Finite Group Databasebeta

← #27449717

Subgroups — #27449717

Order 13568.

816 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810953C211yesmaximal, characteristic
2C2×C4×#11719925C211yesmaximal, characteristic, complemented
3#20278218C211yesmaximal, characteristic
4#20276263C211yesmaximal, characteristic, complemented
5#20276037C211yesmaximal, characteristic, complemented
6#20275908C211yesmaximal, characteristic
7#20275723C211yesmaximal, characteristic
8#20275483C211yesmaximal, characteristic, complemented
9#20275440C211yesmaximal, characteristic, complemented
10#20275349C211yesmaximal, characteristic, complemented
11#20275323C211yesmaximal, characteristic
12#20274983C211yesmaximal, characteristic, complemented
13#20274968C211yesmaximal, characteristic, complemented
14#20274903C211yesmaximal, characteristic
15C53×#2360C211yesmaximal, characteristic
16C22×#11719925C2211yescharacteristic
17C2×#13133196C2211yescharacteristic
18C2×C4×Dic106C2211yescharacteristic
19#16810953—24no—
20C4×#11719925—24no—
21C2×#13133150C2211yescharacteristic
22C2×#13133150C2211yescharacteristic
23C2×#13133148C2211yescharacteristic
24C2×#13133148C2211yescharacteristic
25C2×#13133148C2211yescharacteristic
26C2×C4⋊#88322C2211yescharacteristic
27C2×#13133141C2211yescharacteristic
28C2×#13133141C2211yescharacteristic
29C2×#13133141C2211yescharacteristic
30C2×#13133140C2211yescharacteristic
31C2×#13133140C2211yescharacteristic
32C2×#13133140C2211yescharacteristic
33C2×C22⋊#88322C2211yescharacteristic
34C2×C22⋊#88322C2211yescharacteristic
35C2×C4×#88322C2211yescharacteristic
36#16810564—21no—
37#16810551—21no—
38#16810529C2211yescharacteristic
39#16810521C2211yescharacteristic
40#16810517—21no—
41#16810505—21no—
42#16810489C2211yescharacteristic
43#16810483C2211yescharacteristic
44#16810385C2211yescharacteristic
45#16810385C2211yescharacteristic
46#16810385C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810378C2211yescharacteristic
49#16810378C2211yescharacteristic
50C2×C22⋊Q8×C53C2211yescharacteristic
51C2×C4×Q8×C53C2211yescharacteristic
52C22×C4⋊C4×C53C2211yescharacteristic
53C53×#391C2211yescharacteristic
54C53×#389—21no—
55C53×#389C2211yescharacteristic
56C53×#387—21no—
57C53×#385—21no—
58C4×C4⋊C4×C53C2211yescharacteristic
59C4×C22⋊C4×C53—21no—
60C4×C22⋊C4×C53C2211yescharacteristic
61C22×Dic106C2311yescharacteristic
62C2×#11719925—22no—
63C2×#11719925—22no—
64C2×#11719925—22no—
65C22×#88322C2311yescharacteristic
66C22×#88322C2311yescharacteristic
67C22×#88322C2311yescharacteristic
68C22×#88322C2311yescharacteristic
69#13133196—22no—
70#13133196—41no—
71C4×Dic106—22no—
72C4×Dic106—41no—
73C2×C4⋊Dic53C2311yescharacteristic
74C2×C4⋊Dic53C2311yescharacteristic
75C2×C4⋊Dic53C2311yescharacteristic
