Finite Group Databasebeta

← #27425491

Subgroups — #27425491

Order 13568.

802 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810910C211yesmaximal, characteristic, complemented
2C2×#16810684C211yesmaximal, characteristic, complemented
3#20277408C211yesmaximal, characteristic
4#20276152C211yesmaximal, characteristic, complemented
5#20276151C211yesmaximal, characteristic, complemented
6#20275904C211yesmaximal, characteristic
7#20275897C211yesmaximal, characteristic
8#20275668C211yesmaximal, characteristic, complemented
9#20275481C211yesmaximal, characteristic, complemented
10#20275450C211yesmaximal, characteristic, complemented
11#20275440C211yesmaximal, characteristic, complemented
12#20275411C211yesmaximal, characteristic, complemented
13#20275397C211yesmaximal, characteristic, complemented
14#20274722C211yesmaximal, characteristic, complemented
15C53×#2628C211yesmaximal, characteristic, complemented
16C2×#13133191C2211yescharacteristic
17C2×C4×D212C2211yescharacteristic
18#16810910—24no—
19#16810684—24no—
20C2×#13133150C2211yescharacteristic
21C2×#13133148C2211yescharacteristic
22C2×#13133148C2211yescharacteristic
23C2×#13133148C2211yescharacteristic
24C2×C4⋊#88322C2211yescharacteristic
25C2×C4⋊#88322C2211yescharacteristic
26C2×#13133137C2211yescharacteristic
27C2×#13133137C2211yescharacteristic
28C2×#13133137C2211yescharacteristic
29C2×#13133137C2211yescharacteristic
30C2×C22⋊#88322C2211yescharacteristic
31C2×C22⋊#88322C2211yescharacteristic
32C2×#13133128C2211yescharacteristic
33C2×C4×#88322C2211yescharacteristic
34#16810525C2211yescharacteristic
35#16810523—21no—
36#16810521C2211yescharacteristic
37#16810513—21no—
38#16810499—21no—
39#16810494—21no—
40#16810483C2211yescharacteristic
41C4×C4⋊Dic53C2211yescharacteristic
42#16810385C2211yescharacteristic
43#16810385C2211yescharacteristic
44#16810385C2211yescharacteristic
45#16810385C2211yescharacteristic
46#16810385C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810385C2211yescharacteristic
49#16810385C2211yescharacteristic
50C2×C4⋊Q8×C53C2211yescharacteristic
51C2×#117×C53C2211yescharacteristic
52C53×#400—21no—
53C53×#399—21no—
54C53×#395—21no—
55C53×#393—21no—
56C53×#389C2211yescharacteristic
57C53×#389C2211yescharacteristic
58C53×#382C2211yescharacteristic
59C53×#380C2211yescharacteristic
60C53×#380C2211yescharacteristic
61C22×D212C2311yescharacteristic
62C22×C22⋊D53—21no—
63C22×C4×D53—21no—
64C22×#88322C2311yescharacteristic
65C22×#88322C2311yescharacteristic
66C22×#88322C2311yescharacteristic
67C22×#88322C2311yescharacteristic
68#13133191—22no—
69#13133191—22no—
70C4×D212—22no—
71C4×D212—22no—
72C2×C4⋊Dic53C2311yescharacteristic
73C2×C4⋊Dic53C2311yescharacteristic
74C2×C4⋊Dic53C2311yescharacteristic
75C2×#11719919—21no—
76C2×C22⋊Dic53—21no—
77C2×#11719921—21no—
78C2×#11719921—21no—
79C2×#11719921—21no—
80C2×#11719921—21no—
81C2×C4×Dic53—21no—
82#13133150—22no—
83#13133148—22no—
