×Dic53—21no—
246C2×Dic53—21no—
247C2×Dic53—22no—
248C2×Dic53—22no—
249C2×Dic53—44no—
250C2×Dic53—44no—
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—41no—
261C2×Dic53#12312yes—
262C2×Dic53#12312yes—
263C2×Dic53#13512yes—
264C2×Dic53#13712yes—
265Dic106—22no—
266Dic106—22no—
267Dic106—41no—
268Dic106—41no—
269C8×C53—41no—
270C8×C53—41no—
271#88322—41no—
272#88322—41no—
273#27652—531nomaximal, Sylow
274C22×C53—22no—
275C22×C53—22no—
276C22×C53—22no—
277C22×C53—41no—
278C22×C53—41no—
279C22×C53—41no—
280C22×C53—41no—
281C22×C53C2×#12111yescharacteristic
282C22×C53C2×#12311yescharacteristic
283C22×C53#53611yescharacteristic
284C22×C53C2×#13511yescharacteristic
285C22×C53C2×#13711yescharacteristic
286C22×C53#57511yescharacteristic
287C22×C53#57511yescharacteristic
288C4×C53—22no—
289C4×C53—81no—
290C4×C53—81no—
291C4×C53—81no—
292C4×C53—81no—
293C4×C53#54711yescharacteristic
294C4×C53#54711yescharacteristic
295C4×C53#55211yescharacteristic
296C4×C53#55211yescharacteristic
297Dic53—22no—
298Dic53—22no—
299Dic53—22no—
300Dic53—22no—
301Dic53—42no—
302Dic53—42no—
303Dic53—81no—
304Dic53—81no—
305C2×#485—531no—
306C2×#485—531no—
307C4×C22⋊Q8—531no—
308#2285—531no—
309#2084—531no—
310#2083—531no—
311#2072—531no—
312#2072—531no—
313#1949—531no—
314#1949—531no—
315#1825—531no—
316#1825D5311yescharacteristic
317#1805—531no—
318#1805—531no—
319#1797—531no—
320C2×C53—42no—
321C2×C53#248611yescharacteristic
322C2×C53#303511yescharacteristic
323C2×C53#303711yescharacteristic
324C2×C53#309111yescharacteristic
325C2×C53#309511yescharacteristic
326C2×C53#323111yescharacteristic
327C2×C53#324011yescharacteristic
328C2×C22⋊Q8—531no—
329C2×C4×Q8—531no—
330C2×#117—531no—
331C2×#117C2×D5311yescharacteristic
332C22×C42—531no—
333#485—1064no—
334#485—1064no—
335C2×#107—531no—
336C2×#107—531no—
337C2×#106—531no—
338C2×#106—531no—
339C2×C4⋊C8—531no—
340C2×C4⋊C8—531no—
341C2×#103—531no—
342C2×#103—531no—
343C2×#103—531no—
344C2×#103—531no—
345C2×C22⋊C8—531no—
346C2×C22⋊C8C2×D5311yescharacteristic
347#391—531no—
348#389—531no—
349#387—1061no—
350#385—1061no—
351#384—531no—
352#384—531no—
353#382—531no—
354#382—531no—
355#380—1061no—
356#380—1061no—
357#380C2×D5311yescharacteristic
358C4×C4⋊C4—531no—
359C4×C4⋊C4—1061no—
360C4×C4⋊C4—1061no—
361C4×C4⋊C4—1061no—
362C4×C22⋊C4—1061no—
363#340—531no—
364#340—531no—
365#340—531no—
366#340—531no—
367#340C2×D5311yescharacteristic
368#340C2×D5311yescharacteristic
369#340C2×D5311yescharacteristic
370#340C2×D5311yescharacteristic
371#336—531no—
372#336—531no—
373C53#2765211yescharacteristic, complemented, Sylow
374C22×Q8—531no—
375C23×C4—531no—
376C23×C4—531no—
377C23×C4C22×D5311yescharacteristic
378C22×C8—531no—
379C22×C8—531no—
380C22×C8C22×D5311yescharacteristic
381C22×C8C22×D5311yescharacteristic
382C22⋊Q8—534no—
383C22⋊Q8—1062no—
384C4×Q8—1062no—
385C4×Q8—1062no—
386#117—22no—
387#117—22no—
388#117—1062no—
389#117—1062no—
390C2×C4⋊C4—531no—
391C2×C4⋊C4—531no—
392C2×C4⋊C4—531no—
393C2×C4⋊C4—1061no—
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—21no—
406C2×C22⋊C4—1061no—
407C2×C22⋊C4—1061no—
408C2×C42—21no—
409C2×C42—531no—
410C2×C42—531no—
411C2×C42—531no—
412C2×C42—531no—
413C2×C42—1062no—
414C2×C42—1062no—
415C2×C42—1061no—
416C2×C42—1061no—
417C2×C42—1061no—
418#107—1062no—
419#107—1062no—
420#106—1062no—
421#106—1062no—
422C4⋊C8—1062no—
423C4⋊C8—1062no—
424#103—1062no—
425#103—1062no—
426#103—1062no—
427#103—1062no—
428C22⋊C8—22no—
429C22⋊C8—1062no—
430#95—21no—
431#95—1061no—
432#95—1061no—
433#95—1061no—
434#95—1061no—
435#95—1061no—
436#95—1061no—
437#95—1061no—
438#95—1061no—
439C24D4×D5311yescharacteristic
440C2×Q8—532no—
441C2×Q8—1061no—
442C2×Q8—1061no—
443C2×Q8—1061no—