76C2×#11719919—21no—
77C2×#11719919—21no—
78C2×#11719919—21no—
79C2×C22⋊Dic53—21no—
80C2×C22⋊Dic53—21no—
81C2×C4×Dic53—21no—
82#13133150—22no—
83#13133150—41no—
84#13133148—22no—
85#13133148—22no—
86#13133148—22no—
87C4⋊#88322—22no—
88#13133141—22no—
89#13133141—22no—
90#13133141—41no—
91#13133140—22no—
92#13133140—22no—
93#13133140D414yes—
94C22⋊#88322—22no—
95C22⋊#88322—22no—
96C4×#88322—22no—
97#13133124—21no—
98#13133124—21no—
99#13133124—21no—
100#13133124—21no—
101#13133124—21no—
102#13133124—21no—
103C22×Q8×C53C2311yescharacteristic
104C23×C4×C53—21no—
105C23×C4×C53C2311yescharacteristic
106C22⋊Q8×C53—22no—
107C22⋊Q8×C53—22no—
108C4×Q8×C53—22no—
109C4×Q8×C53—41no—
110C2×C4⋊C4×C53—21no—
111C2×C4⋊C4×C53—21no—
112C2×C4⋊C4×C53—21no—
113C2×C4⋊C4×C53—21no—
114C2×C4⋊C4×C53—22no—
115C2×C4⋊C4×C53—41no—
116C2×C4⋊C4×C53C2311yescharacteristic
117C2×C4⋊C4×C53C2311yescharacteristic
118C2×C4⋊C4×C53C2311yescharacteristic
119C2×C22⋊C4×C53—21no—
120C2×C22⋊C4×C53D411yescharacteristic
121C2×C22⋊C4×C53D411yescharacteristic
122C2×C42×C53—21no—
123C2×C42×C53—21no—
124C2×C42×C53C2311yescharacteristic
125C2×C42×C53C2311yescharacteristic
126#95×C53—21no—
127#95×C53—21no—
128#95×C53—21no—
129#95×C53—21no—
130#95×C53D411yescharacteristic
131#95×C53D411yescharacteristic
132C24×C53C2×D411yescharacteristic
133C2×Q8×C53—21no—
134C2×Q8×C53—21no—
135C2×Q8×C53—21no—
136C2×Q8×C53—21no—
137C2×Q8×C53—21no—
138C2×Q8×C53#4112yes—
139C22×C4×C53—21no—
140C22×C4×C53—21no—
141C22×C4×C53—21no—
142C22×C4×C53—21no—
143C22×C4×C53—21no—
144C22×C4×C53—21no—
145C22×C4×C53—21no—
146C22×C4×C53—21no—
147C22×C4×C53—21no—
148C22×C4×C53—21no—
149C22×C4×C53—42no—
150C22×C4×C53—42no—
151C22×C4×C53—81no—
152C22×C4×C53C2×D411yescharacteristic
153C22×C4×C53C2×D411yescharacteristic
154C22×C4×C53C2×D411yescharacteristic
155C22×C4×C53C2×D411yescharacteristic
156C22×C4×C53C2×D411yescharacteristic
157C22×C4×C53C2411yescharacteristic
158C22×Dic53—21no—
159C22×Dic53—21no—
160C22×Dic53—21no—
161C22×Dic53—21no—
162C2×Dic106—21no—
163C2×Dic106—21no—
164C2×Dic106—21no—
165C2×Dic106—21no—
166C2×Dic106—21no—
167C2×Dic106C2×D412yes—
168C4⋊C4×C53—24no—
169C4⋊C4×C53—21no—
170C4⋊C4×C53—21no—
171C4⋊C4×C53—21no—
172C4⋊C4×C53—21no—
173C4⋊C4×C53—42no—
174C4⋊C4×C53—42no—
175C4⋊C4×C53—81no—
176C4⋊C4×C53C2×D412yes—
177C4⋊C4×C53#4112yes—
178C22⋊C4×C53—24no—
179C22⋊C4×C53—22no—
180C22⋊C4×C53—22no—
181C42×C53—21no—
182C42×C53—21no—
183C42×C53—21no—
184C42×C53—21no—
185C42×C53—42no—
186C42×C53—81no—
187C22⋊Dic53—42no—
188C22⋊Dic53—42no—
189#11719925—28no—
190C4⋊Dic53—21no—
191C4⋊Dic53—21no—
192C4⋊Dic53—21no—
193C4⋊Dic53—21no—
194C4⋊Dic53—21no—
195C4⋊Dic53—21no—
196#11719919—42no—
197#11719919—42no—
198#11719919—42no—
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×#88322#4112yes—
