Finite Group Databasebeta

← #27403479

Subgroups — #27403479

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#20277678C211yesmaximal, characteristic, complemented
3#20275404C211yesmaximal, characteristic, complemented
4#20275390C211yesmaximal, characteristic, complemented
5#20275304C211yesmaximal, characteristic, complemented
6#20275283C211yesmaximal, characteristic, complemented
7#20275049C211yesmaximal, characteristic
8#20275049C211yesmaximal, characteristic
9#20274850C211yesmaximal, characteristic, complemented
10#20274725C211yesmaximal, characteristic
11#20274717C211yesmaximal, characteristic, complemented
12#20274590C211yesmaximal, characteristic, complemented
13#20274540C211yesmaximal, characteristic
14C4×#13133138C211yesmaximal, characteristic
15D4×C4⋊C4×C53C211yesmaximal, characteristic
16C22×#11719923C2211yescharacteristic
17C2×C4×Dic106C2211yescharacteristic
18#16810723C418yescomplemented
19C2×#13133150C2211yescharacteristic
20C2×C4⋊#88322C2211yescharacteristic
21C2×#13133140C2211yescharacteristic
22C2×#13133140C2211yescharacteristic
23C2×#13133140C2211yescharacteristic
24C2×#13133138C2211yescharacteristic
25C2×#13133138C2211yescharacteristic
26C2×#13133138C2211yescharacteristic
27C2×#13133128C2211yescharacteristic
28C2×#13133128C2211yescharacteristic
29C2×C4×#88322C2211yescharacteristic
30#16810529C2211yescharacteristic
31#16810529C2211yescharacteristic
32#16810504—21no—
33#16810504—21no—
34#16810490—21no—
35#16810489C2211yescharacteristic
36C4×C4⋊Dic53C2211yescharacteristic
37C4×C4⋊Dic53C2211yescharacteristic
38#16810468—21no—
39#16810385C2211yescharacteristic
40#16810385C2211yescharacteristic
41#16810385C2211yescharacteristic
42#16810385C2211yescharacteristic
43#16810385C2211yescharacteristic
44#16810385C2211yescharacteristic
45#16810385C2211yescharacteristic
46#16810385C2211yescharacteristic
47#16810378C2211yescharacteristic
48#16810378C2211yescharacteristic
49C2×C4×D4×C53C2211yescharacteristic
50C2×C4×D4×C53C2211yescharacteristic
51C2×C4×D4×C53C2211yescharacteristic
52C22×C4⋊C4×C53—21no—
53C53×#389C2211yescharacteristic
54C53×#389C2211yescharacteristic
55C53×#385—21no—
56C53×#385—21no—
57C53×#384C2211yescharacteristic
58C53×#380—21no—
59C4×C4⋊C4×C53C2211yescharacteristic
60C22×Dic106C2311yescharacteristic
61C2×#11719923—22no—
62C2×#11719923—22no—
63C2×#11719923C2×C414yes—
64C22×#88322C2311yescharacteristic
65C22×#88322C2311yescharacteristic
66C22×#88322C2311yescharacteristic
67C22×#88322C2311yescharacteristic
68C4×Dic106—22no—
69C4×Dic106C2×C414yescomplemented
70C2×C4⋊Dic53C2311yescharacteristic
71C2×C4⋊Dic53C2311yescharacteristic
72C2×C4⋊Dic53C2311yescharacteristic
73C2×#11719919—21no—
74C2×C4×Dic53—21no—
75C2×C4×Dic53—21no—
76C2×C4×Dic53—21no—
77#13133150C2×C414yescomplemented
78C4⋊#88322—22no—
79#13133140—41no—
80#13133140—41no—
81#13133140C2×C414yescomplemented
82#13133138—22no—
83#13133138—22no—
84#13133138C2×C414yes—
85#13133128—41no—
86#13133128C2×C414yescomplemented
87C4×#88322—41no—
88#13133124—21no—
89#13133124—21no—
90#13133124—21no—
91#13133124—21no—
92#13133124—21no—
93#13133124—21no—
94#13133124—21no—
95#13133124—21no—
96C22×D4×C53C2311yescharacteristic
