Finite Group Databasebeta

← #27436165

Subgroups — #27436165

Order 13568.

782 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810920C211yesmaximal, characteristic
2C2×#16810754C211yesmaximal, characteristic, complemented
3#20279633C211yesmaximal, characteristic
4#20276216C211yesmaximal, characteristic, complemented
5#20276206C211yesmaximal, characteristic, complemented
6#20276086C211yesmaximal, characteristic
7#20276086C211yesmaximal, characteristic
8#20275685C211yesmaximal, characteristic, complemented
9#20275478C211yesmaximal, characteristic, complemented
10#20275455C211yesmaximal, characteristic, complemented
11#20275440C211yesmaximal, characteristic, complemented
12#20275411C211yesmaximal, characteristic
13#20275397C211yesmaximal, characteristic
14#20274543C211yesmaximal, characteristic
15C53×#2584C211yesmaximal, characteristic, complemented
16C2×#13133193—21no—
17C2×C4×D212C2211yescharacteristic
18#16810920—24no—
19#16810754—24no—
20C2×#13133150C2211yescharacteristic
21C2×#13133148C2211yescharacteristic
22C2×#13133148C2211yescharacteristic
23C2×#13133148C2211yescharacteristic
24C2×C4⋊#88322C2211yescharacteristic
25C2×C4⋊#88322C2211yescharacteristic
26C2×C4⋊#88322C2211yescharacteristic
27C2×#13133137C2211yescharacteristic
28C2×#13133137C2211yescharacteristic
29C2×#13133137C2211yescharacteristic
30C2×#13133137C2211yescharacteristic
31C2×#13133128C2211yescharacteristic
32C2×#13133128C2211yescharacteristic
33C2×C4×#88322C2211yescharacteristic
34#16810545—21no—
35#16810545—21no—
36#16810525C2211yescharacteristic
37#16810525C2211yescharacteristic
38#16810521C2211yescharacteristic
39#16810521C2211yescharacteristic
40#16810493C2211yescharacteristic
41#16810480—21no—
42#16810385C2211yescharacteristic
43#16810385C2211yescharacteristic
44#16810385C2211yescharacteristic
45#16810385C2211yescharacteristic
46#16810385C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810385C2211yescharacteristic
49#16810385C2211yescharacteristic
50C2×#125×C53C2211yescharacteristic
51C2×C4×Q8×C53C2211yescharacteristic
52C53×#398—21no—
53C53×#395—21no—
54C53×#389—21no—
55C53×#389—21no—
56C53×#389C2211yescharacteristic
57C53×#389C2211yescharacteristic
58C53×#389C2211yescharacteristic
59C53×#389C2211yescharacteristic
60C4×C4⋊C4×C53C2211yescharacteristic
61C22×D212C2311yescharacteristic
62C22×C4×D53—21no—
63C22×#88322C2311yescharacteristic
64C22×#88322C2311yescharacteristic
65C22×#88322C2311yescharacteristic
66C22×#88322C2311yescharacteristic
67#13133193—44no—
68C4×D212—22no—
69C4×D212—22no—
70C2×C4⋊Dic53—21no—
71C2×C4⋊Dic53—21no—
72C2×C4⋊Dic53C2311yescharacteristic
73C2×C4⋊Dic53C2311yescharacteristic
74C2×C4⋊Dic53C2311yescharacteristic
75C2×#11719919—21no—
76C2×#11719919—21no—
77C2×#11719921—21no—
78C2×#11719921—21no—
79C2×#11719921—21no—
80C2×#11719921—21no—
