Finite Group Databasebeta

← #27447202

Subgroups — #27447202

Order 13568.

828 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810932C211yesmaximal, characteristic, complemented
2#20278581C211yesmaximal, characteristic
3Q8×#11719919C211yesmaximal, characteristic, complemented
4#20276241C211yesmaximal, characteristic
5#20276194C211yesmaximal, characteristic, complemented
6#20276101C211yesmaximal, characteristic
7#20276055C211yesmaximal, characteristic, complemented
8#20275448C211yesmaximal, characteristic, complemented
9#20275272C211yesmaximal, characteristic, complemented
10#20275072C211yesmaximal, characteristic
11#20275072C211yesmaximal, characteristic
12#20274976C211yesmaximal, characteristic
13#20274971C211yesmaximal, characteristic, complemented
14#20274969C211yesmaximal, characteristic, complemented
15C2×C4×SD16×C53C211yesmaximal, characteristic
16C2×#13133208C2211yescharacteristic
17C2×Q8×Dic53C2211yescharacteristic
18C2×Q8×Dic53C2211yescharacteristic
19#16810932—24no—
20C2×#13133151C2211yescharacteristic
21C2×#13133151C2211yescharacteristic
22C2×#13133150C2211yescharacteristic
23C2×#13133148C2211yescharacteristic
24C2×#13133146C2211yescharacteristic
25C2×#13133141C2211yescharacteristic
26C2×#13133141C2211yescharacteristic
27C2×#13133138C2211yescharacteristic
28C2×#13133138C2211yescharacteristic
29#16810559—21no—
30#16810546—21no—
31#16810526C2211yescharacteristic
32#16810526C2211yescharacteristic
33#16810522C2211yescharacteristic
34#16810521—21no—
35#16810521C2211yescharacteristic
36#16810521C2211yescharacteristic
37#16810519—21no—
38#16810519C2211yescharacteristic
39#16810496—21no—
40#16810492—21no—
41#16810492C2211yescharacteristic
42#16810483—21no—
43C4×#11719919—21no—
44C4×#11719919C2211yescharacteristic
45#16810386C2211yescharacteristic
46#16810386C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810385C2211yescharacteristic
49#16810378C2211yescharacteristic
50#16810377C2211yescharacteristic
51#16810377C2211yescharacteristic
52#16810377C2211yescharacteristic
53C22×SD16×C53C2211yescharacteristic
54C2×C4×Q8×C53C2211yescharacteristic
55C2×C4×D4×C53C2211yescharacteristic
56C4×SD16×C53—24no—
57C2×#106×C53C2211yescharacteristic
58C2×#103×C53C2211yescharacteristic
59C2×#102×C53C2211yescharacteristic
60C2×C4×C8×C53C2211yescharacteristic
61C22×#88322C2311yescharacteristic
62C22×#88322C2311yescharacteristic
63#13133208—22no—
64#13133208—41no—
65Q8×Dic53—22no—
66Q8×Dic53—22no—
67Q8×Dic53—22no—
68Q8×Dic53—41no—
69C2×C4⋊Dic53—21no—
70C2×C4⋊Dic53—21no—
71C2×C4⋊Dic53—21no—
72C2×C4⋊Dic53—21no—
73C2×C4⋊Dic53C2311yescharacteristic
74C2×C4⋊Dic53C2311yescharacteristic
75C2×C4⋊Dic53C2311yescharacteristic
76C2×C4⋊Dic53C2311yescharacteristic
77C2×#11719919—21no—
78C2×#11719919—21no—
79C2×#11719919—21no—
80C2×#11719919—21no—
81C2×#11719919—21no—
82C2×#11719919—21no—
83C2×#11719919D411yescharacteristic
84C2×#11719919D411yescharacteristic
85C2×C22⋊Dic53—21no—
86C2×C22⋊Dic53—21no—