212C23×C53—21no—
213C23×C53—21no—
214C23×C53—21no—
215C23×C53—21no—
216C23×C53—21no—
217C23×C53—21no—
218C23×C53#13512yes—
219C23×C53C22×D411yescharacteristic
220Q8×C53—22no—
221Q8×C53—22no—
222Q8×C53—41no—
223Q8×C53—41no—
224C2×C4×C53—21no—
225C2×C4×C53—21no—
226C2×C4×C53—21no—
227C2×C4×C53—22no—
228C2×C4×C53—22no—
229C2×C4×C53—21no—
230C2×C4×C53—22no—
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—41no—
250C2×C4×C53—41no—
251C2×C4×C53—41no—
252C2×C4×C53—41no—
253C2×C4×C53—41no—
254C2×C4×C53—84no—
255C2×C4×C53#13712yes—
256C2×C4×C53C22×D411yescharacteristic
257C2×C4×C53C2×#4111yescharacteristic
258C2×C4×C53C2×#4111yescharacteristic
259C2×C4×C53C2×#4111yescharacteristic
260C2×C4×C53H(5,𝔽2)11yescharacteristic
261C2×C4×C532−1+411yescharacteristic
262C2×Dic53—22no—
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—41no—
274Dic106—22no—
275Dic106—22no—
276Dic106—41no—
277Dic106—41no—
278#88322—22no—
279#88322—22no—
280#88322—41no—
281#88322—41no—
282#26558—531nomaximal, Sylow
283C22×C53—22no—
284C22×C53—22no—
285C22×C53—22no—
286C22×C53—41no—
287C22×C53—41no—
288C22×C53—41no—
289C22×C53—41no—
290C22×C53#52511yescharacteristic
291C22×C53D4211yescharacteristic
292C22×C53#54711yescharacteristic
293C22×C53C2×#13511yescharacteristic
294C22×C53C2×#13711yescharacteristic
295C22×C53#57711yescharacteristic
296C22×C53#57811yescharacteristic
297C4×C53—22no—
298C4×C53—41no—
299C4×C53—41no—
300C4×C53—42no—
301C4×C53—41no—
302C4×C53—42no—
303C4×C53—41no—
304C4×C53—42no—
305C4×C53—81no—
306C4×C53#52511yescharacteristic
307C4×C53#53911yescharacteristic
308C4×C53#54711yescharacteristic
309C4×C53#55911yescharacteristic
310Dic53—42no—
311Dic53—81no—
312Dic53—81no—
313Dic53—81no—
314C2×#483—531no—
315C2×C4×Q16—531no—
316#2421—531no—
317#2360D5311yescharacteristic
318#2097—531no—
319#2056—531no—
320#2035—531no—
321#2001—531no—
322#1949—531no—
323#1941—531no—
324#1927—531no—
325#1923—531no—
326#1865—531no—
327#1863—531no—
328#1852—531no—
329C2×C53—42no—
330C2×C53#245511yescharacteristic
331C2×C53#310811yescharacteristic
332C2×C53#312011yescharacteristic
333C2×C53#327011yescharacteristic
334C2×C53#328211yescharacteristic
335C2×C53#333611yescharacteristic
336C2×C53#335611yescharacteristic
337C22×Q16—531no—
338C2×C22⋊Q8—531no—
339C2×C22⋊Q8C2×D5311yescharacteristic
340C2×C4×Q8—531no—
341C2×C4×Q8C2×D5311yescharacteristic
342C22×C4⋊C4C2×D5311yescharacteristic
343#483—1064no—
344C4×Q16—1064no—
345C2×#107—531no—
346C2×#107—531no—
347C2×#106—531no—
348C2×#106—531no—
349C2×#106—531no—
350C2×C4⋊C8—531no—
351C2×#103—531no—
352C2×#103—531no—
353C2×#103—531no—
354C2×#103—531no—
355C2×#103—531no—
356C2×#103—531no—
357C2×C22⋊C8—531no—
358C2×C22⋊C8—531no—
359C2×C4×C8—531no—
360#400—1061no—
