Finite Group Databasebeta

← #27351653

Subgroups — #27351653

Order 13568.

982 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20288764C211yesmaximal, characteristic
2#20283750C211yesmaximal, characteristic, complemented
3#20273286C212yesmaximal, complemented
4#20273269C212yesmaximal, complemented
5#20273256C212yesmaximal, complemented
6#20273247C212yesmaximal, complemented
7#20273238C211yesmaximal, characteristic, complemented
8#20273226C211yesmaximal, characteristic, complemented
9#20273213C211yesmaximal, characteristic, complemented
10#20273199C211yesmaximal, characteristic, complemented
11C53×#2319C211yesmaximal, characteristic, complemented
12#16811343C411yescharacteristic, complemented
13#16811328—21no—
14#16811313C412yescomplemented
15#16811299—22no—
16D4×Dic106C411yescharacteristic, complemented
17#16811282C412yescomplemented
18#16811278C411yescharacteristic, complemented
19#16811277C411yescharacteristic, complemented
20#16811267—21no—
21#16811238—21no—
22#16811215—22no—
23#16811180C2211yescharacteristic
24#16811157C2211yescharacteristic
25#16811134C2212yes—
26C2×#13133196C2212yes—
27#16811102—21no—
28#16810701—21no—
29#16810695—22no—
30D4×#88322—21no—
31#16810680C2211yescharacteristic
32#16810676C2211yescharacteristic
33#16810587C2212yes—
34C2×C22⋊#88322C2212yes—
35#16810359C2211yescharacteristic
36#16810359C2211yescharacteristic
37#16810359C2212yes—
38#16810359C2212yes—
39#16810359C2212yes—
40#16810359C2212yes—
41#16810359C2212yes—
42#16810356C2211yescharacteristic
43#16810356C2211yescharacteristic
44#16810356C2212yes—
45C2×C4×D4×C53C2211yescharacteristic
46C53×#390—21no—
47C53×#389—21no—
48C53×#388—22no—
49C53×#381C2212yes—
50C53×#380C2212yes—
51C4×C4⋊C4×C53C2211yescharacteristic
52C53×#376C2211yescharacteristic
53C2×#11719945C2×C412yes—
54C2×#11719947—21no—
55C22×Dic106C2×C412yes—
56C2×#11719917—22no—
57C22×#88322—22no—
58#13133217C2×C411yescharacteristic
59#13133217C2×C411yescharacteristic
60#13133216—21no—
61#13133211—21no—
62#13133210—22no—
63#13133210—22no—
64#13133209—21no—
65#13133209C2×C411yescharacteristic
66#13133207—22no—
67#13133205C2×C411yescharacteristic
68#13133204—21no—
69#13133203—22no—
70#13133201—22no—
71#13133200—21no—
72#13133200—22no—
73#13133199—22no—
74#13133198—22no—
75#13133197—21no—
76#13133196—21no—
77#13133196—22no—
78#13133196—22no—
79#13133196C2×C412yes—
80#13133196C2×C412yes—
81#13133196C2×C412yes—
82#13133196C2311yescharacteristic
83#13133196C2312yes—
84#13133195—22no—
85#13133194—22no—
86#13133191C2×C412yes—
87#13133191C2311yescharacteristic
88#13133189—22no—
89#13133188—21no—
90#13133186—22no—
91C4×Dic106—21no—
92C4×Dic106C2×C412yes—
93C4×Dic106C2×C411yescharacteristic
94#13133182—21no—
95C4×D212C2×C411yescharacteristic
96D4×Dic53—21no—
97C4×C22⋊D53—21no—
98C4×C22⋊D53—22no—
99C4×C22⋊D53—22no—
100#13133177—22no—
101#13133176—21no—
102#13133176C2×C412yes—
103#13133174—21no—
104#13133173—22no—
105C2×C4⋊Dic53C2×C412yes—
106C2×#11719919—22no—
107C4⋊C4×D53—21no—
