Finite Group Databasebeta

← #27447103

Subgroups — #27447103

Order 13568.

864 conjugacy classes of subgroups.

The subgroup lattice has 642 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810903C211yesmaximal, characteristic
2C2×C4×#11719923C211yesmaximal, characteristic
3#20278554C211yesmaximal, characteristic, complemented
4#20276266C211yesmaximal, characteristic, complemented
5#20276197C211yesmaximal, characteristic, complemented
6#20276107C211yesmaximal, characteristic, complemented
7#20276048C211yesmaximal, characteristic, complemented
8#20275445C211yesmaximal, characteristic, complemented
9#20275274C211yesmaximal, characteristic, complemented
10#20275049C211yesmaximal, characteristic
11#20275049C211yesmaximal, characteristic
12#20274976C211yesmaximal, characteristic
13#20274968C211yesmaximal, characteristic, complemented
14#20274962C211yesmaximal, characteristic
15C53×#2349C211yesmaximal, characteristic
16C22×#11719923C2211yescharacteristic
17C2×#13133217C2211yescharacteristic
18C2×C4×Dic106C2211yescharacteristic
19#16810903—24no—
20C4×#11719923—24no—
21C2×#13133150C2211yescharacteristic
22C2×#13133150C2211yescharacteristic
23C2×#13133150C2211yescharacteristic
24C2×#13133148C2211yescharacteristic
25C2×#13133148C2211yescharacteristic
26C2×C4⋊#88322C2211yescharacteristic
27C2×#13133140C2211yescharacteristic
28C2×#13133140C2211yescharacteristic
29C2×#13133140C2211yescharacteristic
30C2×#13133138C2211yescharacteristic
31C2×#13133138C2211yescharacteristic
32C2×#13133138C2211yescharacteristic
33C2×C4×#88322C2211yescharacteristic
34#16810557—21no—
35#16810543—21no—
36#16810529C2211yescharacteristic
37#16810529C2211yescharacteristic
38#16810527—21no—
39#16810521C2211yescharacteristic
40#16810521C2211yescharacteristic
41#16810509—21no—
42#16810385C2211yescharacteristic
43#16810385C2211yescharacteristic
44#16810385C2211yescharacteristic
45#16810385C2211yescharacteristic
46#16810378C2211yescharacteristic
47#16810378C2211yescharacteristic
48#16810378C2211yescharacteristic
49#16810378C2211yescharacteristic
50C2×C4×D4×C53C2211yescharacteristic
51C2×C4×D4×C53C2211yescharacteristic
52C2×C4×D4×C53C2211yescharacteristic
53C53×#389C2211yescharacteristic
54C53×#389C2211yescharacteristic
55C53×#385—21no—
56C53×#385—21no—
57C53×#382C2211yescharacteristic
58C53×#380—21no—
59C4×C4⋊C4×C53C2211yescharacteristic
60C2×#95×C53—21no—
61C22×Dic106—21no—
62C22×Dic106C2311yescharacteristic
63C2×#11719923—22no—
64C2×#11719923—22no—
65C2×#11719923Q814yescomplemented
66C22×#88322C2311yescharacteristic
67C22×#88322C2311yescharacteristic
68C22×#88322C2311yescharacteristic
69C22×#88322C2311yescharacteristic
70#13133217—22no—
71#13133217—41no—
72C4×Dic106—22no—
73C4×Dic106—41no—
74C2×C4⋊Dic53—21no—
75C2×C4⋊Dic53C2311yescharacteristic
76C2×C4⋊Dic53C2311yescharacteristic
77C2×C4⋊Dic53C2311yescharacteristic
78C2×#11719919—21no—
79C2×#11719919—21no—
80C2×#11719919—21no—
81C2×C4×Dic53—21no—
82#13133150—22no—
