Finite Group Databasebeta

← #27446692

Subgroups — #27446692

Order 13568.

858 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810923C211yesmaximal, characteristic, complemented
2C2×#16810659C211yesmaximal, characteristic, complemented
3#20278428C211yesmaximal, characteristic, complemented
4#20276248C211yesmaximal, characteristic, complemented
5#20276216C211yesmaximal, characteristic, complemented
6#20276103C211yesmaximal, characteristic, complemented
7#20276053C211yesmaximal, characteristic, complemented
8#20275473C211yesmaximal, characteristic, complemented
9#20275293C211yesmaximal, characteristic, complemented
10#20275144C211yesmaximal, characteristic
11#20275038C211yesmaximal, characteristic
12#20274962C211yesmaximal, characteristic, complemented
13#20274914C211yesmaximal, characteristic, complemented
14#20274895C211yesmaximal, characteristic, complemented
15C53×#2363C211yesmaximal, characteristic
16C23×#88322C2211yescharacteristic
17C2×#13133209C2211yescharacteristic
18C2×#13133176C2211yescharacteristic
19#16810923—24no—
20#16810659—24no—
21C2×#13133150C2211yescharacteristic
22C2×#13133150C2211yescharacteristic
23C2×#13133150C2211yescharacteristic
24C2×#13133148C2211yescharacteristic
25C2×#13133148C2211yescharacteristic
26C2×#13133148C2211yescharacteristic
27C2×C4⋊#88322C2211yescharacteristic
28C2×C4⋊#88322C2211yescharacteristic
29C2×#13133138C2211yescharacteristic
30C2×#13133138C2211yescharacteristic
31C2×#13133138C2211yescharacteristic
32C2×#13133138C2211yescharacteristic
33C2×C22⋊#88322C2211yescharacteristic
34#16810559—21no—
35#16810546—21no—
36#16810521C2211yescharacteristic
37#16810521C2211yescharacteristic
38#16810520—21no—
39#16810506—21no—
40#16810493C2211yescharacteristic
41#16810483C2211yescharacteristic
42#16810385C2211yescharacteristic
43#16810385C2211yescharacteristic
44#16810385C2211yescharacteristic
45#16810385C2211yescharacteristic
46#16810385C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810378C2211yescharacteristic
49#16810378C2211yescharacteristic
50C2×#121×C53C2211yescharacteristic
51C2×C4×D4×C53C2211yescharacteristic
52C2×#117×C53C2211yescharacteristic
53C53×#390—21no—
54C53×#390C2211yescharacteristic
55C53×#389C2211yescharacteristic
56C53×#388—21no—
57C53×#386—21no—
58C4×C4⋊C4×C53C2211yescharacteristic
59C4×C22⋊C4×C53—21no—
60C4×C22⋊C4×C53C2211yescharacteristic
61C22×#88322—22no—
62C22×#88322—22no—
63C22×#88322—22no—
64C22×#88322C2311yescharacteristic
65C22×#88322C2311yescharacteristic
66C22×#88322C2311yescharacteristic
67C22×#88322C2311yescharacteristic
68#13133209—22no—
69#13133209—41no—
70#13133176—22no—
71#13133176—41no—
72C2×C4⋊Dic53—21no—
73C2×C4⋊Dic53—21no—
74C2×C4⋊Dic53C2311yescharacteristic
75C2×C4⋊Dic53C2311yescharacteristic
76C2×C4⋊Dic53C2311yescharacteristic
77C2×C4⋊Dic53C2311yescharacteristic
78C2×#11719919—21no—
79C2×#11719919—21no—
80C2×#11719919—21no—
