Finite Group Databasebeta

← #27413888

Subgroups — #27413888

Order 13568.

1126 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810661C211yesmaximal, characteristic, complemented
2C2×#16810659C211yesmaximal, characteristic, complemented
3C2×#16810619C211yesmaximal, characteristic
4#20277206C211yesmaximal, characteristic
5#20277118C211yesmaximal, characteristic, complemented
6#20276265C211yesmaximal, characteristic, complemented
7#20276261C211yesmaximal, characteristic, complemented
8#20276256C211yesmaximal, characteristic, complemented
9#20276251C211yesmaximal, characteristic, complemented
10#20276246C211yesmaximal, characteristic, complemented
11#20276241C211yesmaximal, characteristic, complemented
12#20276238C211yesmaximal, characteristic, complemented
13#20276234C211yesmaximal, characteristic, complemented
14C2×#16810377C211yesmaximal, characteristic, complemented
15C2×C53×#416C211yesmaximal, characteristic
16C23×#88322C2211yescharacteristic
17C2×#13133176C2211yescharacteristic
18C2×#13133176C2211yescharacteristic
19C22×C4×Dic53C2211yescharacteristic
20#16810661—24no—
21#16810659—24no—
22C2×#13133151C2211yescharacteristic
23C2×#13133151C2211yescharacteristic
24C2×#13133150C2211yescharacteristic
25C2×#13133150C2211yescharacteristic
26C2×#13133149C2211yescharacteristic
27C2×#13133149C2211yescharacteristic
28C2×#13133148C2211yescharacteristic
29C2×#13133148C2211yescharacteristic
30#16810619—24no—
31C2×#13133141C2211yescharacteristic
32C2×#13133141C2211yescharacteristic
33C2×#13133138C2211yescharacteristic
34C2×#13133138C2211yescharacteristic
35#16810530C2211yescharacteristic
36#16810530C2211yescharacteristic
37#16810526C2211yescharacteristic
38#16810526C2211yescharacteristic
39#16810503—22no—
40#16810492C2211yescharacteristic
41#16810492C2211yescharacteristic
42#16810488C2211yescharacteristic
43#16810488C2211yescharacteristic
44#16810481—22no—
45C4×#11719919—22no—
46C4×C22⋊Dic53—22no—
47#16810377C2211yescharacteristic
48#16810377C2211yescharacteristic
49#16810377C2211yescharacteristic
50#16810377C2211yescharacteristic
51C22×#41×C53C2211yescharacteristic
52C23×C8×C53C2211yescharacteristic
53C2×#117×C53C2211yescharacteristic
54C53×#416—24no—
55C2×#103×C53C2211yescharacteristic
56C2×#103×C53C2211yescharacteristic
57C2×#102×C53C2211yescharacteristic
58C2×#102×C53C2211yescharacteristic
59C23×Dic53D412yes—
60C22×#88322—22no—
61C22×#88322—22no—
62C22×#88322D414yes—
63C22×#88322C2311yescharacteristic
64C22×#88322C2311yescharacteristic
65#13133176—22no—
66#13133176—22no—
67#13133176—22no—
68#13133176—22no—
69C2×C4⋊Dic53C2311yescharacteristic
70C2×C4⋊Dic53C2311yescharacteristic
71C2×C4⋊Dic53C2311yescharacteristic
72C2×C4⋊Dic53C2311yescharacteristic
73C2×#11719919—22no—
74C2×#11719919—22no—
75C2×C22⋊Dic53—21no—
76C2×C22⋊Dic53—21no—
77C2×C22⋊Dic53—22no—
78C2×C22⋊Dic53—22no—
79C2×C4×Dic53—21no—
80C2×C4×Dic53—21no—
81C2×C4×Dic53—22no—
82C2×C4×Dic53—22no—
83C2×C4×Dic53D412yes—
