Finite Group Databasebeta

← #27391237

Subgroups — #27391237

Order 13568.

976 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
1C2×#16810957C211yesmaximal, characteristic
2#20277253C211yesmaximal, characteristic
3C4×C4⋊C4×D53C211yesmaximal, characteristic
4#20275353C211yesmaximal, characteristic, complemented
5#20275308C211yesmaximal, characteristic, complemented
6#20275307C211yesmaximal, characteristic, complemented
7#20275019C211yesmaximal, characteristic
8#20274901C211yesmaximal, characteristic
9#20274804C211yesmaximal, characteristic
10#107×Dic53C211yesmaximal, characteristic, complemented
11#20274607C211yesmaximal, characteristic, complemented
12#103×Dic53C211yesmaximal, characteristic, complemented
13#103×Dic53C211yesmaximal, characteristic, complemented
14#20274472C211yesmaximal, characteristic
15C53×#1928C211yesmaximal, characteristic, complemented
16C2×#13133193C2211yescharacteristic
17C2×Q8×Dic53C2211yescharacteristic
18C2×C4⋊C4×D53C2211yescharacteristic
19C2×C4⋊C4×D53C2211yescharacteristic
20C2×C42×D53—21no—
21C2×C42×D53C2211yescharacteristic
22#16810957C418yescomplemented
23C2×#13133151C2211yescharacteristic
24C2×#13133150C2211yescharacteristic
25C2×#13133150C2211yescharacteristic
26C2×#13133141C2211yescharacteristic
27C2×#13133141C2211yescharacteristic
28C2×#13133129C2211yescharacteristic
29C2×#13133129C2211yescharacteristic
30C2×C8×Dic53C2211yescharacteristic
31C2×C8×Dic53C2211yescharacteristic
32#16810521C2211yescharacteristic
33#16810510—21no—
34#16810497—21no—
35#16810480C2211yescharacteristic
36C4⋊C4×Dic53C2211yescharacteristic
37C4⋊C4×Dic53C2211yescharacteristic
38C4⋊C4×Dic53C2211yescharacteristic
39C4×#11719919—21no—
40C4×C4⋊Dic53C2211yescharacteristic
41#16810474—21no—
42#16810474—21no—
43#16810470—21no—
44#95×D53—21no—
45#16810386C2211yescharacteristic
46#16810386C2211yescharacteristic
47#16810386C2211yescharacteristic
48#16810385C2211yescharacteristic
49#16810385C2211yescharacteristic
50#16810385C2211yescharacteristic
51#16810385C2211yescharacteristic
52#16810378C2211yescharacteristic
53C2×#107×C53C2211yescharacteristic
54C2×#103×C53C2211yescharacteristic
55C2×#103×C53C2211yescharacteristic
56C53×#391C2211yescharacteristic
57C4×C4⋊C4×C53C2211yescharacteristic
58C53×#340C2211yescharacteristic
59C53×#336C2211yescharacteristic
60C22×C4×D53—21no—
61C22×C4×D53—21no—
62C22×C4×D53D411yescharacteristic
63C22×C4×D53D411yescharacteristic
64C22×C4×D53C2311yescharacteristic
65C22×#88322C2311yescharacteristic
66C22×#88322C2311yescharacteristic
67#13133193—22no—
68#13133193C2×C414yes—
69Q8×Dic53—22no—
70Q8×Dic53—22no—
71C2×C4⋊Dic53—21no—
72C2×C4⋊Dic53C2311yescharacteristic
73C2×C4⋊Dic53C2311yescharacteristic
74C2×C4⋊Dic53C2311yescharacteristic
75C2×C4⋊Dic53C2311yescharacteristic
76C2×#11719919—21no—
77C2×#11719919—21no—
78C2×#11719919—21no—
79C2×#11719919—21no—
80C4⋊C4×D53—22no—
81C4⋊C4×D53—22no—
82C4⋊C4×D53—22no—
83C4⋊C4×D53C2×C414yes—
84C2×#11719921—21no—
85C2×#11719921—21no—
86C2×C4×Dic53—21no—
87C2×C4×Dic53—21no—
