Finite Group Databasebeta

← #27403480

Subgroups — #27403480

Order 13568.

894 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810725C211yesmaximal, characteristic
2#20277682C211yesmaximal, characteristic, complemented
3#20277615C211yesmaximal, characteristic
4#20275403C211yesmaximal, characteristic, complemented
5#20275389C211yesmaximal, characteristic, complemented
6#20275306C211yesmaximal, characteristic, complemented
7#20275285C211yesmaximal, characteristic, complemented
8#20275050C211yesmaximal, characteristic
9#20275050C211yesmaximal, characteristic
10#20274722C211yesmaximal, characteristic
11#20274717C211yesmaximal, characteristic, complemented
12#20274587C211yesmaximal, characteristic, complemented
13#20274539C211yesmaximal, characteristic
14#20274529C211yesmaximal, characteristic
15C53×#1836C211yesmaximal, characteristic, complemented
16C22×#11719913C2211yescharacteristic
17C2×C4×Dic106C2211yescharacteristic
18C2×C4×D212C2211yescharacteristic
19C2×#13133179C2211yescharacteristic
20C2×#13133179C2211yescharacteristic
21C2×C4⋊C4×D53—21no—
22#16810725C418yescomplemented
23C2×#13133151C2211yescharacteristic
24C2×C4⋊#88322C2211yescharacteristic
25C2×#13133142C2211yescharacteristic
26C2×#13133140C2211yescharacteristic
27C2×#13133140C2211yescharacteristic
28C2×#13133139C2211yescharacteristic
29C2×#13133137C2211yescharacteristic
30C2×#13133137C2211yescharacteristic
31C2×#13133127C2211yescharacteristic
32C2×C8×Dic53C2211yescharacteristic
33#16810528C2211yescharacteristic
34#16810528C2211yescharacteristic
35#16810519C2211yescharacteristic
36#16810519C2211yescharacteristic
37#16810510—21no—
38#16810510—21no—
39#16810494—21no—
40#16810494—21no—
41#16810490—21no—
42#16810480—21no—
43C4⋊C4×Dic53C2211yescharacteristic
44C4⋊C4×Dic53C2211yescharacteristic
45C4×C4⋊Dic53C2211yescharacteristic
46#16810468—21no—
47#16810385C2211yescharacteristic
48#16810385C2211yescharacteristic
49#16810385C2211yescharacteristic
50#16810385C2211yescharacteristic
51#16810377C2211yescharacteristic
52#16810377C2211yescharacteristic
53C2×#97×C53C2211yescharacteristic
54C53×#384C2211yescharacteristic
55C53×#382C2211yescharacteristic
56C53×#340C2211yescharacteristic
57C53×#340C2211yescharacteristic
58C53×#340C2211yescharacteristic
59C53×#340C2211yescharacteristic
60C22×Dic106C2311yescharacteristic
61C22×D212C2311yescharacteristic
62C22×C4×D53—21no—
63C22×C4×D53—21no—
64C22×C4×D53—21no—
65C2×#11719913—22no—
66C2×#11719913—22no—
67C2×#11719913C2×C414yes—
68C22×#88322C2311yescharacteristic
69C22×#88322C2311yescharacteristic
70C4×Dic106—22no—
71C4×Dic106C2×C414yescomplemented
72C4×D212—22no—
73C4×D212C2×C414yes—
74#13133179—22no—
75#13133179—22no—
76#13133179—41no—
77#13133179—41no—
78C2×C4⋊Dic53D411yescharacteristic
79C2×C4⋊Dic53D411yescharacteristic
80C2×C4⋊Dic53C2311yescharacteristic
81C2×C4⋊Dic53C2311yescharacteristic
82C2×#11719919—21no—
83C2×#11719919—21no—
