Finite Group Databasebeta

← #27484091

Subgroups — #27484091

Order 13568.

784 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20283023C211yesmaximal, characteristic
2C22⋊Q8×Dic53C211yesmaximal, characteristic
3#20276286C211yesmaximal, characteristic, complemented
4#20276265C211yesmaximal, characteristic, complemented
5#20276052C211yesmaximal, characteristic, complemented
6#20275864C211yesmaximal, characteristic, complemented
7#20275299C211yesmaximal, characteristic, complemented
8#20275289C211yesmaximal, characteristic, complemented
9#20275139C211yesmaximal, characteristic
10#20274985C211yesmaximal, characteristic, complemented
11#20274967C211yesmaximal, characteristic, complemented
12#20274742C211yesmaximal, characteristic
13#20274740C211yesmaximal, characteristic
14C2×#16810377C211yesmaximal, characteristic
15C53×#1892C211yesmaximal, characteristic
16C23×#88322C2211yescharacteristic
17C2×#13133208C2211yescharacteristic
18C2×Q8×Dic53C2211yescharacteristic
19C22×C4×Dic53C2211yescharacteristic
20C2×#13133150C2211yescharacteristic
21C2×#13133150C2211yescharacteristic
22C2×#13133149C2211yescharacteristic
23C2×#13133149C2211yescharacteristic
24C2×#13133141C2211yescharacteristic
25C2×#13133141C2211yescharacteristic
26#16810530C2211yescharacteristic
27#16810522C2211yescharacteristic
28#16810522C2211yescharacteristic
29#16810519C2211yescharacteristic
30#16810510—21no—
31#16810509—21no—
32#16810506—21no—
33#16810498—21no—
34#16810498—21no—
35#16810495—21no—
36#16810488C2211yescharacteristic
37#16810483C2211yescharacteristic
38#16810483C2211yescharacteristic
39C4⋊C4×Dic53—21no—
40C4⋊C4×Dic53C2211yescharacteristic
41C22⋊C4×Dic53—21no—
42#16810386C2211yescharacteristic
43#16810386C2211yescharacteristic
44#16810385C2211yescharacteristic
45#16810385C2211yescharacteristic
46#16810385C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810377C2211yescharacteristic
49#16810377C2211yescharacteristic
50#16810377C2211yescharacteristic
51#16810377C2211yescharacteristic
52C23×C8×C53C2211yescharacteristic
53C2×C22⋊Q8×C53C2211yescharacteristic
54C2×#103×C53C2211yescharacteristic
55C2×#103×C53C2211yescharacteristic
56C53×#380C2211yescharacteristic
57C53×#340C2211yescharacteristic
58C53×#340C2211yescharacteristic
59C23×Dic53D411yescharacteristic
60C23×Dic53D411yescharacteristic
61C22×#88322—22no—
62C22×#88322—22no—
63C22×#88322—41no—
64C22×#88322C2311yescharacteristic
65C22×#88322C2311yescharacteristic
66#13133208—41no—
67#13133208—41no—
68Q8×Dic53—41no—
69Q8×Dic53—41no—
70C2×C4⋊Dic53—21no—
71C2×C4⋊Dic53—21no—
72C2×C4⋊Dic53—21no—
73C2×C4⋊Dic53—21no—
74C2×C4⋊Dic53C2311yescharacteristic
75C2×C4⋊Dic53C2311yescharacteristic
76C2×C4⋊Dic53C2311yescharacteristic
77C2×C4⋊Dic53C2311yescharacteristic
78C2×#11719919—21no—
79C2×#11719919—21no—
80C2×C22⋊Dic53—21no—
81C2×C22⋊Dic53—21no—
82C2×C4×Dic53—21no—
