Finite Group Databasebeta

← #27425396

Subgroups — #27425396

Order 13568.

912 conjugacy classes of subgroups.

The subgroup lattice has 566 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810903C211yesmaximal, characteristic, complemented
2C2×#16810680C211yesmaximal, characteristic, complemented
3#20283067C211yesmaximal, characteristic, complemented
4#20276146C212yesmaximal, complemented
5#20275901C212yesmaximal
6#20275667C211yesmaximal, characteristic, complemented
7#20275478C211yesmaximal, characteristic, complemented
8#20275455C212yesmaximal, complemented
9#20275415C212yesmaximal, complemented
10#20274715C211yesmaximal, characteristic, complemented
11C4×#121×C53C211yesmaximal, characteristic
12C2×#13133217C2211yescharacteristic
13C2×#13133196—21no—
14#16810903—24no—
15#16810680—24no—
16C2×#13133150C2212yes—
17C2×#13133148C2212yes—
18C2×C4⋊#88322C2211yescharacteristic
19C2×C4⋊#88322C2211yescharacteristic
20C2×#13133138C2212yes—
21C2×#13133138C2212yes—
22C2×C22⋊#88322C2212yes—
23C2×#13133128C2212yes—
24#16810529—21no—
25#16810521—21no—
26#16810521C2212yes—
27#16810493C2211yescharacteristic
28#16810483—21no—
29#16810483C2212yes—
30#16810385C2212yes—
31#16810385C2212yes—
32#16810385C2212yes—
33#16810385C2212yes—
34C2×#121×C53C2211yescharacteristic
35C2×C4×D4×C53—21no—
36C2×C4×D4×C53C2211yescharacteristic
37C22×C42×C53C2211yescharacteristic
38C53×#390C2211yescharacteristic
39C53×#389C2211yescharacteristic
40C53×#388—21no—
41C53×#386—21no—
42C53×#380C2211yescharacteristic
43C4×C4⋊C4×C53C2211yescharacteristic
44C4×C22⋊C4×C53—21no—
45C22×Dic106—21no—
46C22×#88322C2312yes—
47C22×#88322C2312yes—
48#13133217—22no—
49#13133217—22no—
50#13133196—44no—
51C2×C4⋊Dic53—21no—
52C2×C4⋊Dic53—21no—
53C2×C4⋊Dic53—21no—
54C2×C4⋊Dic53C2312yes—
55C2×C4⋊Dic53C2312yes—
56C2×#11719919—22no—
57C2×C22⋊Dic53—22no—
58#13133150—24no—
59#13133148—24no—
60C4⋊#88322—22no—
61C4⋊#88322D414yescomplemented
62#13133138—24no—
63#13133138—24no—
64C22⋊#88322—24no—
65#13133128—24no—
66#13133124—22no—
67#13133124—22no—
68C22×D4×C53—21no—
69C22×D4×C53C2311yescharacteristic
70C23×C4×C53—21no—
71C23×C4×C53D411yescharacteristic
72C23×C4×C53D411yescharacteristic
73C23×C4×C53C2311yescharacteristic
74#121×C53—22no—
75#121×C53—22no—
76C4×D4×C53—22no—
77C4×D4×C53—22no—
78C4×D4×C53—44no—
79C2×C4⋊C4×C53—21no—
80C2×C4⋊C4×C53C2311yescharacteristic
81C2×C4⋊C4×C53C2311yescharacteristic
82C2×C4⋊C4×C53C2311yescharacteristic
83C2×C22⋊C4×C53—21no—
84C2×C22⋊C4×C53—21no—
85C2×C22⋊C4×C53—21no—
86C2×C22⋊C4×C53—21no—
87C2×C22⋊C4×C53—21no—
88C2×C42×C53—22no—
89C2×C42×C53—22no—
90C2×C42×C53—21no—
91C2×C42×C53D411yescharacteristic
92C2×C42×C53D411yescharacteristic
93C2×C42×C53C2311yescharacteristic
94C2×C42×C53C2311yescharacteristic
95#95×C53—21no—
96#95×C53—21no—
97#95×C53—21no—
98C24×C53—21no—
99C24×C53C2×D411yescharacteristic
100C2×D4×C53—24no—
101C2×D4×C53—21no—
102C2×D4×C53—21no—
103C2×D4×C53—21no—
104C2×D4×C53—21no—
105C2×D4×C53—21no—
106C2×D4×C53—42no—
107C2×D4×C53—42no—
108C2×D4×C53#4112yes—
109C22×C4×C53—22no—
110C22×C4×C53—21no—
111C22×C4×C53—22no—
112C22×C4×C53—22no—
113C22×C4×C53—22no—
114C22×C4×C53—22no—
115C22×C4×C53—21no—
116C22×C4×C53—21no—
117C22×C4×C53—21no—