83#13133150—22no—
84#13133150—22no—
85#13133148—22no—
86#13133148—41no—
87C4⋊#88322—22no—
88#13133140—22no—
89#13133140—22no—
90#13133140—41no—
91#13133138—22no—
92#13133138—22no—
93#13133138D414yes—
94C4×#88322—22no—
95#13133124—21no—
96#13133124—21no—
97#13133124—21no—
98#13133124—21no—
99#13133124—21no—
100#13133124—21no—
101C22×D4×C53C2311yescharacteristic
102C23×C4×C53—21no—
103C23×C4×C53—21no—
104C23×C4×C53—21no—
105C4×D4×C53—22no—
106C4×D4×C53—22no—
107C4×D4×C53—22no—
108C4×D4×C53—22no—
109C4×D4×C53—41no—
110C4×D4×C53—41no—
111C2×C4⋊C4×C53—21no—
112C2×C4⋊C4×C53—21no—
113C2×C4⋊C4×C53D411yescharacteristic
114C2×C4⋊C4×C53D411yescharacteristic
115C2×C4⋊C4×C53C2311yescharacteristic
116C2×C4⋊C4×C53C2311yescharacteristic
117C2×C4⋊C4×C53C2311yescharacteristic
118C2×C22⋊C4×C53—21no—
119C2×C22⋊C4×C53—21no—
120C2×C22⋊C4×C53—21no—
121C2×C42×C53C2311yescharacteristic
122C2×C42×C53C2311yescharacteristic
123C2×C42×C53C2311yescharacteristic
124#95×C53—21no—
125#95×C53—21no—
126#95×C53—21no—
127#95×C53—21no—
128#95×C53D411yescharacteristic
129#95×C53D411yescharacteristic
130C24×C53—21no—
131C2×D4×C53—21no—
132C2×D4×C53—21no—
133C2×D4×C53—21no—
134C2×D4×C53C2×D412yes—
135C2×D4×C53C2×Q812yes—
136C2×D4×C53#4112yes—
137C22×C4×C53—21no—
138C22×C4×C53—24no—
139C22×C4×C53—21no—
140C22×C4×C53—21no—
141C22×C4×C53—42no—
142C22×C4×C53—42no—
143C22×C4×C53—42no—
144C22×C4×C53—42no—
145C22×C4×C53—81no—
146C22×C4×C53—81no—
147C22×C4×C53—81no—
148C22×C4×C53—81no—
149C22×C4×C53C2×D411yescharacteristic
150C22×C4×C53C2×D411yescharacteristic
151C22×C4×C53C2×D411yescharacteristic
152C22×C4×C53C2×D411yescharacteristic
153C22×C4×C53C2×D411yescharacteristic
154C22×C4×C53C2×D411yescharacteristic
155C22×C4×C53C2411yescharacteristic
156C22×Dic53—21no—
157C22×Dic53—21no—
158C22×Dic53—21no—
159C22×Dic53—21no—
160C2×Dic106—21no—
161C2×Dic106—21no—
162C2×Dic106—21no—
163C2×Dic106—21no—
164C2×Dic106—21no—
165C2×Dic106—42no—
166C2×Dic106—42no—
167C2×Dic106—81no—
168C2×Dic106C2×Q812yes—
169C4⋊C4×C53—22no—
170C4⋊C4×C53—22no—
171C4⋊C4×C53—21no—
172C4⋊C4×C53—21no—
173C4⋊C4×C53—21no—
174C4⋊C4×C53—21no—
175C4⋊C4×C53—21no—
176C4⋊C4×C53—81no—
177C4⋊C4×C53—81no—
178C4⋊C4×C53#4112yes—
179C22⋊C4×C53—24no—
180C22⋊C4×C53—42no—
181C22⋊C4×C53—42no—
182C42×C53—21no—
183C42×C53—21no—
184C42×C53—21no—
185C42×C53—21no—
186C42×C53—21no—
187C42×C53—21no—
188#11719923—28no—
189C4⋊Dic53—21no—
190C4⋊Dic53—21no—
191C4⋊Dic53—21no—
192C4⋊Dic53—21no—
193C4⋊Dic53—21no—
194C4⋊Dic53—42no—
195C4⋊Dic53C2×D412yes—
196#11719919—42no—
197#11719919—81no—
198#11719919—81no—
199C4×Dic53—42no—
200C2×#88322—21no—
201C2×#88322—21no—
202C2×#88322—21no—
203C2×#88322—21no—
204C2×#88322—21no—
205C2×#88322—21no—
206C2×#88322—21no—
207C2×#88322—21no—
208C2×#88322—21no—
209C2×#88322—21no—
210C2×#88322C2×D412yes—
211C2×#88322C2×Q812yes—
212C23×C53—22no—
213C23×C53—22no—
214C23×C53—22no—
215C23×C53—41no—
216C23×C53—41no—
217C23×C53—41no—
218C23×C53—41no—
