Finite Group Databasebeta

← #27351340

Subgroups — #27351340

Order 13568.

1024 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811159C211yesmaximal, characteristic
2#20283344C211yesmaximal, characteristic, complemented
3#20277106C211yesmaximal, characteristic, complemented
4#20273252C214yesmaximal, complemented
5#20273248C211yesmaximal, characteristic, complemented
6#20273245C211yesmaximal, characteristic
7#20273243C211yesmaximal, characteristic
8#20273200C214yesmaximal, complemented
9C53×#1511C211yesmaximal, characteristic, complemented
10#16811159—21no—
11#16811159—21no—
12#16811159—21no—
13#16811159—21no—
14#16811159C412yescomplemented
15#16811159C412yescomplemented
16#16811159C2212yes—
17#16811159C2212yes—
18C2×#13133212—21no—
19C2×#13133208C411yescharacteristic, complemented
20C2×#13133206C411yescharacteristic, complemented
21C2×#13133201—21no—
22C2×#13133191C2211yescharacteristic
23C2×Q8×Dic53C411yescharacteristic, complemented
24C2×#13133179C411yescharacteristic, complemented
25C2×#13133177—21no—
26C2×#13133175—21no—
27C22×C4×Dic53C2211yescharacteristic
28#16810677—21no—
29#16810677C2212yescomplemented
30#16810673—21no—
31#16810673C2212yescomplemented
32#16810646—21no—
33#16810587C2211yescharacteristic
34M16×Dic53—21no—
35#16810490—21no—
36#16810487C2212yescomplemented
37#16810486C2211yescharacteristic
38#16810481—21no—
39#16810474—21no—
40C22⋊C4×Dic53—21no—
41#16810468C2212yescomplemented
42#16810360C2214yescomplemented
43#16810358C2214yes—
44#16810356C2214yes—
45C2×C22⋊Q8×C53C2211yescharacteristic
46C53×#407C2211yescharacteristic
47C53×#381C2211yescharacteristic
48C53×#328C2214yescomplemented
49C22×D212C2×C411yescharacteristic
50C22×C22⋊D53—21no—
51C23×Dic53D411yescharacteristic
52C23×Dic53D411yescharacteristic
53C22×C4×D53—21no—
54C2×#11719917—21no—
55#13133212—22no—
56#13133212—22no—
57#13133212—22no—
58#13133212—22no—
59#13133208—21no—
60#13133208—21no—
61#13133208C2×C412yes—
62#13133208C2×C412yes—
63#13133206—21no—
64#13133206—21no—
65#13133206C2×C412yes—
66#13133206C2×C412yes—
67#13133201—22no—
68#13133201—22no—
69#13133201—22no—
70#13133201—22no—
71#13133191—21no—
72#13133191—21no—
73#13133191C2×C412yes—
74#13133191C2312yes—
75Q8×Dic53—21no—
76Q8×Dic53—21no—
77Q8×Dic53C2×C412yes—
78Q8×Dic53C2×C412yes—
79#13133179—21no—
80#13133179—21no—
81#13133179C2×C412yes—
82#13133179C2×C412yes—
83#13133177—22no—
84#13133177—22no—
85#13133177—22no—
86#13133177—22no—
87#13133175—22no—
88#13133175—22no—
89#13133175—22no—
90#13133175—22no—
91C2×C4⋊Dic53—21no—
92C2×C4⋊Dic53—21no—
93C2×C4⋊Dic53C2×C411yescharacteristic
94C2×#11719919—21no—
95C2×#11719919—22no—
96C2×C22⋊Dic53—21no—
97C2×#11719921—21no—
98C2×#11719921—21no—
99C2×#11719921—21no—
100C2×#11719921—21no—
101C2×C4×Dic53—21no—
102C2×C4×Dic53—22no—
103C2×C4×Dic53C2×C411yescharacteristic
104C2×C4×Dic53C2×C411yescharacteristic
105C2×C4×Dic53C2×C411yescharacteristic
106C2×C4×Dic53C2×C411yescharacteristic
