Finite Group Databasebeta

← #27403487

Subgroups — #27403487

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×#16810723C211yesmaximal, characteristic
2#20277682C211yesmaximal, characteristic, complemented
3D4×C4⋊Dic53C211yesmaximal, characteristic
4#20275406C211yesmaximal, characteristic, complemented
5#20275392C211yesmaximal, characteristic, complemented
6#20275310C211yesmaximal, characteristic, complemented
7#20275289C211yesmaximal, characteristic, complemented
8#20275047C211yesmaximal, characteristic
9#20275047C211yesmaximal, characteristic
10#20274848C211yesmaximal, characteristic, complemented
11#20274721C211yesmaximal, characteristic, complemented
12#20274592C211yesmaximal, characteristic, complemented
13#20274542C211yesmaximal, characteristic
14#102×Dic53C211yesmaximal, characteristic
15C53×#1804C211yesmaximal, characteristic
16C22×#11719923C2211yescharacteristic
17C2×C4×Dic106C2211yescharacteristic
18C2×D4×Dic53C2211yescharacteristic
19C2×D4×Dic53C2211yescharacteristic
20C22×C4⋊Dic53—21no—
21#16810723C418yescomplemented
22C2×#13133150C2211yescharacteristic
23C2×#13133142C2211yescharacteristic
24C2×#13133142C2211yescharacteristic
25C2×#13133140C2211yescharacteristic
26C2×#13133138C2211yescharacteristic
27C2×#13133128C2211yescharacteristic
28C2×#13133127C2211yescharacteristic
29C2×C8×Dic53C2211yescharacteristic
30#16810528C2211yescharacteristic
31#16810528C2211yescharacteristic
32#16810522C2211yescharacteristic
33#16810522C2211yescharacteristic
34#16810510—21no—
35#16810510—21no—
36#16810498—21no—
37#16810498—21no—
38#16810493C2211yescharacteristic
39#16810490—21no—
40#16810483—21no—
41C4⋊C4×Dic53C2211yescharacteristic
42C4⋊C4×Dic53C2211yescharacteristic
43C4×C4⋊Dic53C2211yescharacteristic
44#16810468—21no—
45#16810386C2211yescharacteristic
46#16810386C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810385C2211yescharacteristic
49#16810385C2211yescharacteristic
50#16810385C2211yescharacteristic
51#16810377C2211yescharacteristic
52#16810377C2211yescharacteristic
53C2×C4×D4×C53C2211yescharacteristic
54C2×C4⋊C8×C53C2211yescharacteristic
55C2×#102×C53C2211yescharacteristic
56C2×#102×C53C2211yescharacteristic
57C53×#382C2211yescharacteristic
58C53×#340C2211yescharacteristic
59C53×#340C2211yescharacteristic
60C22×Dic106C2311yescharacteristic
61C23×Dic53—21no—
62C23×Dic53—21no—
63C2×#11719923—22no—
64C2×#11719923—22no—
65C2×#11719923C2×C414yes—
66C22×#88322C2311yescharacteristic
67C22×#88322C2311yescharacteristic
68C4×Dic106—22no—
69C4×Dic106C2×C414yescomplemented
70D4×Dic53—22no—
71D4×Dic53—22no—
72D4×Dic53—41no—
73D4×Dic53—41no—
74C2×C4⋊Dic53—21no—
75C2×C4⋊Dic53—21no—
76C2×C4⋊Dic53—21no—
77C2×C4⋊Dic53—21no—
78C2×C4⋊Dic53—44no—
79C2×C4⋊Dic53D411yescharacteristic
80C2×C4⋊Dic53D411yescharacteristic
81C2×C4⋊Dic53C2311yescharacteristic
82C2×C4⋊Dic53C2311yescharacteristic
83C2×C4⋊Dic53C2311yescharacteristic
84C2×#11719919—21no—
85C2×C22⋊Dic53—21no—
86C2×C22⋊Dic53—21no—
87C2×C4×Dic53—21no—
88C2×C4×Dic53—21no—
89C2×C4×Dic53—21no—
90C2×C4×Dic53C2311yescharacteristic
91C2×C4×Dic53C2311yescharacteristic
92#13133150C2×C414yescomplemented
93#13133142—41no—
94#13133142—41no—
95#13133140C2×C414yescomplemented
96#13133138C2×C414yes—
97#13133128C2×C414yescomplemented
98#13133127—41no—
99C8×Dic53—41no—
100#13133124—21no—
101#13133124—21no—
102#13133124—21no—