84C2×C4×Dic53C2311yescharacteristic
85C2×C4×Dic53C2311yescharacteristic
86#13133151—22no—
87#13133151—22no—
88#13133150—22no—
89#13133150—22no—
90#13133149—22no—
91#13133149—22no—
92#13133148—22no—
93#13133148—22no—
94#13133141—22no—
95#13133141—22no—
96#13133138—22no—
97#13133138—22no—
98#13133124—22no—
99#13133124—22no—
100#13133124—22no—
101#13133124—22no—
102C2×#41×C53—22no—
103C2×#41×C53—22no—
104C2×#41×C53—22no—
105C2×#41×C53—22no—
106C2×#41×C53—22no—
107C2×#41×C53Q814yes—
108C22×Q8×C53C2311yescharacteristic
109C22×D4×C53—21no—
110C22×D4×C53C2311yescharacteristic
111C23×C4×C53—21no—
112C23×C4×C53C2311yescharacteristic
113C22×C8×C53—22no—
114C22×C8×C53—22no—
115C22×C8×C53D414yes—
116C22×C8×C53C2311yescharacteristic
117C22×C8×C53C2311yescharacteristic
118#117×C53—22no—
119#117×C53—22no—
120C2×C4⋊C4×C53C2311yescharacteristic
121C2×C4⋊C4×C53C2311yescharacteristic
122C2×C22⋊C4×C53—21no—
123C2×C42×C53—21no—
124#103×C53—22no—
125#103×C53—22no—
126#102×C53—22no—
127#102×C53—22no—
128C24×C53—21no—
129C24×C53C2×D411yescharacteristic
130#41×C53—28no—
131#41×C53—28no—
132#41×C53—44no—
133#41×C53—44no—
134C2×Q8×C53—21no—
135C2×Q8×C53—21no—
136C2×Q8×C53—21no—
137C2×Q8×C53—21no—
138C2×Q8×C53—21no—
139C2×Q8×C53C2×Q812yes—
140C2×D4×C53—24no—
141C2×D4×C53—21no—
142C2×D4×C53—21no—
143C2×D4×C53—21no—
144C2×D4×C53—21no—
145C2×D4×C53—21no—
146C2×D4×C53—42no—
147C2×D4×C53—42no—
148C2×D4×C53C2×Q812yes—
149C22×C4×C53—21no—
150C22×C4×C53—24no—
151C22×C4×C53—21no—
152C22×C4×C53—21no—
153C22×C4×C53—42no—
154C22×C4×C53—42no—
155C22×C4×C53C2×D411yescharacteristic
156C22×C4×C53C2×D412yes—
157C22×C4×C53C2×D412yes—
158C22×C4×C53C2×Q812yes—
159C22×C4×C53#4112yes—
160C22×C4×C53#4112yes—
161C22×C4×C53#4112yes—
162C22×C4×C53C2411yescharacteristic
163C22×Dic53—24no—
164C22×Dic53—24no—
165C22×Dic53—21no—
166C22×Dic53—21no—
167C22×Dic53—21no—
168C22×Dic53—21no—
169C22×Dic53—42no—
170C22×Dic53C2×D412yes—
171C22×Dic53C2×D412yes—
172C2×C8×C53—28no—
173C2×C8×C53—21no—
174C2×C8×C53—21no—
175C2×C8×C53—21no—
176C2×C8×C53—21no—
177C2×C8×C53C2×D412yes—
178C2×C8×C53C2×D412yes—
179C4⋊C4×C53—21no—
180C4⋊C4×C53—21no—
181C4⋊C4×C53—21no—
182C4⋊C4×C53—21no—
183C22⋊C4×C53—42no—
184C42×C53—42no—
185C22⋊Dic53—44no—
186C22⋊Dic53—42no—
187C22⋊Dic53—42no—
188C22⋊Dic53—44no—
189C4⋊Dic53—21no—
190C4⋊Dic53—21no—
191C4⋊Dic53—21no—
192C4⋊Dic53—21no—
193C4⋊Dic53—21no—
194C4⋊Dic53—21no—
195C4⋊Dic53—21no—
196C4⋊Dic53—21no—
197#11719919—44no—
198#11719919—44no—
199C4×Dic53—24no—
200C4×Dic53—21no—
201C4×Dic53—21no—
202C4×Dic53—21no—
203C4×Dic53—21no—
204C4×Dic53—42no—
205C4×Dic53—42no—
206C2×#88322—28no—
207C2×#88322—21no—
208C2×#88322—21no—
209C2×#88322—21no—
210C2×#88322—21no—
211C2×#88322C2×D412yes—
212C2×#88322C2×D412yes—
213C23×C53—22no—
214C23×C53—22no—
215C23×C53—22no—
216C23×C53—41no—
217C23×C53—41no—
218C23×C53—42no—
219C23×C53—42no—
220C23×C53C22⋊Q812yes—
221C23×C53#12312yes—
222C23×C53#12712yes—
