Finite Group Databasebeta

← #27446974

Subgroups — #27446974

Order 13568.

860 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810921C211yesmaximal, characteristic, complemented
2C2×#16810665C211yesmaximal, characteristic, complemented
3#20278440C211yesmaximal, characteristic, complemented
4#20277757C211yesmaximal, characteristic
5#20276238C211yesmaximal, characteristic, complemented
6#20276215C211yesmaximal, characteristic, complemented
7#20276087C211yesmaximal, characteristic, complemented
8#20276052C211yesmaximal, characteristic, complemented
9#20275486C211yesmaximal, characteristic, complemented
10#20275285C211yesmaximal, characteristic, complemented
11#20275033C211yesmaximal, characteristic
12#20274964C211yesmaximal, characteristic, complemented
13#20274912C211yesmaximal, characteristic, complemented
14#20274898C211yesmaximal, characteristic, complemented
15C53×#1895C211yesmaximal, characteristic
16C2×#13133208C2211yescharacteristic
17C2×D4×Dic53C2211yescharacteristic
18C2×#13133176C2211yescharacteristic
19C22×C4⋊Dic53C2211yescharacteristic
20#16810921—24no—
21#16810665—24no—
22C2×#13133151C2211yescharacteristic
23C2×#13133151C2211yescharacteristic
24C2×#13133150C2211yescharacteristic
25C2×#13133149C2211yescharacteristic
26C2×#13133149C2211yescharacteristic
27C2×#13133148C2211yescharacteristic
28C2×#13133146C2211yescharacteristic
29C2×#13133146C2211yescharacteristic
30C2×#13133138C2211yescharacteristic
31C2×#13133138C2211yescharacteristic
32C2×C22⋊#88322C2211yescharacteristic
33#16810565—21no—
34#16810555—21no—
35#16810526—21no—
36#16810526C2211yescharacteristic
37#16810522C2211yescharacteristic
38#16810522C2211yescharacteristic
39#16810519C2211yescharacteristic
40#16810516—21no—
41#16810503—21no—
42#16810498—21no—
43#16810488C2211yescharacteristic
44#16810481—21no—
45C4⋊C4×Dic53C2211yescharacteristic
46C22⋊C4×Dic53—21no—
47C4×C22⋊Dic53C2211yescharacteristic
48#16810385C2211yescharacteristic
49#16810385C2211yescharacteristic
50#16810385C2211yescharacteristic
51#16810385C2211yescharacteristic
52#16810377C2211yescharacteristic
53#16810377C2211yescharacteristic
54C22×M16×C53C2211yescharacteristic
55C2×#121×C53C2211yescharacteristic
56C2×#102×C53C2211yescharacteristic
57C2×#102×C53C2211yescharacteristic
58C53×#380C2211yescharacteristic
59C53×#340C2211yescharacteristic
60C53×#340C2211yescharacteristic
61C23×Dic53—21no—
62C23×Dic53—21no—
63C22×#88322C2311yescharacteristic
64C22×#88322C2311yescharacteristic
65#13133208—22no—
66#13133208—41no—
67D4×Dic53—22no—
68D4×Dic53—41no—
69#13133176—22no—
70#13133176—41no—
71C2×C4⋊Dic53—21no—
72C2×C4⋊Dic53—21no—
73C2×C4⋊Dic53—21no—
74C2×C4⋊Dic53—21no—
75C2×C4⋊Dic53—22no—
76C2×C4⋊Dic53—41no—
77C2×C4⋊Dic53C2311yescharacteristic
78C2×C4⋊Dic53C2311yescharacteristic
79C2×C4⋊Dic53C2311yescharacteristic
80C2×C4⋊Dic53C2311yescharacteristic
81C2×#11719919—21no—
82C2×#11719919—21no—
83C2×C22⋊Dic53—21no—
84C2×C22⋊Dic53—21no—
85C2×C22⋊Dic53—21no—
86C2×C22⋊Dic53—21no—
87C2×C22⋊Dic53D411yescharacteristic
88C2×C22⋊Dic53D411yescharacteristic