97C23×C4×C53—21no—
98C23×C4×C53—21no—
99C23×C4×C53—21no—
100C4×D4×C53—22no—
101C4×D4×C53—22no—
102C4×D4×C53—22no—
103C4×D4×C53—41no—
104C4×D4×C53—41no—
105C4×D4×C53C2×C414yes—
106C2×C4⋊C4×C53—21no—
107C2×C4⋊C4×C53—21no—
108C2×C4⋊C4×C53—21no—
109C2×C4⋊C4×C53—21no—
110C2×C4⋊C4×C53—44no—
111C2×C4⋊C4×C53D411yescharacteristic
112C2×C4⋊C4×C53D411yescharacteristic
113C2×C4⋊C4×C53C2311yescharacteristic
114C2×C4⋊C4×C53C2311yescharacteristic
115C2×C4⋊C4×C53C2311yescharacteristic
116C2×C22⋊C4×C53—21no—
117C2×C22⋊C4×C53—21no—
118C2×C22⋊C4×C53—21no—
119C2×C42×C53C2311yescharacteristic
120C2×C42×C53C2311yescharacteristic
121C2×C42×C53C2311yescharacteristic
122#95×C53—21no—
123#95×C53—21no—
124#95×C53D411yescharacteristic
125#95×C53D411yescharacteristic
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×C53—42no—
140C22×C4×C53—42no—
141C22×C4×C53—42no—
142C22×C4×C53—42no—
143C22×C4×C53—81no—
144C22×C4×C53—81no—
145C22×C4×C53C2×D411yescharacteristic
146C22×C4×C53C2×D411yescharacteristic
147C22×C4×C53C2×D411yescharacteristic
148C22×C4×C53C2×D411yescharacteristic
149C22×C4×C53C2×D411yescharacteristic
150C22×C4×C53C2×D411yescharacteristic
151C22×C4×C53C2411yescharacteristic
152C22×Dic53—21no—
153C22×Dic53—21no—
154C22×Dic53—21no—
155C22×Dic53—21no—
156C2×Dic106—21no—
157C2×Dic106—21no—
158C2×Dic106—21no—
159C2×Dic106—21no—
160C2×Dic106—21no—
161C2×Dic106C22×C412yes—
162C4⋊C4×C53—22no—
163C4⋊C4×C53—21no—
164C4⋊C4×C53—21no—
165C4⋊C4×C53—21no—
166C4⋊C4×C53—21no—
167C4⋊C4×C53—21no—
168C4⋊C4×C53—41no—
169C4⋊C4×C53—42no—
170C4⋊C4×C53—42no—
171C4⋊C4×C53—42no—
172C4⋊C4×C53—81no—
173C4⋊C4×C53C22×C412yes—
174C22⋊C4×C53—24no—
175C22⋊C4×C53—42no—
176C22⋊C4×C53—42no—
177C42×C53—21no—
178C42×C53—21no—
179C42×C53—21no—
180C42×C53—21no—
181C42×C53C22×C412yes—
182C42×C53#4112yes—
183#11719923—28no—
184C4⋊Dic53—21no—
185C4⋊Dic53—21no—
186C4⋊Dic53—21no—
187C4⋊Dic53—21no—
188C4⋊Dic53—21no—
189C4⋊Dic53C22×C412yes—
190#11719919—24no—
191C4×Dic53—24no—
192C4×Dic53—42no—
193C4×Dic53—42no—
194C2×#88322—21no—
195C2×#88322—21no—
196C2×#88322—21no—
197C2×#88322—21no—
198C2×#88322—21no—
199C2×#88322—21no—
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—21no—
223C2×C4×C53—21no—
224C2×C4×C53—22no—
225C2×C4×C53—21no—
226C2×C4×C53—21no—
227C2×C4×C53—21no—
228C2×C4×C53—21no—
229C2×C4×C53—21no—
230C2×C4×C53—21no—
231C2×C4×C53—21no—
232C2×C4×C53—21no—
233C2×C4×C53—21no—
234C2×C4×C53—48no—
235C2×C4×C53—41no—
236C2×C4×C53—41no—
237C2×C4×C53—41no—
238C2×C4×C53—41no—
239C2×C4×C53—41no—
240C2×C4×C53—41no—
241C2×C4×C53—41no—
242C2×C4×C53—41no—
243C2×C4×C53—84no—
244C2×C4×C53—84no—
245C2×C4×C53C4×D412yes—
246C2×C4×C53C4×D412yes—
247C2×C4×C53AGL(1,ℤ/8)12yes—
248C2×C4×C53C23×C411yescharacteristic
249C2×C4×C53C2×#4111yescharacteristic
250C2×C4×C53C2×#4111yescharacteristic
251C2×C4×C53C2×#4111yescharacteristic
252C2×C4×C53H(5,𝔽2)11yescharacteristic
253C2×C4×C532−1+411yescharacteristic
254C2×Dic53—22no—
255C2×Dic53—22no—
256C2×Dic53—41no—
257C2×Dic53—41no—
258C2×Dic53—41no—
259C2×Dic53—41no—
260C2×Dic53—41no—
261C2×Dic53—41no—
262C2×Dic53—41no—
263C2×Dic53—41no—
264C2×Dic53—41no—
265C2×Dic53—41no—
266Dic106—22no—
267Dic106—22no—
268Dic106—41no—
