Finite Group Databasebeta

← #27351467

Subgroups — #27351467

Order 13568.

1058 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#533×D53C211yesmaximal, characteristic
2#431×D53C211yesmaximal, characteristic
3#20277121C211yesmaximal, characteristic
4#20273285C214yesmaximal
5#20273280C211yesmaximal, characteristic, complemented
6#20273279C211yesmaximal, characteristic, complemented
7#20273278C211yesmaximal, characteristic, complemented
8#20273212C214yesmaximal, complemented
9C53×#1518C211yesmaximal, characteristic, complemented
10#16811180C2211yescharacteristic
11#16811179—21no—
12#16811178C411yescharacteristic, complemented
13#16811178C411yescharacteristic, complemented
14#16811177—21no—
15#16811177C412yescomplemented
16#16811176—21no—
17#16811175—21no—
18#16811175C2212yes—
19C4⋊Q8×D53C411yescharacteristic, complemented
20C4⋊Q8×D53C411yescharacteristic, complemented
21#125×D53—21no—
22C22⋊Q8×D53—21no—
23C22⋊Q8×D53C2212yes—
24C4×Q8×D53—21no—
25C4×Q8×D53C412yescomplemented
26C2×#13133175—21no—
27#117×D53—21no—
28C2×C42×D53C2211yescharacteristic
29#16810676C2211yescharacteristic
30#16810675—21no—
31#16810674—21no—
32#16810673C2212yes—
33C4⋊C8×D53—21no—
34C22⋊C8×D53C2212yes—
35C4×C8×D53—21no—
36#16810492—21no—
37#16810489C2211yescharacteristic
38#16810482—21no—
39#16810482C2212yes—
40#16810474—21no—
41#16810474C2212yes—
42#16810361C2214yes—
43#16810360C2214yes—
44#16810359C2214yes—
45C53×#533C2211yescharacteristic
46C53×#431C2211yescharacteristic
47C53×#382C2211yescharacteristic
48C53×#328C2214yes—
49C2×Q8×D53—21no—
50C2×Q8×D53C2×C412yes—
51C22×C4×D53D411yescharacteristic
52C22×C4×D53D411yescharacteristic
53C22×C4×D53C2311yescharacteristic
54C2×C8×D53—22no—
55#13133217C2×C411yescharacteristic
56#13133217C2×C411yescharacteristic
57#13133216—21no—
58#13133216—21no—
59#13133216—21no—
60#13133216C2×C412yes—
61#13133215—21no—
62#13133215C2×C412yes—
63#13133209—21no—
64#13133208—21no—
65#13133208—21no—
66#13133207—22no—
67#13133207—22no—
68#13133197—21no—
69#13133197C2×C412yes—
70#13133196—21no—
71#13133196C2312yes—
72#13133195—22no—
73#13133195—22no—
74#13133194—22no—
75#13133194—22no—
76#13133193—21no—
77#13133193C2×C412yes—
78Q8×Dic53—21no—
79Q8×Dic53C2×C412yes—
80C4×Dic106—21no—
81C4×Dic106—22no—
82C4×Dic106—22no—
83C4×Dic106C2×C412yes—
84#13133182—21no—
85#13133182—22no—
86#13133182—22no—
87#13133182C2×C412yes—
88#13133176—21no—
89#13133175—21no—
90#13133175—21no—
91#13133175—44no—
92#13133174—22no—
93#13133173—22no—
94C2×C4⋊Dic53—21no—
95C2×#11719919—22no—
96C2×#11719919—22no—
97C4⋊C4×D53—21no—
98C4⋊C4×D53—21no—
99C4⋊C4×D53—21no—
100C4⋊C4×D53—22no—
101C4⋊C4×D53—22no—
102C4⋊C4×D53C2×C412yes—
103C2×#11719921—22no—
104C22⋊C4×D53—22no—
105C2×C4×Dic53—22no—
106C2×C4×Dic53D411yescharacteristic
107C2×C4×Dic53D411yescharacteristic
108C2×C4×Dic53C2311yescharacteristic
109C42×D53—22no—
110C42×D53C2×C411yescharacteristic
111C42×D53C2×C411yescharacteristic
