Finite Group Databasebeta

← #27393783

Subgroups — #27393783

Order 13568.

1166 conjugacy classes of subgroups.

The subgroup lattice has 558 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811160C211yesmaximal, characteristic
2C2×#16810675C211yesmaximal, characteristic, complemented
3#382×D53C211yesmaximal, characteristic, complemented
4#20275678C212yesmaximal, complemented
5#20275523C212yesmaximal, complemented
6#20275031C212yesmaximal
7#20275019C212yesmaximal, complemented
8#20274730C211yesmaximal, characteristic, complemented
9#20274726C211yesmaximal, characteristic, complemented
10#20274722C211yesmaximal, characteristic, complemented
11C53×#1805C211yesmaximal, characteristic, complemented
12#16811160—24no—
13#16811160C418yescomplemented
14C2×#13133212—21no—
15C2×#13133208C2211yescharacteristic
16C2×#13133206C2211yescharacteristic
17C2×#13133202—21no—
18C2×#13133193C2211yescharacteristic
19C2×#13133192C2211yescharacteristic
20C2×C4×D212—21no—
21C2×C4×D212C2211yescharacteristic
22C2×#13133174—21no—
23C2×C4⋊C4×D53C2212yes—
24C2×C42×D53C2211yescharacteristic
25#16810675—24no—
26C2×C4⋊#88322C2211yescharacteristic
27C2×#13133141C2212yes—
28C2×#13133139C2212yes—
29C2×#13133137C2212yes—
30C2×#13133129C2212yes—
31C2×C8×Dic53C2212yes—
32#16810489C2211yescharacteristic
33#16810488C2212yes—
34#16810487—21no—
35#16810487—21no—
36#95×D53—21no—
37#95×D53—21no—
38#16810386C2212yes—
39#16810385C2212yes—
40#16810385C2212yes—
41C2×C4×Q8×C53C2211yescharacteristic
42C2×C4⋊C8×C53C2211yescharacteristic
43C2×#103×C53C2212yes—
44C53×#382C2211yescharacteristic
45C53×#340C2212yes—
46C22×D212—21no—
47C22×D212C2311yescharacteristic
48C22×C4×D53—21no—
49C22×C4×D53—22no—
50C22×C4×D53D411yescharacteristic
51C22×C4×D53D411yescharacteristic
52C22×C4×D53C2311yescharacteristic
53C22×#88322C2312yes—
54#13133212—28no—
55#13133208—22no—
56#13133208C2×C414yescomplemented
57#13133206—22no—
58#13133206C2×C414yescomplemented
59#13133202—28no—
60#13133193—22no—
61#13133193C2×C414yes—
62#13133192—22no—
63#13133192C2×C414yes—
64C4×D212—22no—
65C4×D212—28no—
66C4×D212C2×C414yescomplemented
67#13133174—28no—
68C2×C4⋊Dic53—21no—
69C2×C4⋊Dic53C2312yes—
70C2×C4⋊Dic53C2311yescharacteristic
71C2×#11719919—21no—
72C2×#11719919—21no—
73C2×#11719919—22no—
74C4⋊C4×D53—42no—
75C4⋊C4×D53—42no—
76C2×#11719921—21no—
77C2×#11719921—21no—
78C2×#11719921—21no—
79C2×#11719921—21no—
80C2×#11719921—21no—
81C2×C4×Dic53D411yescharacteristic
82C2×C4×Dic53D411yescharacteristic
83C2×C4×Dic53C2311yescharacteristic
84C42×D53C2×C414yes—
85C42×D53D414yes—
86C4⋊#88322—22no—
