Finite Group Databasebeta

← #27388966

Subgroups — #27388966

Order 13568.

938 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×#16810679C211yesmaximal, characteristic
2#20277108C211yesmaximal, characteristic
3C4×C4⋊C4×D53C211yesmaximal, characteristic
4#20275485C211yesmaximal, characteristic, complemented
5#20275483C211yesmaximal, characteristic, complemented
6#20275481C211yesmaximal, characteristic, complemented
7#20274909C211yesmaximal, characteristic
8#20274906C211yesmaximal, characteristic
9#20274904C211yesmaximal, characteristic
10#20274901C211yesmaximal, characteristic
11#20274615C211yesmaximal, characteristic, complemented
12#20274614C211yesmaximal, characteristic, complemented
13#20274611C211yesmaximal, characteristic, complemented
14#20274611C211yesmaximal, characteristic, complemented
15C53×#1949C211yesmaximal, characteristic, complemented
16C2×C4⋊C4×D53C2211yescharacteristic
17C2×C4⋊C4×D53C2211yescharacteristic
18C2×C42×D53—21no—
19C2×C42×D53C2211yescharacteristic
20#16810679C418yescomplemented
21C2×#13133151C2211yescharacteristic
22C2×#13133150C2211yescharacteristic
23C2×#13133150C2211yescharacteristic
24C2×#13133149C2211yescharacteristic
25C2×#13133148C2211yescharacteristic
26C2×#13133148C2211yescharacteristic
27C2×C4⋊#88322C2211yescharacteristic
28C2×#13133129C2211yescharacteristic
29C2×#13133129C2211yescharacteristic
30C2×#13133127C2211yescharacteristic
31C2×#13133127C2211yescharacteristic
32#16810490—21no—
33#16810489C2211yescharacteristic
34#16810485—21no—
35#16810480C2211yescharacteristic
36#16810480C2211yescharacteristic
37C4⋊C4×Dic53C2211yescharacteristic
38C4⋊C4×Dic53C2211yescharacteristic
39C4⋊C4×Dic53C2211yescharacteristic
40C4⋊C4×Dic53C2211yescharacteristic
41C4×#11719919—21no—
42C4×C4⋊Dic53C2211yescharacteristic
43#16810474—21no—
44#16810470—21no—
45#16810468—21no—
46#95×D53—21no—
47#16810386C2211yescharacteristic
48#16810386C2211yescharacteristic
49#16810385C2211yescharacteristic
50#16810385C2211yescharacteristic
51#16810385C2211yescharacteristic
52#16810385C2211yescharacteristic
53C2×#107×C53C2211yescharacteristic
54C2×#106×C53C2211yescharacteristic
55C2×C4⋊C8×C53C2211yescharacteristic
56C53×#382C2211yescharacteristic
57C4×C4⋊C4×C53C2211yescharacteristic
58C53×#340C2211yescharacteristic
59C53×#340C2211yescharacteristic
60C22×C4×D53—21no—
61C22×C4×D53—21no—
62C22×C4×D53D411yescharacteristic
63C22×C4×D53D411yescharacteristic
64C22×C4×D53C2311yescharacteristic
65C22×#88322C2311yescharacteristic
66C22×#88322C2311yescharacteristic
67C2×C4⋊Dic53C2311yescharacteristic
68C2×C4⋊Dic53C2311yescharacteristic
69C2×C4⋊Dic53C2311yescharacteristic
70C2×C4⋊Dic53C2311yescharacteristic
71C2×#11719919—21no—
72C2×#11719919—21no—
73C4⋊C4×D53—22no—
74C4⋊C4×D53—22no—
75C4⋊C4×D53—22no—
76C4⋊C4×D53—22no—
77C2×#11719921—21no—
78C2×#11719921—21no—
79C2×C4×Dic53—21no—
80C2×C4×Dic53—21no—
81C2×C4×Dic53—21no—
82C2×C4×Dic53—21no—
