Finite Group Databasebeta

← #27350637

Subgroups — #27350637

Order 13568.

988 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811177C211yesmaximal, characteristic
2#20283343C211yesmaximal, characteristic
3#431×D53C211yesmaximal, characteristic
4#20273134C214yesmaximal, complemented
5#20273116C214yesmaximal, complemented
6#20273111C211yesmaximal, characteristic, complemented
7#20273110C211yesmaximal, characteristic, complemented
8#20273109C211yesmaximal, characteristic, complemented
9C53×#1479C211yesmaximal, characteristic, complemented
10#16811177—21no—
11#16811177—21no—
12#16811177—21no—
13#16811177—21no—
14#16811177C412yes—
15#16811177C412yes—
16#16811177C2212yes—
17#16811177C2212yes—
18C2×#13133216C411yescharacteristic
19C2×#13133215C411yescharacteristic
20C2×#13133207—21no—
21C2×#13133197C411yescharacteristic
22C2×#13133195—21no—
23C2×#13133193C411yescharacteristic
24C2×C4×Dic106—21no—
25C2×C4×Dic106C2211yescharacteristic
26C2×#13133174—21no—
27C2×C42×D53C2211yescharacteristic
28#16810679—21no—
29#16810677—21no—
30#16810676C2211yescharacteristic
31#16810676C2211yescharacteristic
32#16810675C2212yes—
33#16810674—21no—
34#16810674C2212yes—
35#16810673—21no—
36#16810645—21no—
37C4⋊C8×D53C2212yes—
38#16810585—21no—
39C22⋊C8×D53—21no—
40C4×#11719912C2212yes—
41C4×C8×D53—21no—
42#16810355C2214yescomplemented
43Q8⋊#88322C2214yescomplemented
44#16810353C2214yescomplemented
45C2×C4×Q8×C53C2211yescharacteristic
46C53×#431C2211yescharacteristic
47C53×#431C2211yescharacteristic
48C53×Q8⋊C8C2214yescomplemented
49C22×Dic106—21no—
50C22×Dic106C2×C411yescharacteristic
51C22×C4×D53C2×C411yescharacteristic
52C22×C4×D53D411yescharacteristic
53C22×C4×D53D411yescharacteristic
54C2×#11719912—22no—
55C2×C8×D53—22no—
56#13133216—21no—
57#13133216—21no—
58#13133216C2×C412yes—
59#13133216C2×C412yes—
60#13133215—21no—
61#13133215—21no—
62#13133215C2×C412yes—
63#13133215C2×C412yes—
64#13133207—22no—
65#13133207—22no—
66#13133207—22no—
67#13133207—22no—
68#13133197—21no—
69#13133197—21no—
70#13133197C2×C412yes—
71#13133197C2×C412yes—
72#13133195—22no—
73#13133195—22no—
74#13133195—22no—
75#13133195—22no—
76#13133193—21no—
77#13133193—21no—
78#13133193C2×C412yes—
79#13133193C2×C412yes—
80C4×Dic106—21no—
81C4×Dic106—21no—
82C4×Dic106—22no—
83C4×Dic106—22no—
84C4×Dic106—22no—
85C4×Dic106—22no—
86C4×Dic106C2×C412yes—
87C4×Dic106C2312yes—
88#13133174—22no—
89#13133174—22no—
90#13133174—22no—
91#13133174—22no—
92C2×C4⋊Dic53—21no—
93C2×C4⋊Dic53C2×C411yescharacteristic
94C2×#11719919—21no—
95C2×#11719919—21no—
96C2×#11719919—21no—
97C2×#11719919—21no—
98C2×#11719919—21no—
99C2×#11719921—21no—
100C2×#11719921—21no—
101C2×C4×Dic53—21no—
102C2×C4×Dic53C2×C411yescharacteristic
103C2×C4×Dic53D411yescharacteristic
104C2×C4×Dic53D411yescharacteristic
105C42×D53C2×C411yescharacteristic
106C42×D53C2×C411yescharacteristic
107C42×D53D411yescharacteristic
108C42×D53D411yescharacteristic
109C42×D53D411yescharacteristic