107C2×C4×Dic53D411yescharacteristic
108C2×C4×Dic53D411yescharacteristic
109C2×C4×Dic53D411yescharacteristic
110C2×C4×Dic53D411yescharacteristic
111C2×C4×Dic53D411yescharacteristic
112C2×C4×Dic53D411yescharacteristic
113C2×C4×Dic53C2311yescharacteristic
114C2×C4×Dic53C2311yescharacteristic
115#13133146—22no—
116#13133146—22no—
117C22⋊#88322—21no—
118C22⋊#88322C2312yes—
119#13133127—22no—
120C8×Dic53—22no—
121#13133124—21no—
122#13133124—22no—
123#13133124—22no—
124#13133124C2312yes—
125C22×Q8×C53C2×C411yescharacteristic
126C23×C4×C53C2311yescharacteristic
127C2×M16×C53—21no—
128C22⋊Q8×C53—21no—
129C22⋊Q8×C53—21no—
130C22⋊Q8×C53C2×C412yes—
131C22⋊Q8×C53C2312yes—
132C2×C4⋊C4×C53—21no—
133C2×C4⋊C4×C53C2×C411yescharacteristic
134C2×C22⋊C4×C53—21no—
135C2×C22⋊C4×C53—21no—
136C22⋊C8×C53—21no—
137C22⋊C8×C53C2312yes—
138#95×C53—21no—
139#95×C53C2312yes—
140C24×C53C2×D411yescharacteristic
141C23×D53—21no—
142C2×Q8×C53—21no—
143C2×Q8×C53—42no—
144C2×Q8×C53C22×C412yes—
145C22×C4×C53—21no—
146C22×C4×C53—21no—
147C22×C4×C53—22no—
148C22×C4×C53—42no—
149C22×C4×C53C22×C411yescharacteristic
150C22×C4×C53C22×C411yescharacteristic
151C22×C4×C53C2×D411yescharacteristic
152C22×C4×C53C2×D411yescharacteristic
153C22×C4×C53C2×D411yescharacteristic
154C22×C4×C53C2411yescharacteristic
155C2×C22⋊D53—22no—
156C2×C22⋊D53—22no—
157C2×C22⋊D53—44no—
158C22×Dic53—21no—
159C22×Dic53—22no—
160C22×Dic53—42no—
161C22×Dic53—42no—
162C22×Dic53C22⋊C411yescharacteristic
163C22×Dic53C22⋊C411yescharacteristic
164C22×Dic53C22⋊C411yescharacteristic
165C22×Dic53C22⋊C411yescharacteristic
166C22×Dic53C22⋊C411yescharacteristic
167C22×Dic53C22⋊C411yescharacteristic
168C22×Dic53C22⋊C411yescharacteristic
169C22×Dic53C22⋊C411yescharacteristic
170C22×Dic53C2×D411yescharacteristic
171C22×Dic53C2×D411yescharacteristic
172C22×Dic53C2×D411yescharacteristic
173C22×Dic53C2×D411yescharacteristic
174C2×D212—21no—
175C2×D212—42no—
176C2×D212C22×C412yes—
177C2×C4×D53—22no—
178C2×C4×D53—22no—
179C2×C4×D53—44no—
180M16×C53—42no—
181C2×C8×C53—22no—
182C4⋊C4×C53—22no—
183C4⋊C4×C53—21no—
184C4⋊C4×C53—22no—
185C4⋊C4×C53C22×C412yes—
186C22⋊C4×C53—22no—
187C22⋊C4×C53—22no—
188C22⋊C4×C53—42no—
189C22⋊Dic53—42no—
190#11719921—22no—
191#11719921—22no—
192#11719921—22no—
193#11719921—22no—
194#11719921—22no—
195#11719921—22no—
196#11719921—22no—
197#11719921—22no—
198C4⋊Dic53—21no—
199C4⋊Dic53—22no—
200C4⋊Dic53—22no—
201C4⋊Dic53—22no—
202C4⋊Dic53—22no—
203C4⋊Dic53C22×C412yes—
204#11719919—22no—
205#11719919—22no—
206#11719919—44no—
207C4×Dic53—22no—
208C4×Dic53—22no—
209C4×Dic53—44no—
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×Dic53C22×C411yescharacteristic
219C4×Dic53C22×C411yescharacteristic
220C4×Dic53C22×C411yescharacteristic
221C4×Dic53C22×C411yescharacteristic
222C4×Dic53C2×D411yescharacteristic
223C4×Dic53C2×D411yescharacteristic
224C4×Dic53C2×D411yescharacteristic
225C4×Dic53C2×D411yescharacteristic