81C2×#11719919—21no—
82C2×C22⋊Dic53—21no—
83C2×C22⋊Dic53—21no—
84C2×C4×Dic53—21no—
85#13133150—22no—
86#13133150—22no—
87#13133150—22no—
88#13133148—22no—
89#13133148—41no—
90#13133148D414yescomplemented
91C4⋊#88322—22no—
92C4⋊#88322—22no—
93#13133138—22no—
94#13133138—22no—
95#13133138—22no—
96#13133138—41no—
97C22⋊#88322—22no—
98#13133124—21no—
99#13133124—21no—
100#13133124—21no—
101C22×D4×C53—21no—
102C22×D4×C53C2311yescharacteristic
103C23×C4×C53—21no—
104C23×C4×C53C2311yescharacteristic
105#121×C53—22no—
106#121×C53Q814yes—
107C4×D4×C53—22no—
108C4×D4×C53—41no—
109#117×C53—22no—
110#117×C53—41no—
111C2×C4⋊C4×C53—21no—
112C2×C4⋊C4×C53C2311yescharacteristic
113C2×C4⋊C4×C53C2311yescharacteristic
114C2×C4⋊C4×C53C2311yescharacteristic
115C2×C22⋊C4×C53—21no—
116C2×C22⋊C4×C53—21no—
117C2×C22⋊C4×C53—21no—
118C2×C22⋊C4×C53—21no—
119C2×C22⋊C4×C53—21no—
120C2×C22⋊C4×C53D411yescharacteristic
121C2×C22⋊C4×C53D411yescharacteristic
122C2×C42×C53—21no—
123C2×C42×C53—21no—
124C2×C42×C53C2311yescharacteristic
125C2×C42×C53C2311yescharacteristic
126#95×C53—21no—
127#95×C53—21no—
128#95×C53D411yescharacteristic
129#95×C53D411yescharacteristic
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×C53—42no—
149C22×C4×C53—42no—
150C22×C4×C53—81no—
151C22×C4×C53C2×D411yescharacteristic
152C22×C4×C53C2×D411yescharacteristic
153C22×C4×C53C2×D411yescharacteristic
154C22×C4×C53C2×D411yescharacteristic
155C22×C4×C53C2×D411yescharacteristic
156C22×C4×C53C2×Q812yes—
157C22×C4×C53#4112yes—
158C22×C4×C53C2411yescharacteristic
159C22×Dic53—21no—
160C22×Dic53—21no—
161C22×Dic53—21no—
162C22×Dic53—21no—
163C4⋊C4×C53—21no—
164C4⋊C4×C53—21no—
165C4⋊C4×C53—21no—
166C4⋊C4×C53—21no—
167C4⋊C4×C53—81no—
168C4⋊C4×C53C2×D412yes—
169C4⋊C4×C53C2×Q812yes—
170C22⋊C4×C53—24no—
171C22⋊C4×C53—22no—
172C22⋊C4×C53—41no—
173C22⋊C4×C53—42no—
174C22⋊C4×C53—42no—
175C22⋊C4×C53—42no—
176C22⋊C4×C53—81no—
177C42×C53—21no—
178C42×C53—21no—
179C42×C53—21no—
180C42×C53—21no—
181C42×C53—42no—
182C42×C53—81no—
183C22⋊Dic53—42no—
184C22⋊Dic53—81no—
185C4⋊Dic53—21no—
186C4⋊Dic53—21no—
187C4⋊Dic53—21no—
188C4⋊Dic53—21no—
189C4⋊Dic53—21no—
190C4⋊Dic53—21no—
191C4⋊Dic53—21no—
192C4⋊Dic53—42no—
193C4⋊Dic53—81no—
194C4⋊Dic53C2×D412yes—
195#11719919—42no—
196#11719919—42no—
197#11719919—42no—
198#11719919—81no—
199C4×Dic53—42no—
200C2×#88322—28no—
201C2×#88322—21no—
202C2×#88322—21no—
203C2×#88322—21no—
204C2×#88322—21no—
205C2×#88322—21no—
206C2×#88322—21no—
207C2×#88322—21no—
208C2×#88322—21no—
209C2×#88322—21no—
210C2×#88322—21no—
211C2×#88322C2×D412yes—
212C2×#88322#4112yes—
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—21no—
236C2×C4×C53—21no—
237C2×C4×C53—22no—
238C2×C4×C53—22no—
239C2×C4×C53—22no—
240C2×C4×C53—22no—
241C2×C4×C53—22no—
242C2×C4×C53—22no—
243C2×C4×C53—21no—
244C2×C4×C53—21no—
245C2×C4×C53—21no—
246C2×C4×C53—21no—
247C2×C4×C53—21no—
248C2×C4×C53—21no—
