Finite Group Databasebeta

← #27466313

Subgroups — #27466313

Order 13568.

752 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20279173C211yesmaximal, characteristic
2#20278400C211yesmaximal, characteristic, complemented
3#20276286C211yesmaximal, characteristic, complemented
4#20276280C211yesmaximal, characteristic, complemented
5#20276132C211yesmaximal, characteristic, complemented
6#20275850C211yesmaximal, characteristic, complemented
7#20275674C211yesmaximal, characteristic, complemented
8#20275399C211yesmaximal, characteristic, complemented
9#20275396C211yesmaximal, characteristic, complemented
10#20275143C211yesmaximal, characteristic
11#20275037C211yesmaximal, characteristic
12#20274844C211yesmaximal, characteristic, complemented
13#20274810C211yesmaximal, characteristic, complemented
14C8×C22⋊Dic53C211yesmaximal, characteristic, complemented
15C53×#1735C211yesmaximal, characteristic, complemented
16C2×#13133216C2211yescharacteristic
17C2×#13133191C2211yescharacteristic
18C2×#13133139C2211yescharacteristic
19C2×#13133139C2211yescharacteristic
20C2×#13133137C2211yescharacteristic
21C2×#13133137C2211yescharacteristic
22C2×C22⋊#88322C2211yescharacteristic
23C2×C8×Dic53C2211yescharacteristic
24C2×C8×Dic53C2211yescharacteristic
25#16810546—21no—
26#16810538—21no—
27#16810538—21no—
28#16810535—21no—
29#16810529—21no—
30#16810524C2211yescharacteristic
31#16810524C2211yescharacteristic
32#16810522C2211yescharacteristic
33#16810519C2211yescharacteristic
34#16810514—21no—
35#16810514—21no—
36#16810507—21no—
37#16810492C2211yescharacteristic
38#16810488C2211yescharacteristic
39#16810488C2211yescharacteristic
40#16810483C2211yescharacteristic
41C4×C22⋊Dic53C2211yescharacteristic
42#16810386C2211yescharacteristic
43#16810386C2211yescharacteristic
44#16810385C2211yescharacteristic
45#16810385C2211yescharacteristic
46#16810385C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810385C2211yescharacteristic
49#16810385C2211yescharacteristic
50#16810377C2211yescharacteristic
51#16810377C2211yescharacteristic
52C23×C8×C53C2211yescharacteristic
53C2×#117×C53C2211yescharacteristic
54C2×#117×C53C2211yescharacteristic
55C53×#340C2211yescharacteristic
56C53×#340C2211yescharacteristic
57C53×#340C2211yescharacteristic
58C53×#340C2211yescharacteristic
59C22×Dic106—21no—
60C22×D212C2311yescharacteristic
61C22×C22⋊D53—21no—
62C22×#88322C2311yescharacteristic
63C22×#88322C2311yescharacteristic
64#13133216—22no—
65#13133216—41no—
66#13133191—22no—
67#13133191—22no—
68C2×C4⋊Dic53—21no—
69C2×C4⋊Dic53C2311yescharacteristic
70C2×C4⋊Dic53C2311yescharacteristic
71C2×C4⋊Dic53C2311yescharacteristic
72C2×#11719919—21no—
73C2×#11719919—21no—
74C2×#11719919—21no—
75C2×#11719919—21no—
76C2×C22⋊Dic53—21no—
77C2×C22⋊Dic53D411yescharacteristic