84#13133148—22no—
85#13133148—22no—
86C4⋊#88322—22no—
87C4⋊#88322D414yescomplemented
88#13133137—22no—
89#13133137—22no—
90#13133137—22no—
91#13133137—22no—
92C22⋊#88322—22no—
93C22⋊#88322—22no—
94#13133128—22no—
95C4×#88322—22no—
96#13133124—21no—
97#13133124—21no—
98#13133124—21no—
99C22×Q8×C53—21no—
100C23×C4×C53C2311yescharacteristic
101C4⋊Q8×C53—22no—
102C4⋊Q8×C53—22no—
103#117×C53—22no—
104#117×C53—22no—
105C2×C4⋊C4×C53—21no—
106C2×C4⋊C4×C53—21no—
107C2×C4⋊C4×C53—21no—
108C2×C4⋊C4×C53D411yescharacteristic
109C2×C4⋊C4×C53D411yescharacteristic
110C2×C4⋊C4×C53C2311yescharacteristic
111C2×C4⋊C4×C53C2311yescharacteristic
112C2×C4⋊C4×C53C2311yescharacteristic
113C2×C4⋊C4×C53C2311yescharacteristic
114C2×C22⋊C4×C53—21no—
115C2×C22⋊C4×C53—21no—
116C2×C22⋊C4×C53D411yescharacteristic
117C2×C22⋊C4×C53D411yescharacteristic
118C2×C42×C53C2311yescharacteristic
119C2×C42×C53C2311yescharacteristic
120#95×C53—21no—
121#95×C53—21no—
122#95×C53—21no—
123#95×C53—21no—
124#95×C53—21no—
125#95×C53—21no—
126C24×C53C2×D411yescharacteristic
127C23×D53—21no—
128C2×Q8×C53—42no—
129C2×Q8×C53—81no—
130C2×Q8×C53—81no—
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—21no—
140C22×C4×C53—21no—
141C22×C4×C53C2×D411yescharacteristic
142C22×C4×C53C2×D411yescharacteristic
143C22×C4×C53C2×D411yescharacteristic
144C22×C4×C53C2×D411yescharacteristic
145C22×C4×C53C2×D411yescharacteristic
146C22×C4×C53C2411yescharacteristic
147C2×C22⋊D53—24no—
148C2×C22⋊D53—81no—
149C2×C22⋊D53—81no—
150C22×Dic53—21no—
151C22×Dic53—21no—
152C22×Dic53—21no—
153C2×D212—21no—
154C2×D212—21no—
155C2×D212—21no—
156C2×D212—21no—
157C2×D212—21no—
158C2×D212#4112yes—
159C2×C4×D53—42no—
160C2×C4×D53—81no—
161C2×C4×D53—81no—
162C4⋊C4×C53—22no—
163C4⋊C4×C53—22no—
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—21no—
171C4⋊C4×C53—21no—
172C4⋊C4×C53—42no—
173C4⋊C4×C53—81no—
174C4⋊C4×C53—81no—
175C22⋊C4×C53—22no—
176C22⋊C4×C53—22no—
177C22⋊C4×C53—81no—
178C22⋊C4×C53—81no—
179C42×C53—21no—
180C42×C53—21no—
181C42×C53C2×D412yes—
182C42×C53#4112yes—
183C22⋊Dic53—81no—
184#11719921—24no—
185#11719921—42no—
186#11719921—81no—
187#11719921—81no—
188C4⋊Dic53—21no—
189C4⋊Dic53—21no—
190C4⋊Dic53—21no—
191C4⋊Dic53—21no—
192C4⋊Dic53—21no—
193C4⋊Dic53#4112yes—
194#11719919—81no—
195C4×Dic53—81no—
196C2×#88322—21no—
197C2×#88322—21no—
198C2×#88322—21no—
199C2×#88322—21no—
200C2×#88322—21no—
201C2×#88322—21no—
202C2×#88322—21no—
203C2×#88322—21no—
204C2×#88322—21no—
205C2×#88322—21no—
206C2×#88322C2×D412yes—
207C2×#88322C2×D412yes—
208C23×C53—21no—
209C23×C53—21no—
210C23×C53—21no—
211C23×C53—21no—
212C23×C53—21no—
213C23×C53—21no—
214C23×C53—21no—
215C23×C53C22×D411yescharacteristic
216C22×D53—22no—
217C22×D53—41no—
218C22×D53—41no—
219C22×D53—41no—
220C22×D53—41no—
221C22×D53—41no—
222C22×D53—41no—
223Q8×C53—84no—
224C2×C4×C53—21no—
225C2×C4×C53—21no—
226C2×C4×C53—21no—
227C2×C4×C53—21no—
228C2×C4×C53—21no—