83C2×C4×Dic53—21no—
84C2×C4×Dic53—41no—
85C2×C4×Dic53—41no—
86C2×C4×Dic53D411yescharacteristic
87C2×C4×Dic53D411yescharacteristic
88C2×C4×Dic53C2311yescharacteristic
89C2×C4×Dic53C2311yescharacteristic
90#13133150—22no—
91#13133150—41no—
92#13133149—22no—
93#13133149—41no—
94#13133141—22no—
95#13133141—41no—
96#13133124—21no—
97#13133124—21no—
98#13133124—21no—
99#13133124—21no—
100#13133124—21no—
101#13133124—21no—
102#13133124—21no—
103#13133124—21no—
104#13133124—21no—
105#13133124—21no—
106C22×Q8×C53C2311yescharacteristic
107C23×C4×C53C2311yescharacteristic
108C22×C8×C53—22no—
109C22×C8×C53—22no—
110C22×C8×C53—41no—
111C22×C8×C53C2311yescharacteristic
112C22×C8×C53C2311yescharacteristic
113C22⋊Q8×C53—22no—
114C22⋊Q8×C53Q814yes—
115C2×C4⋊C4×C53—21no—
116C2×C4⋊C4×C53C2311yescharacteristic
117C2×C4⋊C4×C53C2311yescharacteristic
118C2×C4⋊C4×C53C2311yescharacteristic
119C2×C22⋊C4×C53—21no—
120C2×C22⋊C4×C53—21no—
121#103×C53—22no—
122#103×C53—41no—
123#95×C53—21no—
124C24×C53C2×D411yescharacteristic
125C2×Q8×C53—21no—
126C2×Q8×C53—21no—
127C2×Q8×C53—21no—
128C2×Q8×C53—21no—
129C2×Q8×C53—21no—
130C2×Q8×C53C2×Q812yes—
131C22×C4×C53—21no—
132C22×C4×C53—21no—
133C22×C4×C53—21no—
134C22×C4×C53—21no—
135C22×C4×C53—21no—
136C22×C4×C53—21no—
137C22×C4×C53—21no—
138C22×C4×C53—21no—
139C22×C4×C53C2×D411yescharacteristic
140C22×C4×C53C2×Q812yes—
141C22×C4×C53#4112yes—
142C22×C4×C53C2411yescharacteristic
143C22×Dic53—22no—
144C22×Dic53—22no—
145C22×Dic53—22no—
146C22×Dic53—21no—
147C22×Dic53—21no—
148C22×Dic53—21no—
149C22×Dic53—21no—
150C22×Dic53—41no—
151C22×Dic53—41no—
152C22×Dic53—41no—
153C22×Dic53C2×D411yescharacteristic
154C22×Dic53C2×D411yescharacteristic
155C22×Dic53C2×D411yescharacteristic
156C22×Dic53C2×D411yescharacteristic
157C2×C8×C53—21no—
158C2×C8×C53—21no—
159C2×C8×C53—21no—
160C2×C8×C53—21no—
161C2×C8×C53—21no—
162C2×C8×C53—44no—
163C2×C8×C53#4112yes—
164C4⋊C4×C53—24no—
165C4⋊C4×C53—21no—
166C4⋊C4×C53—21no—
167C4⋊C4×C53—21no—
168C4⋊C4×C53—21no—
169C4⋊C4×C53—21no—
170C4⋊C4×C53C2×Q812yes—
171C22⋊C4×C53—24no—
172C22⋊C4×C53—42no—
173C22⋊Dic53—42no—
174C22⋊Dic53—42no—
175C4⋊Dic53—21no—
176C4⋊Dic53—21no—
177C4⋊Dic53—21no—
178C4⋊Dic53—21no—
179C4⋊Dic53—21no—
180C4⋊Dic53—21no—
181C4⋊Dic53—21no—
182C4⋊Dic53—21no—
183C4⋊Dic53—42no—
184C4⋊Dic53—42no—
185C4⋊Dic53—42no—
186C4⋊Dic53—42no—
187#11719919—42no—
188#11719919—42no—
189C4×Dic53—21no—
190C4×Dic53—21no—
191C4×Dic53—21no—
192C4×Dic53—21no—
193C4×Dic53—41no—
194C4×Dic53—41no—
195C4×Dic53—42no—
196C4×Dic53—42no—
197C2×#88322—21no—
198C2×#88322—21no—
199C2×#88322—21no—
200C2×#88322—21no—
201C2×#88322—21no—
202C2×#88322—44no—
203C2×#88322#4112yes—
204C23×C53—21no—
205C23×C53—21no—
206C23×C53—21no—
207C23×C53—21no—
208C23×C53C22⋊Q812yes—
209C23×C53#13512yes—
210C23×C53#13712yes—
211C23×C53C22×D411yescharacteristic
212Q8×C53—22no—