112C42×D53C2×C411yescharacteristic
113C42×D53C2×C411yescharacteristic
114C42×D53D411yescharacteristic
115C42×D53D411yescharacteristic
116C42×D53D411yescharacteristic
117C42×D53D411yescharacteristic
118C4⋊#88322—21no—
119#13133146—22no—
120C22⋊#88322C2312yes—
121#13133129—22no—
122C4×#88322—21no—
123C8×Dic53—22no—
124#13133124—21no—
125#13133124C2312yes—
126C4⋊Q8×C53C2×C411yescharacteristic
127C4⋊Q8×C53C2×C411yescharacteristic
128#125×C53—21no—
129C22⋊Q8×C53—21no—
130C22⋊Q8×C53C2312yes—
131C4×Q8×C53—21no—
132C4×Q8×C53C2×C412yes—
133#117×C53—21no—
134C2×C4⋊C4×C53—21no—
135C2×C42×C53C2311yescharacteristic
136C4⋊C8×C53—21no—
137C22⋊C8×C53C2312yes—
138C4×C8×C53—21no—
139#95×C53—21no—
140#95×C53C2312yes—
141C23×D53C2×D411yescharacteristic
142C2×Q8×C53—21no—
143C2×Q8×C53C22×C412yes—
144C22×C4×C53—22no—
145C22×C4×C53C2×D411yescharacteristic
146C22×C4×C53C2×D411yescharacteristic
147C22×C4×C53C2411yescharacteristic
148C22×Dic53—22no—
149C22×Dic53C2×D411yescharacteristic
150C22×Dic53C2×D411yescharacteristic
151C22×Dic53C2×D411yescharacteristic
152Q8×D53—44no—
153Q8×D53—44no—
154C2×C4×D53—22no—
155C2×C4×D53—22no—
156C2×C4×D53—42no—
157C2×C4×D53—42no—
158C2×C4×D53C22⋊C411yescharacteristic
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×D53SD1614yes—
167C2×C4×D53Q1614yes—
168C2×C4×D53C22×C411yescharacteristic
169C2×C4×D53C22×C411yescharacteristic
170C2×C4×D53C2×D411yescharacteristic
171C2×C4×D53C2×D411yescharacteristic
172C2×Dic106—21no—
173C2×Dic106—22no—
174C2×Dic106—22no—
175C2×Dic106C22×C412yes—
176C2×C8×C53—22no—
177C4⋊C4×C53—21no—
178C4⋊C4×C53—21no—
179C4⋊C4×C53—21no—
180C4⋊C4×C53—22no—
181C4⋊C4×C53—22no—
182C4⋊C4×C53—42no—
183C4⋊C4×C53C22×C412yes—
184C22⋊C4×C53—22no—
185C42×C53—22no—
186C42×C53C22×C411yescharacteristic
187C42×C53C22×C411yescharacteristic
188C42×C53C2×D411yescharacteristic
189C42×C53C2×D411yescharacteristic
190C22⋊Dic53—22no—
191#11719921—22no—
192#11719921—22no—
193#11719921—44no—
194C4⋊Dic53—21no—
195C4⋊Dic53—21no—
196C4⋊Dic53—21no—
197C4⋊Dic53—22no—
198C4⋊Dic53—22no—
199C4⋊Dic53—42no—
200C4⋊Dic53C22×C412yes—
201#11719919—22no—
202#11719919—22no—
203#11719919—22no—
204#11719919—22no—
205#11719919—22no—
206#11719919—22no—
207#11719919—22no—
208#11719919—22no—
209#11719919—44no—
210#11719919—44no—
211C4×Dic53—22no—
212C4×Dic53—22no—
213C4×Dic53—22no—
214C4×Dic53—44no—
215C4×Dic53C22⋊C411yescharacteristic
216C4×Dic53C22⋊C411yescharacteristic
217C4×Dic53C22⋊C411yescharacteristic
218C4×Dic53C22⋊C411yescharacteristic
219C4×Dic53C22⋊C411yescharacteristic
220C4×Dic53C22⋊C411yescharacteristic
221C4×Dic53C22⋊C411yescharacteristic
222C4×Dic53C22⋊C411yescharacteristic
223C4×Dic53C22×C411yescharacteristic
224C4×Dic53C22×C411yescharacteristic
225C4×Dic53C2×D411yescharacteristic
226C4×Dic53C2×D411yescharacteristic
227C2×#88322—22no—
228C8×D53—44no—
229C23×C53C22×D411yescharacteristic
230C22×D53—41no—
231C22×D53—41no—
232C22×D53C2×C22⋊C411yescharacteristic