444C2×Q8—1061no—
445C2×Q8—1061no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—531no—
455C22×C4—531no—
456C22×C4—531no—
457C22×C4—531no—
458C22×C4—1062no—
459C22×C4—1062no—
460C22×C4—1062no—
461C22×C4—1062no—
462C22×C4—1061no—
463C22×C4—1061no—
464C22×C4—1061no—
465C22×C4—1061no—
466C22×C4—2121no—
467C22×C4—2121no—
468C22×C4#1171994512yes—
469C22×C4D4×D5311yescharacteristic
470C22×C4D4×D5312yes—
471C22×C4C23×D5311yescharacteristic
472C2×C8—21no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—1061no—
479C2×C8—1061no—
480C2×C8—1061no—
481C2×C8—1061no—
482C2×C8—1061no—
483C2×C8—1061no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—532no—
493C4⋊C4—1064no—
494C4⋊C4—1061no—
495C4⋊C4—1061no—
496C4⋊C4—1061no—
497C4⋊C4—1061no—
498C4⋊C4—1061no—
499C4⋊C4—2122no—
500C4⋊C4—2122no—
501C4⋊C4—2122no—
502C4⋊C4—2122no—
503C4⋊C4—2122no—
504C4⋊C4—2122no—
505C22⋊C4—42no—
506C22⋊C4—42no—
507C22⋊C4—1064no—
508C22⋊C4—2122no—
509C42—42no—
510C42—532no—
511C42—532no—
512C42—1062no—
513C42—1062no—
514C42—1061no—
515C42—1061no—
516C42—2122no—
517C42—2122no—
518C42—2122no—
519C23—21no—
520C23—21no—
521C23—21no—
522C23—21no—
523C23—21no—
524C23—21no—
525C23#1313319012yes—
526C23C2×D4×D5311yescharacteristic
527Q8—1062no—
528Q8—1062no—
529Q8—2121no—
530Q8—2121no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—532no—
550C2×C4—532no—
551C2×C4—532no—
552C2×C4—532no—
553C2×C4—1061no—
554C2×C4—1061no—
555C2×C4—1061no—
556C2×C4—1061no—
557C2×C4—1061no—
558C2×C4—1061no—
559C2×C4—1061no—
560C2×C4—1061no—
561C2×C4—1062no—
562C2×C4—1062no—
563C2×C4—2124no—
564C2×C4—2124no—
565C2×C4—2121no—
566C2×C4—2121no—
567C2×C4—2121no—
568C2×C4—2121no—
569C2×C4—2121no—
570C2×C4—2121no—
571C2×C4—2121no—
572C2×C4—2121no—
573C2×C4—2121no—
574C2×C4—2121no—
575C2×C4#1313319012yes—
576C2×C4C2×D4×D5311yescharacteristic
577C2×C4#41×D5311yescharacteristic
578C2×C4#41×D5311yescharacteristic
579C2×C4C2×#1171994511yescharacteristic
580C2×C4#1313326411yescharacteristic
581C2×C4#1313326411yescharacteristic
582C8—41no—
583C8—41no—
584C8—2121no—
585C8—2121no—
586C22—22no—
587C22—22no—
588C22—22no—
589C22—41no—
590C22—41no—
591C22—41no—
592C22—41no—
593C22C2×#1313319011yescharacteristic
594C22#123×D5311yescharacteristic
595C22#1681119511yescharacteristic
596C22#135×D5311yescharacteristic
597C22#137×D5311yescharacteristic
598C22#1681141911yescharacteristic
599C22#1681142011yescharacteristic
600C4—22no—
601C4—81no—
602C4—81no—
603C4—81no—
604C4—81no—
605C4—1062no—
606C4—1062no—
607C4—1062no—
608C4—1062no—
609C4—4241no—
610C4—4241no—
611C4—2122no—
612C4—2122no—
613C4#1681127111yescharacteristic
614C4#1681127111yescharacteristic
615C4#1681130611yescharacteristic
616C4#1681130611yescharacteristic
617C2—42no—
618C2#2027902211yescharacteristic
619C2#2028416811yescharacteristic
620C2#2028418611yescharacteristic
621C2#2028462711yescharacteristic
622C2#2028466211yescharacteristic
623C2#2028587211yescharacteristic
624C2#2028595311yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#1825C2
socleC23×C53C22×D4
prime radicalC24×C53C2×D4
hypercenter#1825D53
hypocenterC53#27652
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1825

Complements

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

NH ≅ G/Ntype
5. #20276195C2semidirect
6. #20276190C2semidirect
7. #20276127C2semidirect
8. #20276127C2semidirect
9. #20275486C2semidirect
10. #20275482C2semidirect
12. #20274729C2semidirect
13. #20274729C2semidirect
373. C53#27652semidirect

Maximal subgroups

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

subgroupindex
C2×#168109652
C2×#168109642
#202772432
#202771182
#202761952
#202761902
#202761272
#202754862
#202754822
#202748072
#202747292
#202746932
C53×#18252
#2765253