89C2×C4×Dic53—21no—
90C2×C4×Dic53—21no—
91C2×C4×Dic53C2311yescharacteristic
92C2×C4×Dic53C2311yescharacteristic
93#13133151—22no—
94#13133151D414yescomplemented
95#13133150—41no—
96#13133149—22no—
97#13133149—22no—
98#13133148—22no—
99#13133146—22no—
100#13133146—22no—
101#13133138—22no—
102#13133138—41no—
103C22⋊#88322—22no—
104#13133124—21no—
105#13133124—21no—
106#13133124—21no—
107#13133124—21no—
108#13133124—21no—
109#13133124—21no—
110#13133124D411yescharacteristic
111#13133124D411yescharacteristic
112C22×D4×C53—21no—
113C22×D4×C53C2311yescharacteristic
114C23×C4×C53C2311yescharacteristic
115C2×M16×C53—22no—
116C2×M16×C53—22no—
117C2×M16×C53—22no—
118C22×C8×C53C2311yescharacteristic
119C22×C8×C53C2311yescharacteristic
120#121×C53—22no—
121#121×C53Q814yes—
122C2×C4⋊C4×C53C2311yescharacteristic
123C2×C4⋊C4×C53C2311yescharacteristic
124C2×C4⋊C4×C53C2311yescharacteristic
125C2×C22⋊C4×C53—21no—
126C2×C22⋊C4×C53—21no—
127#102×C53—22no—
128#102×C53—22no—
129#95×C53—21no—
130C24×C53—21no—
131C24×C53C2×D411yescharacteristic
132C2×D4×C53—24no—
133C2×D4×C53—21no—
134C2×D4×C53—21no—
135C2×D4×C53—21no—
136C2×D4×C53—21no—
137C2×D4×C53—21no—
138C2×D4×C53—81no—
139C2×D4×C53—81no—
140C2×D4×C53C2×Q812yes—
141C22×C4×C53—21no—
142C22×C4×C53—21no—
143C22×C4×C53—21no—
144C22×C4×C53—21no—
145C22×C4×C53—21no—
146C22×C4×C53—21no—
147C22×C4×C53—21no—
148C22×C4×C53C2×D411yescharacteristic
149C22×C4×C53C2×Q812yes—
150C22×C4×C53#4112yes—
151C22×C4×C53C2411yescharacteristic
152C22×Dic53—21no—
153C22×Dic53—21no—
154C22×Dic53—21no—
155C22×Dic53—21no—
156C22×Dic53—42no—
157C22×Dic53—42no—
158C22×Dic53—42no—
159C22×Dic53—42no—
160C22×Dic53—81no—
161C22×Dic53—81no—
162C22×Dic53C2×D411yescharacteristic
163C22×Dic53C2×D411yescharacteristic
164C22×Dic53C2×D411yescharacteristic
165C22×Dic53C2×D411yescharacteristic
166M16×C53—28no—
167C2×C8×C53—21no—
168C2×C8×C53—21no—
169C2×C8×C53—21no—
170C2×C8×C53—21no—
171C2×C8×C53C2×D412yes—
172C2×C8×C53#4112yes—
173C4⋊C4×C53—21no—
174C4⋊C4×C53—21no—
175C4⋊C4×C53—21no—
176C4⋊C4×C53—21no—
177C4⋊C4×C53—21no—
178C4⋊C4×C53C2×Q812yes—
179C22⋊C4×C53—24no—
180C22⋊C4×C53—42no—
181C22⋊Dic53—41no—
182C22⋊Dic53—41no—
183C22⋊Dic53—42no—
184C22⋊Dic53—42no—
185C22⋊Dic53—42no—
186C22⋊Dic53—81no—
187C4⋊Dic53—21no—
188C4⋊Dic53—21no—
189C4⋊Dic53—21no—
190C4⋊Dic53—21no—
191C4⋊Dic53—21no—
192C4⋊Dic53—21no—
193C4⋊Dic53—42no—
194C4⋊Dic53—42no—
195C4⋊Dic53—81no—
196C4⋊Dic53—81no—
197C4⋊Dic53C2×D412yes—
198C4⋊Dic53C2×D412yes—
199#11719919—42no—
200#11719919—81no—
201C4×Dic53—21no—
202C4×Dic53—21no—
203C4×Dic53—21no—
204C4×Dic53—21no—
205C4×Dic53—42no—
206C4×Dic53—81no—
207C2×#88322—21no—
208C2×#88322—21no—
209C2×#88322—21no—
210C2×#88322—21no—
211C2×#88322—21no—
212C2×#88322—21no—
213C23×C53—21no—
214C23×C53—21no—
215C23×C53—21no—
216C23×C53—21no—
217C23×C53—21no—
218C23×C53—21no—
219C23×C53—22no—
220C23×C53—41no—
221C23×C53—41no—
222C23×C53—41no—
223C23×C53—41no—
224C23×C53—41no—
225C23×C53—41no—
226C23×C53C22⋊Q812yes—
227C23×C53C22×D411yescharacteristic
228D4×C53—22no—