108C2×C22⋊Dic53—22no—
109C4⋊#88322—21no—
110C4⋊#88322—21no—
111C22⋊#88322—22no—
112C22⋊#88322—22no—
113C22⋊#88322C2311yescharacteristic
114C22⋊#88322C2312yes—
115C22⋊#88322C2311yescharacteristic
116#13133128—21no—
117C4×#88322—21no—
118C22×D4×C53D411yescharacteristic
119C23×C4×C53C2312yes—
120C4×D4×C53C2×C411yescharacteristic
121C4×D4×C53C2×C411yescharacteristic
122C4×D4×C53C2×C412yes—
123C4×D4×C53D411yescharacteristic
124C4×D4×C53D411yescharacteristic
125C4×D4×C53D412yes—
126C2×C4⋊C4×C53—21no—
127C2×C4⋊C4×C53—22no—
128C2×C4⋊C4×C53D411yescharacteristic
129C2×C22⋊C4×C53—22no—
130C2×C22⋊C4×C53—22no—
131C2×C22⋊C4×C53D412yes—
132C2×C42×C53—21no—
133C2×C42×C53—21no—
134C2×C42×C53C2311yescharacteristic
135#95×C53—21no—
136#95×C53—22no—
137#95×C53C2311yescharacteristic
138#95×C53C2311yescharacteristic
139#95×C53C2312yes—
140C24×C53C2×D412yes—
141C2×D4×C53—42no—
142C2×D4×C53C22⋊C411yescharacteristic
143C2×D4×C53C22⋊C411yescharacteristic
144C2×D4×C53C22⋊C412yes—
145C22×C4×C53—21no—
146C22×C4×C53—21no—
147C22×C4×C53—22no—
148C22×C4×C53—82no—
149C22×C4×C53C22×C412yes—
150C22×C4×C53C22×C412yes—
151C22×C4×C53C2×D411yescharacteristic
152C22×C4×C53C2×D411yescharacteristic
153C22×C4×C53C2×D412yes—
154C22×C4×C53C2×D412yes—
155C22×C4×C53C2×D412yes—
156C22×C4×C53C2411yescharacteristic
157C2×C22⋊D53—22no—
158C2×C22⋊D53—21no—
159C2×C22⋊D53—22no—
160C22×Dic53—22no—
161C22×Dic53—22no—
162#11719947—44no—
163#11719945—82no—
164C2×D212C22×C411yescharacteristic
165C2×C4×D53—21no—
166C2×C4×D53—22no—
167C2×Dic106—21no—
168C2×Dic106—22no—
169C2×Dic106—82no—
170C2×Dic106C22×C411yescharacteristic
171C2×Dic106C22×C412yes—
172C4⋊C4×C53—42no—
173C4⋊C4×C53—82no—
174C4⋊C4×C53C22⋊C411yescharacteristic
175C4⋊C4×C53C22⋊C412yes—
176C4⋊C4×C53C22⋊C411yescharacteristic
177C22⋊C4×C53—82no—
178C22⋊C4×C53—82no—
179C22⋊C4×C53C22⋊C412yes—
180C22⋊C4×C53C22⋊C412yes—
181C22⋊C4×C53C22⋊C412yes—
182C22⋊C4×C53C22⋊C412yes—
183C42×C53—42no—
184C42×C53—81no—
185C42×C53C22×C411yescharacteristic
186C42×C53C22×C411yescharacteristic
187C42×C53C2×D411yescharacteristic
188C42×C53C2×D411yescharacteristic
189C22⋊Dic53—22no—
190C22⋊Dic53—22no—
191C22⋊Dic53—22no—
192C22⋊Dic53—22no—
193C22⋊Dic53—21no—
194C22⋊Dic53—22no—
195#11719921—21no—
196#11719921—21no—
197#11719921—22no—
198#11719921—21no—
199#11719921—21no—
200C4⋊Dic53—21no—
201C4⋊Dic53—22no—
202C4⋊Dic53—21no—
203C4⋊Dic53—21no—
204C4⋊Dic53C22×C411yescharacteristic
205C4⋊Dic53C22×C412yes—
206C4⋊Dic53C22×C411yescharacteristic
207#11719919—22no—
208#11719919—22no—
209#11719919—21no—
210#11719919—22no—
211#11719919—21no—
212#11719919—22no—
213#11719919—21no—
214#11719919—21no—
215#11719919—22no—
216C4×Dic53—22no—
217C4×Dic53—22no—
218C4×Dic53—21no—
219#11719917—82no—
220C2×#88322—21no—
221C2×#88322—21no—
222C2×#88322—22no—
223C2×#88322—82no—
224C23×C53—22no—
225C23×C53—22no—
226C23×C53—42no—
227C23×C53—42no—
228C23×C53C2×C22⋊C412yes—
229C23×C53C2×C22⋊C412yes—
230C23×C53C22×D411yescharacteristic
231C22×D53—21no—