83C2×C4×Dic53—21no—
84C2×C4×Dic53D411yescharacteristic
85C2×C4×Dic53D411yescharacteristic
86C2×C4×Dic53C2311yescharacteristic
87C42×D53—44no—
88C42×D53C2×C414yes—
89C42×D53Q814yes—
90#13133151—22no—
91#13133150—22no—
92#13133150—22no—
93#13133149—22no—
94#13133148—22no—
95#13133148—22no—
96C4⋊#88322C2×C414yescomplemented
97#13133129C2×C414yes—
98#13133129C2×C414yes—
99#13133127C2×C414yescomplemented
100#13133127C2×C414yescomplemented
101#13133124—21no—
102#13133124—21no—
103#13133124—21no—
104#13133124—21no—
105#13133124—21no—
106#13133124—21no—
107#13133124—21no—
108#13133124—21no—
109#13133124—21no—
110C22×C8×C53C2311yescharacteristic
111C22×C8×C53C2311yescharacteristic
112C2×C4⋊C4×C53C2311yescharacteristic
113C2×C4⋊C4×C53C2311yescharacteristic
114C2×C4⋊C4×C53C2311yescharacteristic
115C2×C4⋊C4×C53C2311yescharacteristic
116C2×C42×C53—21no—
117C2×C42×C53C2311yescharacteristic
118#107×C53—22no—
119#106×C53—22no—
120C4⋊C8×C53C2×C414yescomplemented
121#95×C53—21no—
122#95×C53—21no—
123#95×C53—21no—
124C23×D53C2×D411yescharacteristic
125C22×C4×C53—21no—
126C22×C4×C53—21no—
127C22×C4×C53—21no—
128C22×C4×C53—21no—
129C22×C4×C53C2×D411yescharacteristic
130C22×C4×C53C2×D411yescharacteristic
131C22×C4×C53C2411yescharacteristic
132C22×Dic53—21no—
133C22×Dic53—21no—
134C22×Dic53—21no—
135C22×Dic53—21no—
136C22×Dic53C2×D411yescharacteristic
137C22×Dic53C2×D411yescharacteristic
138C22×Dic53C2×D411yescharacteristic
139C2×C4×D53—22no—
140C2×C4×D53—22no—
141C2×C4×D53—21no—
142C2×C4×D53—21no—
143C2×C4×D53—21no—
144C2×C4×D53—21no—
145C2×C4×D53—24no—
146C2×C4×D53—24no—
147C2×C4×D53—41no—
148C2×C4×D53—41no—
149C2×C4×D53—42no—
150C2×C4×D53—42no—
151C2×C4×D53—42no—
152C2×C4×D53—42no—
153C2×C4×D53C4⋊C414yes—
154C2×C4×D53C4⋊C414yes—
155C2×C4×D53C22×C412yes—
156C2×C4×D53C2×Q812yes—
157C2×C8×C53—21no—
158C2×C8×C53—21no—
159C2×C8×C53—21no—
160C2×C8×C53—21no—
161C2×C8×C53C22×C412yes—
162C2×C8×C53C22×C412yes—
163C4⋊C4×C53—21no—
164C4⋊C4×C53—21no—
165C4⋊C4×C53—21no—
166C4⋊C4×C53—21no—
167C4⋊C4×C53—21no—
168C4⋊C4×C53—21no—
169C4⋊C4×C53#4112yes—
170C4⋊C4×C53#4112yes—
171C42×C53—42no—
172C42×C53C22×C412yes—
173C42×C53C2×Q812yes—
174#11719921—42no—
175#11719921—42no—
176C4⋊Dic53—21no—
177C4⋊Dic53—21no—
178C4⋊Dic53—21no—
179C4⋊Dic53—21no—
180C4⋊Dic53—21no—
181C4⋊Dic53—21no—
182C4⋊Dic53#4112yes—
183C4⋊Dic53#4112yes—
184#11719919—24no—
185#11719919—24no—
186C4×Dic53—42no—
187C4×Dic53—42no—
188C4×Dic53—42no—
189C4×Dic53—42no—
190C4×Dic53—42no—
191C4×Dic53C4⋊C414yes—
192C4×Dic53C4⋊C414yes—
193C4×Dic53C22×C412yes—
194C4×Dic53C2×Q812yes—
195C2×#88322—21no—
196C2×#88322—21no—
197C2×#88322—21no—
198C2×#88322—21no—
199C2×#88322C22×C412yes—
200C2×#88322C22×C412yes—
201C23×C53C22×D411yescharacteristic
202C22×D53—21no—
203C22×D53—21no—
204C22×D53—21no—
205C22×D53—21no—
206C22×D53C2×C4⋊C412yes—
207C22×D53AGL(1,ℤ/8)12yes—
208C22×D53#13712yes—
209C2×C4×C53—21no—
210C2×C4×C53—21no—