219C23×C53C22×D411yescharacteristic
220D4×C53—22no—
221D4×C53—22no—
222D4×C53—22no—
223D4×C53C22⋊Q814yes—
224C2×C4×C53—21no—
225C2×C4×C53—21no—
226C2×C4×C53—21no—
227C2×C4×C53—21no—
228C2×C4×C53—21no—
229C2×C4×C53—22no—
230C2×C4×C53—21no—
231C2×C4×C53—21no—
232C2×C4×C53—21no—
233C2×C4×C53—21no—
234C2×C4×C53—21no—
235C2×C4×C53—21no—
236C2×C4×C53—21no—
237C2×C4×C53—21no—
238C2×C4×C53—21no—
239C2×C4×C53—21no—
240C2×C4×C53—21no—
241C2×C4×C53—41no—
242C2×C4×C53—41no—
243C2×C4×C53—41no—
244C2×C4×C53—41no—
245C2×C4×C53—41no—
246C2×C4×C53—41no—
247C2×C4×C53—41no—
248C2×C4×C53—41no—
249C2×C4×C53—84no—
250C2×C4×C53—84no—
251C2×C4×C53—84no—
252C2×C4×C53#13512yes—
253C2×C4×C53#13712yes—
254C2×C4×C53C22×D411yescharacteristic
255C2×C4×C53C22×Q811yescharacteristic
256C2×C4×C53C2×#4111yescharacteristic
257C2×C4×C53C2×#4111yescharacteristic
258C2×C4×C53H(5,𝔽2)11yescharacteristic
259C2×C4×C532−1+411yescharacteristic
260C2×Dic53—22no—
261C2×Dic53—22no—
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—
272Dic106—22no—
273Dic106—22no—
274Dic106—41no—
275Dic106—41no—
276Dic106—84no—
277#88322—22no—
278#88322—22no—
279#88322—41no—
280#88322—41no—
281#26372—531nomaximal, Sylow
282C22×C53—24no—
283C22×C53—42no—
284C22×C53—42no—
285C22×C53—42no—
286C22×C53—42no—
287C22×C53—42no—
288C22×C53—42no—
289C22×C53#52511yescharacteristic
290C22×C53D4211yescharacteristic
291C22×C53D4×Q811yescharacteristic
292C22×C53C2×#13511yescharacteristic
293C22×C53C2×#13711yescharacteristic
294C22×C53#57611yescharacteristic
295C22×C53#57711yescharacteristic
296C4×C53—22no—
297C4×C53—22no—
298C4×C53—41no—
299C4×C53—41no—
300C4×C53—42no—
301C4×C53—41no—
302C4×C53—41no—
303C4×C53—81no—
304C4×C53—81no—
305C4×C53C2×C22⋊Q811yescharacteristic
306C4×C53#53911yescharacteristic
307C4×C53#54711yescharacteristic
308C4×C53#55411yescharacteristic
309Dic53—42no—
310Dic53—42no—
311Dic53—81no—
312Dic53—81no—
313C2×#474—531no—
314C2×C4×SD16—531no—
315#2451—531no—
316#2349D5311yescharacteristic
317#2097—531no—
318#2085—531no—
319#2068—531no—
320#2058—531no—
321#1942—531no—
322#1916—531no—
323#1877—531no—
324#1877—531no—
325#1864—531no—
326#1863—531no—
327#1862—531no—
328C2×C53—44no—
329C2×C53#244911yescharacteristic
330C2×C53#310711yescharacteristic
331C2×C53#311811yescharacteristic
332C2×C53#326611yescharacteristic
333C2×C53#328011yescharacteristic
334C2×C53#338111yescharacteristic
335C2×C53#338511yescharacteristic
336C22×SD16—531no—
337C2×C4⋊Q8—531no—
338C2×C4×Q8—531no—
339C2×C4×D4C2×D5311yescharacteristic
340C2×C4×D4C2×D5311yescharacteristic
341C2×C4×D4C2×D5311yescharacteristic
342#474—1064no—
343C4×SD16—1064no—
344C2×#107—531no—
345C2×#107—531no—
346C2×#107—531no—
347C2×#106—531no—
348C2×#106—531no—
349C2×C4⋊C8—531no—
350C2×#103—531no—
351C2×#103—531no—
352C2×#103—531no—
353C2×#102—531no—
354C2×#102—531no—
355C2×#102—531no—
356C2×C4×C8—531no—
357#398—1061no—
358#395—1061no—
359#391—531no—
360#391—531no—
361#391—1061no—
362#389—531no—