226#11719917—42no—
227C2×#88322—22no—
228C23×C53—22no—
229C23×C53—22no—
230C23×C53—41no—
231C23×C53C2×C22⋊C411yescharacteristic
232C23×C53C2×C22⋊C411yescharacteristic
233C23×C53C22×D411yescharacteristic
234C22×D53—22no—
235C22×D53—42no—
236C22×D53—42no—
237C22×D53—42no—
238Q8×C53—42no—
239Q8×C53—42no—
240C2×C4×C53—22no—
241C2×C4×C53—22no—
242C2×C4×C53—22no—
243C2×C4×C53—22no—
244C2×C4×C53—41no—
245C2×C4×C53—41no—
246C2×C4×C53—41no—
247C2×C4×C53—41no—
248C2×C4×C53—42no—
249C2×C4×C53—42no—
250C2×C4×C53—42no—
251C2×C4×C53—42no—
252C2×C4×C53—42no—
253C2×C4×C53C2×C22⋊C411yescharacteristic
254C2×C4×C53C2×C22⋊C411yescharacteristic
255C2×C4×C53C23×C411yescharacteristic
256C2×C4×C53C22×D411yescharacteristic
257C22⋊D53—44no—
258C22⋊D53—44no—
259C2×Dic53—22no—
260C2×Dic53—22no—
261C2×Dic53—22no—
262C2×Dic53—22no—
263C2×Dic53—22no—
264C2×Dic53—22no—
265C2×Dic53—41no—
266C2×Dic53—41no—
267C2×Dic53—41no—
268C2×Dic53—41no—
269C2×Dic53—41no—
270C2×Dic53—41no—
271C2×Dic53—41no—
272C2×Dic53—42no—
273C2×Dic53—42no—
274C2×Dic53—42no—
275C2×Dic53—42no—
276C2×Dic53C2×C22⋊C411yescharacteristic
277C2×Dic53C2×C22⋊C411yescharacteristic
278C2×Dic53C2×C22⋊C411yescharacteristic
279C2×Dic53C2×C22⋊C411yescharacteristic
280C2×Dic53C2×C22⋊C411yescharacteristic
281C2×Dic53C2×C22⋊C411yescharacteristic
282C2×Dic53C2×C22⋊C411yescharacteristic
283C2×Dic53C2×C22⋊C411yescharacteristic
284D212—42no—
285D212—42no—
286C4×D53—44no—
287C4×D53—44no—
288C8×C53—42no—
289#88322—42no—
290#17554—531nomaximal, Sylow
291C22×C53—21no—
292C22×C53—21no—
293C22×C53—21no—
294C22×C53—21no—
295C22×C53—21no—
296C22×C53—21no—
297C22×C53—21no—
298C22×C53—44no—
299C22×C53#41012yes—
300C22×C53C22×C22⋊C411yescharacteristic
301C22×C53#57512yes—
302C2×D53—42no—
303C2×D53—42no—
304C2×D53—42no—
305C2×D53—84no—
306C4×C53—21no—
307C4×C53—21no—
308C4×C53—21no—
309C4×C53—21no—
310C4×C53—42no—
311C4×C53—42no—
312C4×C53—82no—
313Dic53—21no—
314Dic53—21no—
315Dic53—21no—
316Dic53—21no—
317Dic53—21no—
318Dic53—21no—
319Dic53—21no—
320Dic53—42no—
321Dic53—82no—
322Dic53#42112yes—
323C2×#529—531no—
324#2920—531no—
325#2283—531no—
326#1512—534no—
327#1511—531no—
328#1511D5311yescharacteristic, complemented
329#1510—531no—
330#1510—531no—
331#1500—534no—
332C2×C53—22no—
333C2×C53—22no—
334C2×C53—41no—
335C2×C53C2×#41011yescharacteristic
336C2×C53#289611yescharacteristic
337C2×C53C2×#42111yescharacteristic
338D53—82no—
339#529—532no—
340#529—532no—
341#529—532no—
342#529—532no—
343#529—1061no—
344#529—1061no—
345#529—1061no—
346#529—1061no—
347C2×#126—1061no—
348C2×#125—531no—
349C2×#124—531no—
350C2×#123—1061no—
351C2×C22⋊Q8C2×D5311yescharacteristic
352C2×#121—531no—
353C2×C4×Q8—531no—
354C2×C4×D4—531no—
355C2×C4×D4—1061no—
356C2×#117—1061no—
357C22×C42—531no—
358#432—532no—
359#432—1061no—
360#431—532no—
361#431—1061no—
362#423—1061no—
363#407—531no—
364#407C2×D5311yescharacteristic
365C4×M16—1061no—
366#383—1061no—
367#382—532no—
368#381—531no—