88C2×C4×Dic53—21no—
89C2×C4×Dic53—21no—
90C2×C4×Dic53—21no—
91C2×C4×Dic53D411yescharacteristic
92C2×C4×Dic53D411yescharacteristic
93C2×C4×Dic53C2311yescharacteristic
94C42×D53—22no—
95C42×D53—22no—
96C42×D53—44no—
97#13133151C2×C414yescomplemented
98#13133150—22no—
99#13133150C2×C414yescomplemented
100#13133141—22no—
101#13133141C2×C414yescomplemented
102#13133129—22no—
103#13133129C2×C414yes—
104C8×Dic53—22no—
105C8×Dic53—22no—
106#13133124—21no—
107#13133124—21no—
108#13133124—21no—
109#13133124—21no—
110#13133124—21no—
111#13133124—21no—
112C22×Q8×C53C2311yescharacteristic
113C22×C8×C53C2311yescharacteristic
114C22×C8×C53C2311yescharacteristic
115C2×C4⋊C4×C53C2311yescharacteristic
116C2×C4⋊C4×C53C2311yescharacteristic
117C2×C4⋊C4×C53C2311yescharacteristic
118C2×C42×C53—21no—
119C2×C42×C53C2311yescharacteristic
120#107×C53—22no—
121#103×C53—22no—
122#103×C53C2×C414yescomplemented
123#95×C53—21no—
124#95×C53—21no—
125#95×C53—21no—
126C23×D53C2×D411yescharacteristic
127C2×Q8×C53—21no—
128C2×Q8×C53—21no—
129C2×Q8×C53—21no—
130C2×Q8×C53—21no—
131C2×Q8×C53—21no—
132C2×Q8×C53C22×C412yes—
133C22×C4×C53—21no—
134C22×C4×C53—21no—
135C22×C4×C53—21no—
136C22×C4×C53—21no—
137C22×C4×C53C2×D411yescharacteristic
138C22×C4×C53C2×D411yescharacteristic
139C22×C4×C53C2411yescharacteristic
140C22×Dic53—21no—
141C22×Dic53—21no—
142C22×Dic53—21no—
143C22×Dic53—21no—
144C22×Dic53C2×D411yescharacteristic
145C22×Dic53C2×D411yescharacteristic
146C22×Dic53C2×D411yescharacteristic
147C2×C4×D53—21no—
148C2×C4×D53—21no—
149C2×C4×D53—21no—
150C2×C4×D53—21no—
151C2×C4×D53—22no—
152C2×C4×D53—22no—
153C2×C4×D53—24no—
154C2×C4×D53—24no—
155C2×C4×D53—22no—
156C2×C4×D53—22no—
157C2×C4×D53—41no—
158C2×C4×D53—42no—
159C2×C4×D53—42no—
160C2×C4×D53—42no—
161C2×C4×D53—42no—
162C2×C4×D53Q1614yes—
163C2×C4×D53C22×C412yes—
164C2×C4×D53#4112yes—
165C2×C8×C53—21no—
166C2×C8×C53—21no—
167C2×C8×C53—21no—
168C2×C8×C53—21no—
169C2×C8×C53—21no—
170C2×C8×C53C22×C412yes—
171C4⋊C4×C53—21no—
172C4⋊C4×C53—21no—
173C4⋊C4×C53—21no—
174C4⋊C4×C53—21no—
175C4⋊C4×C53C22×C412yes—
176C4⋊C4×C53#4112yes—
177C42×C53—21no—
178C42×C53—21no—
179C42×C53—42no—
180#11719921—24no—
181#11719921—42no—
182C4⋊Dic53—21no—
183C4⋊Dic53—21no—
184C4⋊Dic53—21no—
185C4⋊Dic53—21no—
186C4⋊Dic53—21no—
187C4⋊Dic53—42no—
188C4⋊Dic53C22×C412yes—
189C4⋊Dic53C22×C412yes—
190C4⋊Dic53#4112yes—
191#11719919—24no—
192#11719919—24no—
193#11719919—24no—
194#11719919—42no—
195C4×Dic53—22no—
196C4×Dic53—22no—
197C4×Dic53—42no—
198C4×Dic53—42no—
199C4×Dic53—42no—
200C4×Dic53—42no—
201C4×Dic53—42no—
202C4×Dic53#4112yes—
203C4×Dic53#4112yes—
204C2×#88322—21no—
205C2×#88322—21no—
206C2×#88322—21no—
207C2×#88322—21no—
208C2×#88322—21no—
209C2×#88322C22×C412yes—
210C23×C53C22×D411yescharacteristic
211C22×D53—21no—
212C22×D53—21no—
213C22×D53—21no—
214C22×D53—21no—
215C22×D53C4×D412yes—
216C22×D53C2×Q1612yes—
217C22×D53#13512yes—
218Q8×C53—22no—
219Q8×C53—22no—
220Q8×C53—41no—
221Q8×C53—41no—
222C2×C4×C53—21no—
223C2×C4×C53—21no—