232D4×C53—22no—
233D4×C53—22no—
234D4×C53—42no—
235C2×C4×C53—21no—
236C2×C4×C53—21no—
237C2×C4×C53—22no—
238C2×C4×C53—42no—
239C2×C4×C53—42no—
240C2×C4×C53—41no—
241C2×C4×C53—42no—
242C2×C4×C53—42no—
243C2×C4×C53—42no—
244C2×C4×C53—41no—
245C2×C4×C53—41no—
246C2×C4×C53—41no—
247C2×C4×C53—42no—
248C2×C4×C53—42no—
249C2×C4×C53—41no—
250C2×C4×C53—42no—
251C2×C4×C53—41no—
252C2×C4×C53—41no—
253C2×C4×C53C2×C22⋊C412yes—
254C2×C4×C53C2×C22⋊C412yes—
255C2×C4×C53C2×C22⋊C411yescharacteristic
256C2×C4×C53C2×C22⋊C411yescharacteristic
257C2×C4×C53C2×C22⋊C411yescharacteristic
258C2×C4×C53C2×C22⋊C411yescharacteristic
259C2×C4×C53C23×C411yescharacteristic
260C2×C4×C53C22×D411yescharacteristic
261C22⋊D53—44no—
262C22⋊D53—41no—
263C22⋊D53—41no—
264C22⋊D53—82no—
265C2×Dic53—21no—
266C2×Dic53—22no—
267C2×Dic53—22no—
268C2×Dic53—21no—
269C2×Dic53—21no—
270C2×Dic53—44no—
271C2×Dic53—82no—
272D212—41no—
273C4×D53—42no—
274C4×D53—82no—
275Dic106—42no—
276Dic106—41no—
277Dic106—42no—
278Dic106—82no—
279#88322—41no—
280#88322—42no—
281#88322—41no—
282#17600—531nomaximal, Sylow
283C22×C53—21no—
284C22×C53—21no—
285C22×C53—22no—
286C22×C53—22no—
287C22×C53—22no—
288C22×C53—22no—
289C22×C53—22no—
290C22×C53—22no—
291C22×C53—82no—
292C22×C53—82no—
293C22×C53C22×C22⋊C411yescharacteristic
294C22×C53#57711yescharacteristic
295C22×C53#57811yescharacteristic
296C2×D53—41no—
297C2×D53—41no—
298C2×D53—41no—
299C4×C53—21no—
300C4×C53—21no—
301C4×C53—22no—
302C4×C53—22no—
303C4×C53—21no—
304C4×C53—21no—
305C4×C53—21no—
306C4×C53—81no—
307C4×C53—82no—
308C4×C53—81no—
309C4×C53#42112yes—
310Dic53—41no—
311Dic53—42no—
312Dic53—41no—
313Dic53—42no—
314Dic53—41no—
315#3520—531no—
316#2974—531no—
317#2319D5311yescharacteristic, complemented
318#1520—532no—
319#1516—532no—
320#1513—532no—
321#1511—532no—
322#1509—531no—
323#1506—531no—
324#1503—531no—
325#1500—531no—
326C2×C53—21no—
327C2×C53—21no—
328C2×C53—42no—
329C2×C53—42no—
330C2×C53#288311yescharacteristic
331C2×C53#289811yescharacteristic
332C2×C53C2×#42111yescharacteristic
333D53—81no—
334#556—531no—
335#554—1061no—
336#552—532no—
337#551—1062no—
338D4×Q8—531no—
339#548—532no—
340#547—531no—
341#547—531no—
342#546—1061no—
343#542—1061no—
344#539—1062no—
345#533—531no—
346#529—531no—
347#525—532no—
348C2×C22⋊Q8—532no—
349#520—1061no—
350C2×C4×D4C2×D5311yescharacteristic
351#436—1061no—
352#435—1062no—
353C8×D4—1061no—
354#432—531no—
355#431—531no—
356#407—532no—
357C2×C22⋊C8—532no—
358#390—21no—
359#389—21no—
360#388—22no—
361#381C2×D5312yes—
362#380C2×D5312yes—
363C4×C4⋊C4C2×D5311yescharacteristic
364#376C2×D5311yescharacteristic
365#328—531no—
366#328—531no—
367#328—532no—
368#328—532no—
369#328—532no—
370#328—532no—
371#328—532no—
372#327—531no—
373#327—531no—
374#327—532no—
375C53#1760011yescharacteristic, complemented, Sylow
376C2×#41—532no—
377C2×#41—1061no—
378C22×Q8—532no—
379C22×D4C22⋊D5311yescharacteristic
380C23×C4C22×D5312yes—
381C2×M16—1062no—
382C22×C8—1062no—
383C4⋊Q8—531no—