81C2×#11719921—21no—
82#13133150—22no—
83#13133148—22no—
84#13133148—22no—
85#13133148—22no—
86C4⋊#88322—22no—
87C4⋊#88322—22no—
88C4⋊#88322D414yes—
89#13133137—22no—
90#13133137—22no—
91#13133137—22no—
92#13133137Q814yescomplemented
93#13133128—22no—
94#13133128—22no—
95C4×#88322—22no—
96#13133124—21no—
97#13133124—21no—
98C22×Q8×C53D411yescharacteristic
99#125×C53—22no—
100#125×C53—22no—
101C4×Q8×C53—22no—
102C4×Q8×C53—22no—
103C2×C4⋊C4×C53—21no—
104C2×C4⋊C4×C53—21no—
105C2×C4⋊C4×C53—21no—
106C2×C4⋊C4×C53—21no—
107C2×C4⋊C4×C53—21no—
108C2×C4⋊C4×C53—21no—
109C2×C4⋊C4×C53D411yescharacteristic
110C2×C4⋊C4×C53D411yescharacteristic
111C2×C4⋊C4×C53D411yescharacteristic
112C2×C4⋊C4×C53C2311yescharacteristic
113C2×C4⋊C4×C53C2311yescharacteristic
114C2×C4⋊C4×C53C2311yescharacteristic
115C2×C4⋊C4×C53C2311yescharacteristic
116C2×C42×C53—21no—
117C2×C42×C53C2311yescharacteristic
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—
125C23×D53—21no—
126C2×Q8×C53—22no—
127C2×Q8×C53—22no—
128C2×Q8×C53—41no—
129C22×C4×C53—21no—
130C22×C4×C53—21no—
131C22×C4×C53—21no—
132C22×C4×C53—21no—
133C22×C4×C53C2×D411yescharacteristic
134C22×C4×C53C2×D411yescharacteristic
135C22×C4×C53C2×D411yescharacteristic
136C22×C4×C53C2×D411yescharacteristic
137C22×C4×C53C2×D411yescharacteristic
138C22×C4×C53C2×D411yescharacteristic
139C22×C4×C53C2411yescharacteristic
140C22×Dic53—21no—
141C22×Dic53—21no—
142C22×Dic53—21no—
143C2×D212—21no—
144C2×D212—21no—
145C2×D212—21no—
146C2×D212—21no—
147C2×D212—21no—
148C2×D212C2×Q812yes—
149C2×C4×D53—42no—
150C2×C4×D53—42no—
151C2×C4×D53—42no—
152C4⋊C4×C53—22no—
153C4⋊C4×C53—22no—
154C4⋊C4×C53—22no—
155C4⋊C4×C53—21no—
156C4⋊C4×C53—21no—
157C4⋊C4×C53—21no—
158C4⋊C4×C53—21no—
159C4⋊C4×C53—21no—
160C4⋊C4×C53—21no—
161C4⋊C4×C53—42no—
162C4⋊C4×C53—42no—
163C4⋊C4×C53—42no—
164C4⋊C4×C53—42no—
165C4⋊C4×C53—81no—
166C4⋊C4×C53—81no—
167C4⋊C4×C53C2×Q812yes—
168C4⋊C4×C53#4112yes—
169C42×C53—21no—
170C42×C53—21no—
171C42×C53—21no—
172C42×C53—21no—
173C42×C53—81no—
174C42×C53C2×D412yes—
175C42×C53#4112yes—
176#11719921—42no—
177#11719921—42no—
178#11719921—42no—
179#11719921—81no—
180#11719921—81no—
181C4⋊Dic53—21no—
182C4⋊Dic53—21no—
183C4⋊Dic53—21no—
184C4⋊Dic53—21no—
185C4⋊Dic53—21no—
186C4⋊Dic53—21no—
187C4⋊Dic53—42no—
188C4⋊Dic53—81no—
189#11719919—42no—
190#11719919—42no—
191C2×#88322—21no—
192C2×#88322—21no—
193C2×#88322—21no—
194C2×#88322—21no—
195C2×#88322—21no—
196C2×#88322—21no—
197C2×#88322—21no—
198C2×#88322—21no—
199C2×#88322—21no—
200C2×#88322C2×D412yes—
201C2×#88322C2×D412yes—
202C2×#88322C2×Q812yes—
203C23×C53C22×D411yescharacteristic
204C22×D53—22no—
205C22×D53—41no—
206C22×D53—41no—
207C22×D53—41no—
208C22×D53—41no—
209C22×D53—41no—
210C22×D53—41no—
211Q8×C53—44no—
212C2×C4×C53—21no—
213C2×C4×C53—21no—
214C2×C4×C53—21no—
215C2×C4×C53—21no—