84C2×#11719919—21no—
85C4⋊C4×D53—44no—
86C2×#11719921—21no—
87C2×#11719921—21no—
88C2×#11719921—21no—
89C2×C4×Dic53—21no—
90C2×C4×Dic53—21no—
91C2×C4×Dic53—21no—
92C2×C4×Dic53C2311yescharacteristic
93C2×C4×Dic53C2311yescharacteristic
94#13133151C2×C414yescomplemented
95C4⋊#88322—22no—
96#13133142C2×C414yescomplemented
97#13133140—41no—
98#13133140—41no—
99#13133139C2×C414yes—
100#13133137—22no—
101#13133137—22no—
102#13133127—41no—
103C8×Dic53—41no—
104#13133124—21no—
105#13133124—21no—
106#13133124—21no—
107#13133124—21no—
108#13133124—21no—
109#13133124—21no—
110#13133124—21no—
111#13133124—21no—
112#13133124D411yescharacteristic
113#13133124D411yescharacteristic
114C22×C8×C53C2311yescharacteristic
115C22×C8×C53C2311yescharacteristic
116C2×C4⋊C4×C53—21no—
117C2×C4⋊C4×C53—21no—
118C2×C4⋊C4×C53C2311yescharacteristic
119C2×C4⋊C4×C53C2311yescharacteristic
120C2×C4⋊C4×C53C2311yescharacteristic
121C2×C4⋊C4×C53C2311yescharacteristic
122C2×C42×C53C2311yescharacteristic
123#97×C53C2×C414yescomplemented
124#95×C53—21no—
125#95×C53—21no—
126C23×D53—21no—
127C22×C4×C53—21no—
128C22×C4×C53—21no—
129C22×C4×C53—21no—
130C22×C4×C53—21no—
131C22×C4×C53C2×D411yescharacteristic
132C22×C4×C53C2×D411yescharacteristic
133C22×C4×C53C2411yescharacteristic
134C22×Dic53—21no—
135C22×Dic53—21no—
136C22×Dic53—21no—
137C22×Dic53C2×D411yescharacteristic
138C22×Dic53C2×D411yescharacteristic
139C22×Dic53C2×D411yescharacteristic
140C22×Dic53C2×D411yescharacteristic
141C2×D212—21no—
142C2×D212—21no—
143C2×D212—21no—
144C2×D212C22×C412yes—
145C2×D212#4112yes—
146C2×D212#4112yes—
147C2×C4×D53—24no—
148C2×C4×D53—42no—
149C2×C4×D53—42no—
150C2×C4×D53—42no—
151C2×C4×D53—42no—
152C2×C4×D53—42no—
153C2×C4×D53—42no—
154C2×C4×D53—81no—
155C2×C4×D53—81no—
156C2×Dic106—21no—
157C2×Dic106—21no—
158C2×Dic106—21no—
159C2×Dic106—21no—
160C2×Dic106—21no—
161C2×Dic106C22×C412yes—
162C2×C8×C53—21no—
163C2×C8×C53—21no—
164C2×C8×C53—21no—
165C2×C8×C53—21no—
166C2×C8×C53C22×C412yes—
167C2×C8×C53C22×C412yes—
168C4⋊C4×C53—21no—
169C4⋊C4×C53—21no—
170C4⋊C4×C53—21no—
171C4⋊C4×C53—21no—
172C4⋊C4×C53—21no—
173C4⋊C4×C53—21no—
174C4⋊C4×C53—21no—
175C4⋊C4×C53—21no—
176C4⋊C4×C53—42no—
177C4⋊C4×C53—81no—
178C42×C53C22×C412yes—
179C42×C53#4112yes—
180#11719921—24no—
181#11719921—42no—
182#11719921—42no—
183C4⋊Dic53—21no—
184C4⋊Dic53—21no—
185C4⋊Dic53—22no—
186C4⋊Dic53—41no—
187C4⋊Dic53C22×C412yes—
188C4⋊Dic53C22×C412yes—
189#11719919—24no—
190#11719919—42no—
191#11719919—42no—
192C4×Dic53—24no—
193C4×Dic53—21no—
194C4×Dic53—21no—
195C4×Dic53—21no—
196C4×Dic53—21no—
197C4×Dic53—42no—
198C4×Dic53—42no—
199C2×#88322—21no—
200C2×#88322—21no—
201C2×#88322—21no—
202C2×#88322—21no—
203C2×#88322—21no—
204C2×#88322—21no—
205#11719913—28no—
206C23×C53C22×D411yescharacteristic
207C22×D53—22no—
208C22×D53—22no—
209C22×D53—22no—
210C22×D53—41no—
211C22×D53—41no—
212C22×D53—41no—
213C22×D53—41no—
214C2×C4×C53—21no—
