Finite Group Databasebeta

← #27400849

Subgroups — #27400849

Order 13568.

1172 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×#16811319C211yesmaximal, characteristic
2#20277594C212yesmaximal, complemented
3#20275550C212yesmaximal, complemented
4#20275549C214yesmaximal, complemented
5#20274702C212yesmaximal, complemented
6C4×#13133132C211yesmaximal, characteristic, complemented
7#20274502C212yesmaximal, complemented
8C53×#1799C211yesmaximal, characteristic, complemented
9#16811319—24no—
10#16811319—22no—
11#16811319—22no—
12C2×#13133212C412yescomplemented
13C2×#13133209C411yescharacteristic, complemented
14C2×#13133199C412yescomplemented
15C2×#13133191C411yescharacteristic, complemented
16C2×#13133191C2211yescharacteristic
17C2×#13133189C2212yes—
18C2×C4×D212C411yescharacteristic, complemented
19C2×#13133177C2212yes—
20C2×#13133176C411yescharacteristic, complemented
21C22×C4⋊Dic53C2211yescharacteristic
22C2×#13133132C2212yes—
23C2×#13133131C2214yes—
24C2×#13133131C2214yes—
25#16810519—22no—
26#16810508—24no—
27#16810481C2214yes—
28C4⋊C4×Dic53—22no—
29C22⋊C4×Dic53C2214yes—
30C4×C22⋊Dic53C2212yes—
31#16810390C2211yescharacteristic
32#16810390C2211yescharacteristic
33C2×C4×D4×C53C2211yescharacteristic
34C2×#99×C53C2212yes—
35C53×#380C2212yes—
36C53×#342C2212yes—
37C22×D212C2×C411yescharacteristic
38C22×C22⋊D53C2×C412yes—
39C22×C22⋊D53C2312yes—
40C23×Dic53C2312yes—
41C22×C4×D53C2×C412yes—
42#13133212—24no—
43#13133212—24no—
44#13133209—22no—
45#13133209—22no—
46#13133199—24no—
47#13133199—24no—
48#13133191—22no—
49#13133191—22no—
50#13133191—22no—
51#13133191—22no—
52#13133189—24no—
53#13133189—24no—
54C4×D212—22no—
55C4×D212—21no—
56C4×D212—21no—
57#13133177—24no—
58#13133177—24no—
59#13133176—22no—
60#13133176—21no—
61#13133176—21no—
62C2×C4⋊Dic53C2×C411yescharacteristic
63C2×C4⋊Dic53C2×C411yescharacteristic
64C2×C4⋊Dic53C2×C411yescharacteristic
65C2×C4⋊Dic53C2×C412yes—
66C2×C4⋊Dic53D414yes—
67C2×C4⋊Dic53D412yes—
68C2×C4⋊Dic53D412yes—
69C2×#11719919—24no—
70C2×#11719919C2×C412yes—
71C2×#11719919C2×C412yes—
72C2×C22⋊Dic53C2×C412yes—
73C2×C22⋊Dic53C2312yes—
74C2×C22⋊Dic53C2312yes—
75C2×#11719921C2×C412yes—
76C2×#11719921C2×C412yes—
77C2×#11719921C2×C412yes—
78C2×C4×Dic53—22no—
79C2×C4×Dic53—24no—
80C2×C4×Dic53C2×C412yes—
81#13133132—24no—
82#13133131—28no—
83#13133131—44no—
84#13133124—22no—
85#13133124—24no—
86#13133124—24no—
87C22×D4×C53C2311yescharacteristic
88C23×C4×C53C2×C411yescharacteristic
89C23×C4×C53C2311yescharacteristic
90C4×D4×C53—22no—