110C42×D53D411yescharacteristic
111C42×D53C2311yescharacteristic
112C42×D53C2311yescharacteristic
113C4⋊#88322—21no—
114C4⋊#88322C2312yes—
115#13133146—22no—
116#13133146—22no—
117C22⋊#88322—21no—
118C22⋊#88322—21no—
119#13133129—22no—
120#13133129—22no—
121#13133127—22no—
122C4×#88322—21no—
123C4×#88322C2312yes—
124C8×Dic53—22no—
125C22×Q8×C53C2×C411yescharacteristic
126C4×Q8×C53—21no—
127C4×Q8×C53—21no—
128C4×Q8×C53C2×C412yes—
129C4×Q8×C53C2312yes—
130C2×C4⋊C4×C53—21no—
131C2×C4⋊C4×C53C2×C411yescharacteristic
132C2×C42×C53—21no—
133C2×C42×C53C2311yescharacteristic
134C4⋊C8×C53—21no—
135C4⋊C8×C53C2312yes—
136C22⋊C8×C53—21no—
137C22⋊C8×C53—21no—
138C4×C8×C53—21no—
139C4×C8×C53C2312yes—
140C23×D53C22⋊C411yescharacteristic
141C2×Q8×C53—21no—
142C2×Q8×C53—22no—
143C2×Q8×C53—22no—
144C2×Q8×C53C22×C412yes—
145C22×C4×C53—21no—
146C22×C4×C53—21no—
147C22×C4×C53C22×C411yescharacteristic
148C22×C4×C53C2×D411yescharacteristic
149C22×C4×C53C2×D411yescharacteristic
150C22×Dic53—21no—
151C22×Dic53—21no—
152C22×Dic53C22⋊C411yescharacteristic
153C22×Dic53C22⋊C411yescharacteristic
154C22×Dic53C22⋊C411yescharacteristic
155C2×C4×D53—22no—
156C2×C4×D53—22no—
157C2×C4×D53—24no—
158C2×C4×D53—42no—
159C2×C4×D53C22⋊C411yescharacteristic
160C2×C4×D53C22⋊C411yescharacteristic
161C2×C4×D53C22⋊C411yescharacteristic
162C2×C4×D53C22⋊C411yescharacteristic
163C2×C4×D53C22⋊C411yescharacteristic
164C2×C4×D53C22⋊C411yescharacteristic
165C2×C4×D53C22×C411yescharacteristic
166C2×C4×D53C22×C411yescharacteristic
167C2×C4×D53C2×D411yescharacteristic
168C2×C4×D53C2×D411yescharacteristic
169C2×C4×D53C2×D411yescharacteristic
170C2×C4×D53C2×D411yescharacteristic
171C2×Dic106—22no—
172C2×Dic106—22no—
173C2×Dic106—21no—
174C2×Dic106—22no—
175C2×Dic106—22no—
176C2×Dic106—44no—
177C2×Dic106C22×C412yes—
178C2×C8×C53—22no—
179C2×C8×C53—22no—
180C4⋊C4×C53—22no—
181C4⋊C4×C53—21no—
182C4⋊C4×C53—22no—
183C4⋊C4×C53C22×C412yes—
184C42×C53—22no—
185C42×C53—22no—
186C42×C53C22×C411yescharacteristic
187C42×C53C2×D411yescharacteristic
188C42×C53C2×D411yescharacteristic
189C42×C53C2411yescharacteristic
190#11719921—22no—
191#11719921—22no—
192#11719921—22no—
193#11719921—22no—
194C4⋊Dic53—21no—
195C4⋊Dic53—22no—
196C4⋊Dic53—22no—
197C4⋊Dic53C22×C412yes—
198#11719919—22no—
199#11719919—22no—
200#11719919—22no—
201#11719919—22no—
202#11719919—22no—
203#11719919—22no—
204#11719919—22no—
205#11719919—22no—
206#11719919—22no—
207#11719919—22no—
208C4×Dic53—22no—
209C4×Dic53—22no—
210C4×Dic53C22⋊C411yescharacteristic
211C4×Dic53C22⋊C411yescharacteristic
212C4×Dic53C22⋊C411yescharacteristic
213C4×Dic53C22⋊C411yescharacteristic
214C4×Dic53C22⋊C411yescharacteristic
215C4×Dic53C22⋊C411yescharacteristic
216C4×Dic53C22×C411yescharacteristic
217C4×Dic53C22×C411yescharacteristic
218C4×Dic53C2×D411yescharacteristic
219C4×Dic53C2×D411yescharacteristic
220C4×Dic53C2×D411yescharacteristic
221C4×Dic53C2×D411yescharacteristic
222C2×#88322—22no—
223C2×#88322—22no—