211C2×C4×C53—21no—
212C2×C4×C53—21no—
213C2×C4×C53—22no—
214C2×C4×C53—22no—
215C2×C4×C53—41no—
216C2×C4×C53—41no—
217C2×C4×C53—41no—
218C2×C4×C53—41no—
219C2×C4×C53—41no—
220C2×C4×C53—41no—
221C2×C4×C53—41no—
222C2×C4×C53—41no—
223C2×C4×C53—41no—
224C2×C4×C53—41no—
225C2×C4×C53C2×C4⋊C412yes—
226C2×C4×C53C2×C4⋊C412yes—
227C2×C4×C53C23×C411yescharacteristic
228C2×C4×C53C22×Q811yescharacteristic
229C2×C4×C53C2×#4111yescharacteristic
230C2×C4×C53C2×#4111yescharacteristic
231C2×C4×C53C2×#4111yescharacteristic
232C2×C4×C53C2×#4111yescharacteristic
233C2×Dic53—21no—
234C2×Dic53—21no—
235C2×Dic53—21no—
236C2×Dic53—21no—
237C2×Dic53—21no—
238C2×Dic53—21no—
239C2×Dic53—22no—
240C2×Dic53—22no—
241C2×Dic53—41no—
242C2×Dic53—41no—
243C2×Dic53—41no—
244C2×Dic53—41no—
245C2×Dic53—41no—
246C2×Dic53—41no—
247C2×Dic53—41no—
248C2×Dic53—41no—
249C2×Dic53—41no—
250C2×Dic53—41no—
251C2×Dic53C2×C4⋊C412yes—
252C2×Dic53C2×C4⋊C412yes—
253C2×Dic53C2×C4⋊C412yes—
254C4×D53—28no—
255C4×D53—22no—
256C4×D53—22no—
257C4×D53—22no—
258C4×D53—22no—
259C4×D53—44no—
260C4×D53—48no—
261C4×D53—48no—
262C8×C53—22no—
263C8×C53—22no—
264#88322—22no—
265#88322—22no—
266#21363—531nomaximal, Sylow
267C22×C53C22×C4⋊C411yescharacteristic
268C22×C53#52511yescharacteristic
269C22×C53#52511yescharacteristic
270C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
271C22×C53C2×#13711yescharacteristic
272C22×C53#57511yescharacteristic
273C22×C53#57511yescharacteristic
274C2×D53—22no—
275C2×D53—22no—
276C2×D53—22no—
277C2×D53—22no—
278C2×D53—22no—
279C2×D53—22no—
280C2×D53#42814yes—
281C4×C53—22no—
282C4×C53—22no—
283C4×C53—42no—
284C4×C53—42no—
285C4×C53—81no—
286C4×C53—81no—
287C4×C53C4×#4111yescharacteristic
288C4×C53C4×#4111yescharacteristic
289C4×C53#53311yescharacteristic
290C4×C53#53311yescharacteristic
291Dic53—22no—
292Dic53—22no—
293Dic53—22no—
294Dic53—42no—
295Dic53—42no—
296Dic53—81no—
297Dic53—81no—
298C2×#432—531no—
299#2283—531no—
300C2×C4×C4⋊C4—531no—
301#1949—531no—
302#1949—531no—
303#1949—531no—
304#1949D5311yescharacteristic, complemented
305#1853—531no—
306#1853—531no—
307#1852—531no—
308#1852—531no—
309#1776—531no—
310#1776—531no—
311#1776—531no—
312#1776—531no—
313C2×C53#229611yescharacteristic
314C2×C53C2×#42811yescharacteristic
315C2×C53#291111yescharacteristic
316C2×C53#311411yescharacteristic
317C2×C53#311411yescharacteristic
318C2×C53#311711yescharacteristic
319C2×C53#311711yescharacteristic
320D53—24no—
321C22×C4⋊C4—531no—
322C22×C4⋊C4—531no—
323C22×C42—531no—
324C22×C42—1061no—
325#432—538no—
326C2×#107—531no—
327C2×#107—531no—
328C2×#107—531no—
329C2×#107C2×D5311yescharacteristic
330C2×#106—531no—
331C2×#106—531no—
332C2×#106—531no—
333C2×#106C2×D5311yescharacteristic
334C2×C4⋊C8—531no—
335C2×C4⋊C8C2×D5311yescharacteristic
336C2×C22⋊C8—531no—
337C2×C22⋊C8—531no—
338C2×#97—531no—
339C2×#97—531no—
340#383—1061no—
341#382—531no—
342#382C2×D5311yescharacteristic