87C2×C22⋊Dic53—21no—
88C2×C22⋊Dic53—21no—
89C2×C4×Dic53—21no—
90C2×C4×Dic53—21no—
91C2×C4×Dic53C2311yescharacteristic
92C2×C4×Dic53C2311yescharacteristic
93#13133151—22no—
94#13133151—22no—
95#13133150—41no—
96#13133148—22no—
97#13133146—22no—
98#13133141—22no—
99#13133141—22no—
100#13133138—22no—
101#13133138—41no—
102#13133124—21no—
103#13133124—21no—
104#13133124—21no—
105#13133124—21no—
106#13133124D411yescharacteristic
107#13133124D411yescharacteristic
108C22×Q8×C53C2311yescharacteristic
109C22×D4×C53C2311yescharacteristic
110C23×C4×C53—21no—
111C2×SD16×C53—22no—
112C2×SD16×C53—22no—
113C2×SD16×C53Q814yes—
114C22×C8×C53C2311yescharacteristic
115C22×C8×C53C2311yescharacteristic
116C4×Q8×C53—22no—
117C4×Q8×C53—41no—
118C4×D4×C53—22no—
119C4×D4×C53—41no—
120C2×C4⋊C4×C53—21no—
121C2×C4⋊C4×C53C2311yescharacteristic
122C2×C4⋊C4×C53C2311yescharacteristic
123C2×C22⋊C4×C53—21no—
124C2×C42×C53—21no—
125C2×C42×C53C2311yescharacteristic
126#106×C53—22no—
127#103×C53D414yes—
128#102×C53—22no—
129C4×C8×C53—22no—
130C24×C53—21no—
131C2×Q8×C53—21no—
132C2×Q8×C53—21no—
133C2×Q8×C53—21no—
134C2×Q8×C53C2×D412yes—
135C2×Q8×C53C2×Q812yes—
136C2×Q8×C53#4112yes—
137C2×D4×C53—21no—
138C2×D4×C53—21no—
139C2×D4×C53—21no—
140C2×D4×C53—21no—
141C2×D4×C53—21no—
142C2×D4×C53C2×Q812yes—
143C22×C4×C53—21no—
144C22×C4×C53—21no—
145C22×C4×C53—21no—
146C22×C4×C53—42no—
147C22×C4×C53—42no—
148C22×C4×C53—81no—
149C22×C4×C53C2×D411yescharacteristic
150C22×C4×C53C2×D411yescharacteristic
151C22×C4×C53C2411yescharacteristic
152C22×Dic53—21no—
153C22×Dic53—21no—
154C22×Dic53—21no—
155C22×Dic53—21no—
156C22×Dic53C2×D411yescharacteristic
157C22×Dic53C2×D411yescharacteristic
158C22×Dic53C2×D411yescharacteristic
159C22×Dic53C2×D411yescharacteristic
160SD16×C53—28no—
161C2×C8×C53—21no—
162C2×C8×C53—21no—
163C2×C8×C53—21no—
164C2×C8×C53—21no—
165C2×C8×C53C2×D412yes—
166C2×C8×C53C2×Q812yes—
167C4⋊C4×C53—21no—
168C4⋊C4×C53—21no—
169C4⋊C4×C53—21no—
170C4⋊C4×C53—42no—
171C4⋊C4×C53C2×D412yes—
172C22⋊C4×C53—42no—
173C42×C53—21no—
174C42×C53—21no—
175C42×C53—42no—
176C22⋊Dic53—42no—
177C22⋊Dic53—81no—
178C22⋊Dic53—81no—
179C22⋊Dic53—81no—
180C4⋊Dic53—24no—
181C4⋊Dic53—21no—
182C4⋊Dic53—21no—
183C4⋊Dic53—21no—
184C4⋊Dic53—21no—
185C4⋊Dic53—21no—
186C4⋊Dic53—21no—
187C4⋊Dic53—21no—
188C4⋊Dic53—42no—
189C4⋊Dic53—42no—
190C4⋊Dic53—81no—
191C4⋊Dic53#4112yes—
192#11719919—22no—
193#11719919—22no—
194#11719919—42no—
195#11719919—42no—
196#11719919—81no—
197#11719919—81no—
198#11719919—81no—
199#11719919—81no—
200C4×Dic53—24no—
201C4×Dic53—21no—
202C4×Dic53—21no—
203C4×Dic53—21no—
204C4×Dic53—21no—
205C4×Dic53—42no—
206C2×#88322—21no—
207C2×#88322—21no—
208C2×#88322—21no—
209C2×#88322—21no—
210C2×#88322—21no—
211C2×#88322—21no—
212C23×C53—22no—
213C23×C53—41no—
214C23×C53—41no—
215C23×C53—41no—
216C23×C53—41no—
217C23×C53—41no—
218C23×C53—41no—
219C23×C53C22×D411yescharacteristic
220Q8×C53—22no—
221Q8×C53—22no—
222Q8×C53—22no—
223Q8×C53C22⋊Q814yes—
