Finite Group Databasebeta

← #27469824

Subgroups — #27469824

Order 13568.

912 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20283014C211yesmaximal, characteristic
2C2×#16810492C211yesmaximal, characteristic, complemented
3#20276127C212yesmaximal, complemented
4#20276121C212yesmaximal, complemented
5#20275530C212yesmaximal, complemented
6#20275033C211yesmaximal, characteristic
7#20275022C211yesmaximal, characteristic, complemented
8#20275020C211yesmaximal, characteristic, complemented
9#20274847C212yesmaximal, complemented
10#20274479C211yesmaximal, characteristic, complemented
11C53×#1874C211yesmaximal, characteristic
12C2×#13133204C2211yescharacteristic
13C2×#13133196C2211yescharacteristic
14C22×C4⋊Dic53C2211yescharacteristic
15C22×#11719919—21no—
16C22×#11719919—21no—
17C22×C4×Dic53C2211yescharacteristic
18C2×#13133142C2212yes—
19C2×#13133140C2212yes—
20C2×#13133138C2212yes—
21C2×C22⋊#88322C2211yescharacteristic
22C2×#13133127C2212yes—
23#16810528C2211yescharacteristic
24#16810526C2211yescharacteristic
25#16810517—21no—
26#16810517—21no—
27#16810503—21no—
28#16810503—21no—
29#16810492—21no—
30#16810492—21no—
31#16810492C2212yes—
32#16810492C2212yes—
33#16810488C2211yescharacteristic
34#16810386C2212yes—
35#16810385C2212yes—
36#16810385C2212yes—
37#16810377C2212yes—
38C2×#121×C53C2211yescharacteristic
39C22×C4⋊C4×C53C2211yescharacteristic
40C2×#102×C53C2212yes—
41C2×C22⋊C8×C53C2211yescharacteristic
42C53×#340C2212yes—
43C22×Dic106C2311yescharacteristic
44C23×Dic53—21no—
45C23×Dic53D411yescharacteristic
46C23×Dic53D411yescharacteristic
47C22×#88322C2312yes—
48#13133204—24no—
49#13133196—22no—
50#13133196—22no—
51C2×C4⋊Dic53—21no—
52C2×C4⋊Dic53—42no—
53C2×C4⋊Dic53C2311yescharacteristic
54C2×C4⋊Dic53C2312yes—
55C2×#11719919—21no—
56C2×#11719919—22no—
57C2×#11719919—22no—
58C2×#11719919—22no—
59C2×#11719919—22no—
60C2×#11719919—44no—
61C2×#11719919—44no—
62C2×C22⋊Dic53—21no—
63C2×C22⋊Dic53—21no—
64C2×C22⋊Dic53—21no—
65C2×C22⋊Dic53—21no—
66C2×C22⋊Dic53—21no—
67C2×C4×Dic53—24no—
68C2×C4×Dic53D411yescharacteristic
69C2×C4×Dic53D411yescharacteristic
70C2×C4×Dic53C2311yescharacteristic
71C2×C4×Dic53C2311yescharacteristic
72#13133142—42no—
73#13133140—42no—
74#13133138—42no—
75C22⋊#88322—22no—
76#13133127—24no—
77#13133124—21no—
78#13133124—21no—
79#13133124—21no—
80#13133124—21no—
81C22×D4×C53—21no—
82C22×D4×C53C2311yescharacteristic
83C23×C4×C53—21no—
84C23×C4×C53C2311yescharacteristic
85C22×C8×C53C2312yes—
86#121×C53—22no—
87#121×C53—22no—
88C2×C4⋊C4×C53—21no—
89C2×C4⋊C4×C53—42no—
90C2×C4⋊C4×C53C2312yes—
91C2×C4⋊C4×C53C2311yescharacteristic
92C2×C22⋊C4×C53—21no—
93#102×C53—42no—
94C22⋊C8×C53—22no—
95C24×C53—21no—
96C24×C53C2×D411yescharacteristic
97C2×D4×C53—21no—
98C2×D4×C53—21no—
99C2×D4×C53—22no—
100C2×D4×C53—22no—
101C2×D4×C53—42no—
102C2×D4×C53—42no—
103C2×D4×C53—42no—
104C22×C4×C53—22no—
105C22×C4×C53—22no—
106C22×C4×C53—21no—
107C22×C4×C53—22no—
108C22×C4×C53—42no—
109C22×C4×C53—42no—
110C22×C4×C53—42no—
111C22×C4×C53C2×D411yescharacteristic
112C22×C4×C53#4112yes—