343#381—1061no—
344#380—531no—
345#380—531no—
346C4×C4⋊C4—531no—
347C4×C4⋊C4—531no—
348C4×C4⋊C4—531no—
349C4×C4⋊C4—531no—
350C4×C4⋊C4—531no—
351C4×C4⋊C4—1061no—
352C4×C4⋊C4—1061no—
353C4×C4⋊C4C2×D5311yescharacteristic
354C4×C22⋊C4—1061no—
355#376—1061no—
356C2×#95—1061no—
357#340—531no—
358#340—531no—
359#340—531no—
360#340—531no—
361#340—531no—
362#340—531no—
363#340C2×D5311yescharacteristic
364#340C2×D5311yescharacteristic
365C53#2136311yescharacteristic, complemented, Sylow
366C23×C4—531no—
367C23×C4—531no—
368C23×C4—531no—
369C23×C4—1061no—
370C23×C4—1061no—
371C22×C8—531no—
372C22×C8—531no—
373C22×C8C22×D5311yescharacteristic
374C22×C8C22×D5311yescharacteristic
375C2×C4⋊C4—531no—
376C2×C4⋊C4—531no—
377C2×C4⋊C4—531no—
378C2×C4⋊C4—531no—
379C2×C4⋊C4—1062no—
380C2×C4⋊C4—1062no—
381C2×C4⋊C4—1061no—
382C2×C4⋊C4—1061no—
383C2×C4⋊C4—1062no—
384C2×C4⋊C4—1062no—
385C2×C4⋊C4C22×D5311yescharacteristic
386C2×C4⋊C4C22×D5311yescharacteristic
387C2×C4⋊C4C22×D5311yescharacteristic
388C2×C4⋊C4C22×D5311yescharacteristic
389C2×C22⋊C4—1061no—
390C2×C22⋊C4—1061no—
391C2×C42—21no—
392C2×C42—531no—
393C2×C42—531no—
394C2×C42—531no—
395C2×C42—534no—
396C2×C42—534no—
397C2×C42—1061no—
398C2×C42—1061no—
399C2×C42—1061no—
400C2×C42—1061no—
401C2×C42—1061no—
402C2×C42—2124no—
403C2×C42C22×D5311yescharacteristic
404#107—22no—
405#107—1062no—
406#107—1062no—
407#107—1062no—
408#106—22no—
409#106—1062no—
410#106—1062no—
411#106—1062no—
412C4⋊C8—534no—
413C4⋊C8C4×D5314yescomplemented
414C22⋊C8—534no—
415C22⋊C8—534no—
416#97—534no—
417#97—534no—
418#95—21no—
419#95—21no—
420#95—21no—
421#95—1061no—
422#95—1061no—
423#95—1061no—
424#95—1061no—
425#95—1061no—
426#95—1061no—
427#95—1061no—
428#95—1061no—
429#95—1061no—
430C24—531no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—531no—
436C22×C4—531no—
437C22×C4—531no—
438C22×C4—534no—
439C22×C4—532no—
440C22×C4—532no—
441C22×C4—534no—
442C22×C4—1062no—
443C22×C4—1062no—
444C22×C4—1064no—
445C22×C4—1064no—
446C22×C4—1061no—
447C22×C4—1061no—
448C22×C4—1061no—
449C22×C4—1061no—
450C22×C4—1061no—
451C22×C4—1061no—
452C22×C4—1061no—
453C22×C4—1061no—
454C22×C4—2121no—
455C22×C4—2121no—
456C22×C4—2122no—
457C22×C4—2122no—
458C22×C4—2122no—
459C22×C4—2122no—
460C22×C4D4×D5311yescharacteristic
461C22×C4D4×D5311yescharacteristic
462C22×C4C23×D5311yescharacteristic
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—532no—
468C2×C8—532no—
469C2×C8—1061no—
470C2×C8—1061no—
471C2×C8—1061no—
472C2×C8—1061no—
473C2×C8C2×C4×D5312yes—
474C2×C8C2×C4×D5312yes—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—532no—
482C4⋊C4—532no—
483C4⋊C4—1064no—
484C4⋊C4—1064no—
485C4⋊C4—1061no—
486C4⋊C4—1061no—
487C4⋊C4—1061no—
488C4⋊C4—1061no—
489C4⋊C4—1061no—
490C4⋊C4—1061no—
491C4⋊C4#1171994712yes—
492C4⋊C4#1171994912yes—
493C22⋊C4—2122no—
494C22⋊C4—2122no—
495C42—42no—
496C42—534no—
497C42—534no—
498C42—532no—
499C42—532no—
500C42—2122no—