224C2×C4×C53—22no—
225C2×C4×C53—22no—
226C2×C4×C53—21no—
227C2×C4×C53—22no—
228C2×C4×C53—21no—
229C2×C4×C53—21no—
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×C53C2×Q1612yes—
240C2×C4×C53C23×C411yescharacteristic
241C2×C4×C53C2×#4111yescharacteristic
242C2×C4×C53C2×#4111yescharacteristic
243C2×C4×C53C2×#4111yescharacteristic
244C2×C4×C53C2×#4111yescharacteristic
245C2×C4×C53C2×#4111yescharacteristic
246C2×Dic53—21no—
247C2×Dic53—21no—
248C2×Dic53—21no—
249C2×Dic53—21no—
250C2×Dic53—22no—
251C2×Dic53—22no—
252C2×Dic53—22no—
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—41no—
262C2×Dic53C4×D412yes—
263C2×Dic53#12312yes—
264C2×Dic53#12312yes—
265C2×Dic53C2×Q1612yes—
266C2×Dic53AGL(1,ℤ/8)12yes—
267C4×D53—22no—
268C4×D53—22no—
269C4×D53—22no—
270C4×D53—22no—
271C4×D53—28no—
272C4×D53—44no—
273C4×D53—48no—
274C4×D53—48no—
275C8×C53—22no—
276C8×C53—41no—
277#88322—22no—
278#88322—41no—
279C4×#484—531nomaximal, Sylow
280C22×C53C2×C4×D411yescharacteristic
281C22×C53C2×#12311yescharacteristic
282C22×C53#52511yescharacteristic
283C22×C53C22×Q1611yescharacteristic
284C22×C53C2×#13511yescharacteristic
285C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
286C22×C53#57511yescharacteristic
287C2×D53—22no—
288C2×D53—22no—
289C2×D53—22no—
290C2×D53—22no—
291C2×D53—22no—
292C2×D53—22no—
293C2×D53C4×Q1614yes—
294C4×C53—22no—
295C4×C53—41no—
296C4×C53—42no—
297C4×C53—42no—
298C4×C53—42no—
299C4×C53—81no—
300C4×C53C4×#4111yescharacteristic
301C4×C53C4×#4111yescharacteristic
302C4×C53#52911yescharacteristic
303C4×C53#52911yescharacteristic
304Dic53—22no—
305Dic53—22no—
306Dic53—42no—
307Dic53—42no—
308Dic53—42no—
309Dic53—81no—
310Dic53#48414yescomplemented
311C2×#484—531no—
312C4×C22⋊Q8—531no—
313C2×C4×C4⋊C4—531no—
314#1928—531no—
315#1928D5311yescharacteristic, complemented
316#1921—531no—
317#1921—531no—
318#1872—531no—
319#1852—531no—
320#1824—531no—
321C4×#107—531no—
322C4×#107—531no—
323C4×#103—531no—
324C4×#103—531no—
325C4×C22⋊C8—531no—
326C2×C53C4×#12311yescharacteristic
327C2×C53C2×C4×Q1611yescharacteristic
328C2×C53C4×AGL(1,ℤ/8)11yescharacteristic
329C2×C53C2×#48411yescharacteristic
330C2×C53#309111yescharacteristic
331C2×C53#310411yescharacteristic
332C2×C53#310511yescharacteristic
333D53—24no—
334C2×C22⋊Q8—531no—
335C2×C4×Q8—531no—
336C22×C4⋊C4—531no—
337C22×C4⋊C4—531no—
338C22×C42—531no—
339C22×C42—1061no—
340#484—538no—
341C2×#107—531no—
342C2×#107—531no—
343C2×#107—531no—
344C2×#107C2×D5311yescharacteristic
345C2×#103—531no—
346C2×#103—531no—
347C2×#103C2×D5311yescharacteristic
348C2×#103C2×D5311yescharacteristic
349C2×C22⋊C8—531no—
350C2×C22⋊C8—531no—
351C2×C4×C8—531no—
352C2×C4×C8—531no—
353#391C2×D5311yescharacteristic
354#389—531no—
355#387—1061no—
356#385—1061no—
357#380—531no—
358C4×C4⋊C4—531no—
359C4×C4⋊C4—531no—
360C4×C4⋊C4—531no—
361C4×C4⋊C4—531no—
362C4×C4⋊C4—1061no—
363C4×C4⋊C4—1061no—
364C4×C4⋊C4—1061no—
365C4×C4⋊C4C2×D5311yescharacteristic
366C4×C22⋊C4—1061no—
367C2×#95—1061no—
368#340—531no—