91C4×D4×C53—22no—
92C2×C4⋊C4×C53—22no—
93C2×C4⋊C4×C53C2×C411yescharacteristic
94C2×C22⋊C4×C53C2×C411yescharacteristic
95C2×C22⋊C4×C53C2314yes—
96C2×C22⋊C4×C53C2311yescharacteristic
97C2×C42×C53C2×C411yescharacteristic
98#99×C53—42no—
99#95×C53—22no—
100C24×C53C22×C411yescharacteristic
101C24×C53C2411yescharacteristic
102C23×D53C22×C412yes—
103C2×D4×C53—22no—
104C2×D4×C53—22no—
105C2×D4×C53—21no—
106C2×D4×C53—21no—
107C22×C4×C53—21no—
108C22×C4×C53—21no—
109C22×C4×C53—24no—
110C22×C4×C53—42no—
111C22×C4×C53—42no—
112C22×C4×C53C22×C411yescharacteristic
113C22×C4×C53C22×C411yescharacteristic
114C22×C4×C53C22×C411yescharacteristic
115C22×C4×C53C22×C411yescharacteristic
116C22×C4×C53C22×C411yescharacteristic
117C22×C4×C53C2×D412yes—
118C22×C4×C53C2×D412yes—
119C2×C22⋊D53—24no—
120C2×C22⋊D53—24no—
121C2×C22⋊D53—22no—
122C2×C22⋊D53—22no—
123C2×C22⋊D53—24no—
124C2×C22⋊D53—82no—
125C22×Dic53—22no—
126C22×Dic53—22no—
127C22×Dic53—82no—
128C22×Dic53C22×C412yes—
129C22×Dic53C22×C412yes—
130C22×Dic53C22×C412yes—
131C22×Dic53C2×D414yes—
132C22×Dic53C2×D414yes—
133C2×D212—21no—
134C2×D212—21no—
135C2×D212—42no—
136C2×C4×D53—22no—
137C2×C4×D53—22no—
138C2×C4×D53—82no—
139C4⋊C4×C53—22no—
140C4⋊C4×C53—44no—
141C22⋊C4×C53—22no—
142C22⋊C4×C53—22no—
143C22⋊C4×C53—24no—
144C22⋊C4×C53—24no—
145C42×C53—21no—
146C42×C53—21no—
147C22⋊Dic53—22no—
148C22⋊Dic53—22no—
149C22⋊Dic53—22no—
150C22⋊Dic53—22no—
151C22⋊Dic53—22no—
152C22⋊Dic53#4114yes—
153#11719921—24no—
154#11719921—24no—
155#11719921—22no—
156#11719921—22no—
157C4⋊Dic53—21no—
158C4⋊Dic53—21no—
159C4⋊Dic53C22⋊C412yes—
160C4⋊Dic53C22⋊C411yescharacteristic
161C4⋊Dic53C22⋊C411yescharacteristic
162C4⋊Dic53C22⋊C412yes—
163C4⋊Dic53C22⋊C412yes—
164C4⋊Dic53C22⋊C414yes—
165C4⋊Dic53C22⋊C414yes—
166#11719919—24no—
167#11719919—24no—
168#11719919—84no—
169C4×Dic53—22no—
170C4×Dic53—22no—
171C4×Dic53—44no—
172C4×Dic53—84no—
173C23×C53—21no—
174C23×C53—21no—
175C23×C53—21no—
176C23×C53—41no—
177C23×C53—41no—
178C23×C53—41no—
179C23×C53—41no—
180C23×C53C23×C411yescharacteristic
181C23×C53C22×D412yes—
182C23×C53C2×#4112yes—
183C23×C53H(5,𝔽2)11yescharacteristic
184C23×C532−1+411yescharacteristic
185C22×D53—22no—
186C22×D53—22no—
187C22×D53—22no—
188C22×D53—42no—
189C22×D53—42no—
190D4×C53—42no—
191D4×C53—42no—
192C2×C4×C53—22no—
193C2×C4×C53—24no—
194C2×C4×C53—21no—
195C2×C4×C53—21no—
196C2×C4×C53—21no—
197C2×C4×C53—21no—
198C2×C4×C53—21no—
199C2×C4×C53—41no—
200C2×C4×C53—42no—
201C2×C4×C53—42no—
202C2×C4×C53—42no—
203C2×C4×C53—41no—