224#11719912—44no—
225C8×D53—44no—
226C23×C53C2×C22⋊C411yescharacteristic
227C22×D53—22no—
228C22×D53—22no—
229C22×D53—41no—
230C22×D53C2×C22⋊C411yescharacteristic
231C22×D53C2×C22⋊C411yescharacteristic
232Q8×C53—22no—
233Q8×C53—22no—
234Q8×C53—42no—
235C2×C4×C53—22no—
236C2×C4×C53—22no—
237C2×C4×C53—22no—
238C2×C4×C53—41no—
239C2×C4×C53—42no—
240C2×C4×C53—42no—
241C2×C4×C53C2×C22⋊C411yescharacteristic
242C2×C4×C53C2×C22⋊C411yescharacteristic
243C2×C4×C53C2×C22⋊C411yescharacteristic
244C2×C4×C53#13112yes—
245C2×C4×C53#13112yes—
246C2×C4×C53C23×C411yescharacteristic
247C2×C4×C53C22×D411yescharacteristic
248C2×C4×C53C22×D411yescharacteristic
249C2×Dic53—22no—
250C2×Dic53—22no—
251C2×Dic53—22no—
252C2×Dic53—22no—
253C2×Dic53—22no—
254C2×Dic53—42no—
255C2×Dic53—42no—
256C2×Dic53C2×C22⋊C411yescharacteristic
257C2×Dic53C2×C22⋊C411yescharacteristic
258C2×Dic53C2×C22⋊C411yescharacteristic
259C2×Dic53C2×C22⋊C411yescharacteristic
260C2×Dic53C2×C22⋊C411yescharacteristic
261C2×Dic53C2×C22⋊C411yescharacteristic
262C4×D53—22no—
263C4×D53—21no—
264C4×D53—21no—
265C4×D53—21no—
266C4×D53—21no—
267C4×D53—22no—
268C4×D53—22no—
269C4×D53—22no—
270C4×D53—22no—
271C4×D53—22no—
272C4×D53—22no—
273C4×D53—41no—
274C4×D53—41no—
275C4×D53#13714yes—
276Dic106—22no—
277Dic106—22no—
278Dic106—44no—
279Dic106—42no—
280Dic106—44no—
281C8×C53—42no—
282C8×C53—42no—
283#88322—42no—
284#88322—42no—
285#17464—531nomaximal, Sylow
286C22×C53—21no—
287C22×C53—21no—
288C22×C53#40812yes—
289C22×C53C22×C22⋊C411yescharacteristic
290C2×D53—21no—
291C2×D53—21no—
292C2×D53—21no—
293C2×D53—21no—
294C2×D53—21no—
295C2×D53—44no—
296C2×D53#41912yes—
297C4×C53—21no—
298C4×C53—21no—
299C4×C53—21no—
300C4×C53—42no—
301C4×C53—42no—
302C4×C53#40812yes—
303C4×C53C2×#13111yescharacteristic
304C4×C53C2×#13111yescharacteristic
305C4×C53C2×#13712yes—
306Dic53—21no—
307Dic53—21no—
308Dic53—21no—
309Dic53—21no—
310Dic53—42no—
311Dic53—42no—
312Dic53#41912yes—
313Dic53#42112yes—
314C2×#533—531no—
315#2920—531no—
316C2×#431—531no—
317#1485—534no—
318#1481—534no—
319#1479—531no—
320#1479—531no—
321#1479—531no—
322#1479D5311yescharacteristic, complemented
323C2×C53—22no—
324C2×C53C2×#40811yescharacteristic
325C2×C53C2×#41911yescharacteristic
326C2×C53C2×#42111yescharacteristic
327D53—22no—
328D53—41no—
329#533—532no—
330#533—532no—
331#533—532no—
332#533—532no—
333#533—1061no—
334#533—1061no—
335#533—1061no—
336#533—1061no—
337C2×C4⋊Q8—531no—
338C2×C4⋊Q8—531no—
339C2×#125—1061no—
340C2×C22⋊Q8—531no—
341C2×C22⋊Q8—531no—
342C2×C22⋊Q8—1061no—
343C2×C4×Q8—531no—
344C2×C4×Q8—1061no—
345C2×C4×Q8C2×D5311yescharacteristic
346C2×#117—1061no—
347C22×C42—531no—
348#432—1061no—
349#432—1061no—
350#431—531no—
351#431—532no—
352#431—531no—
353#431—532no—
354#431—1061no—
355#431—1061no—
356#431C2×D5311yescharacteristic
357#431C2×D5311yescharacteristic
358#423—1061no—
359C2×C4⋊C8—532no—
360#407—1061no—
361C2×C22⋊C8—1061no—
362C4×M16—532no—
363C2×C4×C8—1061no—