223C23×C53C22×D411yescharacteristic
224Q8×C53—22no—
225Q8×C53—22no—
226Q8×C53—41no—
227Q8×C53—41no—
228D4×C53—22no—
229D4×C53—22no—
230D4×C53—48no—
231D4×C53—41no—
232D4×C53—41no—
233C2×C4×C53—22no—
234C2×C4×C53—41no—
235C2×C4×C53—41no—
236C2×C4×C53—48no—
237C2×C4×C53—41no—
238C2×C4×C53—41no—
239C2×C4×C53—42no—
240C2×C4×C53—42no—
241C2×C4×C53C22⋊Q814yes—
242C2×C4×C53C22⋊Q812yes—
243C2×C4×C53C22⋊Q814yes—
244C2×C4×C53#12314yes—
245C2×C4×C53#12314yes—
246C2×C4×C53#12312yes—
247C2×C4×C53#12712yes—
248C2×C4×C53C22×D411yescharacteristic
249C2×C4×C53C22×D411yescharacteristic
250C2×C4×C53C22×Q811yescharacteristic
251C2×C4×C53C2×#4111yescharacteristic
252C2×C4×C53C2×#4111yescharacteristic
253C2×C4×C53C2×#4111yescharacteristic
254C2×Dic53—22no—
255C2×Dic53—22no—
256C2×Dic53—22no—
257C2×Dic53—22no—
258C2×Dic53—22no—
259C2×Dic53—48no—
260C2×Dic53—41no—
261C2×Dic53—41no—
262C2×Dic53—41no—
263C2×Dic53—41no—
264C2×Dic53—42no—
265C2×Dic53—42no—
266C2×Dic53—42no—
267C2×Dic53—42no—
268C2×Dic53#13514yes—
269C8×C53—22no—
270C8×C53—22no—
271#88322—22no—
272#88322—22no—
273#23730—531nomaximal, Sylow
274C22×C53—24no—
275C22×C53—24no—
276C22×C53—22no—
277C22×C53—22no—
278C22×C53—44no—
279C22×C53—42no—
280C22×C53—82no—
281C22×C53—81no—
282C22×C53—81no—
283C22×C53#39914yes—
284C22×C53C2×C22⋊Q811yescharacteristic
285C22×C53C2×#12311yescharacteristic
286C22×C53C2×#12711yescharacteristic
287C22×C53C2×#13512yes—
288C22×C53#57512yes—
289C4×C53—42no—
290C4×C53—81no—
291C4×C53—81no—
292C4×C53#39914yes—
293C4×C53C2×C22⋊Q811yescharacteristic
294C4×C53C2×C22⋊Q811yescharacteristic
295C4×C53C2×#12311yescharacteristic
296C4×C53C2×#12311yescharacteristic
297Dic53—24no—
298Dic53—42no—
299Dic53—81no—
300Dic53—81no—
301Dic53—81no—
302Dic53—81no—
303C2×#427—531no—
304C2×#427—531no—
305C2×#416—531no—
306C2×#416D5311yescharacteristic
307#2297—531no—
308#2285—531no—
309#2097—531no—
310#2097—531no—
311#2096—531no—
312#2095—531no—
313#2094—531no—
314#2093—531no—
315#2092—531no—
316#2092—531no—
317C2×#336—531no—
318C2×C53—24no—
319C2×C53—82no—
320C2×C53C2×#39911yescharacteristic
321C2×C53#307612yes—
322C2×C53#309112yes—
323C2×C53#314711yescharacteristic
324C2×C53#315011yescharacteristic
325C22×#41C2×D5311yescharacteristic
326C23×C8—531no—
327C23×C8C2×D5311yescharacteristic
328C2×#117—531no—
329C2×#117—531no—
330C2×#117C2×D5311yescharacteristic
331C22×C42—531no—
332#427—1064no—
333#427—1064no—
334C2×#107—531no—
335C2×#107—531no—
336C2×#107—531no—
337C2×#107—531no—
338C2×#106—531no—
339C2×#106—531no—
340C2×#106—531no—
341C2×#106—531no—
342#416—24no—
343#416—1064no—
344C2×#103—531no—
345C2×#103—531no—
346C2×#103C2×D5311yescharacteristic
347C2×#103C2×D5311yescharacteristic
348C2×#102—531no—
349C2×#102—531no—
350C2×#102C2×D5311yescharacteristic
351C2×#102C2×D5311yescharacteristic
352#391—531no—
353#391—531no—
354#390—531no—
355#390—531no—
356#386—1062no—
357#384—531no—
358#384—531no—
359#382—531no—
360#382—531no—
