Finite Group Databasebeta

← #27393986

Subgroups — #27393986

Order 13568.

1150 conjugacy classes of subgroups.

The subgroup lattice has 554 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811160C211yesmaximal, characteristic
2C2×C4⋊C8×D53C211yesmaximal, characteristic, complemented
3#20277108C211yesmaximal, characteristic, complemented
4#20275671C212yesmaximal, complemented
5#20275644C212yesmaximal, complemented
6#20275142C212yesmaximal
7#20275135C212yesmaximal, complemented
8#20274730C211yesmaximal, characteristic, complemented
9#20274726C211yesmaximal, characteristic, complemented
10#20274722C211yesmaximal, characteristic, complemented
11C53×#1805C211yesmaximal, characteristic, complemented
12C22×C8×D53C2212yes—
13#16811160—24no—
14#16811160C418yescomplemented
15C2×#13133212—21no—
16C2×#13133208C2211yescharacteristic
17C2×#13133206C2211yescharacteristic
18C2×#13133202—21no—
19C2×#13133193C2211yescharacteristic
20C2×#13133192C2211yescharacteristic
21C2×C4×D212—21no—
22C2×C4×D212C2211yescharacteristic
23C2×#13133174—21no—
24C2×C42×D53C2211yescharacteristic
25C2×C4⋊#88322C2211yescharacteristic
26C2×#13133146C2212yes—
27C4⋊C8×D53—24no—
28C2×#13133141C2212yes—
29C2×#13133139C2212yes—
30C2×#13133137C2212yes—
31#16810490—21no—
32#16810489C2211yescharacteristic
33#16810485—21no—
34#16810480C2212yes—
35C4⋊C4×Dic53C2212yes—
36#16810470—21no—
37#16810468—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×D53D411yescharacteristic
50C22×C4×D53D411yescharacteristic
51C22×C4×D53C2311yescharacteristic
52C22×#88322C2312yes—
53C2×C8×D53—24no—
54C2×C8×D53—42no—
55C2×C8×D53—42no—
56#13133212—28no—
57#13133208—22no—
58#13133208C2×C414yes—
59#13133206—22no—
60#13133206C2×C414yes—
61#13133202—28no—
62#13133193—22no—
63#13133193C2×C414yes—
64#13133192—22no—
65#13133192C2×C414yes—
66C4×D212—22no—
67C4×D212—28no—
68C4×D212C2×C414yes—
69#13133174—28no—
70C2×C4⋊Dic53—21no—
71C2×C4⋊Dic53C2312yes—
72C2×C4⋊Dic53C2311yescharacteristic
73C2×#11719919—21no—
74C2×#11719919—21no—
75C2×#11719921—21no—
76C2×#11719921—22no—
77C2×#11719921—21no—
78C2×#11719921—21no—
79C2×#11719921—21no—
80C2×#11719921—21no—
81C2×C4×Dic53—22no—
82C2×C4×Dic53D411yescharacteristic
83C2×C4×Dic53D411yescharacteristic
84C2×C4×Dic53C2311yescharacteristic
85C42×D53C2×C414yes—
86C42×D53D414yes—
87C4⋊#88322—22no—
88#13133146—24no—
89#13133141—42no—
90#13133139—42no—
91#13133137—42no—
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×C414yes—
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×D53C22⋊C414yes—
145C2×C4×D53C22⋊C414yes—
146C2×C4×D53C22×C412yes—
147C2×C4×D53C2×D412yes—
148C2×C8×C53—22no—
149C2×C8×C53—22no—
150C2×C8×C53—22no—
151C4⋊C4×C53—24no—
152C4⋊C4×C53—21no—