216C2×C4×C53—21no—
217C2×C4×C53—21no—
218C2×C4×C53—21no—
219C2×C4×C53—21no—
220C2×C4×C53—21no—
221C2×C4×C53—22no—
222C2×C4×C53—22no—
223C2×C4×C53—21no—
224C2×C4×C53—21no—
225C2×C4×C53—21no—
226C2×C4×C53—21no—
227C2×C4×C53—21no—
228C2×C4×C53—21no—
229C2×C4×C53—41no—
230C2×C4×C53—41no—
231C2×C4×C53—41no—
232C2×C4×C53—41no—
233C2×C4×C53—41no—
234C2×C4×C53—41no—
235C2×C4×C53—41no—
236C2×C4×C53—41no—
237C2×C4×C53—41no—
238C2×C4×C53—41no—
239C2×C4×C53#12112yes—
240C2×C4×C53#12112yes—
241C2×C4×C53AGL(1,ℤ/8)12yes—
242C2×C4×C53C22×D411yescharacteristic
243C2×C4×C53C22×Q811yescharacteristic
244C2×C4×C53C2×#4111yescharacteristic
245C2×C4×C53C2×#4111yescharacteristic
246C2×C4×C532−1+411yescharacteristic
247C2×C4×C532−1+411yescharacteristic
248C2×Dic53—41no—
249C2×Dic53—41no—
250C2×Dic53—41no—
251C2×Dic53—41no—
252C2×Dic53—41no—
253C2×Dic53—41no—
254C2×Dic53—41no—
255C2×Dic53—41no—
256C2×Dic53—41no—
257D212—22no—
258D212—22no—
259D212—41no—
260D212—41no—
261C4×D53—84no—
262#88322—22no—
263#88322—22no—
264#88322—22no—
265#88322—41no—
266#25486—531nomaximal, Sylow
267C22×C53C2×#12111yescharacteristic
268C22×C53#54811yescharacteristic
269C22×C53D4×Q811yescharacteristic
270C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
271C22×C53#57511yescharacteristic
272C22×C53#57711yescharacteristic
273C22×C53#57711yescharacteristic
274C2×D53—42no—
275C2×D53—42no—
276C2×D53—42no—
277C2×D53—81no—
278C2×D53—81no—
279C2×D53—81no—
280C2×D53—81no—
281C4×C53—22no—
282C4×C53—22no—
283C4×C53—22no—
284C4×C53—42no—
285C4×C53—42no—
286C4×C53—41no—
287C4×C53—41no—
288C4×C53—41no—
289C4×C53—81no—
290C4×C53—81no—
291C4×C53#53311yescharacteristic
292C4×C53#54111yescharacteristic
293C4×C53#54711yescharacteristic
294C4×C53D4×Q811yescharacteristic
295Dic53—81no—
296Dic53—81no—
297Dic53—81no—
298C2×#478—531no—
299C2×#446—531no—
300#2584D5311yescharacteristic, complemented
301#2538—531no—
302#2088—531no—
303#2086—531no—
304#2065—531no—
305#2065—531no—
306#1992—531no—
307#1948—531no—
308#1944—531no—
309#1941—531no—
310#1936—531no—
311#1934—531no—
312#1762—531no—
313C2×C53#242911yescharacteristic
314C2×C53#305511yescharacteristic
315C2×C53#306011yescharacteristic
316C2×C53#336811yescharacteristic
317C2×C53#337611yescharacteristic
318C2×C53#338411yescharacteristic
319C2×C53#338811yescharacteristic
320D53—82no—
321C2×#125C2×D5311yescharacteristic
322C2×C22⋊Q8—1061no—
323C2×C4×Q8C2×D5311yescharacteristic
324C2×C4×D4—531no—
325#478—1064no—
326#446—1064no—
327C2×#107—531no—
328C2×#106—531no—
329C2×#106—531no—
330C2×#106—531no—
331C2×C4⋊C8—531no—
332C2×C4⋊C8—531no—
333C2×C4⋊C8—531no—
334C2×#102—531no—
335C2×#102—531no—
336C2×#102—531no—
337C2×#102—531no—
338C2×#97—531no—
339C2×#97—531no—
340C2×C4×C8—531no—
341#398—21no—
342#396—1061no—
343#396—1061no—
344#395—21no—
345#390—531no—
346#390—531no—
347#389—21no—