384C4⋊Q8—531no—
385C4⋊Q8—1061no—
386#126—1062no—
387#126—1062no—
388#126—1061no—
389#125—531no—
390#125—1062no—
391#125—1061no—
392#124—531no—
393#124—1061no—
394#124—1062no—
395#123—1062no—
396#123—1062no—
397#123—1061no—
398#123—1062no—
399#123—1062no—
400C22⋊Q8—532no—
401C22⋊Q8—532no—
402C22⋊Q8—532no—
403C22⋊Q8—531no—
404C22⋊Q8—532no—
405C22⋊Q8—1062no—
406C22⋊Q8—1062no—
407C22⋊Q8—1061no—
408C22⋊Q8—1062no—
409C22⋊Q8—1061no—
410C22⋊Q8—1062no—
411#121—532no—
412#121—531no—
413#121—1062no—
414#121—1061no—
415#120—1062no—
416C4×Q8—531no—
417C4×Q8—532no—
418C4×Q8—1061no—
419C4×Q8—1061no—
420C4×D4—531no—
421C4×D4—1061no—
422C4×D4—1062no—
423C4×D4—1062no—
424C4×D4—1061no—
425C4×D4—1062no—
426C4×D4C4×D5311yescharacteristic
427C4×D4C4×D5311yescharacteristic
428C4×D4C4×D5312yes—
429C4×D4D21211yescharacteristic
430C4×D4D21212yes—
431C4×D4D21211yescharacteristic
432#117—532no—
433#117—1061no—
434#117—1062no—
435#117—1061no—
436C2×C4⋊C4—21no—
437C2×C4⋊C4—22no—
438C2×C4⋊C4—532no—
439C2×C4⋊C4—1062no—
440C2×C4⋊C4—1061no—
441C2×C4⋊C4C22⋊D5311yescharacteristic
442C2×C22⋊C4—22no—
443C2×C22⋊C4—22no—
444C2×C22⋊C4—1062no—
445C2×C22⋊C4C22⋊D5312yes—
446C2×C42—21no—
447C2×C42—21no—
448C2×C42C22×D5311yescharacteristic
449C4⋊C8—1061no—
450C4⋊C8—1061no—
451C22⋊C8—531no—
452C22⋊C8—532no—
453C22⋊C8—531no—
454C22⋊C8—1062no—
455C22⋊C8—1062no—
456#97—1061no—
457C4×C8—1061no—
458#95—21no—
459#95—22no—
460#95C22×D5311yescharacteristic
461#95C22×D5311yescharacteristic
462#95C22×D5312yes—
463C24C2×C22⋊D5312yes—
464#41—4242no—
465#41—2124no—
466C2×Q8—531no—
467C2×Q8—532no—
468C2×Q8—1062no—
469C2×Q8—1061no—
470C2×Q8—4242no—
471C2×D4—42no—
472C2×D4—531no—
473C2×D4—1062no—
474C2×D4—1062no—
475C2×D4—1061no—
476C2×D4#1171992111yescharacteristic
477C2×D4#1171992111yescharacteristic
478C2×D4#1171992112yes—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—22no—
482C22×C4—82no—
483C22×C4—1062no—
484C22×C4—1062no—
485C22×C4—1062no—
486C22×C4—1061no—
487C22×C4C2×C4×D5312yes—
488C22×C4C2×C4×D5312yes—
489C22×C4C2×D21212yes—
490C22×C4C2×D21212yes—
491C22×C4C2×C22⋊D5311yescharacteristic
492C22×C4C2×C22⋊D5311yescharacteristic
493C22×C4C2×C22⋊D5312yes—
494C22×C4C23×D5311yescharacteristic
495M16—4242no—
496C2×C8—1061no—
497C2×C8—1061no—
498C2×C8—1062no—
499C2×C8—4242no—
500C4⋊C4—42no—
501C4⋊C4—82no—
502C4⋊C4—531no—
503C4⋊C4—532no—
504C4⋊C4—531no—
505C4⋊C4—1062no—
506C4⋊C4—1062no—
507C4⋊C4—1061no—
508C4⋊C4—1062no—
509C4⋊C4—1061no—
510C4⋊C4—1061no—
511C4⋊C4—1061no—
512C4⋊C4—1062no—
513C4⋊C4—1061no—
514C4⋊C4—1062no—
515C4⋊C4—1061no—
516C4⋊C4—1061no—
517C4⋊C4—1062no—
518C4⋊C4#1171992111yescharacteristic
519C4⋊C4#1171992112yes—
520C4⋊C4#1171992111yescharacteristic
521C22⋊C4—82no—
522C22⋊C4—82no—
523C22⋊C4—1062no—
524C22⋊C4—1062no—
525C22⋊C4—1062no—
526C22⋊C4—1062no—
527C22⋊C4—1061no—
528C22⋊C4—1062no—
529C22⋊C4—1061no—
530C22⋊C4—1061no—
531C22⋊C4—1062no—
532C22⋊C4—1061no—
533C22⋊C4—1061no—
534C22⋊C4#1171992112yes—
535C22⋊C4#1171992112yes—
536C22⋊C4#1171992112yes—