153C4⋊C4×C53—22no—
154C4⋊C4×C53—22no—
155C4⋊C4×C53C22×C412yes—
156C42×C53—24no—
157C42×C53C22×C412yes—
158C42×C53C2×D412yes—
159#11719921—24no—
160#11719921—24no—
161#11719921—24no—
162#11719921—24no—
163#11719921—24no—
164#11719921—44no—
165C4⋊Dic53—21no—
166C4⋊Dic53—24no—
167C4⋊Dic53—22no—
168C4⋊Dic53—22no—
169C4⋊Dic53C22×C412yes—
170#11719919—24no—
171#11719919—24no—
172C4×Dic53—44no—
173C4×Dic53C22⋊C414yes—
174C4×Dic53C22⋊C414yes—
175C4×Dic53C22×C412yes—
176C4×Dic53C2×D412yes—
177C2×#88322—22no—
178C2×#88322—22no—
179C2×#88322—22no—
180C8×D53—48no—
181C23×C53C22×D411yescharacteristic
182C22×D53—21no—
183C22×D53—21no—
184C22×D53—22no—
185C22×D53—22no—
186C22×D53—41no—
187C22×D53—41no—
188C22×D53—42no—
189C22×D53—42no—
190C22×D53C2×C22⋊C412yes—
191C22×D53#13514yes—
192Q8×C53—22no—
193Q8×C53—22no—
194Q8×C53—41no—
195Q8×C53—41no—
196C2×C4×C53—22no—
197C2×C4×C53—22no—
198C2×C4×C53—22no—
199C2×C4×C53—22no—
200C2×C4×C53—41no—
201C2×C4×C53—41no—
202C2×C4×C53—42no—
203C2×C4×C53—42no—
204C2×C4×C53—42no—
205C2×C4×C53—42no—
206C2×C4×C53C2×C22⋊C412yes—
207C2×C4×C53C2×C22⋊C412yes—
208C2×C4×C53C23×C411yescharacteristic
209C2×C4×C53C22×D411yescharacteristic
210C2×C4×C53H(5,𝔽2)11yescharacteristic
211C2×C4×C53H(5,𝔽2)11yescharacteristic
212C2×C4×C532−1+411yescharacteristic
213C2×C4×C532−1+411yescharacteristic
214C2×Dic53—21no—
215C2×Dic53—21no—
216C2×Dic53—22no—
217C2×Dic53—22no—
218C2×Dic53—22no—
219C2×Dic53—42no—
220C2×Dic53—42no—
221C2×Dic53—42no—
222C2×Dic53—42no—
223C2×Dic53C2×C22⋊C412yes—
224C2×Dic53C2×C22⋊C412yes—
225C2×Dic53C2×C22⋊C412yes—
226D212—22no—
227D212—22no—
228D212—48no—
229D212—41no—
230D212—41no—
231C4×D53—28no—
232C4×D53—22no—
233C4×D53—22no—
234C4×D53—22no—
235C4×D53—22no—
236C4×D53—44no—
237C4×D53—48no—
238C8×C53—42no—
239#88322—42no—
240#21886—531nomaximal, Sylow
241C22×C53C22×C22⋊C411yescharacteristic
242C22×C53#53711yescharacteristic
243C22×C53#53711yescharacteristic
244C22×C53C2×#13512yes—
245C22×C53#57512yes—
246C2×D53—22no—
247C2×D53—22no—
248C2×D53—24no—
249C2×D53—24no—
250C2×D53—42no—
251C2×D53—42no—
252C2×D53—42no—
253C2×D53—82no—
254C2×D53—82no—
255C2×D53#41614yes—
256C4×C53—22no—
257C4×C53—22no—
258C4×C53—42no—
259C4×C53—42no—
260C4×C53—82no—
261C4×C53#52011yescharacteristic
262C4×C53#52011yescharacteristic
263C4×C53#53511yescharacteristic
264C4×C53#53611yescharacteristic
265Dic53—22no—
266Dic53—22no—
267Dic53—22no—
268Dic53—42no—
269Dic53—82no—
270C2×#529—531no—
271C22×C4⋊C8—531no—
272#2283—531no—
273#1989—532no—
274#1983—532no—
275#1893—532no—
276#1892—532no—
277#1805—531no—
278#1805D5311yescharacteristic, complemented
279#1804—531no—
280#1804—531no—
281C2×C53#231111yescharacteristic
282C2×C53C2×#41611yescharacteristic
