Finite Group Databasebeta

← #27276845

Subgroups — #27276845

Order 13568.

742 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20274695C211yesmaximal, characteristic, complemented
2C2×#16810357C212yesmaximal
3#20272945C211yesmaximal, characteristic, complemented
4#20272593C212yesmaximal, complemented
5C4×#121×C53C211yesmaximal, characteristic
6C22×C4⋊Dic53—21no—
7C2×C4⋊#88322—21no—
8C2×C22⋊#88322C2212yes—
9C2×#13133124—21no—
10C2×#13133124C2212yes—
11#16810378—21no—
12#16810357—24no—
13C2×#121×C53C2211yescharacteristic
14C2×C4×D4×C53—21no—
15C2×C4×D4×C53C411yescharacteristic, complemented
16C22×C42×C53C2211yescharacteristic
17C53×#390C411yescharacteristic, complemented
18C53×#389C411yescharacteristic, complemented
19C53×#388—21no—
20C53×#386—21no—
21C53×#380C2211yescharacteristic
22C4×C4⋊C4×C53C411yescharacteristic, complemented
23C4×C22⋊C4×C53—21no—
24C23×Dic53—22no—
25C22×#88322—22no—
26C2×C4⋊Dic53—24no—
27C2×C4⋊Dic53—42no—
28C2×C4⋊Dic53—42no—
29C4⋊#88322—42no—
30C22⋊#88322—22no—
31C22⋊#88322—22no—
32#13133124—22no—
33#13133124—22no—
34#13133124—42no—
35C22×D4×C53—21no—
36C22×D4×C53C2×C411yescharacteristic
37C23×C4×C53—21no—
38C23×C4×C53D411yescharacteristic
39C23×C4×C53D411yescharacteristic
40C23×C4×C53C2311yescharacteristic
41#121×C53—21no—
42#121×C53—21no—
43#121×C53D412yes—
44#121×C53Q812yes—
45C4×D4×C53—42no—
46C4×D4×C53—42no—
47C4×D4×C53—42no—
48C2×C4⋊C4×C53—21no—
49C2×C4⋊C4×C53C2×C411yescharacteristic
50C2×C4⋊C4×C53C2×C411yescharacteristic
51C2×C4⋊C4×C53C2×C411yescharacteristic
52C2×C22⋊C4×C53—21no—
53C2×C22⋊C4×C53—21no—
54C2×C22⋊C4×C53—21no—
55C2×C22⋊C4×C53—21no—
56C2×C22⋊C4×C53—21no—
57C2×C42×C53—21no—
58C2×C42×C53—42no—
59C2×C42×C53C2×C411yescharacteristic
60C2×C42×C53C2×C411yescharacteristic
61C2×C42×C53D411yescharacteristic
62C2×C42×C53D411yescharacteristic
63#95×C53—21no—
64#95×C53—21no—
65#95×C53—21no—
66C24×C53—21no—
67C24×C53C2×D411yescharacteristic
68C2×D4×C53—22no—
69C2×D4×C53—22no—
70C2×D4×C53—21no—
71C2×D4×C53—41no—
72C2×D4×C53—41no—
73C2×D4×C53—41no—
74C2×D4×C53—41no—
75C2×D4×C53—41no—
76C2×D4×C53—41no—
77C2×D4×C53C4⋊C412yescomplemented
78C22×C4×C53—21no—
79C22×C4×C53—21no—
80C22×C4×C53—22no—
81C22×C4×C53—22no—
82C22×C4×C53—21no—
83C22×C4×C53—21no—
84C22×C4×C53—21no—
85C22×C4×C53—41no—
86C22×C4×C53—41no—
87C22×C4×C53—41no—
88C22×C4×C53—41no—
89C22×C4×C53—42no—
90C22×C4×C53—42no—
91C22×C4×C53—42no—
92C22×C4×C53C22⋊C411yescharacteristic
93C22×C4×C53C22⋊C411yescharacteristic
94C22×C4×C53C22⋊C411yescharacteristic
95C22×C4×C53C22⋊C411yescharacteristic
96C22×C4×C53D812yes—
97C22×C4×C53SD1612yes—
98C22×C4×C53C22×C411yescharacteristic
99C22×C4×C53C2×D411yescharacteristic
100C22×C4×C53C2×D411yescharacteristic
101C22×C4×C53C2×Q811yescharacteristic
102C22×C4×C53#4111yescharacteristic
103C22×C4×C53#4111yescharacteristic
104C22×Dic53—24no—
105C22×Dic53—42no—
106C22×Dic53—42no—
107C22×Dic53—42no—
108C22×Dic53—42no—
109C22×Dic53—42no—
110C22×Dic53—42no—
111C4⋊C4×C53—21no—
112C4⋊C4×C53—41no—