87#13133141—42no—
88#13133139—42no—
89#13133137—42no—
90#13133129—24no—
91C8×Dic53—24no—
92#13133124—21no—
93#13133124—21no—
94#13133124—22no—
95#13133124—22no—
96C22×Q8×C53C2311yescharacteristic
97C22×C8×C53C2312yes—
98C4×Q8×C53—22no—
99C4×Q8×C53C2×C414yescomplemented
100C2×C4⋊C4×C53—21no—
101C2×C4⋊C4×C53C2312yes—
102C2×C4⋊C4×C53C2311yescharacteristic
103C2×C42×C53—21no—
104C2×C42×C53C2311yescharacteristic
105C4⋊C8×C53—22no—
106#103×C53—42no—
107#95×C53—21no—
108#95×C53—21no—
109C23×D53—21no—
110C23×D53C2×D411yescharacteristic
111C2×Q8×C53—21no—
112C2×Q8×C53—21no—
113C2×Q8×C53—21no—
114C2×Q8×C53—21no—
115C2×Q8×C53—21no—
116C2×Q8×C53C22×C412yes—
117C22×C4×C53—21no—
118C22×C4×C53—21no—
119C22×C4×C53—22no—
120C22×C4×C53C2×D411yescharacteristic
121C22×C4×C53C2×D411yescharacteristic
122C22×C4×C53C2411yescharacteristic
123C22×Dic53—21no—
124C22×Dic53—22no—
125C22×Dic53C2×D411yescharacteristic
126C22×Dic53C2×D411yescharacteristic
127C22×Dic53C2×D411yescharacteristic
128C2×D212—21no—
129C2×D212—21no—
130C2×D212—21no—
131C2×D212—21no—
132C2×D212—24no—
133C2×D212—21no—
134C2×D212—44no—
135C2×D212C22×C412yes—
136C2×C4×D53—24no—
137C2×C4×D53—21no—
138C2×C4×D53—21no—
139C2×C4×D53—21no—
140C2×C4×D53—21no—
141C2×C4×D53—24no—
142C2×C4×D53—42no—
143C2×C4×D53—44no—
144C2×C4×D53—44no—
145C2×C4×D53—44no—
146C2×C4×D53—44no—
147C2×C4×D53C22⋊C414yes—
148C2×C4×D53C22⋊C414yes—
149C2×C4×D53C22×C412yes—
150C2×C4×D53C2×D412yes—
151C2×C8×C53—22no—
152C2×C8×C53—22no—
153C2×C8×C53—22no—
154C4⋊C4×C53—24no—
155C4⋊C4×C53—21no—
156C4⋊C4×C53—22no—
157C4⋊C4×C53—22no—
158C4⋊C4×C53C22×C412yes—
159C42×C53—24no—
160C42×C53C22×C412yes—
161C42×C53C2×D412yes—
162#11719921—24no—
163#11719921—24no—
164#11719921—24no—
165#11719921—24no—
166#11719921—24no—
167C4⋊Dic53—21no—
168C4⋊Dic53—24no—
169C4⋊Dic53—22no—
170C4⋊Dic53—22no—
171C4⋊Dic53C22×C412yes—
172#11719919—24no—
173#11719919—24no—
174#11719919—44no—
175C4×Dic53C22⋊C414yes—
176C4×Dic53C22⋊C414yes—
177C4×Dic53C22×C412yes—
178C4×Dic53C2×D412yes—
179C2×#88322—22no—
180C2×#88322—22no—
181C2×#88322—22no—
182C23×C53C22×D411yescharacteristic
183C22×D53—21no—
184C22×D53—21no—
185C22×D53—22no—
186C22×D53—22no—
187C22×D53—41no—
188C22×D53—41no—
189C22×D53—42no—
190C22×D53—42no—
191C22×D53C2×C22⋊C412yes—
192C22×D53#13514yes—
193Q8×C53—22no—
194Q8×C53—22no—
195Q8×C53—41no—
196Q8×C53—41no—
197C2×C4×C53—22no—
198C2×C4×C53—22no—
199C2×C4×C53—22no—
200C2×C4×C53—22no—
201C2×C4×C53—41no—
202C2×C4×C53—41no—
203C2×C4×C53—42no—
204C2×C4×C53—42no—
205C2×C4×C53—42no—
206C2×C4×C53—42no—
207C2×C4×C53C2×C22⋊C412yes—
208C2×C4×C53C2×C22⋊C412yes—