369#340—531no—
370#340—531no—
371#340—531no—
372#340—531no—
373#340—531no—
374#340—531no—
375#340C2×D5311yescharacteristic
376#336—531no—
377#336C2×D5311yescharacteristic
378C53C4×#48411yescharacteristic, complemented, Sylow
379C22×Q8C22×D5311yescharacteristic
380C23×C4—531no—
381C23×C4—531no—
382C23×C4—531no—
383C23×C4—1061no—
384C23×C4—1061no—
385C22×C8—531no—
386C22×C8—531no—
387C22×C8C22×D5311yescharacteristic
388C22×C8C22×D5311yescharacteristic
389C22⋊Q8—534no—
390C22⋊Q8—1062no—
391C4×Q8—1062no—
392C4×Q8—1062no—
393C2×C4⋊C4—531no—
394C2×C4⋊C4—531no—
395C2×C4⋊C4—531no—
396C2×C4⋊C4—534no—
397C2×C4⋊C4—531no—
398C2×C4⋊C4—1062no—
399C2×C4⋊C4—1062no—
400C2×C4⋊C4—1061no—
401C2×C4⋊C4—1061no—
402C2×C4⋊C4—1061no—
403C2×C4⋊C4—1062no—
404C2×C4⋊C4—1061no—
405C2×C4⋊C4—1061no—
406C2×C4⋊C4C22×D5311yescharacteristic
407C2×C4⋊C4C22×D5311yescharacteristic
408C2×C4⋊C4C22×D5311yescharacteristic
409C2×C22⋊C4—1061no—
410C2×C22⋊C4—1061no—
411C2×C42—21no—
412C2×C42—531no—
413C2×C42—531no—
414C2×C42—531no—
415C2×C42—1062no—
416C2×C42—1062no—
417C2×C42—1061no—
418C2×C42—1061no—
419C2×C42—1061no—
420C2×C42—1061no—
421C2×C42—1061no—
422C2×C42—2124no—
423C2×C42C22×D5311yescharacteristic
424#107—22no—
425#107—534no—
426#107—534no—
427#107—1062no—
428#103—22no—
429#103—534no—
430#103—1062no—
431#103C4×D5314yescomplemented
432C22⋊C8—534no—
433C22⋊C8—1062no—
434C4×C8—1062no—
435C4×C8—1062no—
436#95—21no—
437#95—21no—
438#95—21no—
439#95—1061no—
440#95—1061no—
441#95—1061no—
442#95—1061no—
443#95—1061no—
444#95—1061no—
445C24—531no—
446C2×Q8—21no—
447C2×Q8—21no—
448C2×Q8—21no—
449C2×Q8—21no—
450C2×Q8—21no—
451C2×Q8C2×C4×D5312yes—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—531no—
457C22×C4—531no—
458C22×C4—531no—
459C22×C4—534no—
460C22×C4—532no—
461C22×C4—532no—
462C22×C4—1062no—
463C22×C4—1062no—
464C22×C4—1062no—
465C22×C4—1064no—
466C22×C4—1062no—
467C22×C4—1064no—
468C22×C4—1061no—
469C22×C4—1061no—
470C22×C4—1061no—
471C22×C4—1061no—
472C22×C4—1061no—
473C22×C4—1061no—
474C22×C4—1061no—
475C22×C4—1061no—
476C22×C4—2121no—
477C22×C4—2122no—
478C22×C4—2122no—
479C22×C4—2122no—
480C22×C4—2122no—
481C22×C4D4×D5311yescharacteristic
482C22×C4D4×D5311yescharacteristic
483C22×C4C23×D5311yescharacteristic
484C2×C8—21no—
485C2×C8—21no—
486C2×C8—21no—
487C2×C8—21no—
488C2×C8—21no—
489C2×C8—532no—
490C2×C8—1061no—
491C2×C8—1061no—
492C2×C8—1061no—
493C2×C8—1061no—
494C2×C8—1061no—
495C2×C8C2×C4×D5312yes—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—532no—
501C4⋊C4—532no—
502C4⋊C4—532no—
503C4⋊C4—1064no—
504C4⋊C4—1064no—
505C4⋊C4—1064no—
506C4⋊C4—1061no—
507C4⋊C4—1061no—
508C4⋊C4—1061no—
509C4⋊C4—1061no—
510C4⋊C4—1061no—
511C4⋊C4—2122no—
512C4⋊C4—2122no—
513C4⋊C4C2×C4×D5312yes—
514C4⋊C4#1171994712yes—
515C22⋊C4—1064no—
516C22⋊C4—2122no—
517C42—21no—
518C42—21no—
519C42—42no—
520C42—532no—
521C42—532no—
522C42—1062no—
523C42—1062no—
524C42—2122no—
525C42—2122no—
526C42—2122no—
527C42—2122no—
528C42—2122no—
529C23—532no—
530C23—532no—
531C23—532no—