364Q8⋊C8—534no—
365Q8⋊C8—534no—
366Q8⋊C8—534no—
367Q8⋊C8C2×D5314yescomplemented
368C53#1746411yescharacteristic, complemented, Sylow
369C22×Q8—531no—
370C22×Q8—1061no—
371C22×Q8C4×D5311yescharacteristic
372C23×C4—531no—
373C23×C4—531no—
374C23×C4—531no—
375C2×M16—1062no—
376C22×C8—1062no—
377C4⋊Q8—532no—
378C4⋊Q8—532no—
379C4⋊Q8—532no—
380C4⋊Q8—532no—
381C4⋊Q8—1061no—
382C4⋊Q8—1061no—
383C4⋊Q8—1061no—
384C4⋊Q8—1061no—
385#125—1062no—
386#125—1062no—
387#125—1062no—
388#125—1062no—
389C22⋊Q8—532no—
390C22⋊Q8—532no—
391C22⋊Q8—532no—
392C22⋊Q8—532no—
393C22⋊Q8—1061no—
394C22⋊Q8—1061no—
395C22⋊Q8—1062no—
396C22⋊Q8—1062no—
397C22⋊Q8—1062no—
398C22⋊Q8—1062no—
399C22⋊Q8—1061no—
400C22⋊Q8—1061no—
401C4×Q8—21no—
402C4×Q8—21no—
403C4×Q8—532no—
404C4×Q8—532no—
405C4×Q8—1062no—
406C4×Q8—1062no—
407C4×Q8—1062no—
408C4×Q8—1062no—
409C4×Q8—1061no—
410C4×Q8—1061no—
411C4×Q8C4×D5312yes—
412C4×Q8C22×D5312yes—
413#117—1062no—
414#117—1062no—
415#117—1062no—
416#117—1062no—
417C2×C4⋊C4—21no—
418C2×C4⋊C4—531no—
419C2×C4⋊C4—1061no—
420C2×C4⋊C4—1061no—
421C2×C4⋊C4—1061no—
422C2×C4⋊C4—1061no—
423C2×C4⋊C4—1061no—
424C2×C4⋊C4—1061no—
425C2×C4⋊C4C4×D5311yescharacteristic
426C2×C22⋊C4—1061no—
427C2×C22⋊C4—1061no—
428C2×C42—21no—
429C2×C42—531no—
430C2×C42—531no—
431C2×C42—531no—
432C2×C42—531no—
433C2×C42—531no—
434C2×C42—531no—
435C2×C42—531no—
436C2×C42—531no—
437C2×C42—531no—
438C2×C42—531no—
439C2×C42—531no—
440C2×C42—1061no—
441C2×C42C22×D5311yescharacteristic
442C4⋊C8—21no—
443C4⋊C8—532no—
444C4⋊C8—1062no—
445C4⋊C8—1062no—
446C4⋊C8—1061no—
447C4⋊C8C22×D5312yes—
448C22⋊C8—21no—
449C22⋊C8—21no—
450C22⋊C8—1062no—
451C22⋊C8—1062no—
452C22⋊C8—1061no—
453C22⋊C8—1061no—
454#97—1062no—
455C4×C8—21no—
456C4×C8—532no—
457C4×C8—1062no—
458C4×C8—1061no—
459C4×C8C22×D5312yes—
460C24—531no—
461C2×Q8—21no—
462C2×Q8—22no—
463C2×Q8—22no—
464C2×Q8—532no—
465C2×Q8—1062no—
466C2×Q8—1062no—
467C2×Q8—1061no—
468C2×Q8—1062no—
469C2×Q8—1062no—
470C2×Q8—2124no—
471C2×Q8C2×C4×D5312yes—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—531no—
475C22×C4—531no—
476C22×C4—531no—
477C22×C4—531no—
478C22×C4—531no—
479C22×C4—531no—
480C22×C4—531no—
481C22×C4—531no—
482C22×C4—531no—
483C22×C4—531no—
484C22×C4—531no—
485C22×C4—531no—
486C22×C4—531no—
487C22×C4—531no—
488C22×C4—531no—
489C22×C4—1064no—
490C22×C4—1062no—
491C22×C4—1062no—
492C22×C4—1061no—
493C22×C4—1061no—
494C22×C4—2122no—
495C22×C4C2×C4×D5311yescharacteristic
496C22×C4D4×D5311yescharacteristic
497C22×C4D4×D5311yescharacteristic
498M16—2124no—
499C2×C8—22no—
500C2×C8—22no—
501C2×C8—1062no—
502C2×C8—1062no—
503C2×C8—2124no—
504C4⋊C4—22no—
505C4⋊C4—21no—
506C4⋊C4—22no—
507C4⋊C4—532no—
508C4⋊C4—1062no—
509C4⋊C4—1062no—
510C4⋊C4—1062no—
511C4⋊C4—1062no—
512C4⋊C4—1062no—
513C4⋊C4—1062no—
514C4⋊C4—1062no—
515C4⋊C4—1062no—
516C4⋊C4—1062no—
517C4⋊C4—1062no—
518C4⋊C4—1061no—
519C4⋊C4—1062no—
520C4⋊C4—1062no—
521C4⋊C4C2×C4×D5312yes—
522C22⋊C4—1062no—
523C22⋊C4—1062no—
524C22⋊C4—1062no—