113C22×C4×C53#4112yes—
114C22×C4×C53C2411yescharacteristic
115C22×Dic53—22no—
116C22×Dic53—22no—
117C22×Dic53—22no—
118C22×Dic53—22no—
119C22×Dic53—22no—
120C22×Dic53—22no—
121C22×Dic53—21no—
122C22×Dic53—21no—
123C22×Dic53—22no—
124C22×Dic53—42no—
125C22×Dic53—42no—
126C22×Dic53—42no—
127C22×Dic53C2×D411yescharacteristic
128C22×Dic53C2×D411yescharacteristic
129C22×Dic53C2×D411yescharacteristic
130C22×Dic53C2×D411yescharacteristic
131C2×Dic106—21no—
132C2×Dic106—21no—
133C2×Dic106—22no—
134C2×Dic106—22no—
135C2×C8×C53—22no—
136C2×C8×C53—22no—
137C2×C8×C53—22no—
138C4⋊C4×C53—21no—
139C4⋊C4×C53—21no—
140C4⋊C4×C53—22no—
141C4⋊C4×C53—22no—
142C4⋊C4×C53—42no—
143C22⋊C4×C53—42no—
144C22⋊Dic53—42no—
145C22⋊Dic53—42no—
146C22⋊Dic53—42no—
147C22⋊Dic53—42no—
148C22⋊Dic53—42no—
149C4⋊Dic53—21no—
150C4⋊Dic53—21no—
151C4⋊Dic53—22no—
152C4⋊Dic53—22no—
153C4⋊Dic53—42no—
154#11719919—44no—
155#11719919—44no—
156#11719919—44no—
157#11719919—42no—
158#11719919—44no—
159C4×Dic53—22no—
160C4×Dic53—22no—
161C4×Dic53—22no—
162C4×Dic53#4114yes—
163C2×#88322—22no—
164C2×#88322—22no—
165C2×#88322—22no—
166C23×C53—21no—
167C23×C53—21no—
168C23×C53—21no—
169C23×C53—21no—
170C23×C53—41no—
171C23×C53—41no—
172C23×C53—41no—
173C23×C53—41no—
174C23×C53—41no—
175C23×C53—41no—
176C23×C53—41no—
177C23×C53#12412yes—
178C23×C53AGL(1,ℤ/8)12yes—
179C23×C53#13712yes—
180C23×C53C22×D411yescharacteristic
181D4×C53—42no—
182D4×C53—42no—
183D4×C53—84no—
184C2×C4×C53—21no—
185C2×C4×C53—21no—
186C2×C4×C53—24no—
187C2×C4×C53—24no—
188C2×C4×C53—41no—
189C2×C4×C53—42no—
190C2×C4×C53—42no—
191C2×C4×C53—42no—
192C2×C4×C53—41no—
193C2×C4×C53—41no—
194C2×C4×C53—84no—
195C2×C4×C53#12412yes—
196C2×C4×C53C2×#4112yes—
197C2×C4×C53C2×#4111yescharacteristic
198C2×C4×C53C2×#4111yescharacteristic
199C2×C4×C53H(5,𝔽2)12yes—
200C2×Dic53—28no—
201C2×Dic53—28no—
202C2×Dic53—21no—
203C2×Dic53—21no—
204C2×Dic53—21no—
205C2×Dic53—21no—
206C2×Dic53—21no—
207C2×Dic53—21no—
208C2×Dic53—21no—
209C2×Dic53—21no—
210C2×Dic53—41no—
211C2×Dic53—41no—
212C2×Dic53—41no—
213C2×Dic53—41no—
214C2×Dic53—42no—
215C2×Dic53—42no—
216C2×Dic53—42no—
217C2×Dic53—41no—
218C2×Dic53—41no—
219C2×Dic53—84no—
220C2×Dic53#12312yes—
221C2×Dic53#12312yes—
222C2×Dic53AGL(1,ℤ/8)12yes—
223C2×Dic53#13712yes—
224Dic106—42no—
225Dic106—42no—
226C8×C53—42no—
227#88322—42no—
228#27978—531nomaximal, Sylow
229C22×C53—22no—
230C22×C53—22no—
231C22×C53—22no—
232C22×C53—22no—
233C22×C53—22no—
234C22×C53—22no—
235C22×C53—81no—
236C22×C53—81no—
237C22×C53—81no—
238C22×C53—81no—
239C22×C53—81no—
240C22×C53—81no—
241C22×C53—81no—
242C22×C53#48914yes—
243C22×C53C2×#12311yescharacteristic
244C22×C53C2×#12411yescharacteristic
245C22×C53#53511yescharacteristic
246C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
247C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
248C22×C53C2×#13711yescharacteristic
249C22×C53C2×#13711yescharacteristic
250C4×C53—22no—
251C4×C53—81no—
252C4×C53—82no—
253C4×C53#55312yes—
254C4×C53#55312yes—
255Dic53—22no—
256Dic53—22no—
257Dic53—22no—
258Dic53—22no—
259Dic53—81no—