204C2×C4×C53—41no—
205C2×C4×C53—44no—
206C2×C4×C53—44no—
207C2×C4×C53—44no—
208C2×C4×C53C2×C22⋊C411yescharacteristic
209C2×C4×C53C2×C22⋊C412yes—
210C2×C4×C53C2×C22⋊C411yescharacteristic
211C22⋊D53—44no—
212C22⋊D53—44no—
213C22⋊D53—84no—
214C2×Dic53—22no—
215C2×Dic53—22no—
216C2×Dic53—22no—
217C2×Dic53—24no—
218C2×Dic53—42no—
219C2×Dic53—42no—
220C2×Dic53—44no—
221C2×Dic53—44no—
222C2×Dic53—42no—
223C2×Dic53—42no—
224C2×Dic53—42no—
225C2×Dic53—42no—
226C2×Dic53—42no—
227C2×Dic53C2×C22⋊C412yes—
228C2×Dic53C2×C22⋊C412yes—
229C2×Dic53C2×C22⋊C414yes—
230D212—42no—
231D212—42no—
232C4×D53—82no—
233C4×D53—82no—
234#22535—531nomaximal, Sylow
235C22×C53—22no—
236C22×C53—21no—
237C22×C53—22no—
238C22×C53—21no—
239C22×C53—42no—
240C22×C53—41no—
241C22×C53—41no—
242C22×C53—41no—
243C22×C53—41no—
244C22×C53—41no—
245C22×C53—81no—
246C22×C53—81no—
247C22×C53C22×C22⋊C411yescharacteristic
248C22×C53C4×#4111yescharacteristic
249C22×C53#52011yescharacteristic
250C22×C53#54812yes—
251C22×C53#55012yes—
252C2×D53—42no—
253C2×D53—42no—
254C2×D53—42no—
255C2×D53—82no—
256C2×D53—82no—
257C4×C53—21no—
258C4×C53—22no—
259C4×C53—41no—
260C4×C53—41no—
261C4×C53—41no—
262C4×C53—84no—
263C4×C53#41012yes—
264Dic53—22no—
265Dic53—22no—
266Dic53—24no—
267Dic53—44no—
268Dic53—42no—
269Dic53—82no—
270C2×#553—531no—
271#2345—532no—
272#1964—534no—
273#1964—532no—
274#1799—532no—
275#1799D5311yescharacteristic, complemented
276C4×#99—532no—
277C4×#99—531no—
278C2×C53—22no—
279C2×C53—22no—
280C2×C53—21no—
281C2×C53—21no—
282C2×C53—81no—
283C2×C53#234111yescharacteristic
284C2×C53C2×#41011yescharacteristic
285C2×C53#288311yescharacteristic
286D53—82no—
287#553—1062no—
288#553—1064no—
289#553—1062no—
290C2×#126—532no—
291C2×#125—531no—
292C2×#123—532no—
293C2×#121—532no—
294C2×#121—531no—
295C2×#121—531no—
296C2×C4×D4—532no—
297C2×C4×D4—531no—
298C2×C4×D4C2×D5311yescharacteristic
299C2×#117—531no—
300C22×C4⋊C4—531no—
301C2×#99—532no—
302C2×#99—534no—
303C2×#99—534no—
304C2×#99C2×D5312yes—
305#389—1062no—
306#387—1064no—
307#380—534no—
308#380C2×D5312yes—
309C4×C4⋊C4—1062no—
310C4×C22⋊C4—532no—
311C4×C22⋊C4—534no—
312#342—531no—
313#342—531no—
314#342C2×D5312yes—
315C53#2253511yescharacteristic, complemented, Sylow
316C22×D4—532no—
317C22×D4—532no—
318C22×D4—531no—
319C22×D4C22×D5311yescharacteristic
320C23×C4—532no—
321C23×C4—532no—
322C23×C4C4×D5311yescharacteristic
323C23×C4C22×D5311yescharacteristic
324#126—1064no—
325#126—1064no—
326#125—1062no—
327#125—1062no—
328#123—1064no—