249C2×C4×C53—21no—
250C2×C4×C53—21no—
251C2×C4×C53—21no—
252C2×C4×C53—41no—
253C2×C4×C53—41no—
254C2×C4×C53—41no—
255C2×C4×C53—41no—
256C2×C4×C53—41no—
257C2×C4×C53—41no—
258C2×C4×C53—41no—
259C2×C4×C53—84no—
260C2×C4×C53C22⋊Q812yes—
261C2×C4×C53#13512yes—
262C2×C4×C53C22×D411yescharacteristic
263C2×C4×C53C22×Q811yescharacteristic
264C2×C4×C53C2×#4111yescharacteristic
265C2×C4×C53C2×#4111yescharacteristic
266C2×C4×C53H(5,𝔽2)11yescharacteristic
267C2×C4×C532−1+411yescharacteristic
268C2×Dic53—22no—
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—41no—
280#88322—22no—
281#88322—22no—
282#88322—41no—
283#88322—41no—
284#26345—531nomaximal, Sylow
285C22×C53—22no—
286C22×C53—22no—
287C22×C53—22no—
288C22×C53—41no—
289C22×C53—41no—
290C22×C53—41no—
291C22×C53—41no—
292C22×C53—42no—
293C22×C53—42no—
294C22×C53—42no—
295C22×C53—81no—
296C22×C53—81no—
297C22×C53—81no—
298C22×C53—81no—
299C22×C53C2×C22⋊Q811yescharacteristic
300C22×C53D4211yescharacteristic
301C22×C53#54711yescharacteristic
302C22×C53C2×#13511yescharacteristic
303C22×C53#57511yescharacteristic
304C22×C53#57611yescharacteristic
305C22×C53#57711yescharacteristic
306C4×C53—22no—
307C4×C53—22no—
308C4×C53—42no—
309C4×C53—41no—
310C4×C53—41no—
311C4×C53—42no—
312C4×C53—41no—
313C4×C53—81no—
314C4×C53#52511yescharacteristic
315C4×C53#53911yescharacteristic
316C4×C53D4×Q811yescharacteristic
317C4×C53#55411yescharacteristic
318Dic53—42no—
319Dic53—81no—
320Dic53—81no—
321Dic53—81no—
322C2×#478—531no—
323C2×#427—531no—
324#2441—531no—
325#2363D5311yescharacteristic
326#2094—531no—
327#2088—531no—
328#2067—531no—
329#2059—531no—
330#1947—531no—
331#1919—531no—
332#1893—531no—
333#1875—531no—
334#1862—531no—
335#1854—531no—
336#1850—531no—
337C2×C53—42no—
338C2×C53—82no—
339C2×C53#244911yescharacteristic
340C2×C53#307611yescharacteristic
341C2×C53#308111yescharacteristic
342C2×C53#326811yescharacteristic
343C2×C53#328411yescharacteristic
344C2×C53#333911yescharacteristic
345C2×C53#334811yescharacteristic
346C23×C8—531no—
347C2×#125—531no—
348C2×#121C2×D5311yescharacteristic
349C2×C4×D4C2×D5311yescharacteristic
350C2×#117—531no—
351C2×#117C2×D5311yescharacteristic
352#478—1064no—
353#427—1064no—
354C2×#107—531no—
355C2×#107—531no—
356C2×#107—531no—
357C2×#106—531no—
358C2×#106—531no—
359C2×#106—531no—
360C2×C4⋊C8—531no—
361C2×C4⋊C8—531no—
362C2×#102—531no—
363C2×#102—531no—
364C2×#102—531no—
365C2×#102—531no—
366C2×C22⋊C8—531no—
367#399—1061no—
368#396—1061no—
369#390—21no—
370#390C2×D5311yescharacteristic
371#389—531no—
372#389—531no—
373#389—1061no—
374#389C2×D5311yescharacteristic
375#388—21no—
376#387—1061no—
377#386—21no—
378#384—531no—
379#380—531no—
380C4×C4⋊C4C2×D5311yescharacteristic
381C4×C22⋊C4—21no—
382C4×C22⋊C4C2×D5311yescharacteristic
383#340—531no—
384#340—531no—
385#340—531no—
386#340—531no—
387#340—531no—
388#340—531no—
389#336—531no—