269Dic106—41no—
270#88322—22no—
271#88322—22no—
272#88322—41no—
273#88322—41no—
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—22no—
292C4×C53—42no—
293C4×C53—41no—
294C4×C53—41no—
295C4×C53—41no—
296C4×C53—81no—
297C4×C53—81no—
298C4×C53C2×#11711yescharacteristic
299C4×C53#52011yescharacteristic
300C4×C53#55211yescharacteristic
301C4×C53#55311yescharacteristic
302Dic53—42no—
303Dic53—42no—
304Dic53—81no—
305Dic53—81no—
306C2×#440—531no—
307#2356—531no—
308D4×C4⋊C4D5311yescharacteristic
309#1935—531no—
310#1933—531no—
311#1921—531no—
312#1918—531no—
313#1877—531no—
314#1877—531no—
315#1836—531no—
316#1804—531no—
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×D4C2×D5311yescharacteristic
332C2×C4×D4C2×D5311yescharacteristic
333C2×C4×D4C2×D5311yescharacteristic
334C22×C4⋊C4—21no—
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×#97—531no—
346C2×C4×C8—531no—
347#391—531no—
348#391—531no—
349#389C2×D5311yescharacteristic
350#389C2×D5311yescharacteristic
351#387—1061no—
352#387—1061no—
353#385—21no—
354#385—21no—
355#384C2×D5311yescharacteristic
356#383—1061no—
357#382—531no—
358#380—21no—
359C4×C4⋊C4—531no—
360C4×C4⋊C4—531no—
361C4×C4⋊C4C2×D5311yescharacteristic
362#376—1061no—
363#340—531no—
364#340—531no—
365#340—531no—
366#340—531no—
367#340—531no—
368#340—531no—
369#340—531no—
370#340—531no—
371#336—531no—
372#336—531no—
373C53#2277811yescharacteristic, complemented, Sylow
374C22×Q8—531no—
375C22×D4C22×D5311yescharacteristic
376C23×C4—21no—
377C23×C4—21no—
378C23×C4—21no—
379C2×SD16—534no—
380C2×SD16—1062no—
381C2×SD16—1062no—
382C22×C8—531no—
383C22×C8—531no—
384C22×C8—531no—
385C22×C8—531no—
386C4×Q8—534no—
387C4×Q8—1062no—
388C4×D4—22no—
389C4×D4—22no—
390C4×D4—22no—
391C4×D4—41no—
392C4×D4—41no—
393C4×D4C4×D5314yes—
394C2×C4⋊C4—21no—
395C2×C4⋊C4—21no—
396C2×C4⋊C4—21no—
397C2×C4⋊C4—21no—
398C2×C4⋊C4—44no—
399C2×C4⋊C4—531no—
400C2×C4⋊C4—531no—
401C2×C4⋊C4—531no—
402C2×C4⋊C4—1061no—
403C2×C4⋊C4C22⋊D5311yescharacteristic
404C2×C4⋊C4C22⋊D5311yescharacteristic
405C2×C4⋊C4C22×D5311yescharacteristic
406C2×C4⋊C4C22×D5311yescharacteristic
407C2×C4⋊C4C22×D5311yescharacteristic
408C2×C22⋊C4—21no—
409C2×C22⋊C4—21no—
410C2×C22⋊C4—21no—
411C2×C42—1061no—
412C2×C42—1061no—
413C2×C42—1061no—
414C2×C42C22×D5311yescharacteristic
415C2×C42C22×D5311yescharacteristic
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—534no—
426#97—2121no—
427C4×C8—2121no—
428#95—21no—
429#95—21no—
430#95—1061no—
431#95—1061no—
432#95—1061no—
433#95—1061no—
434#95—1061no—
435#95—1061no—
436#95—1061no—
437#95—1061no—
438#95C22⋊D5311yescharacteristic
439#95C22⋊D5311yescharacteristic
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—42no—
460C22×C4—42no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—81no—
464C22×C4—81no—
465C22×C4—1061no—
466C22×C4—1061no—
467C22×C4—1061no—
468C22×C4—1061no—
469C22×C4C2×C22⋊D5311yescharacteristic
470C22×C4C2×C22⋊D5311yescharacteristic
471C22×C4C2×C22⋊D5311yescharacteristic
472C22×C4C2×C22⋊D5311yescharacteristic
473C22×C4C2×C22⋊D5311yescharacteristic
474C22×C4C2×C22⋊D5311yescharacteristic
475C22×C4C23×D5311yescharacteristic
476SD16—1068no—
477C2×C8—532no—
478C2×C8—532no—