78C2×C22⋊Dic53D411yescharacteristic
79C2×#11719921—21no—
80C2×#11719921—21no—
81C2×#11719921—21no—
82C2×#11719921—21no—
83C2×#11719921—21no—
84C2×C4×Dic53C2311yescharacteristic
85C2×C4×Dic53C2311yescharacteristic
86#13133139—22no—
87#13133139—41no—
88#13133137—22no—
89#13133137—41no—
90C22⋊#88322—22no—
91C8×Dic53—22no—
92C8×Dic53—41no—
93#13133124—21no—
94#13133124—21no—
95#13133124—21no—
96#13133124—21no—
97#13133124—21no—
98#13133124—21no—
99#13133124—21no—
100#13133124D411yescharacteristic
101#13133124D411yescharacteristic
102C23×C4×C53C2311yescharacteristic
103C22×C8×C53—22no—
104C22×C8×C53—22no—
105C22×C8×C53—41no—
106C22×C8×C53C2311yescharacteristic
107C22×C8×C53C2311yescharacteristic
108#117×C53—22no—
109#117×C53—22no—
110#117×C53—22no—
111#117×C53—41no—
112C2×C4⋊C4×C53C2311yescharacteristic
113C2×C4⋊C4×C53C2311yescharacteristic
114C2×C4⋊C4×C53C2311yescharacteristic
115C2×C4⋊C4×C53C2311yescharacteristic
116C2×C22⋊C4×C53—21no—
117C2×C22⋊C4×C53—21no—
118C2×C42×C53—21no—
119C2×C42×C53—21no—
120C24×C53C2×D411yescharacteristic
121C23×D53—21no—
122C22×C4×C53—21no—
123C22×C4×C53—21no—
124C22×C4×C53—21no—
125C22×C4×C53—21no—
126C22×C4×C53—21no—
127C22×C4×C53—21no—
128C22×C4×C53—21no—
129C22×C4×C53—21no—
130C22×C4×C53C2×D411yescharacteristic
131C22×C4×C53#4112yes—
132C22×C4×C53#4112yes—
133C22×C4×C53C2411yescharacteristic
134C2×C22⋊D53—42no—
135C2×C22⋊D53—81no—
136C2×C22⋊D53—81no—
137C22×Dic53—21no—
138C22×Dic53—21no—
139C22×Dic53—21no—
140C22×Dic53C2×D411yescharacteristic
141C22×Dic53C2×D411yescharacteristic
142C22×Dic53C2×D411yescharacteristic
143C22×Dic53C2×D411yescharacteristic
144C2×D212—21no—
145C2×D212—21no—
146C2×D212—21no—
147C2×D212—21no—
148C2×D212—21no—
149C2×D212—21no—
150C2×Dic106—42no—
151C2×Dic106—42no—
152C2×Dic106—81no—
153C2×C8×C53—21no—
154C2×C8×C53—21no—
155C2×C8×C53—21no—
156C2×C8×C53—21no—
157C2×C8×C53—21no—
158C2×C8×C53—44no—
159C2×C8×C53#4112yes—
160C4⋊C4×C53—21no—
161C4⋊C4×C53—21no—
162C4⋊C4×C53—21no—
163C4⋊C4×C53—21no—
164C4⋊C4×C53—21no—
165C4⋊C4×C53—21no—
166C4⋊C4×C53—21no—
167C4⋊C4×C53#4112yes—
168C22⋊C4×C53—42no—
169C22⋊C4×C53—42no—
170C42×C53—42no—
171C42×C53—42no—
172C22⋊Dic53—22no—
173C22⋊Dic53—22no—
174C22⋊Dic53—81no—
175#11719921—42no—
176#11719921—42no—
177#11719921—42no—
178#11719921—81no—
179#11719921—81no—
180C4⋊Dic53—21no—
181C4⋊Dic53—21no—
182C4⋊Dic53—21no—
183C4⋊Dic53—21no—
184C4⋊Dic53—21no—
185C4⋊Dic53—21no—
186C4⋊Dic53—81no—
187#11719919—42no—
188#11719919—42no—
189#11719919—81no—
190#11719919—81no—
191C4×Dic53—21no—
192C4×Dic53—21no—
193C4×Dic53—21no—
194C4×Dic53—21no—
195C2×#88322—21no—
196C2×#88322—21no—
197C2×#88322—21no—
198C2×#88322—21no—
199C2×#88322—21no—
200C2×#88322—21no—
201C23×C53—21no—
202C23×C53—21no—
203C23×C53—21no—
204C23×C53—21no—
205C23×C53—21no—