209C2×C4×C53C23×C411yescharacteristic
210C2×C4×C53C22×D411yescharacteristic
211C2×C4×C53H(5,𝔽2)11yescharacteristic
212C2×C4×C53H(5,𝔽2)11yescharacteristic
213C2×C4×C532−1+411yescharacteristic
214C2×C4×C532−1+411yescharacteristic
215C2×Dic53—21no—
216C2×Dic53—21no—
217C2×Dic53—22no—
218C2×Dic53—22no—
219C2×Dic53—22no—
220C2×Dic53—42no—
221C2×Dic53—42no—
222C2×Dic53—42no—
223C2×Dic53—42no—
224C2×Dic53C2×C22⋊C412yes—
225C2×Dic53C2×C22⋊C412yes—
226C2×Dic53C2×C22⋊C412yes—
227D212—22no—
228D212—22no—
229D212—48no—
230D212—41no—
231D212—41no—
232C4×D53—28no—
233C4×D53—22no—
234C4×D53—22no—
235C4×D53—22no—
236C4×D53—22no—
237C4×D53—44no—
238C4×D53—48no—
239C4×D53—88no—
240C8×C53—42no—
241#88322—42no—
242#21861—531nomaximal, Sylow
243C22×C53C22×C22⋊C411yescharacteristic
244C22×C53#53511yescharacteristic
245C22×C53#53611yescharacteristic
246C22×C53C2×#13512yes—
247C22×C53#57512yes—
248C2×D53—22no—
249C2×D53—22no—
250C2×D53—24no—
251C2×D53—24no—
252C2×D53—42no—
253C2×D53—42no—
254C2×D53—42no—
255C2×D53—82no—
256C2×D53—82no—
257C2×D53#41614yes—
258C4×C53—22no—
259C4×C53—22no—
260C4×C53—42no—
261C4×C53—42no—
262C4×C53—82no—
263C4×C53#52011yescharacteristic
264C4×C53#52011yescharacteristic
265C4×C53#53711yescharacteristic
266C4×C53#53711yescharacteristic
267Dic53—22no—
268Dic53—22no—
269Dic53—22no—
270Dic53—42no—
271Dic53—82no—
272C2×#529—531no—
273C2×#431—531no—
274C2×#382—531no—
275#1991—532no—
276#1958—532no—
277#1874—532no—
278#1872—532no—
279#1805—531no—
280#1805D5311yescharacteristic, complemented
281#1804—531no—
282#1804—531no—
283C2×C53#231111yescharacteristic
284C2×C53C2×#41611yescharacteristic
285C2×C53#289611yescharacteristic
286C2×C53#321112yes—
287C2×C53#323212yes—
288D53—24no—
289D53—82no—
290#529—538no—
291#529—1064no—
292C2×#126—1061no—
293C2×#125—531no—
294C2×#124—531no—
295C2×#123—1061no—
296C2×C22⋊Q8—531no—
297C2×#121—531no—
298C2×C4×Q8C2×D5311yescharacteristic
299C2×C4×D4—531no—
300C2×C4×D4—1061no—
301C2×#117—1061no—
302C22×C4⋊C4—532no—
303C22×C42—531no—
304#431—1064no—
305C2×C4⋊C8—531no—
306C2×C4⋊C8C2×D5311yescharacteristic
307C2×#103—532no—
308C2×#103C2×D5312yes—
309C2×#102—532no—
310C2×#102—532no—
311C2×C22⋊C8—532no—
312C2×C4×C8—532no—
313#382—532no—
314#382—531no—
315#382—1061no—
316#382—1061no—
317#382C2×D5311yescharacteristic
318C2×#95—1061no—
319C2×#95—1061no—
320#340—532no—
321#340—532no—
322#340—532no—
323#340C2×D5312yes—
324C53#2186111yescharacteristic, complemented, Sylow
325C22×Q8C22×D5311yescharacteristic
326C22×D4—531no—
327C22×D4—1061no—
328C23×C4—531no—
329C23×C4—531no—
330C23×C4—531no—