260Dic53—82no—
261Dic53—81no—
262#2850—531no—
263C2×#384—531no—
264#2072—532no—
265#2071—532no—
266#1959—532no—
267#1874—531no—
268#1874D5311yescharacteristic
269#1872—531no—
270#1872—531no—
271#1836—532no—
272#1749—531no—
273C2×C53—24no—
274C2×C53—161no—
275C2×C53#250411yescharacteristic
276C2×C53#309211yescharacteristic
277C2×C53#309211yescharacteristic
278C2×C53C2×#48911yescharacteristic
279C2×C53#314111yescharacteristic
280C2×C53#321711yescharacteristic
281C2×C53#321811yescharacteristic
282C2×#124—531no—
283C2×C22⋊Q8—531no—
284C2×#121C2×D5311yescharacteristic
285C22×C4⋊C4—531no—
286C22×C4⋊C4—1061no—
287C22×C4⋊C4—1061no—
288C22×C4⋊C4C2×D5311yescharacteristic
289C22×C42—531no—
290C2×#103—532no—
291C2×#103—532no—
292C2×#102—532no—
293C2×#102C2×D5312yes—
294C2×C22⋊C8—531no—
295C2×C22⋊C8C2×D5311yescharacteristic
296C2×#97—532no—
297#391—531no—
298#390—531no—
299#388—1061no—
300#388—1061no—
301#386—1061no—
302#386—1061no—
303#384—532no—
304#384—532no—
305#384—1061no—
306#384—1061no—
307#382—531no—
308#340—532no—
309#340—532no—
310#340—532no—
311#340C2×D5312yes—
312#336—532no—
313C53#2797811yescharacteristic, complemented, Sylow
314C22×Q8—531no—
315C22×D4—21no—
316C22×D4C22×D5311yescharacteristic
317C23×C4—21no—
318C23×C4—531no—
319C23×C4—531no—
320C23×C4—1061no—
321C23×C4C22×D5311yescharacteristic
322C22×C8—532no—
323C22×C8C22×D5312yes—
324#124—1064no—
325C22⋊Q8—1062no—
326C22⋊Q8—1062no—
327#121—22no—
328#121—22no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—42no—
331C2×C4⋊C4—532no—
332C2×C4⋊C4—531no—
333C2×C4⋊C4—1062no—
334C2×C4⋊C4—1062no—
335C2×C4⋊C4—1062no—
336C2×C4⋊C4—1062no—
337C2×C4⋊C4—1061no—
338C2×C4⋊C4—1061no—
339C2×C4⋊C4—2122no—
340C2×C4⋊C4—2124no—
341C2×C4⋊C4—2124no—
342C2×C4⋊C4C22×D5312yes—
343C2×C4⋊C4C22×D5311yescharacteristic
344C2×C22⋊C4—21no—
345C2×C22⋊C4—1061no—
346C2×C22⋊C4—1061no—
347C2×C22⋊C4—1061no—
348C2×C22⋊C4—1061no—
349C2×C22⋊C4—1061no—
350C2×C42—531no—
351C2×C42—531no—
352C2×C42—531no—
353C2×C42—531no—
354C2×C42—1064no—
355#103—2122no—
356#103—2122no—
357#102—42no—
358#102—2122no—
359C22⋊C8—22no—
360C22⋊C8—1062no—
361#97—1064no—
362#95—1061no—
363#95—1061no—
364#95—1061no—
365#95—1061no—
366C24—21no—
367C24D4×D5311yescharacteristic
368C2×Q8—1062no—
369C2×Q8—1061no—
370C2×Q8—1061no—
371C2×Q8—1062no—
372C2×D4—22no—
373C2×D4—21no—
374C2×D4—21no—
375C2×D4—22no—
376C2×D4—42no—
377C2×D4—42no—
378C2×D4—42no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—21no—
382C22×C4—22no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—531no—
387C22×C4—531no—
388C22×C4—531no—
389C22×C4—531no—
390C22×C4—1062no—
391C22×C4—1062no—
392C22×C4—1062no—
393C22×C4—1062no—
394C22×C4—1062no—
395C22×C4—1062no—
396C22×C4—1061no—
397C22×C4—1061no—
398C22×C4—1062no—
399C22×C4—2122no—
400C22×C4—2122no—
401C22×C4—2122no—
402C22×C4#1171994512yes—
403C22×C4D4×D5311yescharacteristic
404C22×C4#1171994712yes—
405C22×C4C23×D5311yescharacteristic
406C2×C8—22no—
407C2×C8—22no—
408C2×C8—22no—
409C2×C8—1062no—
410C2×C8—1062no—
411C2×C8—1062no—
412C4⋊C4—21no—
413C4⋊C4—21no—