215C2×C4×C53—21no—
216C2×C4×C53—21no—
217C2×C4×C53—21no—
218C2×C4×C53—41no—
219C2×C4×C53—41no—
220C2×C4×C53—41no—
221C2×C4×C53—41no—
222C2×C4×C53—41no—
223C2×C4×C53—41no—
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×C53C4×D412yes—
231C2×C4×C53C4×D412yes—
232C2×C4×C53C23×C411yescharacteristic
233C2×C4×C53C2×#4111yescharacteristic
234C2×C4×C53C2×#4111yescharacteristic
235C2×C4×C53C2×#4111yescharacteristic
236C2×C4×C53H(5,𝔽2)11yescharacteristic
237C2×C4×C532−1+411yescharacteristic
238C2×Dic53—21no—
239C2×Dic53—21no—
240C2×Dic53—22no—
241C2×Dic53—22no—
242C2×Dic53—22no—
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—41no—
253C2×Dic53—41no—
254C2×Dic53—41no—
255C2×Dic53—41no—
256C2×Dic53—41no—
257C2×Dic53—41no—
258C2×Dic53AGL(1,ℤ/8)12yes—
259D212—22no—
260D212—22no—
261D212—22no—
262D212#11714yes—
263C4×D53—48no—
264C4×D53—84no—
265C4×D53—84no—
266Dic106—22no—
267Dic106—22no—
268Dic106—41no—
269Dic106—41no—
270C8×C53—22no—
271C8×C53—22no—
272#88322—41no—
273#88322—41no—
274#22778—531nomaximal, Sylow
275C22×C53C2×C4×D411yescharacteristic
276C22×C53#54611yescharacteristic
277C22×C53#54811yescharacteristic
278C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
279C22×C53#57511yescharacteristic
280C22×C53#57711yescharacteristic
281C22×C53#57811yescharacteristic
282C2×D53—24no—
283C2×D53—42no—
284C2×D53—42no—
285C2×D53—42no—
286C2×D53—42no—
287C2×D53—42no—
288C2×D53—42no—
289C4×C53—22no—
290C4×C53—22no—
291C4×C53—81no—
292C4×C53—81no—
293C4×C53—81no—
294C4×C53—81no—
295C4×C53C2×#11711yescharacteristic
296C4×C53#52011yescharacteristic
297C4×C53#55211yescharacteristic
298C4×C53#55311yescharacteristic
299Dic53—22no—
300Dic53—42no—
301Dic53—42no—
302Dic53—42no—
303Dic53—41no—
304Dic53—41no—
305Dic53—41no—
306C2×#440—531no—
307#2356—531no—
308D4×C4⋊C4—531no—
309#1935—531no—
310#1933—531no—
311#1921—531no—
312#1918—531no—
313#1877—531no—
314#1877—531no—
315#1836D5311yescharacteristic, complemented
316#1804—531no—
317#1803—531no—
318#1771—531no—
319#1762—531no—
320C4×#102—531no—
321C2×C53#235211yescharacteristic
322C2×C53C4×AGL(1,ℤ/8)11yescharacteristic
323C2×C53#294711yescharacteristic
324C2×C53#331011yescharacteristic
325C2×C53#331511yescharacteristic
326C2×C53#336711yescharacteristic
327C2×C53#337611yescharacteristic
328D53—44no—
329C22×SD16—531no—
330C2×C4×Q8—531no—
331C2×C4×D4—531no—
332C2×C4×D4—531no—
333C2×C4×D4—531no—
334C22×C4⋊C4—1061no—
335#440—538no—
336C2×#107—531no—
337C2×C4⋊C8—531no—
338C2×#103—531no—
339C2×#103—531no—
340C2×#103—531no—
341C2×#102—531no—
342C2×#102—531no—
343C2×#102—531no—
344C2×#97—531no—
345C2×#97C2×D5311yescharacteristic
346C2×C4×C8—531no—
347#391—531no—
348#391—531no—
349#389—531no—
350#389—531no—
351#387—1061no—
352#387—1061no—
353#385—1061no—
354#385—1061no—
355#384C2×D5311yescharacteristic
356#383—1061no—
357#382C2×D5311yescharacteristic
358#380—1061no—
359C4×C4⋊C4—531no—
360C4×C4⋊C4—531no—
361C4×C4⋊C4—531no—