103#13133124—21no—
104#13133124—21no—
105#13133124—21no—
106#13133124—21no—
107#13133124—21no—
108#13133124D411yescharacteristic
109#13133124D411yescharacteristic
110C22×D4×C53C2311yescharacteristic
111C23×C4×C53—21no—
112C22×C8×C53C2311yescharacteristic
113C22×C8×C53C2311yescharacteristic
114C4×D4×C53—22no—
115C4×D4×C53C2×C414yes—
116C2×C4⋊C4×C53C2311yescharacteristic
117C2×C4⋊C4×C53C2311yescharacteristic
118C2×C4⋊C4×C53C2311yescharacteristic
119C2×C22⋊C4×C53—21no—
120C2×C42×C53C2311yescharacteristic
121C4⋊C8×C53—22no—
122#102×C53—22no—
123#102×C53—22no—
124#95×C53—21no—
125#95×C53—21no—
126C24×C53—21no—
127C2×D4×C53—21no—
128C2×D4×C53—21no—
129C2×D4×C53—21no—
130C2×D4×C53C22×C412yes—
131C2×D4×C53#4112yes—
132C2×D4×C53#4112yes—
133C22×C4×C53—21no—
134C22×C4×C53—24no—
135C22×C4×C53—21no—
136C22×C4×C53—21no—
137C22×C4×C53—42no—
138C22×C4×C53—42no—
139C22×C4×C53C2×D411yescharacteristic
140C22×C4×C53C2×D411yescharacteristic
141C22×C4×C53C2411yescharacteristic
142C22×Dic53—21no—
143C22×Dic53—21no—
144C22×Dic53—21no—
145C22×Dic53—21no—
146C22×Dic53—42no—
147C22×Dic53—42no—
148C22×Dic53—42no—
149C22×Dic53—42no—
150C22×Dic53—81no—
151C22×Dic53—81no—
152C22×Dic53C2×D411yescharacteristic
153C22×Dic53C2×D411yescharacteristic
154C22×Dic53C2×D411yescharacteristic
155C22×Dic53C2×D411yescharacteristic
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×C53—21no—
167C2×C8×C53—21no—
168C4⋊C4×C53—21no—
169C4⋊C4×C53—21no—
170C4⋊C4×C53—21no—
171C4⋊C4×C53—21no—
172C4⋊C4×C53—21no—
173C4⋊C4×C53C22×C412yes—
174C22⋊C4×C53—24no—
175C42×C53C22×C412yes—
176C42×C53#4112yes—
177C22⋊Dic53—42no—
178C22⋊Dic53—42no—
179#11719923—28no—
180C4⋊Dic53—21no—
181C4⋊Dic53—22no—
182C4⋊Dic53—21no—
183C4⋊Dic53—21no—
184C4⋊Dic53—21no—
185C4⋊Dic53—21no—
186C4⋊Dic53—41no—
187C4⋊Dic53—42no—
188C4⋊Dic53—42no—
189C4⋊Dic53—42no—
190C4⋊Dic53—81no—
191C4⋊Dic53C22×C412yes—
192#11719919—24no—
193C4×Dic53—24no—
194C4×Dic53—21no—
195C4×Dic53—21no—
196C4×Dic53—21no—
197C4×Dic53—21no—
198C4×Dic53—42no—
199C4×Dic53—42no—
200C2×#88322—21no—
201C2×#88322—21no—
202C2×#88322—21no—
203C2×#88322—21no—
204C2×#88322C22×C412yes—
205C2×#88322C22×C412yes—
206C23×C53—22no—
207C23×C53—22no—
208C23×C53—22no—
209C23×C53—41no—
210C23×C53—41no—
211C23×C53—41no—
212C23×C53—41no—
213C23×C53C22×D411yescharacteristic
214D4×C53—22no—
215D4×C53—22no—
216D4×C53—22no—
217D4×C53#11714yes—
218C2×C4×C53—21no—
219C2×C4×C53—21no—
220C2×C4×C53—21no—
221C2×C4×C53—21no—
222C2×C4×C53—22no—
223C2×C4×C53—48no—
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×C53C4×D412yes—
233C2×C4×C53C4×D412yes—
234C2×C4×C53C23×C411yescharacteristic
235C2×C4×C53C2×#4111yescharacteristic
236C2×C4×C53C2×#4111yescharacteristic
237C2×C4×C53C2×#4111yescharacteristic
238C2×C4×C53H(5,𝔽2)11yescharacteristic
239C2×C4×C532−1+411yescharacteristic
240C2×Dic53—21no—
241C2×Dic53—21no—
242C2×Dic53—22no—
243C2×Dic53—22no—
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—21no—
253C2×Dic53—41no—
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—84no—
264C2×Dic53—84no—
265C2×Dic53AGL(1,ℤ/8)12yes—
266Dic106—22no—
267Dic106—22no—
268Dic106—41no—
269Dic106—41no—
270C8×C53—41no—
271C8×C53—41no—
272#88322—22no—
273#88322—22no—
274#22778—531nomaximal, Sylow
275C22×C53—24no—
276C22×C53—42no—
277C22×C53—42no—
278C22×C53—42no—
279C22×C53—42no—
280C22×C53—42no—
281C22×C53—42no—