118C22×C4×C53—21no—
119C22×C4×C53—21no—
120C22×C4×C53—21no—
121C22×C4×C53—21no—
122C22×C4×C53—21no—
123C22×C4×C53—41no—
124C22×C4×C53—42no—
125C22×C4×C53—42no—
126C22×C4×C53—42no—
127C22×C4×C53C2×D411yescharacteristic
128C22×C4×C53C2×D411yescharacteristic
129C22×C4×C53C2×D411yescharacteristic
130C22×C4×C53C2×D411yescharacteristic
131C22×C4×C53C2×D411yescharacteristic
132C22×C4×C53C2411yescharacteristic
133C22×Dic53—22no—
134C22×Dic53—22no—
135C2×Dic106—42no—
136C2×Dic106—42no—
137C2×Dic106—42no—
138C4⋊C4×C53—21no—
139C4⋊C4×C53—21no—
140C4⋊C4×C53—21no—
141C4⋊C4×C53—21no—
142C4⋊C4×C53—21no—
143C4⋊C4×C53—42no—
144C4⋊C4×C53#4112yes—
145C22⋊C4×C53—24no—
146C22⋊C4×C53—42no—
147C22⋊C4×C53—42no—
148C22⋊C4×C53—42no—
149C22⋊C4×C53—42no—
150C42×C53—22no—
151C42×C53—22no—
152C42×C53—21no—
153C42×C53—21no—
154C42×C53—42no—
155C42×C53C2×D412yes—
156C42×C53#4112yes—
157C22⋊Dic53—44no—
158C4⋊Dic53—22no—
159C4⋊Dic53—22no—
160C4⋊Dic53—22no—
161C4⋊Dic53—22no—
162C4⋊Dic53—42no—
163C4⋊Dic53—42no—
164C4⋊Dic53—42no—
165#11719919—44no—
166C2×#88322—22no—
167C2×#88322—22no—
168C2×#88322—22no—
169C2×#88322—22no—
170C2×#88322—22no—
171C2×#88322C2×D414yes—
172C23×C53—21no—
173C23×C53—21no—
174C23×C53—21no—
175C23×C53—21no—
176C23×C53—21no—
177C23×C53—21no—
178C23×C53—22no—
179C23×C53—41no—
180C23×C53—41no—
181C23×C53—41no—
182C23×C53—41no—
183C23×C53—41no—
184C23×C53—41no—
185C23×C53#13712yes—
186C23×C53C22×D411yescharacteristic
187D4×C53—22no—
188D4×C53—22no—
189D4×C53—48no—
190D4×C53—41no—
191D4×C53—41no—
192C2×C4×C53—21no—
193C2×C4×C53—21no—
194C2×C4×C53—21no—
195C2×C4×C53—21no—
196C2×C4×C53—21no—
197C2×C4×C53—21no—
198C2×C4×C53—22no—
199C2×C4×C53—21no—
200C2×C4×C53—21no—
201C2×C4×C53—21no—
202C2×C4×C53—21no—
203C2×C4×C53—21no—
204C2×C4×C53—44no—
205C2×C4×C53—41no—
206C2×C4×C53—41no—
207C2×C4×C53—41no—
208C2×C4×C53—41no—
209C2×C4×C53—41no—
210C2×C4×C53—41no—
211C2×C4×C53—44no—
212C2×C4×C53—41no—
213C2×C4×C53—41no—
214C2×C4×C53—41no—
215C2×C4×C53—41no—
216C2×C4×C53—41no—
217C2×C4×C53—41no—
218C2×C4×C53—84no—
219C2×C4×C53#12112yes—
220C2×C4×C53#12112yes—
221C2×C4×C53AGL(1,ℤ/8)12yes—
222C2×C4×C53AGL(1,ℤ/8)12yes—
223C2×C4×C53C22×D411yescharacteristic
224C2×C4×C53C2×#4111yescharacteristic
225C2×C4×C53C2×#4111yescharacteristic
226C2×C4×C53C2×#4111yescharacteristic
227C2×C4×C532−1+411yescharacteristic
228C2×C4×C532−1+411yescharacteristic
229C2×Dic53—42no—
230C2×Dic53—42no—
231C2×Dic53—42no—
232C2×Dic53—42no—
233C2×Dic53—42no—
234C2×Dic53—42no—
235Dic106—84no—
236#88322—24no—
237#88322—42no—
238#24709—531nomaximal, Sylow
239C22×C53—22no—
240C22×C53—22no—
241C22×C53—22no—
242C22×C53—41no—
243C22×C53—41no—
244C22×C53—41no—
245C22×C53—41no—
246C22×C53—42no—
247C22×C53—42no—
248C22×C53—42no—
249C22×C53—81no—
250C22×C53—81no—
251C22×C53—81no—
252C22×C53—81no—
253C22×C53C2×#12111yescharacteristic
254C22×C53#52511yescharacteristic
255C22×C53#53611yescharacteristic
256C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
257C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
258C22×C53C2×#13711yescharacteristic
259C22×C53#57511yescharacteristic
260C4×C53—22no—
261C4×C53—42no—
262C4×C53—41no—
263C4×C53—41no—
264C4×C53—41no—
265C4×C53—81no—