113C4⋊C4×C53—41no—
114C4⋊C4×C53—42no—
115C4⋊C4×C53C4⋊C412yescomplemented
116C22⋊C4×C53—22no—
117C22⋊C4×C53—22no—
118C22⋊C4×C53—41no—
119C22⋊C4×C53—41no—
120C22⋊C4×C53—42no—
121C22⋊C4×C53—42no—
122C22⋊C4×C53—42no—
123C42×C53—22no—
124C42×C53—22no—
125C42×C53—41no—
126C42×C53—41no—
127C42×C53—42no—
128C4⋊Dic53—48no—
129C2×#88322—42no—
130C2×#88322—42no—
131C2×#88322—42no—
132C23×C53—21no—
133C23×C53—21no—
134C23×C53—21no—
135C23×C53—21no—
136C23×C53—22no—
137C23×C53—41no—
138C23×C53—41no—
139C23×C53—41no—
140C23×C53—41no—
141C23×C53—41no—
142C23×C53—41no—
143C23×C53C2×C22⋊C411yescharacteristic
144C23×C53#12111yescharacteristic
145C23×C53C22⋊Q811yescharacteristic
146C23×C53C2×D811yescharacteristic
147C23×C53C2×SD1611yescharacteristic
148C23×C53AGL(1,ℤ/8)11yescharacteristic
149C23×C53#13711yescharacteristic
150D4×C53—42no—
151D4×C53—42no—
152D4×C53—42no—
153D4×C53—42no—
154D4×C53—42no—
155D4×C53—42no—
156C2×C4×C53—24no—
157C2×C4×C53—24no—
158C2×C4×C53—21no—
159C2×C4×C53—21no—
160C2×C4×C53—21no—
161C2×C4×C53—21no—
162C2×C4×C53—21no—
163C2×C4×C53—21no—
164C2×C4×C53—21no—
165C2×C4×C53—21no—
166C2×C4×C53—21no—
167C2×C4×C53—21no—
168C2×C4×C53—21no—
169C2×C4×C53—21no—
170C2×C4×C53—22no—
171C2×C4×C53—42no—
172C2×C4×C53—42no—
173C2×C4×C53—41no—
174C2×C4×C53—41no—
175C2×C4×C53—41no—
176C2×C4×C53—41no—
177C2×C4×C53—41no—
178C2×C4×C53—41no—
179C2×C4×C53—41no—
180C2×C4×C53—41no—
181C2×C4×C53—41no—
182C2×C4×C53—41no—
183C2×C4×C53—41no—
184C2×C4×C53—41no—
185C2×C4×C53—84no—
186C2×C4×C53#9912yes—
187C2×C4×C53#10212yes—
188C2×C4×C53#10212yes—
189C2×C4×C53#10412yes—
190C2×C4×C53C2×C4⋊C411yescharacteristic
191C2×C4×C53#11711yescharacteristic
192C2×C4×C53#12111yescharacteristic
193C2×C4×C53C22⋊Q811yescharacteristic
194C2×Dic53—44no—
195C2×Dic53—44no—
196C2×Dic53—44no—
197C2×Dic53—82no—
198C2×Dic53—82no—
199C2×Dic53—82no—
200C2×Dic53—82no—
201#88322—82no—
202#8577—531nomaximal, Sylow
203C22×C53—21no—
204C22×C53—21no—
205C22×C53—21no—
206C22×C53—21no—
207C22×C53—21no—
208C22×C53—21no—
209C22×C53—21no—
210C22×C53—21no—
211C22×C53—21no—
212C22×C53—21no—
213C22×C53—21no—
214C22×C53—21no—
215C22×C53—42no—
216C22×C53—42no—
217C22×C53—42no—
218C22×C53—81no—
219C22×C53—81no—
220C22×C53—81no—
221C22×C53—81no—
222C22×C53#38011yescharacteristic
223C22×C53C2×#9911yescharacteristic
224C22×C53#41011yescharacteristic
225C22×C53C2×#10211yescharacteristic
226C22×C53#41711yescharacteristic
227C22×C53C2×#10411yescharacteristic
228C22×C53#42111yescharacteristic
229C22×C53#45911yescharacteristic
230C22×C53#46111yescharacteristic
231C22×C53#47411yescharacteristic
232C22×C53#47611yescharacteristic
233C4×C53—21no—
234C4×C53—21no—
235C4×C53—21no—
236C4×C53—21no—
237C4×C53—22no—
238C4×C53—22no—
239C4×C53—22no—
240C4×C53—22no—
241C4×C53—42no—
242C4×C53—81no—
243C4×C53—81no—
244Dic53—84no—
245C4×#121D5311yescharacteristic
246#1797—531no—
247C2×#327—532no—
248C4⋊#95—531no—
249#1294—532no—
250C2×C53—21no—
251C2×C53—21no—
252C2×C53—21no—
253C2×C53—21no—
254C2×C53—82no—