331C23×C4—1062no—
332C23×C4—1061no—
333C22×C8—532no—
334C22×C8C22×D5312yes—
335#126—1068no—
336#125—534no—
337#125—1062no—
338#124—534no—
339#124—1062no—
340#123—1068no—
341C22⋊Q8—534no—
342C22⋊Q8—1062no—
343#121—534no—
344#121—1062no—
345C4×Q8—22no—
346C4×Q8C4×D5314yescomplemented
347C4×D4—534no—
348C4×D4—1068no—
349C4×D4—1062no—
350#117—1068no—
351C2×C4⋊C4—21no—
352C2×C4⋊C4—531no—
353C2×C4⋊C4—532no—
354C2×C4⋊C4—1061no—
355C2×C4⋊C4—1061no—
356C2×C4⋊C4—1061no—
357C2×C4⋊C4—1062no—
358C2×C4⋊C4—2122no—
359C2×C4⋊C4—2122no—
360C2×C4⋊C4C22×D5312yes—
361C2×C4⋊C4C22×D5311yescharacteristic
362C2×C22⋊C4—1061no—
363C2×C22⋊C4—1061no—
364C2×C22⋊C4—1061no—
365C2×C22⋊C4—1061no—
366C2×C22⋊C4—1061no—
367C2×C42—21no—
368C2×C42—531no—
369C2×C42—531no—
370C2×C42—531no—
371C2×C42—534no—
372C2×C42—534no—
373C2×C42C22×D5311yescharacteristic
374C4⋊C8—22no—
375C4⋊C8—1062no—
376#103—42no—
377#103—2122no—
378#102—2122no—
379#102—2122no—
380C22⋊C8—1064no—
381C4×C8—1064no—
382#95—21no—
383#95—21no—
384#95—1062no—
385#95—1062no—
386#95—1061no—
387#95—1061no—
388C24—531no—
389C24—1061no—
390C2×Q8—21no—
391C2×Q8—21no—
392C2×Q8—21no—
393C2×Q8—21no—
394C2×Q8—21no—
395C2×Q8C2×C4×D5312yes—
396C2×D4—532no—
397C2×D4—1064no—
398C2×D4—1061no—
399C2×D4—1061no—
400C2×D4—1061no—
401C2×D4—1061no—
402C2×D4—1061no—
403C2×D4—2124no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—22no—
407C22×C4—531no—
408C22×C4—531no—
409C22×C4—531no—
410C22×C4—534no—
411C22×C4—532no—
412C22×C4—532no—
413C22×C4—534no—
414C22×C4—1064no—
415C22×C4—1061no—
416C22×C4—1061no—
417C22×C4—1061no—
418C22×C4—1061no—
419C22×C4—1061no—
420C22×C4—1064no—
421C22×C4—1062no—
422C22×C4—2122no—
423C22×C4—2124no—
424C22×C4—2124no—
425C22×C4—2124no—
426C22×C4—2124no—
427C22×C4D4×D5311yescharacteristic
428C22×C4D4×D5311yescharacteristic
429C22×C4C23×D5311yescharacteristic
430C2×C8—22no—
431C2×C8—22no—
432C2×C8—22no—
433C2×C8—1062no—
434C2×C8—1062no—
435C2×C8—1062no—
436C4⋊C4—24no—
437C4⋊C4—21no—
438C4⋊C4—22no—
439C4⋊C4—22no—
440C4⋊C4—532no—
441C4⋊C4—1064no—
442C4⋊C4—1064no—
443C4⋊C4—1064no—
444C4⋊C4—1061no—
445C4⋊C4—1062no—
446C4⋊C4—1062no—
447C4⋊C4—2124no—
448C4⋊C4C2×C4×D5312yes—
449C22⋊C4—1064no—
450C22⋊C4—1064no—
451C22⋊C4—1064no—
452C22⋊C4—1064no—
453C22⋊C4—1064no—
454C42—24no—
455C42—534no—
456C42—534no—
457C42—532no—
458C42—532no—
459C42C2×C4×D5312yes—
460C42D4×D5312yes—
461C23—532no—
462C23—534no—
463C23—1062no—
464C23—1061no—
465C23—1061no—
466C23—1062no—
467C23—2122no—
468C23—2122no—
469C23—2121no—
470C23—2121no—
471C23C2×D4×D5311yescharacteristic
472Q8—22no—
473Q8—22no—
474Q8—41no—