213Q8×C53—22no—
214Q8×C53—41no—
215Q8×C53—41no—
216C2×C4×C53—21no—
217C2×C4×C53—21no—
218C2×C4×C53—22no—
219C2×C4×C53—22no—
220C2×C4×C53—22no—
221C2×C4×C53—22no—
222C2×C4×C53—22no—
223C2×C4×C53—22no—
224C2×C4×C53—41no—
225C2×C4×C53—41no—
226C2×C4×C53—41no—
227C2×C4×C53—41no—
228C2×C4×C53—41no—
229C2×C4×C53—41no—
230C2×C4×C53—41no—
231C2×C4×C53—41no—
232C2×C4×C53—41no—
233C2×C4×C53—41no—
234C2×C4×C53C22⋊Q812yes—
235C2×C4×C53C22×Q811yescharacteristic
236C2×C4×C53C2×#4111yescharacteristic
237C2×C4×C53C2×#4111yescharacteristic
238C2×C4×C53C2×#4111yescharacteristic
239C2×C4×C53H(5,𝔽2)11yescharacteristic
240C2×C4×C53H(5,𝔽2)11yescharacteristic
241C2×Dic53—21no—
242C2×Dic53—21no—
243C2×Dic53—21no—
244C2×Dic53—21no—
245C2×Dic53—21no—
246C2×Dic53—21no—
247C2×Dic53—21no—
248C2×Dic53—21no—
249C2×Dic53—21no—
250C2×Dic53—21no—
251C2×Dic53—21no—
252C2×Dic53—44no—
253C2×Dic53—44no—
254C2×Dic53—41no—
255C2×Dic53—41no—
256C2×Dic53—41no—
257C2×Dic53—41no—
258C2×Dic53—41no—
259C2×Dic53—41no—
260C2×Dic53—41no—
261C2×Dic53—41no—
262C2×Dic53—41no—
263C2×Dic53—41no—
264C2×Dic53—41no—
265C2×Dic53—41no—
266C2×Dic53#13512yes—
267C8×C53—22no—
268C8×C53—41no—
269#88322—22no—
270#88322—41no—
271#29158—531nomaximal, Sylow
272C22×C53—22no—
273C22×C53—22no—
274C22×C53—22no—
275C22×C53—22no—
276C22×C53—22no—
277C22×C53—22no—
278C22×C53#47914yes—
279C22×C53C2×C22⋊Q811yescharacteristic
280C22×C53#53211yescharacteristic
281C22×C53#53411yescharacteristic
282C22×C53C2×#13511yescharacteristic
283C22×C53C2×#13511yescharacteristic
284C22×C53C2×#13711yescharacteristic
285C22×C53#57511yescharacteristic
286C4×C53—22no—
287C4×C53—42no—
288C4×C53—42no—
289C4×C53—81no—
290C4×C53—81no—
291C4×C53#55111yescharacteristic
292C4×C53#55111yescharacteristic
293C4×C53#55411yescharacteristic
294C4×C53#55411yescharacteristic
295Dic53—22no—
296Dic53—41no—
297Dic53—41no—
298Dic53—41no—
299Dic53—81no—
300Dic53—81no—
301Dic53—81no—
302Dic53—81no—
303#2851—531no—
304C4×C22⋊Q8—531no—
305#2101—531no—
306#2097—531no—
307#2059—531no—
308#2026—531no—
309#1920—531no—
310#1918—531no—
311#1892—531no—
312#1892D5311yescharacteristic
313#1865—531no—
314#1863—531no—
315#1808—531no—
316#1807—531no—
317C2×#336—531no—
318C2×C53—24no—
319C2×C53#256411yescharacteristic
320C2×C53C2×#47911yescharacteristic
321C2×C53#307611yescharacteristic
322C2×C53#316811yescharacteristic
323C2×C53#318111yescharacteristic
324C2×C53#319011yescharacteristic
325C2×C53#319111yescharacteristic
326C23×C8—531no—
327C23×C8C2×D5311yescharacteristic
328C2×#125—531no—
329C2×C22⋊Q8C2×D5311yescharacteristic
330C2×C4×Q8—531no—
331C22×C42—531no—
332C2×#107—531no—
333C2×#107—531no—
334C2×#106—531no—
335C2×#106—531no—
336C2×#103—531no—
337C2×#103—531no—
338C2×#103C2×D5311yescharacteristic
339C2×#103C2×D5311yescharacteristic
340#391—531no—
341#389—531no—
342#389—531no—
343#389—531no—
344#387—1061no—