414C4⋊C4—22no—
415C4⋊C4—22no—
416C4⋊C4—42no—
417C4⋊C4—1061no—
418C4⋊C4—1061no—
419C4⋊C4—1062no—
420C4⋊C4—1062no—
421C4⋊C4—2122no—
422C4⋊C4—2124no—
423C4⋊C4—2124no—
424C4⋊C4—2124no—
425C4⋊C4—2124no—
426C4⋊C4—2122no—
427C22⋊C4—42no—
428C22⋊C4—2122no—
429C22⋊C4—2122no—
430C22⋊C4—2122no—
431C22⋊C4—2122no—
432C22⋊C4—2122no—
433C42—534no—
434C42—1062no—
435C42—1062no—
436C42—1062no—
437C23—21no—
438C23—21no—
439C23—21no—
440C23—21no—
441C23—41no—
442C23—41no—
443C23—41no—
444C23—41no—
445C23—41no—
446C23—41no—
447C23—41no—
448C23#1313320312yes—
449C23#1313324112yes—
450C23#1313324612yes—
451C23C2×D4×D5311yescharacteristic
452Q8—2122no—
453Q8—2122no—
454D4—42no—
455D4—42no—
456D4—84no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—24no—
460C2×C4—24no—
461C2×C4—41no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—42no—
467C2×C4—84no—
468C2×C4—532no—
469C2×C4—532no—
470C2×C4—532no—
471C2×C4—532no—
472C2×C4—1068no—
473C2×C4—1068no—
474C2×C4—1061no—
475C2×C4—1061no—
476C2×C4—1061no—
477C2×C4—1061no—
478C2×C4—1061no—
479C2×C4—1061no—
480C2×C4—1061no—
481C2×C4—1061no—
482C2×C4—4244no—
483C2×C4—2121no—
484C2×C4—2121no—
485C2×C4—2121no—
486C2×C4—2121no—
487C2×C4—2122no—
488C2×C4—2122no—
489C2×C4—2121no—
490C2×C4—2122no—
491C2×C4—2121no—
492C2×C4#1313320312yes—
493C2×C4#41×D5312yes—
494C2×C4C2×#1171994711yescharacteristic
495C2×C4C2×#1171994511yescharacteristic
496C2×C4#1313326212yes—
497C8—42no—
498C8—2122no—
499C22—22no—
500C22—22no—
501C22—22no—
502C22—22no—
503C22—22no—
504C22—22no—
505C22—81no—
506C22—81no—
507C22—81no—
508C22—81no—
509C22—81no—
510C22—81no—
511C22—81no—
512C22#1681098114yes—
513C22#123×D5311yescharacteristic
514C22C2×#1313320311yescharacteristic
515C22#1681118811yescharacteristic
516C22AGL(1,ℤ/8)×D5311yescharacteristic
517C22C2×#1313324111yescharacteristic
518C22#137×D5311yescharacteristic
519C22C2×#1313324611yescharacteristic
520C4—22no—
521C4—82no—
522C4—81no—
523C4—1062no—
524C4—1062no—
525C4—1062no—
526C4—1062no—
527C4—4242no—
528C4—4241no—
529C4—4241no—
530C4#1681131612yes—
531C4#1681132212yes—
532C2—24no—
533C2—161no—
534C2#2027920511yescharacteristic
535C2#2028463411yescharacteristic
536C2#2028463711yescharacteristic
537C2C2×#1681098111yescharacteristic
538C2#2028510611yescharacteristic
539C2#2028573111yescharacteristic
540C2#2028574011yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#1874C2
socleC23×C53C22×D4
prime radicalC2×#121×C53C22
hypercenter#1874D53
hypocenterC53#27978
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1874

Complements

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

NH ≅ G/Ntype
2. C2×#16810492C2semidirect
3. #20276127C2semidirect
4. #20276121C2semidirect
5. #20275530C2semidirect
7. #20275022C2semidirect
8. #20275020C2semidirect
9. #20274847C2semidirect
10. #20274479C2semidirect
313. C53#27978semidirect

Maximal subgroups

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

subgroupindex
#202830142
C2×#168104922
#202761272
#202761212
#202755302
#202750332
#202750222
#202750202
#202748472
#202744792
C53×#18742
#2797853