348#389—21no—
349#389—531no—
350#389—531no—
351#389C2×D5311yescharacteristic
352#389C2×D5311yescharacteristic
353#389C2×D5311yescharacteristic
354#389C2×D5311yescharacteristic
355#384—531no—
356#380—1061no—
357C4×C4⋊C4C2×D5311yescharacteristic
358#340—531no—
359#340—531no—
360#340—531no—
361#340—531no—
362#340—531no—
363#340—531no—
364#340—531no—
365#340—531no—
366C53#2548611yescharacteristic, complemented, Sylow
367C22×Q8C22⋊D5311yescharacteristic
368C22×D4—531no—
369C23×C4—1061no—
370C22×C8—531no—
371C22×C8—531no—
372C22×C8—531no—
373C22×C8—531no—
374#125—22no—
375#125—22no—
376C22⋊Q8—2124no—
377C4×Q8—22no—
378C4×Q8—22no—
379C4×D4—1062no—
380C4×D4—1062no—
381C2×C4⋊C4—21no—
382C2×C4⋊C4—21no—
383C2×C4⋊C4—21no—
384C2×C4⋊C4—21no—
385C2×C4⋊C4—21no—
386C2×C4⋊C4—21no—
387C2×C4⋊C4—531no—
388C2×C4⋊C4—531no—
389C2×C4⋊C4—531no—
390C2×C4⋊C4—1061no—
391C2×C4⋊C4—1061no—
392C2×C4⋊C4—1061no—
393C2×C4⋊C4—1061no—
394C2×C4⋊C4C22⋊D5311yescharacteristic
395C2×C4⋊C4C22⋊D5311yescharacteristic
396C2×C4⋊C4C22⋊D5311yescharacteristic
397C2×C4⋊C4C22×D5311yescharacteristic
398C2×C4⋊C4C22×D5311yescharacteristic
399C2×C4⋊C4C22×D5311yescharacteristic
400C2×C4⋊C4C22×D5311yescharacteristic
401C2×C22⋊C4—1061no—
402C2×C22⋊C4—1061no—
403C2×C22⋊C4—1061no—
404C2×C22⋊C4—1061no—
405C2×C22⋊C4—1061no—
406C2×C42—21no—
407C2×C42C22×D5311yescharacteristic
408C2×C42C22×D5311yescharacteristic
409C2×C42C22×D5311yescharacteristic
410#107—1062no—
411#106—1062no—
412#106—1062no—
413#106—1062no—
414C4⋊C8—534no—
415C4⋊C8—1062no—
416C4⋊C8—1062no—
417#102—534no—
418#102—1062no—
419#102—1062no—
420#102—1062no—
421#97—1062no—
422#97—1062no—
423C4×C8—1062no—
424#95—21no—
425#95—21no—
426#95—21no—
427#95—21no—
428#95—21no—
429#95—1061no—
430#95—1061no—
431C24—1061no—
432C2×Q8—22no—
433C2×Q8—22no—
434C2×Q8—41no—
435C2×D4—532no—
436C2×D4—1061no—
437C2×D4—1061no—
438C2×D4—1061no—
439C2×D4—1061no—
440C2×D4—1061no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—1061no—
446C22×C4—1061no—
447C22×C4—1061no—
448C22×C4—2122no—
449C22×C4—2122no—
450C22×C4—2122no—
451C22×C4C2×C22⋊D5311yescharacteristic
452C22×C4C2×C22⋊D5311yescharacteristic
453C22×C4C2×C22⋊D5311yescharacteristic
454C22×C4C2×C22⋊D5311yescharacteristic
455C22×C4C2×C22⋊D5311yescharacteristic
456C22×C4C2×C22⋊D5311yescharacteristic
457C22×C4C23×D5311yescharacteristic
458C2×C8—532no—
459C2×C8—532no—
460C2×C8—532no—
461C2×C8—1061no—
462C2×C8—1061no—
463C2×C8—1061no—
464C2×C8—1061no—
465C2×C8—1061no—
466C2×C8—1061no—
467C2×C8—1061no—
468C2×C8—1061no—
469C2×C8—1061no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—42no—
480C4⋊C4—42no—
481C4⋊C4—42no—
482C4⋊C4—42no—
483C4⋊C4—81no—
484C4⋊C4—81no—
485C4⋊C4—1061no—
486C4⋊C4—1061no—
487C4⋊C4—1061no—
488C4⋊C4—1061no—
489C4⋊C4—1061no—
490C4⋊C4—1061no—
491C4⋊C4—4241no—
492C4⋊C4—2122no—
493C4⋊C4—2122no—