282C22×C53C2×C4×D411yescharacteristic
283C22×C53#54611yescharacteristic
284C22×C53#54811yescharacteristic
285C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
286C22×C53#57511yescharacteristic
287C22×C53#57711yescharacteristic
288C22×C53#57811yescharacteristic
289C4×C53—22no—
290C4×C53—22no—
291C4×C53—42no—
292C4×C53—81no—
293C4×C53—81no—
294C4×C53C2×#11711yescharacteristic
295C4×C53#52011yescharacteristic
296C4×C53#55211yescharacteristic
297C4×C53#55311yescharacteristic
298Dic53—22no—
299Dic53—42no—
300Dic53—42no—
301Dic53—41no—
302Dic53—41no—
303Dic53—41no—
304Dic53—81no—
305Dic53—81no—
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#1836—531no—
316#1804D5311yescharacteristic
317#1803—531no—
318#1771—531no—
319#1762—531no—
320C4×#102—531no—
321C2×C53—44no—
322C2×C53#235211yescharacteristic
323C2×C53C4×AGL(1,ℤ/8)11yescharacteristic
324C2×C53#294711yescharacteristic
325C2×C53#331011yescharacteristic
326C2×C53#331511yescharacteristic
327C2×C53#336711yescharacteristic
328C2×C53#337611yescharacteristic
329C22×SD16—531no—
330C2×C4×Q8—531no—
331C2×C4×D4—531no—
332C2×C4×D4—531no—
333C2×C4×D4C2×D5311yescharacteristic
334C22×C4⋊C4—1061no—
335#440—538no—
336C2×#107—531no—
337C2×C4⋊C8C2×D5311yescharacteristic
338C2×#103—531no—
339C2×#103—531no—
340C2×#103—531no—
341C2×#102—531no—
342C2×#102C2×D5311yescharacteristic
343C2×#102C2×D5311yescharacteristic
344C2×#97—531no—
345C2×#97—531no—
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#384—531no—
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#340—531no—
368#340—531no—
369#340C2×D5311yescharacteristic
370#340C2×D5311yescharacteristic
371#336—531no—
372#336—531no—
373C53#2277811yescharacteristic, complemented, Sylow
374C22×Q8—531no—
375C22×D4C22×D5311yescharacteristic
376C23×C4—21no—
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—22no—
389C4×D4—1062no—
390C4×D4—1062no—
391C4×D4—2121no—
392C4×D4—2121no—
393C4×D4C4×D5314yes—
394C2×C4⋊C4—531no—
395C2×C4⋊C4—531no—
396C2×C4⋊C4—531no—
397C2×C4⋊C4—531no—
398C2×C4⋊C4—531no—
399C2×C4⋊C4—1061no—
400C2×C4⋊C4—1061no—
401C2×C4⋊C4—1061no—
402C2×C4⋊C4—1061no—
403C2×C4⋊C4—1061no—
404C2×C4⋊C4—2124no—
405C2×C4⋊C4C22×D5311yescharacteristic
406C2×C4⋊C4C22×D5311yescharacteristic
407C2×C4⋊C4C22×D5311yescharacteristic
408C2×C22⋊C4—21no—
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—22no—
419#103—534no—
420#103—2121no—
421#103—2121no—
422#102—22no—
423#102—22no—
424#102—534no—
425#97—534no—
426#97—2121no—
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—21no—
441C2×Q8—532no—
442C2×Q8—1061no—
443C2×Q8—1061no—
444C2×Q8—1061no—
445C2×Q8—1061no—
446C2×Q8—1061no—
447C2×D4—21no—
448C2×D4—21no—
449C2×D4—21no—
450C2×D4C2×C4×D5312yes—
451C2×D4#1171994712yes—
452C2×D4#1171994912yes—
453C22×C4—21no—
454C22×C4—24no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—42no—
458C22×C4—42no—
459C22×C4—531no—
460C22×C4—531no—
461C22×C4—531no—
462C22×C4—531no—
463C22×C4—1061no—
464C22×C4—1061no—
465C22×C4—1061no—
466C22×C4—1061no—
467C22×C4—4241no—
468C22×C4—4241no—
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—21no—
482C2×C8—21no—
483C2×C8—532no—
484C2×C8—532no—
485C2×C8—1061no—
486C2×C8—1061no—
487C2×C8—1061no—
488C2×C8—1061no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—532no—
495C4⋊C4—1062no—
496C4⋊C4—1064no—
497C4⋊C4—1061no—
498C4⋊C4—1061no—