390#336—531no—
391C53#2634511yescharacteristic, complemented, Sylow
392C22×D4—21no—
393C22×D4C22×D5311yescharacteristic
394C23×C4—21no—
395C23×C4C22×D5311yescharacteristic
396C22×C8—531no—
397C22×C8—531no—
398C22×C8—531no—
399C22×C8—531no—
400C22×C8—1062no—
401C22×C8—1062no—
402C22×C8—1062no—
403#125—1062no—
404#125—2121no—
405#121—22no—
406#121Dic10614yes—
407C4×D4—22no—
408C4×D4—41no—
409#117—22no—
410#117—41no—
411#117—1062no—
412#117—2121no—
413C2×C4⋊C4—21no—
414C2×C4⋊C4—531no—
415C2×C4⋊C4—531no—
416C2×C4⋊C4—531no—
417C2×C4⋊C4—531no—
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⋊C4C22×D5311yescharacteristic
425C2×C4⋊C4C22×D5311yescharacteristic
426C2×C4⋊C4C22×D5311yescharacteristic
427C2×C22⋊C4—21no—
428C2×C22⋊C4—21no—
429C2×C22⋊C4—21no—
430C2×C22⋊C4—21no—
431C2×C22⋊C4—21no—
432C2×C22⋊C4—1061no—
433C2×C22⋊C4—1061no—
434C2×C22⋊C4C22⋊D5311yescharacteristic
435C2×C22⋊C4C22⋊D5311yescharacteristic
436C2×C42—21no—
437C2×C42—21no—
438C2×C42—1061no—
439C2×C42C22×D5311yescharacteristic
440C2×C42C22×D5311yescharacteristic
441#107—1062no—
442#107—1062no—
443#107—1062no—
444#106—534no—
445#106—1062no—
446#106—2121no—
447C4⋊C8—1062no—
448C4⋊C8—1062no—
449#102—1062no—
450#102—1062no—
451#102—1062no—
452#102—2121no—
453C22⋊C8—1062no—
454#95—21no—
455#95—21no—
456#95—1061no—
457#95—1061no—
458#95—1061no—
459#95C22⋊D5311yescharacteristic
460#95C22⋊D5311yescharacteristic
461C24—21no—
462C24C2×C22⋊D5311yescharacteristic
463C2×D4—24no—
464C2×D4—21no—
465C2×D4—21no—
466C2×D4—21no—
467C2×D4—21no—
468C2×D4—21no—
469C2×D4—81no—
470C2×D4—81no—
471C2×D4C2×Dic10612yes—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—42no—
480C22×C4—42no—
481C22×C4—81no—
482C22×C4—1061no—
483C22×C4—1061no—
484C22×C4—1061no—
485C22×C4—1061no—
486C22×C4C2×Dic10612yes—
487C22×C4#1171994512yes—
488C22×C4C2×C22⋊D5311yescharacteristic
489C22×C4C2×C22⋊D5311yescharacteristic
490C22×C4C2×C22⋊D5311yescharacteristic
491C22×C4C2×C22⋊D5311yescharacteristic
492C22×C4C2×C22⋊D5311yescharacteristic
493C22×C4C23×D5311yescharacteristic
494C2×C8—532no—
495C2×C8—532no—
496C2×C8—1068no—
497C2×C8—1061no—
498C2×C8—1061no—
499C2×C8—1061no—
500C2×C8—1061no—
501C2×C8—1061no—
502C2×C8—1061no—
503C2×C8—1061no—
504C2×C8—1061no—
505C2×C8—1061no—
506C2×C8—1061no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—81no—
512C4⋊C4—532no—
513C4⋊C4—1061no—
514C4⋊C4—1061no—
515C4⋊C4—1061no—
516C4⋊C4—1061no—
517C4⋊C4—1061no—
518C4⋊C4—1061no—
519C4⋊C4—1061no—
520C4⋊C4—4241no—
521C4⋊C4—4241no—
522C4⋊C4—2122no—
523C4⋊C4—2122no—
524C4⋊C4—2122no—
525C4⋊C4—2122no—
526C4⋊C4C2×Dic10612yes—
527C4⋊C4D4×D5312yes—
528C22⋊C4—24no—
529C22⋊C4—22no—
530C22⋊C4—41no—
531C22⋊C4—42no—
532C22⋊C4—42no—
533C22⋊C4—42no—
534C22⋊C4—81no—
535C22⋊C4—4241no—
536C22⋊C4—2122no—
537C42—21no—
538C42—21no—
539C42—21no—
540C42—21no—
541C42—42no—
542C42—81no—
543C42—2122no—
544C23—21no—
545C23—21no—