206C23×C53—21no—
207C23×C53#12412yes—
208C23×C53C22×D411yescharacteristic
209C22×D53—41no—
210C22×D53—41no—
211C22×D53—41no—
212C22×D53—41no—
213C22×D53—41no—
214C22×D53—41no—
215C22×D53—41no—
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—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—41no—
233C2×C4×C53—41no—
234C2×C4×C53#12412yes—
235C2×C4×C53C2×#4111yescharacteristic
236C2×C4×C53C2×#4111yescharacteristic
237C2×C4×C53C2×#4111yescharacteristic
238C2×C4×C53C2×#4111yescharacteristic
239C2×C4×C53H(5,𝔽2)11yescharacteristic
240C2×C4×C532−1+411yescharacteristic
241C22⋊D53—84no—
242C2×Dic53—21no—
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—21no—
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#13512yes—
262C2×Dic53#13512yes—
263D212—41no—
264D212—41no—
265D212—41no—
266D212—41no—
267Dic106—84no—
268C8×C53—22no—
269C8×C53—41no—
270#88322—41no—
271#88322—41no—
272#27696—531nomaximal, Sylow
273C22×C53—22no—
274C22×C53—22no—
275C22×C53—22no—
276C22×C53—41no—
277C22×C53—41no—
278C22×C53—41no—
279C22×C53—41no—
280C22×C53C2×#12411yescharacteristic
281C22×C53#54611yescharacteristic
282C22×C53#54711yescharacteristic
283C22×C53C2×#13511yescharacteristic
284C22×C53C2×#13511yescharacteristic
285C22×C53#57611yescharacteristic
286C22×C53#57711yescharacteristic
287C2×D53—81no—
288C2×D53—81no—
289C2×D53—81no—
290C2×D53—81no—
291C2×D53—81no—
292C2×D53—81no—
293C2×D53—81no—
294C4×C53—22no—
295C4×C53—42no—
296C4×C53—81no—
297C4×C53—81no—
298C4×C53—81no—
299C4×C53#52911yescharacteristic
300C4×C53#53911yescharacteristic
301C4×C53#55111yescharacteristic
302C4×C53#55211yescharacteristic
303Dic53—22no—
304Dic53—22no—
305Dic53—41no—
306Dic53—41no—
307Dic53—81no—
308Dic53—81no—
309Dic53—81no—
310#2500—531no—
311#2439—531no—
312#2101—531no—
313#2100—531no—
314#2073—531no—
315#2023—531no—
316#1990—531no—
317#1934—531no—
318#1934—531no—
319#1893—531no—
320#1875—531no—
321#1835—531no—
322#1825—531no—
323C8×C22⋊C4—531no—
324#1735D5311yescharacteristic, complemented
325C2×C53—42no—
326C2×C53#248811yescharacteristic
327C2×C53#313111yescharacteristic
328C2×C53#313611yescharacteristic
329C2×C53#329711yescharacteristic
330C2×C53#330111yescharacteristic
331C2×C53#334011yescharacteristic
332C2×C53#334111yescharacteristic
333D53—161no—
334C23×C8C2×D5311yescharacteristic
335C2×C4⋊Q8—531no—
336C2×#121—531no—
337C2×#117C2×D5311yescharacteristic
338C2×#117C2×D5311yescharacteristic
339C2×#102—531no—
340C2×#102—531no—
341C2×#102—531no—
342C2×#102—531no—
343C2×C22⋊C8—531no—
344C2×C4×C8—531no—
345C2×C4×C8—531no—
346#396—1061no—
347#394—1061no—
348#394—1061no—
349#393—1061no—
350#391—1061no—
351#390—531no—
352#390—531no—
353#389—531no—
354#389—531no—
355#388—1061no—
356#388—1061no—
357#387—1061no—
358#384—531no—
359#382—531no—
360#382—531no—