345#387—1061no—
346#387—1061no—
347#385—1061no—
348#385—1061no—
349#385—1061no—
350#382—531no—
351#380—531no—
352#380—531no—
353#380C2×D5311yescharacteristic
354C4×C4⋊C4—531no—
355C4×C4⋊C4—1061no—
356C4×C22⋊C4—1061no—
357#340—531no—
358#340—531no—
359#340—531no—
360#340—531no—
361#340—531no—
362#340—531no—
363#340C2×D5311yescharacteristic
364#340C2×D5311yescharacteristic
365#336—531no—
366#336—531no—
367#336—531no—
368#336—531no—
369C53#2915811yescharacteristic, complemented, Sylow
370C22×Q8C22×D5311yescharacteristic
371C23×C4—531no—
372C23×C4—531no—
373C23×C4C22×D5311yescharacteristic
374C22×C8—22no—
375C22×C8—22no—
376C22×C8—41no—
377C22×C8—531no—
378C22×C8—531no—
379C22×C8—1062no—
380C22×C8—1062no—
381C22×C8—2121no—
382C22×C8C22×D5311yescharacteristic
383C22×C8C22×D5311yescharacteristic
384#125—2121no—
385#125—2121no—
386C22⋊Q8—22no—
387C22⋊Q8Dic10614yes—
388C4×Q8—2121no—
389C4×Q8—2121no—
390C2×C4⋊C4—21no—
391C2×C4⋊C4—531no—
392C2×C4⋊C4—531no—
393C2×C4⋊C4—531no—
394C2×C4⋊C4—531no—
395C2×C4⋊C4—1061no—
396C2×C4⋊C4—1061no—
397C2×C4⋊C4—1061no—
398C2×C4⋊C4—1061no—
399C2×C4⋊C4—1061no—
400C2×C4⋊C4—1061no—
401C2×C4⋊C4C22×D5311yescharacteristic
402C2×C4⋊C4C22×D5311yescharacteristic
403C2×C4⋊C4C22×D5311yescharacteristic
404C2×C22⋊C4—21no—
405C2×C22⋊C4—21no—
406C2×C22⋊C4—1061no—
407C2×C22⋊C4—1061no—
408C2×C42—531no—
409C2×C42—531no—
410C2×C42—531no—
411C2×C42—531no—
412C2×C42—1061no—
413C2×C42—1061no—
414C2×C42—2121no—
415C2×C42—2121no—
416#107—1062no—
417#107—2121no—
418#106—1062no—
419#106—2121no—
420#103—22no—
421#103—41no—
422#103—1062no—
423#103—2121no—
424#95—21no—
425#95—1061no—
426#95—1061no—
427#95—1061no—
428#95—1061no—
429#95—1061no—
430#95—1061no—
431#95—1061no—
432#95—1061no—
433#95—1061no—
434#95—1061no—
435C24D4×D5311yescharacteristic
436C2×Q8—21no—
437C2×Q8—21no—
438C2×Q8—21no—
439C2×Q8—21no—
440C2×Q8—21no—
441C2×Q8C2×Dic10612yes—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—531no—
451C22×C4—531no—
452C22×C4—531no—
453C22×C4—531no—
454C22×C4—1062no—
455C22×C4—1062no—
456C22×C4—1062no—
457C22×C4—1061no—
458C22×C4—1061no—
459C22×C4—1061no—
460C22×C4—1061no—
461C22×C4—2121no—
462C22×C4—2121no—
463C22×C4—2121no—
464C22×C4C2×Dic10612yes—
465C22×C4D4×D5311yescharacteristic
466C22×C4#1171994712yes—
467C22×C4C23×D5311yescharacteristic
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—44no—
474C2×C8—532no—
475C2×C8—1061no—
476C2×C8—1061no—
477C2×C8—1061no—
478C2×C8—1061no—
479C2×C8—1061no—
480C2×C8—2124no—
481C2×C8#1171994712yes—
482C4⋊C4—24no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—1061no—
489C4⋊C4—1061no—
490C4⋊C4—1061no—
491C4⋊C4—1061no—
492C4⋊C4—1061no—
493C4⋊C4—1061no—
494C4⋊C4—1061no—
495C4⋊C4—1061no—
496C4⋊C4—2122no—
497C4⋊C4—2122no—
498C4⋊C4—2122no—
499C4⋊C4—2122no—
500C4⋊C4—2122no—
501C4⋊C4—2122no—
502C4⋊C4C2×Dic10612yes—
503C22⋊C4—24no—
504C22⋊C4—42no—