361#380—1062no—
362C4×C4⋊C4—1062no—
363C4×C22⋊C4—1062no—
364#336—531no—
365#336—531no—
366#336—531no—
367#336—531no—
368C53#2373011yescharacteristic, complemented, Sylow
369C2×#41—22no—
370C2×#41—22no—
371C2×#41—22no—
372C2×#41—22no—
373C2×#41—22no—
374C2×#41Dic10614yes—
375C22×Q8C22×D5311yescharacteristic
376C22×D4—21no—
377C22×D4C22×D5311yescharacteristic
378C23×C4—21no—
379C23×C4—532no—
380C23×C4C22×D5311yescharacteristic
381C22×C8—22no—
382C22×C8—22no—
383C22×C8—534no—
384C22×C8—531no—
385C22×C8—531no—
386C22×C8—1062no—
387C22×C8—1062no—
388C22×C8C22⋊D5314yes—
389C22×C8C22×D5311yescharacteristic
390C22×C8C22×D5311yescharacteristic
391#117—22no—
392#117—22no—
393#117—1062no—
394#117—1062no—
395#117—1062no—
396#117—1062no—
397C2×C4⋊C4—531no—
398C2×C4⋊C4—531no—
399C2×C4⋊C4—531no—
400C2×C4⋊C4—531no—
401C2×C4⋊C4—1062no—
402C2×C4⋊C4—1062no—
403C2×C4⋊C4C22×D5311yescharacteristic
404C2×C4⋊C4C22×D5311yescharacteristic
405C2×C22⋊C4—21no—
406C2×C22⋊C4—1062no—
407C2×C22⋊C4—1062no—
408C2×C22⋊C4—1061no—
409C2×C22⋊C4—1061no—
410C2×C42—21no—
411C2×C42—532no—
412C2×C42—531no—
413C2×C42—531no—
414C2×C42—1062no—
415C2×C42—1062no—
416C2×C42—1061no—
417C2×C42—1061no—
418#107—1062no—
419#107—1062no—
420#107—1062no—
421#107—1062no—
422#106—1062no—
423#106—1062no—
424#106—1062no—
425#106—1062no—
426#103—22no—
427#103—22no—
428#103—1062no—
429#103—1062no—
430#102—22no—
431#102—22no—
432#102—1062no—
433#102—1062no—
434#95—1062no—
435#95—1062no—
436#95—1062no—
437#95—1062no—
438C24—21no—
439C24D4×D5311yescharacteristic
440#41—28no—
441#41—28no—
442#41—44no—
443#41—44no—
444C2×Q8—21no—
445C2×Q8—21no—
446C2×Q8—21no—
447C2×Q8—21no—
448C2×Q8—21no—
449C2×Q8C2×Dic10612yes—
450C2×D4—24no—
451C2×D4—21no—
452C2×D4—21no—
453C2×D4—21no—
454C2×D4—21no—
455C2×D4—21no—
456C2×D4—42no—
457C2×D4—42no—
458C2×D4C2×Dic10612yes—
459C22×C4—21no—
460C22×C4—24no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—42no—
464C22×C4—42no—
465C22×C4—532no—
466C22×C4—532no—
467C22×C4—1064no—
468C22×C4—1064no—
469C22×C4—1061no—
470C22×C4—1061no—
471C22×C4—1061no—
472C22×C4—1061no—
473C22×C4—2122no—
474C22×C4C2×Dic10612yes—
475C22×C4#1171994512yes—
476C22×C4D4×D5312yes—
477C22×C4D4×D5311yescharacteristic
478C22×C4#1171994712yes—
479C22×C4#1171994712yes—
480C22×C4C2×C22⋊D5312yes—
481C22×C4C23×D5311yescharacteristic
482C2×C8—28no—
483C2×C8—21no—
484C2×C8—21no—
485C2×C8—21no—
486C2×C8—21no—
487C2×C8—532no—
488C2×C8—532no—
489C2×C8—1068no—
490C2×C8—1061no—
491C2×C8—1061no—
492C2×C8—1061no—
493C2×C8—1061no—
494C2×C8C2×C22⋊D5312yes—
495C2×C8C2×C22⋊D5312yes—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—1061no—
501C4⋊C4—1061no—
502C4⋊C4—1061no—
503C4⋊C4—1061no—
504C4⋊C4—1061no—
505C4⋊C4—1061no—
506C4⋊C4—1061no—
507C4⋊C4—1061no—
508C4⋊C4—2124no—
509C4⋊C4—2124no—
510C22⋊C4—42no—
511C22⋊C4—2124no—
512C22⋊C4—2124no—
513C22⋊C4—2122no—
514C22⋊C4—2122no—
515C42—42no—
516C42—1064no—
517C42—1061no—
518C42—1061no—
519C42—1061no—