499C4⋊C4—1061no—
500C4⋊C4—1061no—
501C4⋊C4—1061no—
502C4⋊C4—4241no—
503C4⋊C4—2121no—
504C4⋊C4—2122no—
505C4⋊C4—2122no—
506C4⋊C4—2122no—
507C4⋊C4C2×C4×D5312yes—
508C22⋊C4—24no—
509C22⋊C4—2122no—
510C22⋊C4—2122no—
511C42—1064no—
512C42—1061no—
513C42—1061no—
514C42—1061no—
515C42—1061no—
516C42—2122no—
517C42—2122no—
518C42C2×C4×D5312yes—
519C42#1171994712yes—
520C23—22no—
521C23—22no—
522C23—22no—
523C23—41no—
524C23—41no—
525C23—41no—
526C23—41no—
527C23C2×D4×D5311yescharacteristic
528Q8—1062no—
529Q8—1062no—
530Q8—2121no—
531Q8—2121no—
532D4—22no—
533D4—22no—
534D4—22no—
535D4#1313317514yes—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—22no—
541C2×C4—48no—
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—532no—
551C2×C4—1061no—
552C2×C4—1061no—
553C2×C4—1061no—
554C2×C4—1061no—
555C2×C4—1061no—
556C2×C4—1062no—
557C2×C4—1062no—
558C2×C4—1061no—
559C2×C4—1061no—
560C2×C4—1061no—
561C2×C4—1061no—
562C2×C4—1061no—
563C2×C4—1061no—
564C2×C4—4244no—
565C2×C4—4244no—
566C2×C4—2121no—
567C2×C4—2121no—
568C2×C4—2121no—
569C2×C4—2121no—
570C2×C4—2121no—
571C2×C4—2121no—
572C2×C4—2121no—
573C2×C4—2121no—
574C2×C4—2121no—
575C2×C4—2121no—
576C2×C4#1313317712yes—
577C2×C4#1313317712yes—
578C2×C4C22×C4×D5311yescharacteristic
579C2×C4C2×#1171994711yescharacteristic
580C2×C4C2×#1171994711yescharacteristic
581C2×C4C2×#1171994911yescharacteristic
582C2×C4#1313326311yescharacteristic
583C2×C4#1313326411yescharacteristic
584C8—41no—
585C8—41no—
586C8—1062no—
587C8—1062no—
588C22—24no—
589C22—42no—
590C22—42no—
591C22—42no—
592C22—42no—
593C22—42no—
594C22—42no—
595C22C2×#1313317711yescharacteristic
596C22#1681126411yescharacteristic
597C22#1681127911yescharacteristic
598C22AGL(1,ℤ/8)×D5311yescharacteristic
599C22#1681141911yescharacteristic
600C22#1681143011yescharacteristic
601C22#1681143711yescharacteristic
602C4—22no—
603C4—22no—
604C4—42no—
605C4—81no—
606C4—81no—
607C4—1062no—
608C4—4241no—
609C4—4241no—
610C4—2121no—
611C4—2121no—
612C4—2121no—
613C4—2122no—
614C4—2122no—
615C4C2×#1313317511yescharacteristic
616C4#1681110311yescharacteristic
617C4#1681131511yescharacteristic
618C4#1681132511yescharacteristic
619C2—44no—
620C2#2027764611yescharacteristic
621C2#2028351911yescharacteristic
622C2#2028356111yescharacteristic
623C2#2028673311yescharacteristic
624C2#2028680811yescharacteristic
625C2#2028742811yescharacteristic
626C2#2028753911yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1804C2
socleC23×C53C22×D4
prime radicalC22×D4×C53C23
cosocleC22×C4×C53C24
hypercenter#1804D53
hypocenterC53#22778
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1804

Complements

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

NH ≅ G/Ntype
2. #20277682C2semidirect
4. #20275406C2semidirect
5. #20275392C2semidirect
6. #20275310C2semidirect
7. #20275289C2semidirect
10. #20274848C2semidirect
11. #20274721C2semidirect
12. #20274592C2semidirect
21. #16810723C4semidirect
69. C4×Dic106C2×C4semidirect
92. #13133150C2×C4semidirect
95. #13133140C2×C4semidirect
97. #13133128C2×C4semidirect
373. C53#22778semidirect

Maximal subgroups

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

subgroupindex
C2×#168107232
#202776822
D4×C4⋊Dic532
#202754062
#202753922
#202753102
#202752892
#202750472
#202748482
#202747212
#202745922
#202745422
#102×Dic532
C53×#18042
#2277853