363#389—531no—
364#389C2×D5311yescharacteristic
365#389C2×D5311yescharacteristic
366#387—1061no—
367#385—21no—
368#385—21no—
369#382C2×D5311yescharacteristic
370#380—21no—
371C4×C4⋊C4C2×D5311yescharacteristic
372C2×#95—21no—
373#340—531no—
374#340—531no—
375#340—531no—
376#340—531no—
377#336—531no—
378#336—531no—
379#336—531no—
380#336—531no—
381C53#2637211yescharacteristic, complemented, Sylow
382C22×Q8—531no—
383C22×Q8—1061no—
384C22×D4C22×D5311yescharacteristic
385C23×C4—21no—
386C23×C4—21no—
387C23×C4—21no—
388C2×SD16—534no—
389C2×SD16—1062no—
390C2×SD16—1062no—
391C22×C8—531no—
392C22×C8—531no—
393C22×C8—531no—
394C22×C8—531no—
395C4⋊Q8—1062no—
396C4⋊Q8—2121no—
397C4×Q8—1062no—
398C4×Q8—2121no—
399C4×D4—22no—
400C4×D4—22no—
401C4×D4—22no—
402C4×D4—22no—
403C4×D4—41no—
404C4×D4—41no—
405C2×C4⋊C4—21no—
406C2×C4⋊C4—21no—
407C2×C4⋊C4—531no—
408C2×C4⋊C4—531no—
409C2×C4⋊C4—531no—
410C2×C4⋊C4—1061no—
411C2×C4⋊C4—1061no—
412C2×C4⋊C4—1061no—
413C2×C4⋊C4—1061no—
414C2×C4⋊C4C22⋊D5311yescharacteristic
415C2×C4⋊C4C22⋊D5311yescharacteristic
416C2×C4⋊C4C22×D5311yescharacteristic
417C2×C4⋊C4C22×D5311yescharacteristic
418C2×C4⋊C4C22×D5311yescharacteristic
419C2×C22⋊C4—21no—
420C2×C22⋊C4—21no—
421C2×C22⋊C4—21no—
422C2×C42—1061no—
423C2×C42C22×D5311yescharacteristic
424C2×C42C22×D5311yescharacteristic
425C2×C42C22×D5311yescharacteristic
426#107—1062no—
427#107—1062no—
428#107—1062no—
429#106—1062no—
430#106—2121no—
431C4⋊C8—1062no—
432#103—1062no—
433#103—1062no—
434#103—2121no—
435#102—534no—
436#102—1062no—
437#102—1062no—
438C4×C8—1062no—
439#95—21no—
440#95—21no—
441#95—21no—
442#95—21no—
443#95—1061no—
444#95—1061no—
445#95—1061no—
446#95—1061no—
447#95—1061no—
448#95—1061no—
449#95C22⋊D5311yescharacteristic
450#95C22⋊D5311yescharacteristic
451C24—21no—
452C2×Q8—532no—
453C2×Q8—1061no—
454C2×Q8—1061no—
455C2×Q8—1061no—
456C2×Q8—1061no—
457C2×Q8—1061no—
458C2×Q8—4241no—
459C2×Q8—2122no—
460C2×Q8—2122no—
461C2×D4—21no—
462C2×D4—21no—
463C2×D4—21no—
464C2×D4#1171994512yes—
465C2×D4D4×D5312yes—
466C2×D4Q8×D5312yes—
467C22×C4—21no—
468C22×C4—24no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—42no—
472C22×C4—42no—
473C22×C4—42no—
474C22×C4—42no—
475C22×C4—81no—
476C22×C4—81no—
477C22×C4—81no—
478C22×C4—81no—
479C22×C4—1061no—
480C22×C4—1061no—
481C22×C4—1061no—
482C22×C4—1061no—
483C22×C4C2×C22⋊D5311yescharacteristic
484C22×C4C2×C22⋊D5311yescharacteristic
485C22×C4C2×C22⋊D5311yescharacteristic
486C22×C4C2×C22⋊D5311yescharacteristic
487C22×C4C2×C22⋊D5311yescharacteristic
488C22×C4C2×C22⋊D5311yescharacteristic
489C22×C4C23×D5311yescharacteristic
490SD16—1068no—
491C2×C8—532no—
492C2×C8—532no—
493C2×C8—1061no—
494C2×C8—1061no—
495C2×C8—1061no—
496C2×C8—1061no—
497C2×C8—1061no—
498C2×C8—1061no—
499C2×C8—1061no—
500C2×C8—1061no—
501C2×C8—1061no—
502C2×C8—1061no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—21no—
510C4⋊C4—81no—
511C4⋊C4—81no—
512C4⋊C4—532no—
513C4⋊C4—1061no—