329#123—1064no—
330#121—1062no—
331#121—1062no—
332#121—1064no—
333#121—1064no—
334#121—1062no—
335#121—1062no—
336C4×D4—22no—
337C4×D4—22no—
338C4×D4—1062no—
339C4×D4—1061no—
340C4×D4—1061no—
341C4×D4—1064no—
342C4×D4—1064no—
343#117—1062no—
344#117—1061no—
345#117—1061no—
346C2×C4⋊C4—22no—
347C2×C4⋊C4—531no—
348C2×C4⋊C4—531no—
349C2×C4⋊C4—532no—
350C2×C4⋊C4—532no—
351C2×C4⋊C4—534no—
352C2×C4⋊C4—532no—
353C2×C4⋊C4—531no—
354C2×C4⋊C4—532no—
355C2×C4⋊C4—532no—
356C2×C4⋊C4—1064no—
357C2×C4⋊C4C4×D5311yescharacteristic
358C2×C22⋊C4—532no—
359C2×C22⋊C4—532no—
360C2×C22⋊C4—532no—
361C2×C22⋊C4—532no—
362C2×C22⋊C4—532no—
363C2×C22⋊C4—532no—
364C2×C22⋊C4C4×D5311yescharacteristic
365C2×C22⋊C4C22×D5314yes—
366C2×C22⋊C4C22×D5311yescharacteristic
367C2×C42—532no—
368C2×C42—1064no—
369C2×C42—1062no—
370C2×C42C4×D5311yescharacteristic
371#99—42no—
372#99—1068no—
373#99—1064no—
374#99—2124no—
375#95—22no—
376#95—1064no—
377#95—1064no—
378#95—1062no—
379C24—532no—
380C24C2×C4×D5311yescharacteristic
381C24C23×D5311yescharacteristic
382C2×D4—22no—
383C2×D4—22no—
384C2×D4—21no—
385C2×D4—21no—
386C2×D4—1064no—
387C2×D4—1064no—
388C2×D4—1064no—
389C2×D4—1062no—
390C2×D4—1062no—
391C2×D4—1061no—
392C2×D4—1061no—
393C2×D4—4242no—
394C2×D4—2122no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—24no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—532no—
401C22×C4—532no—
402C22×C4—534no—
403C22×C4—534no—
404C22×C4—532no—
405C22×C4—1062no—
406C22×C4—1062no—
407C22×C4—1062no—
408C22×C4—1062no—
409C22×C4—4242no—
410C22×C4—4242no—
411C22×C4C2×C4×D5311yescharacteristic
412C22×C4C2×C4×D5311yescharacteristic
413C22×C4C2×C4×D5311yescharacteristic
414C22×C4C2×C4×D5311yescharacteristic
415C22×C4C2×C4×D5311yescharacteristic
416C22×C4D4×D5312yes—
417C22×C4D4×D5312yes—
418C4⋊C4—22no—
419C4⋊C4—44no—
420C4⋊C4—532no—
421C4⋊C4—531no—
422C4⋊C4—531no—
423C4⋊C4—532no—
424C4⋊C4—532no—
425C4⋊C4—534no—
426C4⋊C4—534no—
427C4⋊C4—1064no—
428C4⋊C4—1064no—
429C4⋊C4—1061no—
430C4⋊C4—1061no—
431C4⋊C4—4244no—
432C22⋊C4—22no—
433C22⋊C4—22no—
434C22⋊C4—24no—
435C22⋊C4—24no—
436C22⋊C4—534no—
437C22⋊C4—1062no—
438C22⋊C4—1062no—
439C22⋊C4—1064no—
440C22⋊C4—1062no—
441C22⋊C4—1062no—
442C22⋊C4—1062no—
443C22⋊C4—1064no—
444C22⋊C4—1062no—
445C22⋊C4—1062no—
446C42—21no—
447C42—21no—
448C42—1062no—
449C42—1062no—
450C42—4244no—
451C42—2124no—
452C23—21no—
453C23—21no—
454C23—21no—
455C23—41no—
456C23—41no—
457C23—41no—
458C23—41no—
459C23—1062no—
460C23—1062no—
461C23—1062no—
462C23—2122no—
463C23—2122no—