525C22⋊C4—1062no—
526C42—22no—
527C42—22no—
528C42—531no—
529C42—531no—
530C42—531no—
531C42—531no—
532C42—531no—
533C42—531no—
534C42—531no—
535C42—531no—
536C42—531no—
537C42—531no—
538C42—531no—
539C42—531no—
540C42—1062no—
541C42—1062no—
542C42C2×C4×D5311yescharacteristic
543C42D4×D5311yescharacteristic
544C42D4×D5311yescharacteristic
545C42C23×D5311yescharacteristic
546C23—531no—
547C23—531no—
548C23—1062no—
549C23—1062no—
550C23—2121no—
551C23C22⋊C4×D5311yescharacteristic
552Q8—22no—
553Q8—22no—
554Q8—42no—
555Q8—1062no—
556Q8—1062no—
557Q8—2124no—
558Q8—2124no—
559Q8—2122no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—41no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—531no—
567C2×C4—531no—
568C2×C4—534no—
569C2×C4—531no—
570C2×C4—531no—
571C2×C4—531no—
572C2×C4—531no—
573C2×C4—1062no—
574C2×C4—1062no—
575C2×C4—1062no—
576C2×C4—1062no—
577C2×C4—1062no—
578C2×C4—1062no—
579C2×C4—1062no—
580C2×C4—1062no—
581C2×C4—1062no—
582C2×C4—1061no—
583C2×C4—1061no—
584C2×C4—1061no—
585C2×C4—1061no—
586C2×C4—1062no—
587C2×C4—1062no—
588C2×C4—1062no—
589C2×C4—2121no—
590C2×C4—2121no—
591C2×C4—2122no—
592C2×C4—2122no—
593C2×C4C22⋊C4×D5311yescharacteristic
594C2×C4C22⋊C4×D5311yescharacteristic
595C2×C4C22⋊C4×D5311yescharacteristic
596C2×C4#1313322312yes—
597C2×C4#1313322512yes—
598C2×C4C22×C4×D5311yescharacteristic
599C2×C4C2×D4×D5311yescharacteristic
600C2×C4C2×D4×D5311yescharacteristic
601C8—42no—
602C8—42no—
603C8—2122no—
604C8—2122no—
605C22—21no—
606C22—21no—
607C22—532no—
608C22—1061no—
609C22—1061no—
610C22—1061no—
611C22—1061no—
612C22—1061no—
613C22—2124no—
614C22#1681058812yes—
615C22C2×C22⋊C4×D5311yescharacteristic
616C4—21no—
617C4—21no—
618C4—21no—
619C4—42no—
620C4—42no—
621C4—532no—
622C4—532no—
623C4—1061no—
624C4—1061no—
625C4—1061no—
626C4—1061no—
627C4—2122no—
628C4—2122no—
629C4#1681058812yes—
630C4C2×#1313322311yescharacteristic
631C4C2×#1313322511yescharacteristic
632C4#137×D5312yes—
633C2—22no—
634C2—1062no—
635C2—2121no—
636C2C2×#1681058811yescharacteristic
637C2#419×D5311yescharacteristic
638C2#421×D5311yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC42C23×D53
FittingC53×#1479C2
socleC22×C53C22×C22⋊C4
prime radicalC23×D53C22⋊C4
cosocleC42×C53C24
hypercenter#1479D53
hypocenterC53#17464
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1479

Complements

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

NH ≅ G/Ntype
4. #20273134C2semidirect
5. #20273116C2semidirect
6. #20273111C2semidirect
7. #20273110C2semidirect
8. #20273109C2semidirect
9. C53×#1479C2semidirect
42. #16810355C22semidirect
43. Q8⋊#88322C22semidirect
44. #16810353C22semidirect
48. C53×Q8⋊C8C22semidirect
322. #1479D53semidirect
367. Q8⋊C8C2×D53semidirect
368. C53#17464semidirect

Maximal subgroups

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

subgroupindex
C2×#168111772
#202833432
#431×D532
#202731342
#202731162
#202731112
#202731102
#202731092
C53×#14792
#1746453