283C2×C53#289611yescharacteristic
284C2×C53#325012yes—
285C2×C53#325012yes—
286D53—24no—
287D53—82no—
288C23×C8—532no—
289#529—538no—
290#529—1064no—
291C2×#126—1061no—
292C2×#125—531no—
293C2×#124—531no—
294C2×#123—1061no—
295C2×C22⋊Q8—531no—
296C2×#121—531no—
297C2×C4×Q8C2×D5311yescharacteristic
298C2×C4×D4—531no—
299C2×C4×D4—1061no—
300C2×#117—1061no—
301C22×C42—531no—
302C2×C4⋊C8—532no—
303C2×C4⋊C8—531no—
304C2×C4⋊C8—1064no—
305C2×C4⋊C8C2×D5311yescharacteristic
306C2×#103—532no—
307C2×#103C2×D5312yes—
308C2×#102—532no—
309C2×#102—532no—
310#383—1061no—
311#382—531no—
312#382C2×D5311yescharacteristic
313#381—1061no—
314#380—532no—
315C4×C4⋊C4—532no—
316C4×C22⋊C4—1061no—
317#376—1061no—
318#340—532no—
319#340—532no—
320#340—532no—
321#340C2×D5312yes—
322C53#2188611yescharacteristic, complemented, Sylow
323C22×Q8C22×D5311yescharacteristic
324C22×D4—531no—
325C22×D4—1061no—
326C23×C4—531no—
327C23×C4—531no—
328C23×C4—531no—
329C23×C4—1061no—
330C22×C8—532no—
331C22×C8—1064no—
332C22×C8—2122no—
333C22×C8—2122no—
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×D5314yes—
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⋊C4C22×D5312yes—
358C2×C4⋊C4C22×D5311yescharacteristic
359C2×C22⋊C4—1061no—
360C2×C22⋊C4—1061no—
361C2×C22⋊C4—1061no—
362C2×C22⋊C4—1061no—
363C2×C22⋊C4—1061no—
364C2×C22⋊C4—1062no—
365C2×C42—21no—
366C2×C42—531no—
367C2×C42—531no—
368C2×C42—531no—
369C2×C42—534no—
370C2×C42—534no—
371C2×C42—1062no—
372C2×C42C22×D5311yescharacteristic
373C4⋊C8—22no—
374C4⋊C8—1064no—
375C4⋊C8—1062no—
376#103—42no—
377#103—2122no—
378#102—2122no—
379#102—2122no—
380#95—21no—
381#95—21no—
382#95—1062no—
383#95—1062no—
384#95—1061no—
385#95—1061no—
386C24—531no—
387C24—1061no—
388C2×Q8—21no—
389C2×Q8—21no—
390C2×Q8—21no—
391C2×Q8—21no—
392C2×Q8—21no—
393C2×Q8C2×C4×D5312yes—
394C2×D4—532no—
395C2×D4—1064no—
396C2×D4—1061no—
397C2×D4—1061no—
398C2×D4—1061no—
399C2×D4—1061no—
400C2×D4—1061no—
401C2×D4—2124no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—22no—
405C22×C4—531no—
406C22×C4—531no—
407C22×C4—531no—
408C22×C4—534no—
409C22×C4—532no—
410C22×C4—532no—
411C22×C4—534no—
412C22×C4—1064no—
413C22×C4—1061no—
414C22×C4—1061no—
415C22×C4—1061no—
416C22×C4—1061no—
417C22×C4—1061no—
418C22×C4—1064no—
419C22×C4—1062no—
420C22×C4—2122no—
421C22×C4—2124no—
422C22×C4D4×D5311yescharacteristic
423C22×C4D4×D5311yescharacteristic
424C22×C4C23×D5311yescharacteristic
425C2×C8—22no—
426C2×C8—22no—
427C2×C8—22no—
428C2×C8—1062no—
429C2×C8—1062no—
430C2×C8—1062no—
431C2×C8—2128no—
432C4⋊C4—24no—
433C4⋊C4—21no—
434C4⋊C4—22no—
435C4⋊C4—22no—
436C4⋊C4—532no—
437C4⋊C4—1064no—
438C4⋊C4—1064no—
439C4⋊C4—1064no—
440C4⋊C4—1061no—
441C4⋊C4—1062no—