369#381C2×D5311yescharacteristic
370#380—1061no—
371C4×C4⋊C4—1061no—
372C4×C22⋊C4—1061no—
373#376—532no—
374#328—534no—
375#328C2×D5314yescomplemented
376#327—534no—
377#327—534no—
378C53#1755411yescharacteristic, complemented, Sylow
379C22×Q8C4×D5311yescharacteristic
380C22×D4—531no—
381C22×D4—1061no—
382C23×C4—531no—
383C23×C4—531no—
384C23×C4—1061no—
385C23×C4C22×D5311yescharacteristic
386C2×M16—21no—
387C2×M16—1061no—
388#126—1062no—
389#126—1062no—
390#126—1062no—
391#126—1062no—
392#125—532no—
393#125—532no—
394#125—1061no—
395#125—1061no—
396#124—532no—
397#124—532no—
398#124—1061no—
399#124—1061no—
400#123—1062no—
401#123—1062no—
402#123—1062no—
403#123—1062no—
404C22⋊Q8—21no—
405C22⋊Q8—21no—
406C22⋊Q8C4×D5312yes—
407C22⋊Q8C22×D5312yes—
408#121—532no—
409#121—532no—
410#121—1061no—
411#121—1061no—
412C4×Q8—532no—
413C4×Q8—532no—
414C4×Q8—1061no—
415C4×Q8—1061no—
416C4×D4—532no—
417C4×D4—532no—
418C4×D4—1061no—
419C4×D4—1061no—
420C4×D4—1062no—
421C4×D4—1062no—
422C4×D4—1062no—
423C4×D4—1062no—
424#117—1062no—
425#117—1062no—
426#117—1062no—
427#117—1062no—
428C2×C4⋊C4—21no—
429C2×C4⋊C4—531no—
430C2×C4⋊C4—1061no—
431C2×C4⋊C4—1061no—
432C2×C4⋊C4—1061no—
433C2×C4⋊C4—1062no—
434C2×C4⋊C4C4×D5311yescharacteristic
435C2×C22⋊C4—21no—
436C2×C22⋊C4—21no—
437C2×C22⋊C4—1061no—
438C2×C22⋊C4—1061no—
439C2×C22⋊C4—1061no—
440C2×C22⋊C4—1061no—
441C2×C22⋊C4—1061no—
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—531no—
453C2×C42—531no—
454C2×C42—1061no—
455C2×C42—1062no—
456C4⋊C8—1062no—
457C4⋊C8—1062no—
458C22⋊C8—21no—
459C22⋊C8—532no—
460C22⋊C8—1061no—
461C22⋊C8C22×D5312yes—
462#97—1062no—
463C4×C8—1062no—
464#95—21no—
465#95—532no—
466#95—1062no—
467#95—1062no—
468#95—1061no—
469#95C22×D5312yes—
470C24—1061no—
471C24D4×D5311yescharacteristic
472C2×Q8—21no—
473C2×Q8—42no—
474C2×Q8C2×C4×D5312yes—
475C2×D4—532no—
476C2×D4—1062no—
477C2×D4—1062no—
478C2×D4—1061no—
479C2×D4—2122no—
480C2×D4—2124no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—22no—
484C22×C4—42no—
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—531no—
496C22×C4—531no—
497C22×C4—1062no—
498C22×C4—1062no—
499C22×C4—1061no—
500C22×C4—1062no—
501C22×C4—2124no—
502C22×C4—2122no—
503C22×C4—2122no—
504C22×C4C2×C4×D5311yescharacteristic
505C22×C4C2×C4×D5311yescharacteristic
506C22×C4D4×D5311yescharacteristic
507C22×C4D4×D5311yescharacteristic
508C22×C4D4×D5311yescharacteristic
509C22×C4C23×D5311yescharacteristic
510M16—42no—
511M16—2122no—
512C2×C8—22no—
513C2×C8—1062no—
514C4⋊C4—22no—
515C4⋊C4—21no—
516C4⋊C4—22no—
517C4⋊C4—532no—
518C4⋊C4—1062no—
519C4⋊C4—1062no—
520C4⋊C4—1062no—
521C4⋊C4—1062no—
522C4⋊C4—1062no—
523C4⋊C4—1062no—
524C4⋊C4—1061no—
525C4⋊C4—2124no—
526C4⋊C4C2×C4×D5312yes—
527C22⋊C4—22no—
528C22⋊C4—22no—
529C22⋊C4—42no—
530C22⋊C4—1062no—
531C22⋊C4—1062no—
532C22⋊C4—1062no—
533C22⋊C4—1062no—
534C22⋊C4—1062no—
535C22⋊C4—1062no—
536C22⋊C4—1062no—
537C22⋊C4—1062no—