494C4⋊C4—2122no—
495C4⋊C4Q8×D5312yes—
496C4⋊C4#1171994912yes—
497C22⋊C4—4241no—
498C22⋊C4—4241no—
499C22⋊C4—2122no—
500C22⋊C4—2122no—
501C22⋊C4—2122no—
502C42—21no—
503C42—21no—
504C42—21no—
505C42—21no—
506C42—81no—
507C42D4×D5312yes—
508C42#1171994712yes—
509C23—1062no—
510C23—2121no—
511C23—2121no—
512C23—2121no—
513C23—2121no—
514C23—2121no—
515C23—2121no—
516C23C22×C22⋊D5311yescharacteristic
517Q8—44no—
518D4—1062no—
519D4—1062no—
520D4—2121no—
521D4—2121no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
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—4244no—
550C2×C4—2121no—
551C2×C4—2121no—
552C2×C4—2121no—
553C2×C4—2121no—
554C2×C4—2121no—
555C2×C4—2121no—
556C2×C4—2121no—
557C2×C4—2121no—
558C2×C4—2121no—
559C2×C4#1313318912yes—
560C2×C4#1313318912yes—
561C2×C4#1313324212yes—
562C2×C4C2×D4×D5311yescharacteristic
563C2×C4C2×Q8×D5311yescharacteristic
564C2×C4C2×#1171994711yescharacteristic
565C2×C4C2×#1171994911yescharacteristic
566C2×C4#1313326511yescharacteristic
567C2×C4#1313326511yescharacteristic
568C8—1062no—
569C8—1062no—
570C8—1062no—
571C8—2121no—
572C22—4241no—
573C22—4241no—
574C22—4241no—
575C22—4241no—
576C22—2122no—
577C22—2122no—
578C22—2122no—
579C22C2×#1313318911yescharacteristic
580C22#1681128411yescharacteristic
581C22Q8×C22⋊D5311yescharacteristic
582C22C2×#1313324211yescharacteristic
583C22#1681142111yescharacteristic
584C22#1681143111yescharacteristic
585C22#1681143411yescharacteristic
586C4—22no—
587C4—22no—
588C4—22no—
589C4—42no—
590C4—42no—
591C4—41no—
592C4—41no—
593C4—41no—
594C4—81no—
595C4—81no—
596C4—4241no—
597C4—4241no—
598C4—4241no—
599C4#1681117811yescharacteristic
600C4#1681123411yescharacteristic
601C4#1681127611yescharacteristic
602C4#1681128811yescharacteristic
603C2—4242no—
604C2#2027831711yescharacteristic
605C2#2028433611yescharacteristic
606C2#2028437411yescharacteristic
607C2#2028744411yescharacteristic
608C2#2028754211yescharacteristic
609C2#2028761411yescharacteristic
610C2#2028765411yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2584C2
socleC23×C53C22×D4
prime radicalC22×D212C23
hypercenter#2584D53
hypocenterC53#25486
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ #2584

Complements

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

NH ≅ G/Ntype
2. C2×#16810754C2semidirect
4. #20276216C2semidirect
5. #20276206C2semidirect
8. #20275685C2semidirect
9. #20275478C2semidirect
10. #20275455C2semidirect
11. #20275440C2semidirect
15. C53×#2584C2semidirect
92. #13133137Q8semidirect
300. #2584D53semidirect
366. C53#25486semidirect

Maximal subgroups

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

subgroupindex
C2×#168109202
C2×#168107542
#202796332
#202762162
#202762062
#202760862
#202756852
#202754782
#202754552
#202754402
#202754112
#202753972
#202745432
C53×#25842
#2548653