546C23—21no—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—22no—
551C23—41no—
552C23—41no—
553C23—41no—
554C23—41no—
555C23—41no—
556C23—41no—
557C23#1313319612yes—
558C23C22×C22⋊D5311yescharacteristic
559D4—22no—
560D4—22no—
561D4—41no—
562D4—41no—
563D4—84no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—21no—
573C2×C4—22no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—22no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—84no—
591C2×C4—1062no—
592C2×C4—2121no—
593C2×C4—2121no—
594C2×C4—2121no—
595C2×C4—2121no—
596C2×C4—2121no—
597C2×C4—2121no—
598C2×C4—2121no—
599C2×C4—2121no—
600C2×C4—2121no—
601C2×C4—2121no—
602C2×C4—2121no—
603C2×C4#1313319612yes—
604C2×C4#1313323712yes—
605C2×C4C2×D4×D5311yescharacteristic
606C2×C4C22×Dic10611yescharacteristic
607C2×C4#41×D5311yescharacteristic
608C2×C4C2×#1171994511yescharacteristic
609C2×C4#1313326211yescharacteristic
610C2×C4#1313326411yescharacteristic
611C8—1062no—
612C8—1062no—
613C8—2121no—
614C8—2121no—
615C22—22no—
616C22—22no—
617C22—22no—
618C22—41no—
619C22—41no—
620C22—41no—
621C22—41no—
622C22—42no—
623C22—42no—
624C22—42no—
625C22—81no—
626C22—81no—
627C22—81no—
628C22—81no—
629C22C2×#1313319611yescharacteristic
630C22D4×C22⋊D5311yescharacteristic
631C22#1681127211yescharacteristic
632C22C2×#1313323711yescharacteristic
633C22#1681142111yescharacteristic
634C22#1681142611yescharacteristic
635C22#1681143411yescharacteristic
636C4—22no—
637C4—22no—
638C4—41no—
639C4—42no—
640C4—42no—
641C4—41no—
642C4—41no—
643C4—81no—
644C4—4241no—
645C4—4241no—
646C4—4241no—
647C4—2122no—
648C4#1681113211yescharacteristic
649C4#1681121511yescharacteristic
650C4D4×Dic10611yescharacteristic
651C4#1681133211yescharacteristic
652C2—42no—
653C2—82no—
654C2#2027851811yescharacteristic
655C2#2028450311yescharacteristic
656C2#2028454611yescharacteristic
657C2#2028625111yescharacteristic
658C2#2028644511yescharacteristic
659C2#2028711111yescharacteristic
660C2#2028723011yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2363C2
socleC23×C53C22×D4
prime radicalC2×#121×C53C22
hypercenter#2363D53
hypocenterC53#26345
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2363

Complements

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

NH ≅ G/Ntype
1. C2×#16810923C2semidirect
2. C2×#16810659C2semidirect
3. #20278428C2semidirect
4. #20276248C2semidirect
5. #20276216C2semidirect
6. #20276103C2semidirect
7. #20276053C2semidirect
8. #20275473C2semidirect
9. #20275293C2semidirect
12. #20274962C2semidirect
13. #20274914C2semidirect
14. #20274895C2semidirect
90. #13133148D4semidirect
391. C53#26345semidirect

Maximal subgroups

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

subgroupindex
C2×#168109232
C2×#168106592
#202784282
#202762482
#202762162
#202761032
#202760532
#202754732
#202752932
#202751442
#202750382
#202749622
#202749142
#202748952
C53×#23632
#2634553