464C23C22×C4×D5311yescharacteristic
465C23C2×D4×D5312yes—
466C23#41×D5312yes—
467C23#1313326211yescharacteristic
468C23#1313326511yescharacteristic
469D4—42no—
470D4—42no—
471D4—4244no—
472D4—2122no—
473D4—2124no—
474D4—2124no—
475D4—2122no—
476C2×C4—24no—
477C2×C4—22no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—41no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—44no—
490C2×C4—44no—
491C2×C4—44no—
492C2×C4—532no—
493C2×C4—532no—
494C2×C4—534no—
495C2×C4—1064no—
496C2×C4—1062no—
497C2×C4—1062no—
498C2×C4—1062no—
499C2×C4—4242no—
500C2×C4—4242no—
501C2×C4—2124no—
502C2×C4—2124no—
503C2×C4—2122no—
504C2×C4—2122no—
505C2×C4—2122no—
506C2×C4—2122no—
507C2×C4—2122no—
508C2×C4—2122no—
509C2×C4—2122no—
510C2×C4C22⋊C4×D5311yescharacteristic
511C2×C4C22⋊C4×D5312yes—
512C2×C4C22⋊C4×D5311yescharacteristic
513C22—22no—
514C22—21no—
515C22—22no—
516C22—21no—
517C22—42no—
518C22—41no—
519C22—41no—
520C22—41no—
521C22—41no—
522C22—41no—
523C22—81no—
524C22—81no—
525C22—4242no—
526C22—4242no—
527C22—2122no—
528C22—2122no—
529C22—2122no—
530C22C2×C22⋊C4×D5311yescharacteristic
531C22#1681109111yescharacteristic
532C22#1681110411yescharacteristic
533C22#1681128312yes—
534C22#1681129112yes—
535C4—21no—
536C4—22no—
537C4—41no—
538C4—41no—
539C4—41no—
540C4—84no—
541C4—1062no—
542C4—1062no—
543C4—1064no—
544C4—4242no—
545C4—2124no—
546C4—2122no—
547C4#1681059512yes—
548C2—22no—
549C2—22no—
550C2—21no—
551C2—21no—
552C2—81no—
553C2—4242no—
554C2#2027755211yescharacteristic
555C2C2×#1681059511yescharacteristic
556C2#2028317411yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D53
commutatorC23×C53C23×C4
FrattiniC24C23×D53
FittingC53×#1799C2
socleC22×C53C22×C22⋊C4
prime radicalC2×#16811319C2
cosocleC24×C53C24
hypercenter#1799D53
hypocenterC53#22535
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1799

Complements

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

NH ≅ G/Ntype
2. #20277594C2semidirect
3. #20275550C2semidirect
4. #20275549C2semidirect
5. #20274702C2semidirect
6. C4×#13133132C2semidirect
7. #20274502C2semidirect
8. C53×#1799C2semidirect
12. C2×#13133212C4semidirect
13. C2×#13133209C4semidirect
14. C2×#13133199C4semidirect
15. C2×#13133191C4semidirect
18. C2×C4×D212C4semidirect
20. C2×#13133176C4semidirect
275. #1799D53semidirect
315. C53#22535semidirect

Maximal subgroups

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

subgroupindex
C2×#168113192
#202775942
#202755502
#202755492
#202747022
C4×#131331322
#202745022
C53×#17992
#2253553