255C2×C53#150211yescharacteristic
256C2×C53#150411yescharacteristic
257C2×C53#151211yescharacteristic
258C2×C53#151911yescharacteristic
259C2×C53#179911yescharacteristic
260C2×C53#180211yescharacteristic
261C2×C53#180611yescharacteristic
262C2×#121C2×D5311yescharacteristic
263C2×C4×D4—21no—
264C2×C4×D4Dic5311yescharacteristic, complemented
265C22×C4⋊C4—1061no—
266C22×C42C2×D5311yescharacteristic
267C2×C4⋊C8—1061no—
268C2×C22⋊C8—532no—
269#390Dic5311yescharacteristic, complemented
270#389Dic5311yescharacteristic, complemented
271#388—21no—
272#386—21no—
273#380C2×D5311yescharacteristic
274C4×C4⋊C4Dic5311yescharacteristic, complemented
275C4×C22⋊C4—21no—
276C2×#95—532no—
277C2×#95—1061no—
278#336—1061no—
279#327—1064no—
280C53#857711yescharacteristic, complemented, Sylow
281C22×D4—21no—
282C22×D4C2×Dic5311yescharacteristic
283C23×C4—21no—
284C23×C4—1062no—
285C23×C4C22⋊D5311yescharacteristic
286C23×C4C22⋊D5311yescharacteristic
287C23×C4C22×D5311yescharacteristic
288C22×C8—1062no—
289#121—21no—
290#121—21no—
291#121Dic10612yes—
292#121D21212yes—
293C4×D4—42no—
294C4×D4—42no—
295C4×D4—42no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—1064no—
298C2×C4⋊C4—2122no—
299C2×C4⋊C4—2122no—
300C2×C4⋊C4C2×Dic5311yescharacteristic
301C2×C4⋊C4C2×Dic5311yescharacteristic
302C2×C4⋊C4C2×Dic5311yescharacteristic
303C2×C22⋊C4—21no—
304C2×C22⋊C4—21no—
305C2×C22⋊C4—21no—
306C2×C22⋊C4—21no—
307C2×C22⋊C4—21no—
308C2×C42—21no—
309C2×C42—42no—
310C2×C42C2×Dic5311yescharacteristic
311C2×C42C2×Dic5311yescharacteristic
312C2×C42C22⋊D5311yescharacteristic
313C2×C42C22⋊D5311yescharacteristic
314C4⋊C8—2122no—
315C22⋊C8—1062no—
316C22⋊C8—1062no—
317#95—21no—
318#95—21no—
319#95—21no—
320#95—1062no—
321#95—1062no—
322#95—2122no—
323C24—21no—
324C24C2×C22⋊D5311yescharacteristic
325C2×D4—22no—
326C2×D4—22no—
327C2×D4—21no—
328C2×D4—41no—
329C2×D4—41no—
330C2×D4—41no—
331C2×D4—41no—
332C2×D4—41no—
333C2×D4—41no—
334C2×D4C4⋊Dic5312yescomplemented
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—1064no—
350C22×C4—2122no—
351C22×C4—2122no—
352C22×C4—2122no—
353C22×C4—2122no—
354C22×C4—2122no—
355C22×C4—2122no—
356C22×C4#1171992212yes—
357C22×C4#1171992312yes—
358C22×C4C22⋊Dic5311yescharacteristic
359C22×C4C22⋊Dic5311yescharacteristic
360C22×C4C22⋊Dic5311yescharacteristic
361C22×C4C22⋊Dic5311yescharacteristic
362C22×C4C2×Dic10611yescharacteristic
363C22×C4C2×D21211yescharacteristic
364C22×C4#1171994511yescharacteristic
365C22×C4#1171994511yescharacteristic
366C22×C4C22×Dic5311yescharacteristic
367C22×C4C2×C22⋊D5311yescharacteristic
368C2×C8—2122no—
369C2×C8—2122no—
370C2×C8—2122no—
371C4⋊C4—21no—
372C4⋊C4—41no—
373C4⋊C4—41no—
374C4⋊C4—42no—
375C4⋊C4—2128no—
376C4⋊C4C4⋊Dic5312yescomplemented
377C22⋊C4—22no—
378C22⋊C4—22no—
379C22⋊C4—41no—
380C22⋊C4—41no—
381C22⋊C4—42no—
382C22⋊C4—42no—
383C22⋊C4—42no—
384C42—22no—
385C42—22no—
386C42—41no—
387C42—41no—
388C42—42no—
389C23—21no—
390C23—21no—
391C23—21no—
392C23—21no—
393C23—22no—
394C23—41no—