229C2×C4×C53—21no—
230C2×C4×C53—21no—
231C2×C4×C53—21no—
232C2×C4×C53—21no—
233C2×C4×C53—21no—
234C2×C4×C53—21no—
235C2×C4×C53—21no—
236C2×C4×C53—21no—
237C2×C4×C53—41no—
238C2×C4×C53—41no—
239C2×C4×C53—41no—
240C2×C4×C53—41no—
241C2×C4×C53—41no—
242C2×C4×C53—41no—
243C2×C4×C53—41no—
244C2×C4×C53—41no—
245C2×C4×C53—41no—
246C2×C4×C53—41no—
247C2×C4×C53—41no—
248C2×C4×C53—41no—
249C2×C4×C53—41no—
250C2×C4×C53—41no—
251C2×C4×C53—41no—
252C2×C4×C53—41no—
253C2×C4×C53#12112yes—
254C2×C4×C53#12112yes—
255C2×C4×C53C22×D411yescharacteristic
256C2×C4×C53C2×#4111yescharacteristic
257C2×C4×C53C2×#4111yescharacteristic
258C2×C4×C53C2×#4111yescharacteristic
259C2×C4×C532−1+411yescharacteristic
260C2×C4×C532−1+411yescharacteristic
261C22⋊D53—84no—
262C2×Dic53—22no—
263C2×Dic53—41no—
264C2×Dic53—41no—
265C2×Dic53—41no—
266C2×Dic53—41no—
267C2×Dic53—41no—
268C2×Dic53—41no—
269C2×Dic53—41no—
270C2×Dic53—41no—
271D212—22no—
272D212—22no—
273D212—41no—
274D212—41no—
275C4×D53—84no—
276#88322—22no—
277#88322—22no—
278#88322—41no—
279#88322—41no—
280#24716—531nomaximal, Sylow
281C22×C53—41no—
282C22×C53—41no—
283C22×C53—41no—
284C22×C53—41no—
285C22×C53—41no—
286C22×C53—41no—
287C22×C53—41no—
288C22×C53C2×#12111yescharacteristic
289C22×C53#52511yescharacteristic
290C22×C53#53611yescharacteristic
291C22×C53#57611yescharacteristic
292C22×C53#57711yescharacteristic
293C22×C53#57711yescharacteristic
294C22×C53#57711yescharacteristic
295C2×D53—42no—
296C2×D53—42no—
297C2×D53—42no—
298C2×D53—81no—
299C2×D53—81no—
300C2×D53—81no—
301C2×D53—81no—
302C4×C53—22no—
303C4×C53—22no—
304C4×C53—41no—
305C4×C53—41no—
306C4×C53—41no—
307C4×C53—81no—
308C4×C53—81no—
309C4×C53—81no—
310C4×C53—81no—
311C4×C53#54711yescharacteristic
312C4×C53#54811yescharacteristic
313C4×C53#55211yescharacteristic
314C4×C53#55611yescharacteristic
315Dic53—42no—
316Dic53—81no—
317Dic53—81no—
318C2×#476—531no—
319C2×#433—531no—
320#2628D5311yescharacteristic, complemented
321#2324—531no—
322#2077—531no—
323#2077—531no—
324#2034—531no—
325#2033—531no—
326#1988—531no—
327#1949—531no—
328#1943—531no—
329#1941—531no—
330#1936—531no—
331#1934—531no—
332#1804—531no—
333C2×C53—81no—
334C2×C53#241511yescharacteristic
335C2×C53#305711yescharacteristic
336C2×C53#306211yescharacteristic
337C2×C53#311811yescharacteristic
338C2×C53#312511yescharacteristic
339C2×C53#324111yescharacteristic
340C2×C53#324411yescharacteristic
341D53—82no—
342C2×C4⋊Q8C2×D5311yescharacteristic
343C2×#121—531no—
344C2×C4×D4—531no—
345C2×#117C2×D5311yescharacteristic
346#476—1064no—
347#433—1064no—
348C2×#107—531no—
349C2×#106—531no—
350C2×#106—531no—
351C2×#106—531no—
352C2×C4⋊C8—531no—
353C2×C4⋊C8—531no—
354C2×#102—531no—
355C2×#102—531no—
356C2×#102—531no—
357C2×#102—531no—
358C2×C22⋊C8—531no—
359C2×C22⋊C8—531no—
360C2×#97—531no—
361C2×C4×C8—531no—
362#400—21no—
363#399—21no—
364#395—21no—
365#393—21no—
366#390—531no—
367#390—1061no—