479C2×C8—1061no—
480C2×C8—1061no—
481C2×C8—1061no—
482C2×C8—1061no—
483C2×C8—1061no—
484C2×C8—1061no—
485C2×C8—1061no—
486C2×C8—1061no—
487C2×C8—1061no—
488C2×C8—1061no—
489C4⋊C4—22no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—41no—
496C4⋊C4—42no—
497C4⋊C4—42no—
498C4⋊C4—42no—
499C4⋊C4—81no—
500C4⋊C4—532no—
501C4⋊C4—1064no—
502C4⋊C4—1061no—
503C4⋊C4—1061no—
504C4⋊C4—1061no—
505C4⋊C4—1061no—
506C4⋊C4—1061no—
507C4⋊C4C2×C4×D5312yes—
508C22⋊C4—24no—
509C22⋊C4—42no—
510C22⋊C4—42no—
511C42—21no—
512C42—21no—
513C42—21no—
514C42—21no—
515C42—1064no—
516C42—2122no—
517C42—2122no—
518C42C2×C4×D5312yes—
519C42#1171994512yes—
520C23—22no—
521C23—22no—
522C23—22no—
523C23—41no—
524C23—41no—
525C23—41no—
526C23—41no—
527C23C22×C22⋊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—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—22no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—48no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—84no—
562C2×C4—84no—
563C2×C4—1062no—
564C2×C4—1062no—
565C2×C4—2121no—
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×C4C4×C22⋊D5312yes—
576C2×C4C4×C22⋊D5312yes—
577C2×C4#1313324312yes—
578C2×C4C22×C4×D5311yescharacteristic
579C2×C4C2×#1171994711yescharacteristic
580C2×C4C2×#1171994511yescharacteristic
581C2×C4C2×#1171994911yescharacteristic
582C2×C4#1313326211yescharacteristic
583C2×C4#1313326511yescharacteristic
584C8—1062no—
585C8—1062no—
586C8—2121no—
587C8—2121no—
588C22—24no—
589C22—42no—
590C22—42no—
591C22—42no—
592C22—42no—
593C22—42no—
594C22—42no—
595C22C2×C4×C22⋊D5311yescharacteristic
596C22#1681126711yescharacteristic
597C22#1681128411yescharacteristic
598C22C2×#1313324311yescharacteristic
599C22#1681142111yescharacteristic
600C22#1681143411yescharacteristic
601C22#1681143811yescharacteristic
602C4—22no—
603C4—22no—
604C4—22no—
605C4—41no—
606C4—41no—
607C4—41no—
608C4—42no—
609C4—81no—
610C4—81no—
611C4—4241no—
612C4—4241no—
613C4—2122no—
614C4—2122no—
615C4C2×#1313317511yescharacteristic
616C4#1681110411yescharacteristic
617C4#1681130811yescharacteristic
618C4#1681132111yescharacteristic
619C2—44no—
620C2#2027764011yescharacteristic
621C2C4×#1313324311yescharacteristic
622C2#2028355411yescharacteristic
623C2#2028673611yescharacteristic
624C2#2028681111yescharacteristic
625C2#2028743311yescharacteristic
626C2#2028754211yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingD4×C4⋊C4×C53C2
socleC23×C53C22×D4
prime radicalC22×D4×C53C23
cosocleC22×C4×C53C24
hypercenterD4×C4⋊C4D53
hypocenterC53#22778
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ D4×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. #20277678C2semidirect
3. #20275404C2semidirect
4. #20275390C2semidirect
5. #20275304C2semidirect
6. #20275283C2semidirect
9. #20274850C2semidirect
11. #20274717C2semidirect
12. #20274590C2semidirect
18. #16810723C4semidirect
69. C4×Dic106C2×C4semidirect
77. #13133150C2×C4semidirect
81. #13133140C2×C4semidirect
86. #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
#202776782
#202754042
#202753902
#202753042
#202752832
#202750492
#202748502
#202747252
#202747172
#202745902
#202745402
C4×#131331382
D4×C4⋊C4×C532
#2277853