538C22⋊C4—2122no—
539C42—531no—
540C42—531no—
541C42—531no—
542C42—531no—
543C42—531no—
544C42—531no—
545C42—531no—
546C42—531no—
547C42—531no—
548C42—531no—
549C42—531no—
550C42—531no—
551C42—531no—
552C42—531no—
553C42—531no—
554C42—531no—
555C42—1062no—
556C42—1062no—
557C42—2124no—
558C23—22no—
559C23—22no—
560C23—41no—
561C23—1062no—
562C23—2122no—
563C23—2122no—
564C23—2122no—
565C23C22⋊C4×D5311yescharacteristic
566C23C22⋊C4×D5311yescharacteristic
567C23C2×D4×D5311yescharacteristic
568Q8—42no—
569Q8—42no—
570D4—2122no—
571D4—2122no—
572D4—2124no—
573D4—2124no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—531no—
588C2×C4—531no—
589C2×C4—531no—
590C2×C4—531no—
591C2×C4—531no—
592C2×C4—531no—
593C2×C4—531no—
594C2×C4—531no—
595C2×C4—1062no—
596C2×C4—1062no—
597C2×C4—1062no—
598C2×C4—1062no—
599C2×C4—1062no—
600C2×C4—1062no—
601C2×C4—2124no—
602C2×C4—2124no—
603C2×C4—2121no—
604C2×C4—2121no—
605C2×C4—2121no—
606C2×C4—2121no—
607C2×C4—2121no—
608C2×C4—2121no—
609C2×C4—2121no—
610C2×C4—2122no—
611C2×C4—2122no—
612C2×C4—2122no—
613C2×C4—2122no—
614C2×C4C22⋊C4×D5311yescharacteristic
615C2×C4C22⋊C4×D5311yescharacteristic
616C2×C4C22×C4×D5311yescharacteristic
617C2×C4C2×D4×D5311yescharacteristic
618C8—42no—
619C8—2122no—
620C22—21no—
621C22—21no—
622C22—21no—
623C22—21no—
624C22—21no—
625C22—21no—
626C22—21no—
627C22—44no—
628C22—4244no—
629C22—2122no—
630C22—2122no—
631C22—2122no—
632C22#1681059512yes—
633C22C2×C22⋊C4×D5311yescharacteristic
634C22#1681142212yes—
635C4—21no—
636C4—21no—
637C4—21no—
638C4—21no—
639C4—42no—
640C4—42no—
641C4—82no—
642C4—532no—
643C4—1061no—
644C4—1061no—
645C4—1061no—
646C4—1061no—
647C4—1061no—
648C4—1061no—
649C4—1061no—
650C4—4242no—
651C4—2122no—
652C2—22no—
653C2—22no—
654C2—41no—
655C2—4242no—
656C2C2×#1681059511yescharacteristic
657C2#2028323311yescharacteristic
658C2#421×D5311yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1511C2
socleC22×C53C22×C22⋊C4
prime radicalC2×#13133191C22
cosocleC22×C4×C53C24
hypercenter#1511D53
hypocenterC53#17554
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1511

Complements

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

NH ≅ G/Ntype
2. #20283344C2semidirect
3. #20277106C2semidirect
4. #20273252C2semidirect
5. #20273248C2semidirect
8. #20273200C2semidirect
9. C53×#1511C2semidirect
14. #16811159C4semidirect
15. #16811159C4semidirect
19. C2×#13133208C4semidirect
20. C2×#13133206C4semidirect
23. C2×Q8×Dic53C4semidirect
24. C2×#13133179C4semidirect
29. #16810677C22semidirect
31. #16810673C22semidirect
36. #16810487C22semidirect
41. #16810468C22semidirect
42. #16810360C22semidirect
48. C53×#328C22semidirect
328. #1511D53semidirect
375. #328C2×D53semidirect
378. C53#17554semidirect

Maximal subgroups

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

subgroupindex
C2×#168111592
#202833442
#202771062
#202732522
#202732482
#202732452
#202732432
#202732002
C53×#15112
#1755453