361#397—1061no—
362#391—531no—
363#391C2×D5311yescharacteristic
364#389—21no—
365#389—531no—
366#389C2×D5311yescharacteristic
367#388—1061no—
368#387—21no—
369#387—1061no—
370#385—21no—
371#382—531no—
372#380—531no—
373C4×C4⋊C4C2×D5311yescharacteristic
374C4×C22⋊C4—21no—
375C4×C22⋊C4C2×D5311yescharacteristic
376#340—531no—
377#340—531no—
378#340—531no—
379#340—531no—
380#336—531no—
381#336—531no—
382C53#2655811yescharacteristic, complemented, Sylow
383C22×Q8—531no—
384C22×Q8C22×D5311yescharacteristic
385C23×C4—21no—
386C23×C4C22×D5311yescharacteristic
387C2×Q16—1062no—
388C2×Q16—1062no—
389C2×Q16—1062no—
390C22×C8—531no—
391C22×C8—531no—
392C22×C8—531no—
393C22×C8—531no—
394C22⋊Q8—22no—
395C22⋊Q8—22no—
396C22⋊Q8—1062no—
397C22⋊Q8—2121no—
398C4×Q8—22no—
399C4×Q8—41no—
400C4×Q8—1062no—
401C4×Q8—2121no—
402C2×C4⋊C4—21no—
403C2×C4⋊C4—22no—
404C2×C4⋊C4—21no—
405C2×C4⋊C4—21no—
406C2×C4⋊C4—21no—
407C2×C4⋊C4—41no—
408C2×C4⋊C4—531no—
409C2×C4⋊C4—531no—
410C2×C4⋊C4—531no—
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×C22⋊C4—21no—
418C2×C22⋊C4—1061no—
419C2×C22⋊C4—1061no—
420C2×C22⋊C4C22⋊D5311yescharacteristic
421C2×C22⋊C4C22⋊D5311yescharacteristic
422C2×C42—21no—
423C2×C42—21no—
424C2×C42—1061no—
425C2×C42C22×D5311yescharacteristic
426C2×C42C22×D5311yescharacteristic
427#107—1062no—
428#107—2121no—
429#106—1062no—
430#106—1062no—
431#106—1062no—
432C4⋊C8—1062no—
433#103—534no—
434#103—1062no—
435#103—1062no—
436#103—1062no—
437#103—1062no—
438#103—2121no—
439C22⋊C8—1062no—
440C22⋊C8—1062no—
441C4×C8—1062no—
442#95—21no—
443#95—21no—
444#95—21no—
445#95—21no—
446#95—1061no—
447#95—1061no—
448#95—1061no—
449#95—1061no—
450#95—1061no—
451#95—1061no—
452#95C22⋊D5311yescharacteristic
453#95C22⋊D5311yescharacteristic
454C24C2×C22⋊D5311yescharacteristic
455C2×Q8—21no—
456C2×Q8—21no—
457C2×Q8—21no—
458C2×Q8—21no—
459C2×Q8—21no—
460C2×Q8—532no—
461C2×Q8—1061no—
462C2×Q8—1061no—
463C2×Q8—1061no—
464C2×Q8—1061no—
465C2×Q8—1061no—
466C2×Q8#1171994512yes—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—42no—
478C22×C4—42no—
479C22×C4—81no—
480C22×C4—1061no—
481C22×C4—1061no—
482C22×C4—1061no—
483C22×C4—1061no—
484C22×C4C2×C22⋊D5311yescharacteristic
485C22×C4C2×C22⋊D5311yescharacteristic
486C22×C4C2×C22⋊D5311yescharacteristic
487C22×C4C2×C22⋊D5311yescharacteristic
488C22×C4C2×C22⋊D5311yescharacteristic
489C22×C4C23×D5311yescharacteristic
490Q16—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—24no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—42no—
509C4⋊C4—42no—
510C4⋊C4—81no—
511C4⋊C4—1061no—
512C4⋊C4—1061no—
513C4⋊C4—1061no—
514C4⋊C4—1061no—
515C4⋊C4—1061no—
516C4⋊C4—1061no—
517C4⋊C4—2122no—
518C4⋊C4—2122no—
519C4⋊C4—2122no—
520C4⋊C4#1171994512yes—
521C4⋊C4D4×D5312yes—