362#376—1061no—
363#340—531no—
364#340—531no—
365#340—531no—
366#340—531no—
367#340C2×D5311yescharacteristic
368#340C2×D5311yescharacteristic
369#340C2×D5311yescharacteristic
370#340C2×D5311yescharacteristic
371#336—531no—
372#336—531no—
373C53#2277811yescharacteristic, complemented, Sylow
374C22×Q8—531no—
375C22×D4—531no—
376C23×C4—1061no—
377C23×C4—1061no—
378C23×C4—1061no—
379C2×SD16—534no—
380C2×SD16—1062no—
381C2×SD16—1062no—
382C22×C8—531no—
383C22×C8—531no—
384C22×C8C22×D5311yescharacteristic
385C22×C8C22×D5311yescharacteristic
386C4×Q8—534no—
387C4×Q8—1062no—
388C4×D4—534no—
389C4×D4—1062no—
390C4×D4—1062no—
391C4×D4—1062no—
392C4×D4—2121no—
393C4×D4—2121no—
394C2×C4⋊C4—21no—
395C2×C4⋊C4—21no—
396C2×C4⋊C4—531no—
397C2×C4⋊C4—531no—
398C2×C4⋊C4—531no—
399C2×C4⋊C4—531no—
400C2×C4⋊C4—1061no—
401C2×C4⋊C4—1061no—
402C2×C4⋊C4—1061no—
403C2×C4⋊C4—2124no—
404C2×C4⋊C4C22×D5311yescharacteristic
405C2×C4⋊C4C22×D5311yescharacteristic
406C2×C4⋊C4C22×D5311yescharacteristic
407C2×C4⋊C4C22×D5311yescharacteristic
408C2×C22⋊C4—1061no—
409C2×C22⋊C4—1061no—
410C2×C22⋊C4—1061no—
411C2×C42—531no—
412C2×C42—531no—
413C2×C42—1061no—
414C2×C42—1061no—
415C2×C42—1061no—
416C2×C42C22×D5311yescharacteristic
417#107—534no—
418C4⋊C8—1062no—
419#103—534no—
420#103—2121no—
421#103—2121no—
422#102—534no—
423#102—1062no—
424#102—1062no—
425#97—2121no—
426#97C4×D5314yescomplemented
427C4×C8—2121no—
428#95—21no—
429#95—21no—
430#95—531no—
431#95—531no—
432#95—1061no—
433#95—1061no—
434#95—1061no—
435#95—1061no—
436#95—1061no—
437#95—1061no—
438#95—1061no—
439#95—1061no—
440C24—1061no—
441C2×Q8—532no—
442C2×Q8—1061no—
443C2×Q8—1061no—
444C2×Q8—1061no—
445C2×Q8—1061no—
446C2×Q8—1061no—
447C2×D4—532no—
448C2×D4—532no—
449C2×D4—532no—
450C2×D4—1061no—
451C2×D4—1061no—
452C2×D4—1061no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—531no—
458C22×C4—531no—
459C22×C4—531no—
460C22×C4—531no—
461C22×C4—1061no—
462C22×C4—1061no—
463C22×C4—1061no—
464C22×C4—1064no—
465C22×C4—4241no—
466C22×C4—4241no—
467C22×C4—2122no—
468C22×C4—2122no—
469C22×C4—2122no—
470C22×C4—2122no—
471C22×C4—2122no—
472C22×C4—2122no—
473C22×C4D4×D5311yescharacteristic
474C22×C4D4×D5311yescharacteristic
475C22×C4C23×D5311yescharacteristic
476SD16—1068no—
477C2×C8—21no—
478C2×C8—21no—
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—1061no—
482C2×C8—1061no—
483C2×C8—1061no—
484C2×C8—1061no—
485C2×C8—1061no—
486C2×C8—1061no—
487C2×C8C2×C4×D5312yes—
488C2×C8C2×C4×D5312yes—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—42no—
498C4⋊C4—81no—
499C4⋊C4—532no—
500C4⋊C4—532no—
501C4⋊C4—1062no—
502C4⋊C4—1064no—
503C4⋊C4—1061no—
504C4⋊C4—1061no—
505C4⋊C4—2121no—
506C4⋊C4—2122no—
507C4⋊C4—2122no—
508C22⋊C4—1064no—
509C22⋊C4—2122no—
510C22⋊C4—2122no—
511C42—1064no—
512C42—1061no—
513C42—1061no—
514C42—1061no—
515C42—1061no—
516C42—2122no—
517C42—2122no—
518C42C2×C4×D5312yes—
519C42#1171994912yes—