224D4×C53—22no—
225D4×C53—22no—
226D4×C53—41no—
227D4×C53—41no—
228C2×C4×C53—22no—
229C2×C4×C53—22no—
230C2×C4×C53—21no—
231C2×C4×C53—22no—
232C2×C4×C53—21no—
233C2×C4×C53—22no—
234C2×C4×C53—21no—
235C2×C4×C53—21no—
236C2×C4×C53—21no—
237C2×C4×C53—41no—
238C2×C4×C53—41no—
239C2×C4×C53—41no—
240C2×C4×C53—41no—
241C2×C4×C53—41no—
242C2×C4×C53—84no—
243C2×C4×C53#13712yes—
244C2×C4×C53C22×D411yescharacteristic
245C2×C4×C53C22×Q811yescharacteristic
246C2×C4×C53C2×#4111yescharacteristic
247C2×C4×C53C2×#4111yescharacteristic
248C2×C4×C53H(5,𝔽2)11yescharacteristic
249C2×C4×C532−1+411yescharacteristic
250C2×Dic53—21no—
251C2×Dic53—21no—
252C2×Dic53—22no—
253C2×Dic53—21no—
254C2×Dic53—21no—
255C2×Dic53—21no—
256C2×Dic53—21no—
257C2×Dic53—21no—
258C2×Dic53—21no—
259C2×Dic53—21no—
260C2×Dic53—21no—
261C2×Dic53—21no—
262C2×Dic53—41no—
263C2×Dic53—41no—
264C2×Dic53—41no—
265C2×Dic53—41no—
266C2×Dic53—41no—
267C2×Dic53—41no—
268C2×Dic53—41no—
269C2×Dic53—41no—
270C2×Dic53—41no—
271C2×Dic53—41no—
272C2×Dic53—41no—
273C2×Dic53#13512yes—
274C8×C53—22no—
275C8×C53—22no—
276#88322—41no—
277#88322—41no—
278#26379—531nomaximal, Sylow
279C22×C53—42no—
280C22×C53—42no—
281C22×C53—42no—
282C22×C53—81no—
283C22×C53—81no—
284C22×C53—81no—
285C22×C53—81no—
286C22×C53#52511yescharacteristic
287C22×C53D4211yescharacteristic
288C22×C53D4×Q811yescharacteristic
289C22×C53C2×#13511yescharacteristic
290C22×C53C2×#13711yescharacteristic
291C22×C53#57611yescharacteristic
292C22×C53#57711yescharacteristic
293C4×C53—24no—
294C4×C53—22no—
295C4×C53—42no—
296C4×C53—41no—
297C4×C53—81no—
298C4×C53C2×C22⋊Q811yescharacteristic
299C4×C53#53911yescharacteristic
300C4×C53#54711yescharacteristic
301C4×C53#55411yescharacteristic
302Dic53—22no—
303Dic53—41no—
304Dic53—42no—
305Dic53—41no—
306Dic53—41no—
307Dic53—81no—
308Dic53—81no—
309Dic53—81no—
310C2×#479—531no—
311C2×C4×SD16D5311yescharacteristic
312#2454—531no—
313Q8×C4⋊C4—531no—
314#2093—531no—
315#2084—531no—
316#2067—531no—
317#2059—531no—
318#1942—531no—
319#1916—531no—
320#1880—531no—
321#1880—531no—
322#1864—531no—
323#1863—531no—
324#1863—531no—
325C2×C53—82no—
326C2×C53#244911yescharacteristic
327C2×C53#310811yescharacteristic
328C2×C53#311911yescharacteristic
329C2×C53#326611yescharacteristic
330C2×C53#328711yescharacteristic
331C2×C53#338111yescharacteristic
332C2×C53#338511yescharacteristic
333C22×SD16C2×D5311yescharacteristic
334C2×#125—531no—
335C2×C4×Q8—531no—
336C2×C4×Q8—531no—
337C2×C4×Q8C2×D5311yescharacteristic
338C2×C4×D4C2×D5311yescharacteristic
339#479—1064no—
340C4×SD16—24no—
341C2×#107—531no—
342C2×#107—531no—
343C2×#107—531no—
344C2×#106—531no—
345C2×#106C2×D5311yescharacteristic
346C2×C4⋊C8—531no—
347C2×#103—531no—
348C2×#103—531no—
349C2×#103C2×D5311yescharacteristic
350C2×#102—531no—
351C2×#102—531no—
352C2×#102C2×D5311yescharacteristic
353C2×C4×C8C2×D5311yescharacteristic
354#399—1061no—
355#396—1061no—
356#390—531no—
357#390—531no—
358#389—531no—