475Q8—41no—
476D4—1062no—
477D4—1062no—
478D4—2128no—
479D4—2121no—
480D4—2121no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—42no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—532no—
492C2×C4—532no—
493C2×C4—532no—
494C2×C4—1068no—
495C2×C4—1062no—
496C2×C4—1062no—
497C2×C4—1061no—
498C2×C4—1061no—
499C2×C4—1062no—
500C2×C4—1062no—
501C2×C4—1062no—
502C2×C4—1062no—
503C2×C4—1062no—
504C2×C4—4248no—
505C2×C4—2124no—
506C2×C4—2128no—
507C2×C4—2122no—
508C2×C4—2122no—
509C2×C4—2122no—
510C2×C4—2122no—
511C2×C4C22⋊C4×D5312yes—
512C2×C4C22⋊C4×D5312yes—
513C2×C4C22×C4×D5311yescharacteristic
514C2×C4C2×D4×D5311yescharacteristic
515C2×C4#1313326211yescharacteristic
516C2×C4#1313326311yescharacteristic
517C2×C4#1313326411yescharacteristic
518C2×C4#1313326511yescharacteristic
519C8—42no—
520C8—2122no—
521C22—534no—
522C22—1062no—
523C22—1062no—
524C22—1064no—
525C22—1064no—
526C22—4242no—
527C22—4242no—
528C22—2122no—
529C22—2122no—
530C22—2122no—
531C22C2×C22⋊C4×D5311yescharacteristic
532C22#1681118911yescharacteristic
533C22#1681119611yescharacteristic
534C22#135×D5312yes—
535C22#1681142212yes—
536C4—22no—
537C4—22no—
538C4—42no—
539C4—42no—
540C4—82no—
541C4—1062no—
542C4—1062no—
543C4—1062no—
544C4—4242no—
545C4—2122no—
546C4#1681110311yescharacteristic
547C4#1681110411yescharacteristic
548C4#1681120211yescharacteristic
549C4#1681120311yescharacteristic
550C2—1064no—
551C2—4242no—
552C2#2027730411yescharacteristic
553C2#416×D5311yescharacteristic
554C2#2028323311yescharacteristic
555C2#2028568412yes—
556C2#2028588712yes—
subgroupG/H
centerC23C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1805C2
socleC23×C53C22×D4
prime radicalC2×#13133192C22
cosocleC22×C4×C53C24
hypercenter#1805D53
hypocenterC53#21861
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1805

Complements

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

NH ≅ G/Ntype
2. C2×#16810675C2semidirect
3. #382×D53C2semidirect
4. #20275678C2semidirect
5. #20275523C2semidirect
7. #20275019C2semidirect
8. #20274730C2semidirect
9. #20274726C2semidirect
10. #20274722C2semidirect
11. C53×#1805C2semidirect
13. #16811160C4semidirect
56. #13133208C2×C4semidirect
58. #13133206C2×C4semidirect
66. C4×D212C2×C4semidirect
99. C4×Q8×C53C2×C4semidirect
280. #1805D53semidirect
324. C53#21861semidirect
346. C4×Q8C4×D53semidirect

Maximal subgroups

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

subgroupindex
C2×#168111602
C2×#168106752
#382×D532
#202756782
#202755232
#202750312
#202750192
#202747302
#202747262
#202747222
C53×#18052
#2186153