522C22⋊C4—24no—
523C22⋊C4—22no—
524C22⋊C4—22no—
525C22⋊C4—2122no—
526C22⋊C4—2122no—
527C42—21no—
528C42—21no—
529C42—21no—
530C42—21no—
531C42—42no—
532C42—81no—
533C42—2122no—
534C23—21no—
535C23—21no—
536C23—21no—
537C23—21no—
538C23—21no—
539C23—21no—
540C23#1313323712yes—
541C23C22×C22⋊D5311yescharacteristic
542Q8—22no—
543Q8—22no—
544Q8—41no—
545Q8—41no—
546Q8—1062no—
547Q8—1062no—
548Q8—2121no—
549Q8—2121no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—22no—
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—22no—
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—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—84no—
581C2×C4—1062no—
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—2121no—
591C2×C4—2121no—
592C2×C4—2121no—
593C2×C4#1313324812yes—
594C2×C4C2×D4×D5311yescharacteristic
595C2×C4#41×D5311yescharacteristic
596C2×C4C2×#1171994511yescharacteristic
597C2×C4C2×#1171994511yescharacteristic
598C2×C4#1313326211yescharacteristic
599C2×C4#1313326411yescharacteristic
600C8—1062no—
601C8—1062no—
602C8—2121no—
603C8—2121no—
604C22—22no—
605C22—22no—
606C22—22no—
607C22—41no—
608C22—41no—
609C22—41no—
610C22—41no—
611C22#1681113411yescharacteristic
612C22D4×C22⋊D5311yescharacteristic
613C22#1681127211yescharacteristic
614C22C2×#1313323711yescharacteristic
615C22C2×#1313324811yescharacteristic
616C22#1681143411yescharacteristic
617C22#1681143811yescharacteristic
618C4—22no—
619C4—42no—
620C4—41no—
621C4—42no—
622C4—41no—
623C4—41no—
624C4—41no—
625C4—42no—
626C4—81no—
627C4—4241no—
628C4—4241no—
629C4—4241no—
630C4—2122no—
631C4#1681113211yescharacteristic
632C4#1681121511yescharacteristic
633C4#1681127811yescharacteristic
634C4#1681135611yescharacteristic
635C2—42no—
636C2#2027859211yescharacteristic
637C2#2028479511yescharacteristic
638C2#2028492911yescharacteristic
639C2#2028628611yescharacteristic
640C2#2028641811yescharacteristic
641C2#2028707711yescharacteristic
642C2#2028733011yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2360C2
socleC23×C53C22×D4
prime radicalC24×C53C2×D4
hypercenter#2360D53
hypocenterC53#26558
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2360

Complements

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

NH ≅ G/Ntype
2. C2×C4×#11719925C2semidirect
4. #20276263C2semidirect
5. #20276037C2semidirect
8. #20275483C2semidirect
9. #20275440C2semidirect
10. #20275349C2semidirect
12. #20274983C2semidirect
13. #20274968C2semidirect
382. C53#26558semidirect

Maximal subgroups

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

subgroupindex
C2×#168109532
C2×C4×#117199252
#202782182
#202762632
#202760372
#202759082
#202757232
#202754832
#202754402
#202753492
#202753232
#202749832
#202749682
#202749032
C53×#23602
#2655853