359#389—531no—
360#389—531no—
361#389—531no—
362#389—1061no—
363#389—1061no—
364#385—1061no—
365#384—531no—
366#384—1061no—
367#380—1061no—
368C4×C4⋊C4—531no—
369C4×C4⋊C4—1061no—
370#340—531no—
371#340—531no—
372#340—531no—
373#340—531no—
374#336—531no—
375#336—531no—
376#336—531no—
377#336—531no—
378C53#2637911yescharacteristic, complemented, Sylow
379C22×Q8C22×D5311yescharacteristic
380C22×D4C22×D5311yescharacteristic
381C23×C4—21no—
382C2×SD16—22no—
383C2×SD16—22no—
384C2×SD16Dic10614yes—
385C22×C8—531no—
386C22×C8—531no—
387C22×C8C22×D5311yescharacteristic
388C22×C8C22×D5311yescharacteristic
389#125—1062no—
390#125—2121no—
391C4×Q8—22no—
392C4×Q8—41no—
393C4×Q8—1062no—
394C4×Q8—1062no—
395C4×Q8—1062no—
396C4×Q8—2121no—
397C4×D4—22no—
398C4×D4—41no—
399C2×C4⋊C4—21no—
400C2×C4⋊C4—531no—
401C2×C4⋊C4—531no—
402C2×C4⋊C4—531no—
403C2×C4⋊C4—531no—
404C2×C4⋊C4—531no—
405C2×C4⋊C4—531no—
406C2×C4⋊C4—1061no—
407C2×C4⋊C4—1061no—
408C2×C4⋊C4—1061no—
409C2×C4⋊C4—1061no—
410C2×C4⋊C4—1061no—
411C2×C4⋊C4—1061no—
412C2×C4⋊C4—1061no—
413C2×C4⋊C4—1061no—
414C2×C4⋊C4—1061no—
415C2×C4⋊C4—1061no—
416C2×C4⋊C4C22×D5311yescharacteristic
417C2×C4⋊C4C22×D5311yescharacteristic
418C2×C22⋊C4—21no—
419C2×C22⋊C4—1061no—
420C2×C22⋊C4—1061no—
421C2×C22⋊C4—1061no—
422C2×C22⋊C4—1061no—
423C2×C42—21no—
424C2×C42—531no—
425C2×C42—531no—
426C2×C42—1061no—
427C2×C42—1061no—
428C2×C42C22×D5311yescharacteristic
429#107—1062no—
430#107—1062no—
431#107—2121no—
432#106—22no—
433#106—1062no—
434C4⋊C8—1062no—
435#103—1062no—
436#103—1062no—
437#103C22⋊D5314yes—
438#102—22no—
439#102—1062no—
440#102—2121no—
441C4×C8—22no—
442#95—531no—
443#95—531no—
444#95—1061no—
445#95—1061no—
446#95—1061no—
447#95—1061no—
448C24—21no—
449C2×Q8—21no—
450C2×Q8—21no—
451C2×Q8—21no—
452C2×Q8C2×Dic10612yes—
453C2×Q8#1171994512yes—
454C2×Q8C2×C22⋊D5312yes—
455C2×D4—21no—
456C2×D4—21no—
457C2×D4—21no—
458C2×D4—21no—
459C2×D4—21no—
460C2×D4C2×Dic10612yes—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4—81no—
467C22×C4—531no—
468C22×C4—531no—
469C22×C4—531no—
470C22×C4—531no—
471C22×C4—1061no—
472C22×C4—1061no—
473C22×C4—1061no—
474C22×C4—1061no—
475C22×C4D4×D5311yescharacteristic
476C22×C4D4×D5311yescharacteristic
477C22×C4C23×D5311yescharacteristic
478SD16—28no—
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—1061no—
484C2×C8—1061no—
485C2×C8—1061no—
486C2×C8—1061no—
487C2×C8—1061no—
488C2×C8—1061no—
489C2×C8C2×Dic10612yes—
490C2×C8C2×C22⋊D5312yes—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—42no—
495C4⋊C4—532no—
496C4⋊C4—1062no—
497C4⋊C4—1064no—
498C4⋊C4—1062no—
499C4⋊C4—1061no—
500C4⋊C4—1061no—
501C4⋊C4—1061no—
502C4⋊C4—1061no—
503C4⋊C4—1061no—
504C4⋊C4—1061no—
505C4⋊C4—1061no—
506C4⋊C4—4241no—
507C4⋊C4—4241no—
508C4⋊C4—4241no—
509C4⋊C4—4241no—
510C4⋊C4—4241no—
511C4⋊C4—2122no—
512C4⋊C4—2122no—
513C4⋊C4—2122no—
514C4⋊C4—2122no—
515C4⋊C4C2×C22⋊D5312yes—
516C22⋊C4—42no—
517C22⋊C4—4241no—
518C22⋊C4—4241no—
519C22⋊C4—4241no—