520C23—1062no—
521C23—1062no—
522C23—1062no—
523C23—2121no—
524C23—2121no—
525C23—2121no—
526C23—2121no—
527C23C2×D4×D5311yescharacteristic
528Q8—1062no—
529Q8—1062no—
530Q8—2121no—
531Q8—2121no—
532D4—534no—
533D4—1062no—
534D4—1062no—
535D4—1062no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—532no—
553C2×C4—1061no—
554C2×C4—1061no—
555C2×C4—1061no—
556C2×C4—1061no—
557C2×C4—1061no—
558C2×C4—1062no—
559C2×C4—1062no—
560C2×C4—1062no—
561C2×C4—1061no—
562C2×C4—1061no—
563C2×C4—1061no—
564C2×C4—1061no—
565C2×C4—1061no—
566C2×C4—1061no—
567C2×C4—4244no—
568C2×C4—4244no—
569C2×C4—2128no—
570C2×C4—2121no—
571C2×C4—2121no—
572C2×C4—2121no—
573C2×C4—2121no—
574C2×C4—2121no—
575C2×C4—2121no—
576C2×C4#1313317912yes—
577C2×C4#1313317912yes—
578C2×C4C22×C4×D5311yescharacteristic
579C2×C4#41×D5311yescharacteristic
580C2×C4#41×D5311yescharacteristic
581C2×C4C2×#1171994911yescharacteristic
582C2×C4#1313326311yescharacteristic
583C2×C4#1313326411yescharacteristic
584C8—22no—
585C8—22no—
586C8—2121no—
587C8—2121no—
588C22—1064no—
589C22—2122no—
590C22—2122no—
591C22—2122no—
592C22—2122no—
593C22—2122no—
594C22—2122no—
595C22C2×#1313317911yescharacteristic
596C22#1681126611yescharacteristic
597C22#1681128111yescharacteristic
598C22AGL(1,ℤ/8)×D5311yescharacteristic
599C22#1681142211yescharacteristic
600C22#1681143311yescharacteristic
601C22#1681143911yescharacteristic
602C4—22no—
603C4—22no—
604C4—81no—
605C4—81no—
606C4—81no—
607C4—81no—
608C4—1062no—
609C4—2121no—
610C4—2121no—
611C4—2121no—
612C4—2122no—
613C4—2122no—
614C4—2122no—
615C4#117×D5311yescharacteristic
616C4#1681110311yescharacteristic
617C4#1681131111yescharacteristic
618C4#1681132611yescharacteristic
619C2—2124no—
620C2#2027764111yescharacteristic
621C2#2028352211yescharacteristic
622C2#2028355611yescharacteristic
623C2#2028673511yescharacteristic
624C2#2028681011yescharacteristic
625C2#2028743011yescharacteristic
626C2#2028754111yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1836C2
socleC23×C53C22×D4
prime radicalC22×D212C23
cosocleC22×C4×C53C24
hypercenter#1836D53
hypocenterC53#22778
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1836

Complements

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

NH ≅ G/Ntype
2. #20277682C2semidirect
4. #20275403C2semidirect
5. #20275389C2semidirect
6. #20275306C2semidirect
7. #20275285C2semidirect
11. #20274717C2semidirect
12. #20274587C2semidirect
15. C53×#1836C2semidirect
22. #16810725C4semidirect
71. C4×Dic106C2×C4semidirect
94. #13133151C2×C4semidirect
96. #13133142C2×C4semidirect
123. #97×C53C2×C4semidirect
315. #1836D53semidirect
373. C53#22778semidirect
426. #97C4×D53semidirect

Maximal subgroups

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

subgroupindex
C2×#168107252
#202776822
#202776152
#202754032
#202753892
#202753062
#202752852
#202750502
#202747222
#202747172
#202745872
#202745392
#202745292
C53×#18362
#2277853