229D4×C53—22no—
230D4×C53—41no—
231D4×C53—41no—
232D4×C53—84no—
233C2×C4×C53—21no—
234C2×C4×C53—21no—
235C2×C4×C53—22no—
236C2×C4×C53—22no—
237C2×C4×C53—22no—
238C2×C4×C53—22no—
239C2×C4×C53—22no—
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×C53C22⋊Q812yes—
249C2×C4×C53C22×D411yescharacteristic
250C2×C4×C53C22×Q811yescharacteristic
251C2×C4×C53C2×#4111yescharacteristic
252C2×C4×C53C2×#4111yescharacteristic
253C2×C4×C53H(5,𝔽2)11yescharacteristic
254C2×C4×C532−1+411yescharacteristic
255C2×Dic53—22no—
256C2×Dic53—22no—
257C2×Dic53—21no—
258C2×Dic53—21no—
259C2×Dic53—21no—
260C2×Dic53—21no—
261C2×Dic53—21no—
262C2×Dic53—21no—
263C2×Dic53—21no—
264C2×Dic53—21no—
265C2×Dic53—21no—
266C2×Dic53—21no—
267C2×Dic53—21no—
268C2×Dic53—21no—
269C2×Dic53—41no—
270C2×Dic53—41no—
271C2×Dic53—41no—
272C2×Dic53—41no—
273C2×Dic53—41no—
274C2×Dic53—41no—
275C2×Dic53—41no—
276C2×Dic53—41no—
277C2×Dic53—41no—
278C2×Dic53—41no—
279C2×Dic53—84no—
280C2×Dic53—84no—
281C8×C53—22no—
282C8×C53—22no—
283#88322—41no—
284#88322—41no—
285#26363—531nomaximal, Sylow
286C22×C53—22no—
287C22×C53—22no—
288C22×C53—22no—
289C22×C53—41no—
290C22×C53—41no—
291C22×C53—41no—
292C22×C53—41no—
293C22×C53—42no—
294C22×C53—42no—
295C22×C53—42no—
296C22×C53—81no—
297C22×C53—81no—
298C22×C53—81no—
299C22×C53—81no—
300C22×C53C2×C22⋊Q811yescharacteristic
301C22×C53D4211yescharacteristic
302C22×C53#54711yescharacteristic
303C22×C53#57511yescharacteristic
304C22×C53#57511yescharacteristic
305C22×C53#57611yescharacteristic
306C22×C53#57711yescharacteristic
307C4×C53—22no—
308C4×C53—42no—
309C4×C53—81no—
310C4×C53—81no—
311C4×C53#52511yescharacteristic
312C4×C53#53911yescharacteristic
313C4×C53D4×Q811yescharacteristic
314C4×C53#55411yescharacteristic
315Dic53—41no—
316Dic53—41no—
317Dic53—41no—
318Dic53—41no—
319Dic53—42no—
320Dic53—42no—
321Dic53—81no—
322Dic53—81no—
323C2×#478—531no—
324C2×#428—531no—
325#2442—531no—
326#2363—531no—
327#2092—531no—
328#2088—531no—
329#2065—531no—
330#2059—531no—
331#1949—531no—
332#1918—531no—
333#1895D5311yescharacteristic
334#1874—531no—
335#1862—531no—
336#1854—531no—
337#1851—531no—
338C2×C53—42no—
339C2×C53—82no—
340C2×C53#244911yescharacteristic
341C2×C53#308011yescharacteristic
342C2×C53#308111yescharacteristic
343C2×C53#327111yescharacteristic
344C2×C53#327311yescharacteristic
345C2×C53#334811yescharacteristic
346C2×C53#335511yescharacteristic
347C22×M16C2×D5311yescharacteristic
348C2×#125—531no—
349C2×#121C2×D5311yescharacteristic
350C2×C4×D4—531no—
351C2×#117—531no—
352C22×C4⋊C4—531no—
353#478—1064no—
354#428—1064no—
355C2×#107—531no—
356C2×#107—531no—
357C2×#107—531no—
358C2×#106—531no—
359C2×#106—531no—
360C2×#106—531no—
361C2×C4⋊C8—531no—
362C2×C4⋊C8—531no—
363C2×#102—531no—
364C2×#102—531no—
365C2×#102C2×D5311yescharacteristic
366C2×#102C2×D5311yescharacteristic
367C2×C22⋊C8—531no—
368#400—1061no—
369#398—1061no—
370#390—531no—
371#390—1061no—
372#389—531no—
373#389—531no—
374#389—531no—
375#388—1061no—
376#386—1061no—
377#385—1061no—
378#382—531no—