501C42—2122no—
502C42—2122no—
503C42—2122no—
504C42—2122no—
505C42C2×C4×D5312yes—
506C42Q8×D5312yes—
507C23—532no—
508C23—532no—
509C23—532no—
510C23—1061no—
511C23—1061no—
512C23—1061no—
513C23—1061no—
514C23C2×D4×D5311yescharacteristic
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—532no—
532C2×C4—532no—
533C2×C4—532no—
534C2×C4—1068no—
535C2×C4—1061no—
536C2×C4—1061no—
537C2×C4—1061no—
538C2×C4—1061no—
539C2×C4—1061no—
540C2×C4—1061no—
541C2×C4—1062no—
542C2×C4—1062no—
543C2×C4—1062no—
544C2×C4—1062no—
545C2×C4—1062no—
546C2×C4—1062no—
547C2×C4—2124no—
548C2×C4—2128no—
549C2×C4—2128no—
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—2121no—
560C2×C4C4⋊C4×D5312yes—
561C2×C4C4⋊C4×D5312yes—
562C2×C4C22×C4×D5311yescharacteristic
563C2×C4C2×Q8×D5311yescharacteristic
564C2×C4#41×D5311yescharacteristic
565C2×C4#41×D5311yescharacteristic
566C2×C4C2×#1171994711yescharacteristic
567C2×C4C2×#1171994911yescharacteristic
568C8—22no—
569C8—22no—
570C8—1062no—
571C8—1062no—
572C22—534no—
573C22—1062no—
574C22—1062no—
575C22—1062no—
576C22—1062no—
577C22—1062no—
578C22—1062no—
579C22C2×C4⋊C4×D5311yescharacteristic
580C22#1681113311yescharacteristic
581C22#1681113611yescharacteristic
582C22AGL(1,ℤ/8)×D5311yescharacteristic
583C22#137×D5311yescharacteristic
584C22#1681142011yescharacteristic
585C22#1681142011yescharacteristic
586C4—22no—
587C4—22no—
588C4—42no—
589C4—42no—
590C4—81no—
591C4—81no—
592C4—1062no—
593C4—1062no—
594C4—1062no—
595C4—4241no—
596C4—4241no—
597C4—2122no—
598C4—2122no—
599C4C4×#1171994711yescharacteristic
600C4C4×#1171994911yescharacteristic
601C4#1681117811yescharacteristic
602C4#1681117911yescharacteristic
603C2—1064no—
604C2#2027719611yescharacteristic
605C2#428×D5311yescharacteristic
606C2#2028330311yescharacteristic
607C2#2028485211yescharacteristic
608C2#2028485811yescharacteristic
609C2#2028488511yescharacteristic
610C2#2028489111yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1949C2
socleC23×C53C22×D4
prime radicalC23×D53C2×D4
cosocleC22×C4×C53C24
hypercenter#1949D53
hypocenterC53#21363
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1949

Complements

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

NH ≅ G/Ntype
4. #20275485C2semidirect
5. #20275483C2semidirect
6. #20275481C2semidirect
11. #20274615C2semidirect
12. #20274614C2semidirect
13. #20274611C2semidirect
14. #20274611C2semidirect
15. C53×#1949C2semidirect
20. #16810679C4semidirect
96. C4⋊#88322C2×C4semidirect
99. #13133127C2×C4semidirect
100. #13133127C2×C4semidirect
120. C4⋊C8×C53C2×C4semidirect
304. #1949D53semidirect
365. C53#21363semidirect
413. C4⋊C8C4×D53semidirect

Maximal subgroups

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

subgroupindex
C2×#168106792
#202771082
C4×C4⋊C4×D532
#202754852
#202754832
#202754812
#202749092
#202749062
#202749042
#202749012
#202746152
#202746142
#202746112
C53×#19492
#2136353