537C22⋊C4#1171992112yes—
538C42—42no—
539C42—81no—
540C42—1062no—
541C42—1062no—
542C42—1061no—
543C42C2×C4×D5311yescharacteristic
544C42C2×C4×D5311yescharacteristic
545C42C2×D21211yescharacteristic
546C42C2×D21211yescharacteristic
547C23—22no—
548C23—22no—
549C23—42no—
550C23—42no—
551C23—1061no—
552C23C2×#1171992112yes—
553C23C2×#1171992112yes—
554C23C22×C22⋊D5311yescharacteristic
555Q8—4242no—
556Q8—2121no—
557Q8—2122no—
558Q8—2122no—
559D4—22no—
560D4—22no—
561D4—42no—
562D4—4242no—
563D4—2121no—
564D4—2124no—
565D4—2121no—
566D4—2121no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—22no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—41no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—41no—
582C2×C4—42no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—1061no—
586C2×C4—1062no—
587C2×C4—1061no—
588C2×C4—1062no—
589C2×C4—1061no—
590C2×C4—4242no—
591C2×C4—4242no—
592C2×C4—2124no—
593C2×C4—2122no—
594C2×C4C2×#1171992111yescharacteristic
595C2×C4C2×#1171992112yes—
596C2×C4C2×#1171992111yescharacteristic
597C2×C4C2×#1171992111yescharacteristic
598C2×C4C2×#1171992112yes—
599C2×C4C2×#1171992111yescharacteristic
600C2×C4C22×C4×D5311yescharacteristic
601C2×C4C22×D21211yescharacteristic
602C8—2121no—
603C8—2122no—
604C8—2121no—
605C22—21no—
606C22—21no—
607C22—22no—
608C22—22no—
609C22—22no—
610C22—22no—
611C22—22no—
612C22—22no—
613C22—82no—
614C22—82no—
615C22—2121no—
616C22—2121no—
617C22—2121no—
618C22C22×#1171992111yescharacteristic
619C22#1681143411yescharacteristic
620C22#1681143811yescharacteristic
621C4—21no—
622C4—21no—
623C4—22no—
624C4—22no—
625C4—21no—
626C4—21no—
627C4—21no—
628C4—81no—
629C4—82no—
630C4—81no—
631C4—2122no—
632C4—2121no—
633C4—2121no—
634C4—2121no—
635C4—2122no—
636C4#1681064112yes—
637C2—21no—
638C2—21no—
639C2—42no—
640C2—42no—
641C2—4241no—
642C2#2028317511yescharacteristic
643C2#2028324511yescharacteristic
644C2C2×#1681064111yescharacteristic
subgroupG/H
centerC22C22×#11719921
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#2319C2
socleC22×C53C22×C22⋊C4
prime radical#20288764C2
cosocleC22×C4×C53C24
hypercenter#2319D53
hypocenterC53#17600
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C4×D4 ◁ #2319

Complements

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

NH ≅ G/Ntype
2. #20283750C2semidirect
3. #20273286C2semidirect
4. #20273269C2semidirect
5. #20273256C2semidirect
6. #20273247C2semidirect
7. #20273238C2semidirect
8. #20273226C2semidirect
9. #20273213C2semidirect
10. #20273199C2semidirect
11. C53×#2319C2semidirect
12. #16811343C4semidirect
14. #16811313C4semidirect
16. D4×Dic106C4semidirect
17. #16811282C4semidirect
18. #16811278C4semidirect
19. #16811277C4semidirect
317. #2319D53semidirect
375. C53#17600semidirect

Maximal subgroups

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

subgroupindex
#202887642
#202837502
#202732862
#202732692
#202732562
#202732472
#202732382
#202732262
#202732132
#202731992
C53×#23192
#1760053