379#380—1061no—
380#380C2×D5311yescharacteristic
381C4×C4⋊C4—531no—
382C4×C22⋊C4—531no—
383C4×C22⋊C4—1061no—
384#340—531no—
385#340—531no—
386#340—531no—
387#340—531no—
388#340C2×D5311yescharacteristic
389#340C2×D5311yescharacteristic
390#336—531no—
391#336—531no—
392C53#2636311yescharacteristic, complemented, Sylow
393C22×D4—21no—
394C22×D4C22×D5311yescharacteristic
395C23×C4—1061no—
396C23×C4—1061no—
397C23×C4C22×D5311yescharacteristic
398C2×M16—22no—
399C2×M16—22no—
400C2×M16—22no—
401C22×C8—531no—
402C22×C8—531no—
403C22×C8C22×D5311yescharacteristic
404C22×C8C22×D5311yescharacteristic
405#125—1062no—
406#125—2121no—
407#121—22no—
408#121Dic10614yes—
409C4×D4—1062no—
410C4×D4—2121no—
411#117—1062no—
412#117—2121no—
413C2×C4⋊C4—531no—
414C2×C4⋊C4—531no—
415C2×C4⋊C4—531no—
416C2×C4⋊C4—531no—
417C2×C4⋊C4—1062no—
418C2×C4⋊C4—1061no—
419C2×C4⋊C4—1061no—
420C2×C4⋊C4—1061no—
421C2×C4⋊C4—1061no—
422C2×C4⋊C4—1061no—
423C2×C4⋊C4—1061no—
424C2×C4⋊C4—2121no—
425C2×C4⋊C4C22×D5311yescharacteristic
426C2×C4⋊C4C22×D5311yescharacteristic
427C2×C4⋊C4C22×D5311yescharacteristic
428C2×C22⋊C4—21no—
429C2×C22⋊C4—21no—
430C2×C22⋊C4—531no—
431C2×C22⋊C4—531no—
432C2×C22⋊C4—1061no—
433C2×C22⋊C4—1061no—
434C2×C22⋊C4—1061no—
435C2×C22⋊C4—1061no—
436C2×C42—531no—
437C2×C42—531no—
438C2×C42—1061no—
439C2×C42—1061no—
440#107—534no—
441#107—1062no—
442#107—2121no—
443#106—1062no—
444#106—1062no—
445#106—1062no—
446C4⋊C8—1062no—
447C4⋊C8—1062no—
448#102—22no—
449#102—22no—
450#102—1062no—
451#102—2121no—
452C22⋊C8—1062no—
453#95—21no—
454#95—531no—
455#95—531no—
456#95—1061no—
457#95—1061no—
458#95—1061no—
459#95—1061no—
460#95—1061no—
461#95—1061no—
462C24—21no—
463C24D4×D5311yescharacteristic
464C2×D4—24no—
465C2×D4—21no—
466C2×D4—21no—
467C2×D4—21no—
468C2×D4—21no—
469C2×D4—21no—
470C2×D4—81no—
471C2×D4—81no—
472C2×D4C2×Dic10612yes—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—531no—
481C22×C4—531no—
482C22×C4—531no—
483C22×C4—531no—
484C22×C4—1061no—
485C22×C4—1061no—
486C22×C4—1061no—
487C22×C4—1061no—
488C22×C4—4241no—
489C22×C4—4241no—
490C22×C4—2122no—
491C22×C4—2122no—
492C22×C4—2122no—
493C22×C4—2122no—
494C22×C4C2×Dic10612yes—
495C22×C4D4×D5311yescharacteristic
496C22×C4#1171994712yes—
497C22×C4C23×D5311yescharacteristic
498M16—28no—
499C2×C8—21no—
500C2×C8—21no—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8—1061no—
504C2×C8—1061no—
505C2×C8—1061no—
506C2×C8—1061no—
507C2×C8—1061no—
508C2×C8—1061no—
509C2×C8D4×D5312yes—
510C2×C8#1171994712yes—
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—532no—
517C4⋊C4—532no—
518C4⋊C4—1061no—
519C4⋊C4—1061no—
520C4⋊C4—1061no—
521C4⋊C4—1061no—
522C4⋊C4—1061no—
523C4⋊C4—1061no—
524C4⋊C4—4241no—
525C4⋊C4—4241no—
526C4⋊C4—4241no—
527C4⋊C4—2122no—
528C4⋊C4—2122no—
529C4⋊C4—2122no—
530C4⋊C4C2×Dic10612yes—
531C22⋊C4—24no—
532C22⋊C4—42no—
533C22⋊C4—4241no—
534C22⋊C4—2121no—
535C22⋊C4—2121no—
536C22⋊C4—2122no—
537C22⋊C4—2122no—
538C22⋊C4—2122no—
539C42—1061no—
540C42—1061no—
541C42—1061no—
542C42—1061no—
543C42—4241no—
544C42—2122no—