361#380—531no—
362C4×C22⋊C4—531no—
363#340—531no—
364#340—531no—
365#340—531no—
366#340—531no—
367#340—531no—
368#340—531no—
369#340—531no—
370#340—531no—
371#340C2×D5311yescharacteristic
372#340C2×D5311yescharacteristic
373#340C2×D5311yescharacteristic
374#340C2×D5311yescharacteristic
375#336—531no—
376#336—531no—
377C53#2769611yescharacteristic, complemented, Sylow
378C22×Q8—1061no—
379C22×D4—531no—
380C22×D4—1061no—
381C23×C4C22×D5311yescharacteristic
382C22×C8—22no—
383C22×C8—22no—
384C22×C8—41no—
385C22×C8—531no—
386C22×C8—531no—
387C22×C8C22×D5311yescharacteristic
388C22×C8C22×D5311yescharacteristic
389C4⋊Q8—1062no—
390C4⋊Q8—2121no—
391#121—1062no—
392#121—1062no—
393#117—22no—
394#117—22no—
395#117—22no—
396#117—41no—
397C2×C4⋊C4—531no—
398C2×C4⋊C4—531no—
399C2×C4⋊C4—531no—
400C2×C4⋊C4—1061no—
401C2×C4⋊C4—1061no—
402C2×C4⋊C4—1061no—
403C2×C4⋊C4—1061no—
404C2×C4⋊C4—1061no—
405C2×C4⋊C4C22×D5311yescharacteristic
406C2×C4⋊C4C22×D5311yescharacteristic
407C2×C4⋊C4C22×D5311yescharacteristic
408C2×C4⋊C4C22×D5311yescharacteristic
409C2×C22⋊C4—21no—
410C2×C22⋊C4—21no—
411C2×C22⋊C4—531no—
412C2×C22⋊C4—531no—
413C2×C22⋊C4—1061no—
414C2×C22⋊C4—1061no—
415C2×C22⋊C4—1061no—
416C2×C22⋊C4—1061no—
417C2×C22⋊C4—1061no—
418C2×C22⋊C4—1061no—
419C2×C42—21no—
420C2×C42—21no—
421C2×C42—531no—
422C2×C42—531no—
423#102—1062no—
424#102—1062no—
425#102—2121no—
426#102—2121no—
427C22⋊C8—1062no—
428C4×C8—1062no—
429C4×C8—2121no—
430#95—531no—
431#95—531no—
432#95—1061no—
433#95—1061no—
434#95—1061no—
435#95—1061no—
436#95—1061no—
437#95—1061no—
438#95—1061no—
439C24—1061no—
440C24D4×D5311yescharacteristic
441C2×Q8—4241no—
442C2×Q8—2122no—
443C2×Q8—2122no—
444C2×D4—1061no—
445C2×D4—1061no—
446C2×D4—1061no—
447C2×D4—1061no—
448C2×D4—1061no—
449C2×D4—1061no—
450C2×D4—4241no—
451C2×D4—4241no—
452C2×D4—2122no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—531no—
462C22×C4—531no—
463C22×C4—531no—
464C22×C4—531no—
465C22×C4—1061no—
466C22×C4—1061no—
467C22×C4—1061no—
468C22×C4#1171994512yes—
469C22×C4D4×D5311yescharacteristic
470C22×C4#1171994712yes—
471C22×C4C23×D5311yescharacteristic
472C2×C8—21no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—44no—
478C2×C8—1061no—
479C2×C8—1061no—
480C2×C8—1061no—
481C2×C8—1061no—
482C2×C8—1061no—
483C2×C8—1061no—
484C2×C8#1171994512yes—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—1061no—
493C4⋊C4—1061no—
494C4⋊C4—1061no—
495C4⋊C4—1061no—
496C4⋊C4—1061no—
497C4⋊C4—1061no—
498C4⋊C4—4241no—
499C4⋊C4—4241no—
500C4⋊C4—4241no—
501C4⋊C4—2122no—
502C4⋊C4—2122no—
503C4⋊C4#1171994512yes—
504C22⋊C4—42no—
505C22⋊C4—42no—
506C22⋊C4—1062no—
507C22⋊C4—1062no—
508C22⋊C4—4241no—
509C22⋊C4—4241no—
510C22⋊C4—4241no—
511C22⋊C4—2122no—
512C22⋊C4—2122no—
513C22⋊C4—2122no—
514C42—42no—
515C42—42no—
516C42—1061no—