233C22×D53C2×C22⋊C411yescharacteristic
234C22×D53C2×SD1612yes—
235C22×D53C2×Q1612yes—
236Q8×C53—42no—
237Q8×C53—42no—
238C2×C4×C53—22no—
239C2×C4×C53—22no—
240C2×C4×C53—41no—
241C2×C4×C53—41no—
242C2×C4×C53—42no—
243C2×C4×C53—42no—
244C2×C4×C53—42no—
245C2×C4×C53C2×C22⋊C411yescharacteristic
246C2×C4×C53C2×C22⋊C411yescharacteristic
247C2×C4×C53C2×C22⋊C411yescharacteristic
248C2×C4×C53C2×C22⋊C411yescharacteristic
249C2×C4×C53C2×SD1612yes—
250C2×C4×C53C2×Q1612yes—
251C2×C4×C53C23×C411yescharacteristic
252C2×C4×C53C22×D411yescharacteristic
253C2×Dic53—22no—
254C2×Dic53—22no—
255C2×Dic53—22no—
256C2×Dic53—22no—
257C2×Dic53—42no—
258C2×Dic53—42no—
259C2×Dic53—42no—
260C2×Dic53C2×C22⋊C411yescharacteristic
261C2×Dic53C2×C22⋊C411yescharacteristic
262C2×Dic53C2×C22⋊C411yescharacteristic
263C2×Dic53C2×C22⋊C411yescharacteristic
264C2×Dic53C2×C22⋊C411yescharacteristic
265C2×Dic53C2×C22⋊C411yescharacteristic
266C2×Dic53C2×SD1612yes—
267C2×Dic53C2×Q1612yes—
268C4×D53—22no—
269C4×D53—22no—
270C4×D53—22no—
271C4×D53—22no—
272C4×D53—41no—
273C4×D53—41no—
274C4×D53—41no—
275C4×D53—41no—
276C4×D53—44no—
277C4×D53—44no—
278C4×D53#10314yes—
279C4×D53#10314yes—
280C4×D53#10314yes—
281C4×D53#10314yes—
282Dic106—42no—
283Dic106—44no—
284Dic106—44no—
285Dic106—42no—
286C8×C53—42no—
287#88322—42no—
288#17575—531nomaximal, Sylow
289C22×C53—21no—
290C22×C53—21no—
291C22×C53C22×C22⋊C411yescharacteristic
292C22×C53C22×SD1611yescharacteristic
293C22×C53C22×Q1611yescharacteristic
294C2×D53—21no—
295C2×D53—21no—
296C2×D53—21no—
297C2×D53—21no—
298C2×D53—42no—
299C2×D53—42no—
300C2×D53C2×#10312yes—
301C2×D53C2×#10312yes—
302C4×C53—21no—
303C4×C53—21no—
304C4×C53—21no—
305C4×C53—21no—
306C4×C53—42no—
307C4×C53—42no—
308C4×C53—82no—
309C4×C53C2×#10312yes—
310C4×C53C2×#10312yes—
311Dic53—21no—
312Dic53—21no—
313Dic53—42no—
314Dic53—42no—
315Dic53—82no—
316Dic53#41012yes—
317Dic53C2×#10312yes—
318Dic53C2×#10312yes—
319Dic53#42112yes—
320C2×#533—531no—
321C2×#431—531no—
322#2285—531no—
323#1520—534no—
324#1518—531no—
325#1518—531no—
326#1518—531no—
327#1518D5311yescharacteristic, complemented
328#1503—534no—
329C2×C53—21no—
330C2×C53—21no—
331C2×C53C2×#41011yescharacteristic
332C2×C53C22×#10311yescharacteristic
333C2×C53C2×#42111yescharacteristic
334D53—22no—
335D53—22no—
336#533—531no—
337#533—531no—
338#533—532no—
339#533—532no—
340#533—531no—
341#533—1061no—
342#533—1061no—
343#533—1061no—
344#533—1061no—
345#533C2×D5311yescharacteristic
346C2×C4⋊Q8—531no—
347C2×C4⋊Q8—531no—
348C2×#125—1061no—
349C2×C22⋊Q8—532no—
350C2×C22⋊Q8—1061no—
351C2×C4×Q8—532no—
352C2×C4×Q8—1061no—
353C2×#117—1061no—
354C2×#117—1061no—
355C22×C42—531no—
356#431—531no—
357#431—532no—
358#431—1061no—
359#431—1061no—
360#431C2×D5311yescharacteristic
361C2×C4⋊C8—1061no—
362C2×C22⋊C8—532no—
363C2×C4×C8—1061no—
364#384—1061no—
365#382—531no—
366#382C2×D5311yescharacteristic
367#380—532no—
368#380—1061no—
369C4×C4⋊C4—532no—