442C4⋊C4—1062no—
443C4⋊C4C2×C4×D5312yes—
444C22⋊C4—1064no—
445C22⋊C4—1064no—
446C22⋊C4—1064no—
447C22⋊C4—1064no—
448C22⋊C4—1064no—
449C22⋊C4—2124no—
450C42—24no—
451C42—534no—
452C42—534no—
453C42—532no—
454C42—532no—
455C42—2124no—
456C42C2×C4×D5312yes—
457C42D4×D5312yes—
458C23—532no—
459C23—534no—
460C23—1062no—
461C23—1061no—
462C23—1061no—
463C23—1062no—
464C23—2122no—
465C23—2122no—
466C23—2121no—
467C23—2121no—
468C23C2×D4×D5311yescharacteristic
469Q8—22no—
470Q8—22no—
471Q8—41no—
472Q8—41no—
473D4—1062no—
474D4—1062no—
475D4—2128no—
476D4—2121no—
477D4—2121no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—42no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—532no—
489C2×C4—532no—
490C2×C4—532no—
491C2×C4—1068no—
492C2×C4—1062no—
493C2×C4—1062no—
494C2×C4—1061no—
495C2×C4—1061no—
496C2×C4—1062no—
497C2×C4—1062no—
498C2×C4—1062no—
499C2×C4—1062no—
500C2×C4—1062no—
501C2×C4—2124no—
502C2×C4—2128no—
503C2×C4—2122no—
504C2×C4—2122no—
505C2×C4—2122no—
506C2×C4—2122no—
507C2×C4C22⋊C4×D5312yes—
508C2×C4C22⋊C4×D5312yes—
509C2×C4C22×C4×D5311yescharacteristic
510C2×C4C2×D4×D5311yescharacteristic
511C2×C4#1313326211yescharacteristic
512C2×C4#1313326311yescharacteristic
513C2×C4#1313326411yescharacteristic
514C2×C4#1313326511yescharacteristic
515C8—42no—
516C8—2122no—
517C22—534no—
518C22—1062no—
519C22—1062no—
520C22—1064no—
521C22—1064no—
522C22—4242no—
523C22—4242no—
524C22—2122no—
525C22—2122no—
526C22—2122no—
527C22C2×C22⋊C4×D5311yescharacteristic
528C22#1681120211yescharacteristic
529C22#1681120311yescharacteristic
530C22#135×D5312yes—
531C22#1681142212yes—
532C4—22no—
533C4—22no—
534C4—42no—
535C4—42no—
536C4—82no—
537C4—1062no—
538C4—1062no—
539C4—1062no—
540C4—4242no—
541C4—2122no—
542C4#1681110311yescharacteristic
543C4#1681110411yescharacteristic
544C4#1681118911yescharacteristic
545C4#1681119611yescharacteristic
546C2—1064no—
547C2—4242no—
548C2#2027730411yescharacteristic
549C2#416×D5311yescharacteristic
550C2#2028323311yescharacteristic
551C2#2028606512yes—
552C2#2028606912yes—
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#21886
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×C4⋊C8×D53C2semidirect
3. #20277108C2semidirect
4. #20275671C2semidirect
5. #20275644C2semidirect
7. #20275135C2semidirect
8. #20274730C2semidirect
9. #20274726C2semidirect
10. #20274722C2semidirect
11. C53×#1805C2semidirect
14. #16811160C4semidirect
278. #1805D53semidirect
322. C53#21886semidirect

Maximal subgroups

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

subgroupindex
C2×#168111602
C2×C4⋊C8×D532
#202771082
#202756712
#202756442
#202751422
#202751352
#202747302
#202747262
#202747222
C53×#18052
#2188653