532C23—1061no—
533C23—1061no—
534C23—1061no—
535C23—1061no—
536C23C2×D4×D5311yescharacteristic
537Q8—22no—
538Q8—22no—
539Q8—41no—
540Q8—41no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—21no—
546C2×C4—22no—
547C2×C4—21no—
548C2×C4—21no—
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—532no—
559C2×C4—532no—
560C2×C4—532no—
561C2×C4—532no—
562C2×C4—532no—
563C2×C4—1068no—
564C2×C4—1061no—
565C2×C4—1061no—
566C2×C4—1061no—
567C2×C4—1061no—
568C2×C4—1062no—
569C2×C4—1062no—
570C2×C4—1062no—
571C2×C4—1062no—
572C2×C4—1062no—
573C2×C4—1062no—
574C2×C4—1062no—
575C2×C4—2124no—
576C2×C4—2128no—
577C2×C4—2128no—
578C2×C4—2121no—
579C2×C4—2121no—
580C2×C4—2121no—
581C2×C4—2121no—
582C2×C4—2121no—
583C2×C4—2121no—
584C2×C4—2121no—
585C2×C4—2121no—
586C2×C4—2121no—
587C2×C4Q16×D5312yes—
588C2×C4C22×C4×D5311yescharacteristic
589C2×C4#41×D5311yescharacteristic
590C2×C4#41×D5311yescharacteristic
591C2×C4#41×D5311yescharacteristic
592C2×C4#41×D5311yescharacteristic
593C2×C4C2×#1171994711yescharacteristic
594C8—22no—
595C8—41no—
596C8—1062no—
597C8—2121no—
598C22—534no—
599C22—1062no—
600C22—1062no—
601C22—1062no—
602C22—1062no—
603C22—1062no—
604C22—1062no—
605C22C4×D4×D5311yescharacteristic
606C22#123×D5311yescharacteristic
607C22#1681113311yescharacteristic
608C22C2×Q16×D5311yescharacteristic
609C22#135×D5311yescharacteristic
610C22AGL(1,ℤ/8)×D5311yescharacteristic
611C22#1681142211yescharacteristic
612C4—22no—
613C4—42no—
614C4—42no—
615C4—41no—
616C4—42no—
617C4—81no—
618C4—534no—
619C4—1062no—
620C4—1062no—
621C4—4241no—
622C4—2122no—
623C4—2122no—
624C4—2122no—
625C4C4×#1171994711yescharacteristic
626C4#1681109111yescharacteristic
627C4#1681115211yescharacteristic
628C4#1681115611yescharacteristic
629C2—1064no—
630C2#2027724211yescharacteristic
631C2C4×Q16×D5311yescharacteristic
632C2#2028352011yescharacteristic
633C2#484×D5311yescharacteristic
634C2#2028463311yescharacteristic
635C2#2028475311yescharacteristic
636C2#2028476111yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1928C2
socleC23×C53C22×D4
prime radicalC23×D53C2×D4
cosocleC22×C4×C53C24
hypercenter#1928D53
hypocenterC53C4×#484
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1928

Complements

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

NH ≅ G/Ntype
4. #20275353C2semidirect
5. #20275308C2semidirect
6. #20275307C2semidirect
10. #107×Dic53C2semidirect
11. #20274607C2semidirect
12. #103×Dic53C2semidirect
13. #103×Dic53C2semidirect
15. C53×#1928C2semidirect
22. #16810957C4semidirect
97. #13133151C2×C4semidirect
99. #13133150C2×C4semidirect
101. #13133141C2×C4semidirect
122. #103×C53C2×C4semidirect
310. Dic53#484semidirect
315. #1928D53semidirect
378. C53C4×#484semidirect
431. #103C4×D53semidirect

Maximal subgroups

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

subgroupindex
C2×#168109572
#202772532
C4×C4⋊C4×D532
#202753532
#202753082
#202753072
#202750192
#202749012
#202748042
#107×Dic532
#202746072
#103×Dic532
#202744722
C53×#19282
C4×#48453