368#389—531no—
369#389C2×D5311yescharacteristic
370#389C2×D5311yescharacteristic
371#388—1061no—
372#386—1061no—
373#385—1061no—
374#382C2×D5311yescharacteristic
375#380—531no—
376#380C2×D5311yescharacteristic
377#380C2×D5311yescharacteristic
378C4×C4⋊C4—531no—
379#340—531no—
380#340—531no—
381#340—531no—
382#340—531no—
383#340—531no—
384#340—531no—
385#340—531no—
386#340—531no—
387C53#2471611yescharacteristic, complemented, Sylow
388C22×Q8—21no—
389C22×D4—531no—
390C22×D4—1061no—
391C23×C4—1061no—
392C23×C4C22×D5311yescharacteristic
393C22×C8—531no—
394C22×C8—531no—
395C22×C8—531no—
396C22×C8—531no—
397C4⋊Q8—22no—
398C4⋊Q8—22no—
399#121—1062no—
400#121—1062no—
401C4×D4—1062no—
402C4×D4—1062no—
403#117—22no—
404#117—22no—
405C2×C4⋊C4—21no—
406C2×C4⋊C4—21no—
407C2×C4⋊C4—21no—
408C2×C4⋊C4—531no—
409C2×C4⋊C4—531no—
410C2×C4⋊C4—531no—
411C2×C4⋊C4—1061no—
412C2×C4⋊C4C22⋊D5311yescharacteristic
413C2×C4⋊C4C22⋊D5311yescharacteristic
414C2×C4⋊C4C22×D5311yescharacteristic
415C2×C4⋊C4C22×D5311yescharacteristic
416C2×C4⋊C4C22×D5311yescharacteristic
417C2×C4⋊C4C22×D5311yescharacteristic
418C2×C22⋊C4—21no—
419C2×C22⋊C4—21no—
420C2×C22⋊C4—1061no—
421C2×C22⋊C4—1061no—
422C2×C22⋊C4—1061no—
423C2×C22⋊C4—1061no—
424C2×C22⋊C4—1061no—
425C2×C22⋊C4C22⋊D5311yescharacteristic
426C2×C22⋊C4C22⋊D5311yescharacteristic
427C2×C42—1061no—
428C2×C42C22×D5311yescharacteristic
429C2×C42C22×D5311yescharacteristic
430#107—1062no—
431#106—1062no—
432#106—1062no—
433#106—1062no—
434C4⋊C8—534no—
435C4⋊C8—1062no—
436#102—1062no—
437#102—1062no—
438#102—1062no—
439#102—1062no—
440C22⋊C8—1062no—
441C22⋊C8—1062no—
442#97—1062no—
443C4×C8—1062no—
444#95—21no—
445#95—21no—
446#95—21no—
447#95—21no—
448#95—21no—
449#95—21no—
450#95—1061no—
451#95—1061no—
452#95—1061no—
453C24—1061no—
454C24C2×C22⋊D5311yescharacteristic
455C2×Q8—42no—
456C2×Q8—81no—
457C2×Q8—81no—
458C2×D4—532no—
459C2×D4—1064no—
460C2×D4—1061no—
461C2×D4—1061no—
462C2×D4—1061no—
463C2×D4—1061no—
464C2×D4—1061no—
465C2×D4—4241no—
466C2×D4—4241no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—1061no—
478C22×C4—1061no—
479C22×C4—1061no—
480C22×C4—4241no—
481C22×C4—4241no—
482C22×C4—2122no—
483C22×C4C2×C22⋊D5311yescharacteristic
484C22×C4C2×C22⋊D5311yescharacteristic
485C22×C4C2×C22⋊D5311yescharacteristic
486C22×C4C2×C22⋊D5311yescharacteristic
487C22×C4C2×C22⋊D5311yescharacteristic
488C22×C4C23×D5311yescharacteristic
489C2×C8—532no—
490C2×C8—532no—
491C2×C8—1061no—
492C2×C8—1061no—
493C2×C8—1061no—
494C2×C8—1061no—
495C2×C8—1061no—
496C2×C8—1061no—
497C2×C8—1061no—
498C2×C8—1061no—
499C2×C8—1061no—
500C2×C8—1061no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—42no—
512C4⋊C4—81no—
513C4⋊C4—81no—
514C4⋊C4—532no—
515C4⋊C4—1061no—
516C4⋊C4—1061no—
517C4⋊C4—1061no—
518C4⋊C4—1061no—
519C4⋊C4—1061no—
520C4⋊C4—4241no—
521C22⋊C4—22no—
522C22⋊C4—22no—
523C22⋊C4—81no—
524C22⋊C4—81no—
525C22⋊C4—1064no—
526C22⋊C4—4241no—
527C22⋊C4—4241no—