520C22⋊C4—2122no—
521C42—21no—
522C42—21no—
523C42—42no—
524C42—1064no—
525C42—1061no—
526C42—1061no—
527C42—1061no—
528C42—1061no—
529C42—2122no—
530C23—22no—
531C23—41no—
532C23—41no—
533C23—41no—
534C23—41no—
535C23—41no—
536C23—41no—
537C23C2×D4×D5311yescharacteristic
538Q8—22no—
539Q8—22no—
540Q8—22no—
541Q8#1313319614yes—
542D4—22no—
543D4—22no—
544D4—41no—
545D4—41no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—22no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—22no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—84no—
561C2×C4—532no—
562C2×C4—1061no—
563C2×C4—1061no—
564C2×C4—1062no—
565C2×C4—1061no—
566C2×C4—1061no—
567C2×C4—1061no—
568C2×C4—1061no—
569C2×C4—1061no—
570C2×C4—1061no—
571C2×C4—1061no—
572C2×C4—1061no—
573C2×C4—1061no—
574C2×C4—2121no—
575C2×C4—2121no—
576C2×C4—2121no—
577C2×C4—2121no—
578C2×C4—2121no—
579C2×C4—2121no—
580C2×C4—2121no—
581C2×C4—2121no—
582C2×C4—2121no—
583C2×C4—2121no—
584C2×C4—2121no—
585C2×C4#1313324612yes—
586C2×C4C22×C22⋊D5311yescharacteristic
587C2×C4C22×Dic10611yescharacteristic
588C2×C4#41×D5311yescharacteristic
589C2×C4C2×#1171994511yescharacteristic
590C2×C4#1313326211yescharacteristic
591C2×C4#1313326411yescharacteristic
592C8—22no—
593C8—22no—
594C8—2121no—
595C8—2121no—
596C22—42no—
597C22—42no—
598C22—42no—
599C22—81no—
600C22—81no—
601C22—81no—
602C22—81no—
603C22#1681113211yescharacteristic
604C22D4×C22⋊D5311yescharacteristic
605C22D4×Dic10611yescharacteristic
606C22#135×D5311yescharacteristic
607C22C2×#1313324611yescharacteristic
608C22#1681142511yescharacteristic
609C22#1681142911yescharacteristic
610C4—22no—
611C4—24no—
612C4—41no—
613C4—42no—
614C4—81no—
615C4—1062no—
616C4—4241no—
617C4—4241no—
618C4—4241no—
619C4—2121no—
620C4—2121no—
621C4—2121no—
622C4—2122no—
623C4C2×#1313319611yescharacteristic
624C4#1681121511yescharacteristic
625C4#1681127211yescharacteristic
626C4#1681133211yescharacteristic
627C2—82no—
628C2#2027851811yescharacteristic
629C2#2028479311yescharacteristic
630C2#2028492011yescharacteristic
631C2#2028622511yescharacteristic
632C2#2028646911yescharacteristic
633C2#2028759111yescharacteristic
634C2#2028762311yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC2×C4×SD16×C53C2
socleC23×C53C22×D4
prime radicalC22×D4×C53C23
hypercenterC2×C4×SD16D53
hypocenterC53#26379
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4×SD16

Complements

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

NH ≅ G/Ntype
1. C2×#16810932C2semidirect
3. Q8×#11719919C2semidirect
5. #20276194C2semidirect
7. #20276055C2semidirect
8. #20275448C2semidirect
9. #20275272C2semidirect
13. #20274971C2semidirect
14. #20274969C2semidirect
378. C53#26379semidirect

Maximal subgroups

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

subgroupindex
C2×#168109322
#202785812
Q8×#117199192
#202762412
#202761942
#202761012
#202760552
#202754482
#202752722
#202750722
#202749762
#202749712
#202749692
C2×C4×SD16×C532
#2637953