370C4×C4⋊C4—1061no—
371#328—534no—
372#328—534no—
373#328—534no—
374#328C2×D5314yes—
375C53#1757511yescharacteristic, complemented, Sylow
376C22×Q8—532no—
377C22×Q8—1061no—
378C23×C4—531no—
379C23×C4—531no—
380C23×C4—531no—
381C22×C8—1062no—
382C4⋊Q8—532no—
383C4⋊Q8—532no—
384C4⋊Q8—531no—
385C4⋊Q8—531no—
386C4⋊Q8—1061no—
387C4⋊Q8—1061no—
388C4⋊Q8—1061no—
389C4⋊Q8—1061no—
390C4⋊Q8C4×D5311yescharacteristic
391C4⋊Q8C4×D5311yescharacteristic
392#125—21no—
393#125—1062no—
394#125—1062no—
395#125—1061no—
396#125—1061no—
397#125—1061no—
398C22⋊Q8—21no—
399C22⋊Q8—532no—
400C22⋊Q8—532no—
401C22⋊Q8—532no—
402C22⋊Q8—1062no—
403C22⋊Q8—1062no—
404C22⋊Q8—1062no—
405C22⋊Q8—1062no—
406C22⋊Q8—1061no—
407C22⋊Q8—1061no—
408C22⋊Q8—1061no—
409C22⋊Q8C22×D5312yes—
410C4×Q8—21no—
411C4×Q8—532no—
412C4×Q8—532no—
413C4×Q8—532no—
414C4×Q8—1062no—
415C4×Q8—1062no—
416C4×Q8—1062no—
417C4×Q8—1061no—
418C4×Q8—1062no—
419C4×Q8—1061no—
420C4×Q8—1061no—
421C4×Q8C4×D5312yes—
422#117—21no—
423#117—1062no—
424#117—1062no—
425#117—1061no—
426#117—1061no—
427#117—1061no—
428#117—2124no—
429C2×C4⋊C4—21no—
430C2×C4⋊C4—532no—
431C2×C4⋊C4—1061no—
432C2×C4⋊C4—1061no—
433C2×C4⋊C4—1062no—
434C2×C4⋊C4—1062no—
435C2×C4⋊C4—1061no—
436C2×C4⋊C4—1062no—
437C2×C4⋊C4—1062no—
438C2×C4⋊C4—1061no—
439C2×C22⋊C4—1062no—
440C2×C22⋊C4—1062no—
441C2×C42—531no—
442C2×C42—531no—
443C2×C42—531no—
444C2×C42—531no—
445C2×C42—531no—
446C2×C42—531no—
447C2×C42—531no—
448C2×C42—531no—
449C2×C42—531no—
450C2×C42—531no—
451C2×C42—531no—
452C2×C42—1062no—
453C2×C42—1062no—
454C2×C42C22×D5311yescharacteristic
455C4⋊C8—21no—
456C4⋊C8—1062no—
457C4⋊C8—1061no—
458C22⋊C8—532no—
459C22⋊C8—1062no—
460C22⋊C8C22×D5312yes—
461C4×C8—21no—
462C4×C8—1062no—
463C4×C8—1061no—
464#95—21no—
465#95—532no—
466#95—1061no—
467#95C22×D5312yes—
468C24—531no—
469C2×Q8—21no—
470C2×Q8—532no—
471C2×Q8—1062no—
472C2×Q8—1062no—
473C2×Q8—1061no—
474C2×Q8—2124no—
475C2×Q8—2124no—
476C2×Q8C2×C4×D5312yes—
477C22×C4—22no—
478C22×C4—531no—
479C22×C4—531no—
480C22×C4—531no—
481C22×C4—531no—
482C22×C4—531no—
483C22×C4—534no—
484C22×C4—534no—
485C22×C4—531no—
486C22×C4—531no—
487C22×C4—531no—
488C22×C4—531no—
489C22×C4—531no—
490C22×C4—531no—
491C22×C4—531no—
492C22×C4—531no—
493C22×C4—531no—
494C22×C4—531no—
495C22×C4—1062no—
496C22×C4—1062no—
497C22×C4—1062no—
498C22×C4—2122no—
499C22×C4—2122no—
500C22×C4D4×D5311yescharacteristic
501C22×C4D4×D5311yescharacteristic
502C22×C4C23×D5311yescharacteristic
503C2×C8—22no—
504C2×C8—1062no—
505C2×C8—2124no—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—42no—
512C4⋊C4—532no—
513C4⋊C4—1062no—
514C4⋊C4—1062no—
515C4⋊C4—1062no—
516C4⋊C4—1062no—
517C4⋊C4—1062no—
518C4⋊C4—1062no—
519C4⋊C4—1062no—
520C4⋊C4—1062no—
521C4⋊C4—1061no—
522C4⋊C4—1061no—
523C4⋊C4—1062no—
524C4⋊C4—1061no—
525C4⋊C4—1062no—
526C4⋊C4—2124no—
527C4⋊C4—2124no—
528C4⋊C4—2122no—
529C4⋊C4C2×C4×D5312yes—
530C22⋊C4—22no—
531C22⋊C4—1062no—
532C22⋊C4—1062no—