528C22⋊C4—4241no—
529C22⋊C4—2122no—
530C42—21no—
531C42—21no—
532C42—4241no—
533C42D4×D5312yes—
534C42#1171994712yes—
535C23—21no—
536C23—21no—
537C23—21no—
538C23—21no—
539C23—21no—
540C23—21no—
541C23—21no—
542C23—1062no—
543C23—2121no—
544C23—2121no—
545C23—2121no—
546C23—2121no—
547C23—2121no—
548C23—2121no—
549C23C22×C22⋊D5311yescharacteristic
550Q8—84no—
551D4—1062no—
552D4—1062no—
553D4—4244no—
554D4—2121no—
555D4—2121no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—1062no—
586C2×C4—4244no—
587C2×C4—2121no—
588C2×C4—2121no—
589C2×C4—2121no—
590C2×C4—2121no—
591C2×C4—2121no—
592C2×C4—2121no—
593C2×C4—2121no—
594C2×C4—2121no—
595C2×C4#1313318912yes—
596C2×C4#1313318912yes—
597C2×C4C2×D4×D5311yescharacteristic
598C2×C4#41×D5311yescharacteristic
599C2×C4#41×D5311yescharacteristic
600C2×C4C2×#1171994711yescharacteristic
601C2×C4#1313326511yescharacteristic
602C2×C4#1313326511yescharacteristic
603C8—1062no—
604C8—1062no—
605C8—2121no—
606C8—2121no—
607C22—41no—
608C22—41no—
609C22—41no—
610C22—41no—
611C22—41no—
612C22—41no—
613C22—41no—
614C22—4241no—
615C22—4241no—
616C22—4241no—
617C22—4241no—
618C22—2122no—
619C22—2122no—
620C22—2122no—
621C22C2×#1313318911yescharacteristic
622C22#1681113011yescharacteristic
623C22#1681119711yescharacteristic
624C22#1681142611yescharacteristic
625C22#1681143111yescharacteristic
626C22#1681143111yescharacteristic
627C22#1681143411yescharacteristic
628C4—22no—
629C4—22no—
630C4—41no—
631C4—41no—
632C4—41no—
633C4—81no—
634C4—81no—
635C4—81no—
636C4—81no—
637C4—4241no—
638C4—4241no—
639C4—2122no—
640C4#1681127511yescharacteristic
641C4#1681128311yescharacteristic
642C4#1681130711yescharacteristic
643C4#1681134411yescharacteristic
644C2—81no—
645C2—4242no—
646C2#2027814511yescharacteristic
647C2#2028435611yescharacteristic
648C2#2028439111yescharacteristic
649C2#2028489311yescharacteristic
650C2#2028498611yescharacteristic
651C2#2028596811yescharacteristic
652C2#2028601311yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2628C2
socleC23×C53C22×D4
prime radicalC2×#13133191C22
hypercenter#2628D53
hypocenterC53#24716
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2628

Complements

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

NH ≅ G/Ntype
1. C2×#16810910C2semidirect
2. C2×#16810684C2semidirect
4. #20276152C2semidirect
5. #20276151C2semidirect
8. #20275668C2semidirect
9. #20275481C2semidirect
10. #20275450C2semidirect
11. #20275440C2semidirect
12. #20275411C2semidirect
13. #20275397C2semidirect
14. #20274722C2semidirect
15. C53×#2628C2semidirect
87. C4⋊#88322D4semidirect
320. #2628D53semidirect
387. C53#24716semidirect

Maximal subgroups

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

subgroupindex
C2×#168109102
C2×#168106842
#202774082
#202761522
#202761512
#202759042
#202758972
#202756682
#202754812
#202754502
#202754402
#202754112
#202753972
#202747222
C53×#26282
#2471653