505C22⋊C4—2122no—
506C22⋊C4—2122no—
507C42—1061no—
508C42—1061no—
509C42—1061no—
510C42—1061no—
511C42—2121no—
512C42—2121no—
513C42—2122no—
514C42—2122no—
515C23—21no—
516C23—21no—
517C23—21no—
518C23—21no—
519C23#1313319412yes—
520C23#1313323812yes—
521C23#1313324912yes—
522C23C2×D4×D5311yescharacteristic
523Q8—22no—
524Q8—22no—
525Q8—41no—
526Q8—41no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—532no—
546C2×C4—1061no—
547C2×C4—1061no—
548C2×C4—1061no—
549C2×C4—1061no—
550C2×C4—1061no—
551C2×C4—1061no—
552C2×C4—1061no—
553C2×C4—1061no—
554C2×C4—1061no—
555C2×C4—1061no—
556C2×C4—1061no—
557C2×C4—2124no—
558C2×C4—2121no—
559C2×C4—2121no—
560C2×C4—2121no—
561C2×C4—2121no—
562C2×C4—2121no—
563C2×C4—2121no—
564C2×C4—2121no—
565C2×C4—2121no—
566C2×C4—2121no—
567C2×C4—2121no—
568C2×C4—2121no—
569C2×C4—2121no—
570C2×C4—2124no—
571C2×C4#1313319412yes—
572C2×C4C22×Dic10611yescharacteristic
573C2×C4#41×D5311yescharacteristic
574C2×C4C2×#1171994711yescharacteristic
575C2×C4C2×#1171994711yescharacteristic
576C2×C4#1313326211yescharacteristic
577C2×C4#1313326311yescharacteristic
578C8—22no—
579C8—41no—
580C8—1062no—
581C8—2121no—
582C22—22no—
583C22—22no—
584C22—22no—
585C22—22no—
586C22—22no—
587C22—22no—
588C22#1681092814yes—
589C22C2×#1313319411yescharacteristic
590C22#1681116811yescharacteristic
591C22#1681118111yescharacteristic
592C22#135×D5311yescharacteristic
593C22C2×#1313323811yescharacteristic
594C22C2×#1313324911yescharacteristic
595C22#1681142011yescharacteristic
596C4—22no—
597C4—42no—
598C4—42no—
599C4—81no—
600C4—81no—
601C4—1062no—
602C4—4241no—
603C4—4241no—
604C4—4241no—
605C4—4241no—
606C4—2121no—
607C4—2121no—
608C4—2121no—
609C4#1681129611yescharacteristic
610C4#1681130411yescharacteristic
611C4#1681133011yescharacteristic
612C4#1681133211yescharacteristic
613C2—24no—
614C2#2027989911yescharacteristic
615C2C2×#1681092811yescharacteristic
616C2#2028449911yescharacteristic
617C2#2028527811yescharacteristic
618C2#2028541911yescharacteristic
619C2#2028549711yescharacteristic
620C2#2028550411yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#1892C2
socleC23×C53C22×D4
prime radicalC24×C53C2×D4
hypercenter#1892D53
hypocenterC53#29158
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1892

Complements

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

NH ≅ G/Ntype
3. #20276286C2semidirect
4. #20276265C2semidirect
5. #20276052C2semidirect
6. #20275864C2semidirect
7. #20275299C2semidirect
8. #20275289C2semidirect
10. #20274985C2semidirect
11. #20274967C2semidirect
369. C53#29158semidirect

Maximal subgroups

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

subgroupindex
#202830232
C22⋊Q8×Dic532
#202762862
#202762652
#202760522
#202758642
#202752992
#202752892
#202751392
#202749852
#202749672
#202747422
#202747402
C2×#168103772
C53×#18922
#2915853