395C23—41no—
396C23—41no—
397C23—41no—
398C23—41no—
399C23—41no—
400C23C2×C22⋊Dic5311yescharacteristic
401C23#1313319111yescharacteristic
402C23#1313319611yescharacteristic
403C23C2×#1171992211yescharacteristic
404C23C2×#1171992311yescharacteristic
405C23#1313324211yescharacteristic
406C23#1313324711yescharacteristic
407D4—42no—
408D4—42no—
409D4—42no—
410D4—42no—
411D4—42no—
412D4—42no—
413C2×C4—24no—
414C2×C4—24no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—22no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—84no—
443C2×C4—4242no—
444C2×C4—4242no—
445C2×C4—4242no—
446C2×C4—4242no—
447C2×C4—2124no—
448C2×C4—2124no—
449C2×C4—2124no—
450C2×C4#1313313212yes—
451C2×C4#1313313812yes—
452C2×C4#1313313812yes—
453C2×C4#1313314412yes—
454C2×C4C2×C4⋊Dic5311yescharacteristic
455C2×C4#1313317611yescharacteristic
456C2×C4#1313319111yescharacteristic
457C2×C4#1313319611yescharacteristic
458C8—4242no—
459C22—21no—
460C22—21no—
461C22—21no—
462C22—21no—
463C22—21no—
464C22—21no—
465C22—21no—
466C22—21no—
467C22—21no—
468C22—21no—
469C22—21no—
470C22—21no—
471C22—42no—
472C22—42no—
473C22—42no—
474C22—81no—
475C22—81no—
476C22—81no—
477C22—81no—
478C22#1681048311yescharacteristic
479C22C2×#1313313211yescharacteristic
480C22#1681059711yescharacteristic
481C22C2×#1313313811yescharacteristic
482C22#1681062411yescharacteristic
483C22C2×#1313314411yescharacteristic
484C22#1681064011yescharacteristic
485C22#1681081811yescharacteristic
486C22#1681082811yescharacteristic
487C22#1681090311yescharacteristic
488C22#1681091311yescharacteristic
489C4—21no—
490C4—21no—
491C4—21no—
492C4—21no—
493C4—22no—
494C4—22no—
495C4—22no—
496C4—22no—
497C4—42no—
498C4—81no—
499C4—81no—
500C4—4244no—
501C2—21no—
502C2—21no—
503C2—21no—
504C2—21no—
505C2—82no—
506C2#2027321011yescharacteristic
507C2#2027321911yescharacteristic
508C2#2027325311yescharacteristic
509C2#2027328311yescharacteristic
510C2#2027470311yescharacteristic
511C2#2027471511yescharacteristic
512C2#2027473511yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic53
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC4×#121×C53C2
socleC23×C53C2×C22⋊C4
prime radicalC2×#121×C53C22
cosocleC23×C4×C53C23
hypercenterC4×#121D53
hypocenterC53#8577
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C22×C42 ◁ C4×#121

Complements

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

NH ≅ G/Ntype
1. #20274695C2semidirect
3. #20272945C2semidirect
4. #20272593C2semidirect
15. C2×C4×D4×C53C4semidirect
17. C53×#390C4semidirect
18. C53×#389C4semidirect
22. C4×C4⋊C4×C53C4semidirect
77. C2×D4×C53C4⋊C4semidirect
115. C4⋊C4×C53C4⋊C4semidirect
264. C2×C4×D4Dic53semidirect
269. #390Dic53semidirect
270. #389Dic53semidirect
274. C4×C4⋊C4Dic53semidirect
280. C53#8577semidirect
334. C2×D4C4⋊Dic53semidirect
376. C4⋊C4C4⋊Dic53semidirect

Maximal subgroups

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

subgroupindex
#202746952
C2×#168103572
#202729452
#202725932
C4×#121×C532
#857753