533C22⋊C4—1062no—
534C22⋊C4—2124no—
535C42—22no—
536C42—531no—
537C42—531no—
538C42—531no—
539C42—531no—
540C42—531no—
541C42—531no—
542C42—531no—
543C42—531no—
544C42—531no—
545C42—531no—
546C42—531no—
547C42—531no—
548C42—1062no—
549C42—1062no—
550C42—1062no—
551C42—2124no—
552C42C2×C4×D5311yescharacteristic
553C42C2×C4×D5311yescharacteristic
554C42D4×D5311yescharacteristic
555C42D4×D5311yescharacteristic
556C23—531no—
557C23—531no—
558C23—532no—
559C23—532no—
560C23—2121no—
561C23—2121no—
562C23C2×D4×D5311yescharacteristic
563Q8—42no—
564Q8—42no—
565Q8—2122no—
566Q8—2122no—
567Q8—2124no—
568Q8—2124no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—531no—
577C2×C4—531no—
578C2×C4—531no—
579C2×C4—531no—
580C2×C4—532no—
581C2×C4—532no—
582C2×C4—534no—
583C2×C4—534no—
584C2×C4—534no—
585C2×C4—534no—
586C2×C4—531no—
587C2×C4—531no—
588C2×C4—1062no—
589C2×C4—1062no—
590C2×C4—1062no—
591C2×C4—1062no—
592C2×C4—1062no—
593C2×C4—1062no—
594C2×C4—1062no—
595C2×C4—1062no—
596C2×C4—2121no—
597C2×C4—2121no—
598C2×C4—2121no—
599C2×C4—2121no—
600C2×C4—2124no—
601C2×C4—2124no—
602C2×C4—2122no—
603C2×C4—2122no—
604C2×C4—2122no—
605C2×C4C22⋊C4×D5311yescharacteristic
606C2×C4C22⋊C4×D5311yescharacteristic
607C2×C4C22⋊C4×D5311yescharacteristic
608C2×C4C22⋊C4×D5311yescharacteristic
609C2×C4SD16×D5312yes—
610C2×C4Q16×D5312yes—
611C2×C4C22×C4×D5311yescharacteristic
612C2×C4C2×D4×D5311yescharacteristic
613C8—42no—
614C8—2122no—
615C22—21no—
616C22—21no—
617C22—532no—
618C22—532no—
619C22—1061no—
620C22—1061no—
621C22—1061no—
622C22—1061no—
623C22—2122no—
624C22—2122no—
625C22C2×C22⋊C4×D5311yescharacteristic
626C22C2×SD16×D5311yescharacteristic
627C22C2×Q16×D5311yescharacteristic
628C4—21no—
629C4—21no—
630C4—21no—
631C4—21no—
632C4—42no—
633C4—42no—
634C4—82no—
635C4—532no—
636C4—532no—
637C4—532no—
638C4—532no—
639C4—1061no—
640C4—1061no—
641C4—4242no—
642C4—2122no—
643C4—2122no—
644C4#103×D5312yes—
645C4#103×D5312yes—
646C2—21no—
647C2—21no—
648C2—1062no—
649C2—1062no—
650C2#410×D5311yescharacteristic
651C2C2×#103×D5311yescharacteristic
652C2#421×D5311yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1518C2
socleC22×C53C22×C22⋊C4
prime radicalC23×D53C2×D4
cosocleC22×C4×C53C24
hypercenter#1518D53
hypocenterC53#17575
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1518

Complements

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

NH ≅ G/Ntype
5. #20273280C2semidirect
6. #20273279C2semidirect
7. #20273278C2semidirect
8. #20273212C2semidirect
9. C53×#1518C2semidirect
12. #16811178C4semidirect
13. #16811178C4semidirect
15. #16811177C4semidirect
19. C4⋊Q8×D53C4semidirect
20. C4⋊Q8×D53C4semidirect
25. C4×Q8×D53C4semidirect
327. #1518D53semidirect
375. C53#17575semidirect

Maximal subgroups

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

subgroupindex
#533×D532
#431×D532
#202771212
#202732852
#202732802
#202732792
#202732782
#202732122
C53×#15182
#1757553