520C42—1061no—
521C42—2122no—
522C42—2122no—
523C23—22no—
524C23—22no—
525C23—22no—
526C23—41no—
527C23—42no—
528C23—41no—
529C23—42no—
530C23#1313319412yes—
531C23#1313319912yes—
532C23#1313321312yes—
533C23C2×D4×D5311yescharacteristic
534Q8—22no—
535Q8—22no—
536Q8—41no—
537Q8—41no—
538D4—22no—
539D4—22no—
540D4—48no—
541D4—41no—
542D4—41no—
543C2×C4—22no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—48no—
547C2×C4—42no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—42no—
551C2×C4—534no—
552C2×C4—1062no—
553C2×C4—1062no—
554C2×C4—1062no—
555C2×C4—1062no—
556C2×C4—1062no—
557C2×C4—2128no—
558C2×C4—2122no—
559C2×C4—2122no—
560C2×C4—2121no—
561C2×C4—2121no—
562C2×C4—2121no—
563C2×C4—2121no—
564C2×C4—2122no—
565C2×C4—2122no—
566C2×C4#1313319412yes—
567C2×C4#1313319414yes—
568C2×C4#1313319614yes—
569C2×C4#1313319814yes—
570C2×C4#1313319912yes—
571C2×C4#1313319914yes—
572C2×C4#1313321312yes—
573C2×C4C2×D4×D5311yescharacteristic
574C2×C4C22×C22⋊D5311yescharacteristic
575C2×C4C22×Dic10611yescharacteristic
576C2×C4C2×#1171994711yescharacteristic
577C2×C4C2×#1171994711yescharacteristic
578C2×C4C2×#1171994511yescharacteristic
579C8—22no—
580C8—22no—
581C8—1062no—
582C8—1062no—
583C22—24no—
584C22—24no—
585C22—22no—
586C22—22no—
587C22—44no—
588C22—42no—
589C22—82no—
590C22—81no—
591C22—81no—
592C22#1681055914yes—
593C22C2×#1313319411yescharacteristic
594C22C2×#1313319911yescharacteristic
595C22C2×#1313321311yescharacteristic
596C22#135×D5312yes—
597C22#1681142012yes—
598C4—42no—
599C4—81no—
600C4—81no—
601C4—1064no—
602C4—4241no—
603C4—4241no—
604C4—4241no—
605C4—4241no—
606C4—2122no—
607C4#1681055914yes—
608C4C2×#1313319411yescharacteristic
609C4C2×#1313319611yescharacteristic
610C4C2×#1313319811yescharacteristic
611C4C2×#1313319911yescharacteristic
612C2—24no—
613C2—82no—
614C2C2×#1681055911yescharacteristic
615C2#2028449912yes—
616C2#2028462612yes—
617C2#2028513911yescharacteristic
618C2#2028515811yescharacteristic
subgroupG/H
centerC22×C4D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC2×C53×#416C2
socleC23×C53C22×D4
prime radicalC22×#41×C53C22
hypercenterC2×#416D53
hypocenterC53#23730
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22×C4 ◁ C23×C4 ◁ C2×#416

Complements

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

NH ≅ G/Ntype
1. C2×#16810661C2semidirect
2. C2×#16810659C2semidirect
5. #20277118C2semidirect
6. #20276265C2semidirect
7. #20276261C2semidirect
8. #20276256C2semidirect
9. #20276251C2semidirect
10. #20276246C2semidirect
11. #20276241C2semidirect
12. #20276238C2semidirect
13. #20276234C2semidirect
14. C2×#16810377C2semidirect
368. C53#23730semidirect

Maximal subgroups

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

subgroupindex
C2×#168106612
C2×#168106592
C2×#168106192
#202772062
#202771182
#202762652
#202762612
#202762562
#202762512
#202762462
#202762412
#202762382
#202762342
C2×#168103772
C2×C53×#4162
#2373053