545C23—21no—
546C23—21no—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23—22no—
552C23—41no—
553C23—41no—
554C23—41no—
555C23—41no—
556C23—41no—
557C23—41no—
558C23#1313319412yes—
559C23C2×D4×D5311yescharacteristic
560D4—22no—
561D4—22no—
562D4—41no—
563D4—41no—
564D4—84no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
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—1061no—
581C2×C4—1061no—
582C2×C4—1061no—
583C2×C4—1061no—
584C2×C4—1061no—
585C2×C4—1061no—
586C2×C4—1061no—
587C2×C4—1061no—
588C2×C4—1061no—
589C2×C4—1061no—
590C2×C4—1061no—
591C2×C4—1061no—
592C2×C4—1062no—
593C2×C4—1062no—
594C2×C4—4244no—
595C2×C4—4244no—
596C2×C4—2121no—
597C2×C4—2121no—
598C2×C4—2121no—
599C2×C4—2121no—
600C2×C4—2121no—
601C2×C4—2121no—
602C2×C4—2121no—
603C2×C4—2121no—
604C2×C4—2121no—
605C2×C4—2121no—
606C2×C4#1313319412yes—
607C2×C4C2×D4×D5311yescharacteristic
608C2×C4C22×Dic10611yescharacteristic
609C2×C4C2×#1171994711yescharacteristic
610C2×C4C2×#1171994711yescharacteristic
611C2×C4#1313326311yescharacteristic
612C2×C4#1313326411yescharacteristic
613C8—22no—
614C8—22no—
615C8—2121no—
616C8—2121no—
617C22—22no—
618C22—22no—
619C22—22no—
620C22—41no—
621C22—41no—
622C22—41no—
623C22—41no—
624C22—42no—
625C22—42no—
626C22—42no—
627C22—81no—
628C22—81no—
629C22—81no—
630C22—81no—
631C22C2×#1313319411yescharacteristic
632C22#1681125711yescharacteristic
633C22#1681127311yescharacteristic
634C22#1681141911yescharacteristic
635C22#1681142211yescharacteristic
636C22#1681142711yescharacteristic
637C22#1681143311yescharacteristic
638C4—22no—
639C4—42no—
640C4—81no—
641C4—81no—
642C4—4241no—
643C4—4241no—
644C4—2121no—
645C4—2121no—
646C4—2121no—
647C4—2122no—
648C4—2122no—
649C4—2121no—
650C4#1681112911yescharacteristic
651C4#1681121711yescharacteristic
652C4D4×Dic10611yescharacteristic
653C4#1681133011yescharacteristic
654C2—42no—
655C2—82no—
656C2#2027853011yescharacteristic
657C2#2028453511yescharacteristic
658C2#2028454811yescharacteristic
659C2#2028630711yescharacteristic
660C2#2028632311yescharacteristic
661C2#2028722511yescharacteristic
662C2#2028731611yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#1895C2
socleC23×C53C22×D4
prime radicalC2×#121×C53C22
hypercenter#1895D53
hypocenterC53#26363
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1895

Complements

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

NH ≅ G/Ntype
1. C2×#16810921C2semidirect
2. C2×#16810665C2semidirect
3. #20278440C2semidirect
5. #20276238C2semidirect
6. #20276215C2semidirect
7. #20276087C2semidirect
8. #20276052C2semidirect
9. #20275486C2semidirect
10. #20275285C2semidirect
12. #20274964C2semidirect
13. #20274912C2semidirect
14. #20274898C2semidirect
94. #13133151D4semidirect
392. C53#26363semidirect

Maximal subgroups

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

subgroupindex
C2×#168109212
C2×#168106652
#202784402
#202777572
#202762382
#202762152
#202760872
#202760522
#202754862
#202752852
#202750332
#202749642
#202749122
#202748982
C53×#18952
#2636353