517C42—1061no—
518C42—1061no—
519C42—1061no—
520C23—21no—
521C23—21no—
522C23—21no—
523C23—21no—
524C23—21no—
525C23—21no—
526C23—2121no—
527C23—2121no—
528C23—2121no—
529C23—2121no—
530C23—2121no—
531C23—2121no—
532C23—2121no—
533C23#1313320312yes—
534C23C2×D4×D5311yescharacteristic
535Q8—4244no—
536D4—4244no—
537D4—2121no—
538D4—2121no—
539D4—2121no—
540D4—2121no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—532no—
560C2×C4—532no—
561C2×C4—1061no—
562C2×C4—1061no—
563C2×C4—1061no—
564C2×C4—1061no—
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#1313320312yes—
581C2×C4C2×#1171994711yescharacteristic
582C2×C4C2×#1171994511yescharacteristic
583C2×C4C2×#1171994511yescharacteristic
584C2×C4C2×#1171994511yescharacteristic
585C2×C4#1313326211yescharacteristic
586C2×C4#1313326511yescharacteristic
587C8—22no—
588C8—41no—
589C8—2121no—
590C8—2121no—
591C22—22no—
592C22—22no—
593C22—22no—
594C22—41no—
595C22—41no—
596C22—41no—
597C22—41no—
598C22—4241no—
599C22—4241no—
600C22—4241no—
601C22—4241no—
602C22—4241no—
603C22—4241no—
604C22—4241no—
605C22C2×#1313320311yescharacteristic
606C22#1681126311yescharacteristic
607C22#1681127411yescharacteristic
608C22#135×D5311yescharacteristic
609C22#135×D5311yescharacteristic
610C22#1681142511yescharacteristic
611C22#1681143011yescharacteristic
612C4—22no—
613C4—42no—
614C4—81no—
615C4—81no—
616C4—81no—
617C4—1062no—
618C4—1062no—
619C4—4241no—
620C4—4241no—
621C4—4241no—
622C4—2121no—
623C4—2121no—
624C4#1681115711yescharacteristic
625C4#1681122011yescharacteristic
626C4#1681129911yescharacteristic
627C4#1681130811yescharacteristic
628C2—42no—
629C2—8481no—
630C2#2027904111yescharacteristic
631C2#2028503511yescharacteristic
632C2#2028507311yescharacteristic
633C2#2028656611yescharacteristic
634C2#2028661811yescharacteristic
635C2#2028712411yescharacteristic
636C2#2028713511yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#1735C2
socleC23×C53C22×D4
prime radicalC2×#13133191C22
hypercenter#1735D53
hypocenterC53#27696
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1735

Complements

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

NH ≅ G/Ntype
2. #20278400C2semidirect
3. #20276286C2semidirect
4. #20276280C2semidirect
5. #20276132C2semidirect
6. #20275850C2semidirect
7. #20275674C2semidirect
8. #20275399C2semidirect
9. #20275396C2semidirect
12. #20274844C2semidirect
13. #20274810C2semidirect
14. C8×C22⋊Dic53C2semidirect
15. C53×#1735C2semidirect
324. #1735D53semidirect
377. C53#27696semidirect

Maximal subgroups

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

subgroupindex
#202791732
#202784002
#202762862
#202762802
#202761322
#202758502
#202756742
#202753992
#202753962
#202751432
#202750372
#202748442
#202748102
C8×C22⋊Dic532
C53×#17352
#2769653