514C4⋊C4—1061no—
515C4⋊C4—1061no—
516C4⋊C4—1061no—
517C4⋊C4—1061no—
518C4⋊C4—4241no—
519C4⋊C4—4241no—
520C4⋊C4—2122no—
521C4⋊C4—2122no—
522C4⋊C4#1171994512yes—
523C22⋊C4—24no—
524C22⋊C4—42no—
525C22⋊C4—42no—
526C42—21no—
527C42—21no—
528C42—21no—
529C42—21no—
530C42—21no—
531C42—21no—
532C42—2122no—
533C23—22no—
534C23—22no—
535C23—22no—
536C23—41no—
537C23—41no—
538C23—41no—
539C23—41no—
540C23C22×C22⋊D5311yescharacteristic
541Q8—1062no—
542Q8—1062no—
543Q8—4244no—
544Q8—2121no—
545Q8—2121no—
546D4—22no—
547D4—22no—
548D4—22no—
549D4#1313319514yes—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—22no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—84no—
576C2×C4—84no—
577C2×C4—84no—
578C2×C4—1062no—
579C2×C4—1062no—
580C2×C4—2121no—
581C2×C4—2121no—
582C2×C4—2121no—
583C2×C4—2121no—
584C2×C4—2121no—
585C2×C4—2121no—
586C2×C4—2121no—
587C2×C4—2121no—
588C2×C4—2121no—
589C2×C4—2121no—
590C2×C4#1313323712yes—
591C2×C4#1313324712yes—
592C2×C4C2×D4×D5311yescharacteristic
593C2×C4C2×Q8×D5311yescharacteristic
594C2×C4C2×#1171994511yescharacteristic
595C2×C4C2×#1171994511yescharacteristic
596C2×C4#1313326211yescharacteristic
597C2×C4#1313326511yescharacteristic
598C8—1062no—
599C8—1062no—
600C8—2121no—
601C8—2121no—
602C22—24no—
603C22—42no—
604C22—42no—
605C22—42no—
606C22—42no—
607C22—42no—
608C22—42no—
609C22#1681113411yescharacteristic
610C22D4×C22⋊D5311yescharacteristic
611C22Q8×C22⋊D5311yescharacteristic
612C22C2×#1313323711yescharacteristic
613C22C2×#1313324711yescharacteristic
614C22#1681142611yescharacteristic
615C22#1681143411yescharacteristic
616C4—22no—
617C4—22no—
618C4—41no—
619C4—41no—
620C4—41no—
621C4—41no—
622C4—42no—
623C4—81no—
624C4—81no—
625C4—4241no—
626C4—4241no—
627C4—2122no—
628C4—2122no—
629C4C2×#1313319511yescharacteristic
630C4#1681122011yescharacteristic
631C4#1681127411yescharacteristic
632C4#1681132811yescharacteristic
633C2—44no—
634C2#2027852411yescharacteristic
635C2#2028478611yescharacteristic
636C2#2028489911yescharacteristic
637C2#2028622111yescharacteristic
638C2#2028639611yescharacteristic
639C2#2028759011yescharacteristic
640C2#2028762511yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2349C2
socleC23×C53C22×D4
prime radicalC22×D4×C53C23
hypercenter#2349D53
hypocenterC53#26372
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2349

Complements

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

NH ≅ G/Ntype
3. #20278554C2semidirect
4. #20276266C2semidirect
5. #20276197C2semidirect
6. #20276107C2semidirect
7. #20276048C2semidirect
8. #20275445C2semidirect
9. #20275274C2semidirect
13. #20274968C2semidirect
65. C2×#11719923Q8semidirect
381. C53#26372semidirect

Maximal subgroups

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

subgroupindex
C2×#168109032
C2×C4×#117199232
#202785542
#202762662
#202761972
#202761072
#202760482
#202754452
#202752742
#202750492
#202749762
#202749682
#202749622
C53×#23492
#2637253