266C4×C53—81no—
267C4×C53#46914yes—
268C4×C53#54711yescharacteristic
269C4×C53#54811yescharacteristic
270C4×C53#55211yescharacteristic
271C4×C53#55611yescharacteristic
272Dic53—82no—
273Dic53—82no—
274C2×#474—531no—
275C2×#432—531no—
276#2857—531no—
277C4×#121D5311yescharacteristic
278#2076—532no—
279#2034—532no—
280#1988—531no—
281#1948—531no—
282#1944—532no—
283#1936—532no—
284#1802—531no—
285C2×C53—42no—
286C2×C53—82no—
287C2×C53#241511yescharacteristic
288C2×C53C2×#46911yescharacteristic
289C2×C53#305611yescharacteristic
290C2×C53#311011yescharacteristic
291C2×C53#311511yescharacteristic
292C2×C53#323511yescharacteristic
293C2×C53#323911yescharacteristic
294C2×C4⋊Q8—531no—
295C2×C22⋊Q8—1061no—
296C2×#121C2×D5311yescharacteristic
297C2×C4×D4—21no—
298C2×C4×D4C2×D5311yescharacteristic
299C22×C42C2×D5311yescharacteristic
300#474—1064no—
301#432—1064no—
302C2×#107—532no—
303C2×#106—532no—
304C2×C4⋊C8—531no—
305C2×C4⋊C8—531no—
306C2×#102—532no—
307C2×#102—532no—
308C2×C22⋊C8—532no—
309C2×#97—532no—
310#391—1061no—
311#390C2×D5311yescharacteristic
312#389—532no—
313#389—1061no—
314#389C2×D5311yescharacteristic
315#388—21no—
316#386—21no—
317#384—531no—
318#380—532no—
319#380—1061no—
320#380C2×D5311yescharacteristic
321C4×C4⋊C4C2×D5311yescharacteristic
322C4×C22⋊C4—21no—
323#340—532no—
324#340—532no—
325#340—532no—
326#340—532no—
327C53#2470911yescharacteristic, complemented, Sylow
328C22×Q8—1061no—
329C22×D4—21no—
330C22×D4C22×D5311yescharacteristic
331C23×C4—21no—
332C23×C4C22⋊D5311yescharacteristic
333C23×C4C22⋊D5311yescharacteristic
334C23×C4C22×D5311yescharacteristic
335C22×C8—532no—
336C22×C8—532no—
337C4⋊Q8—1062no—
338C4⋊Q8—1062no—
339C22⋊Q8—2124no—
340#121—22no—
341#121—22no—
342C4×D4—22no—
343C4×D4—22no—
344C4×D4—44no—
345C2×C4⋊C4—21no—
346C2×C4⋊C4—532no—
347C2×C4⋊C4—532no—
348C2×C4⋊C4—1062no—
349C2×C4⋊C4—1061no—
350C2×C4⋊C4—1061no—
351C2×C4⋊C4—1061no—
352C2×C4⋊C4C22×D5311yescharacteristic
353C2×C4⋊C4C22×D5311yescharacteristic
354C2×C4⋊C4C22×D5311yescharacteristic
355C2×C22⋊C4—21no—
356C2×C22⋊C4—21no—
357C2×C22⋊C4—21no—
358C2×C22⋊C4—21no—
359C2×C22⋊C4—21no—
360C2×C22⋊C4—1062no—
361C2×C42—22no—
362C2×C42—22no—
363C2×C42—21no—
364C2×C42C22⋊D5311yescharacteristic
365C2×C42C22⋊D5311yescharacteristic
366C2×C42C22×D5311yescharacteristic
367C2×C42C22×D5311yescharacteristic
368#107—1064no—
369#106—1064no—
370C4⋊C8—534no—
371C4⋊C8—1062no—
372#102—1064no—
373#102—1064no—
374C22⋊C8—1064no—
375#97—1064no—
376#95—21no—
377#95—21no—
378#95—21no—
379#95—1062no—
380#95—1062no—
381C24—21no—
382C24C2×C22⋊D5311yescharacteristic
383C2×Q8—2122no—
384C2×Q8—2122no—
385C2×Q8—2122no—
386C2×D4—24no—
387C2×D4—21no—
388C2×D4—21no—
389C2×D4—21no—
390C2×D4—21no—
391C2×D4—21no—
392C2×D4—42no—
393C2×D4—42no—
394C2×D4#1171994512yes—
395C22×C4—22no—
396C22×C4—21no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—41no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—1062no—
414C22×C4—1062no—
415C22×C4C2×C22⋊D5311yescharacteristic
416C22×C4C2×C22⋊D5311yescharacteristic
417C22×C4C2×C22⋊D5311yescharacteristic
418C22×C4C2×C22⋊D5311yescharacteristic
419C22×C4C2×C22⋊D5311yescharacteristic
420C22×C4C23×D5311yescharacteristic
421C2×C8—534no—
422C2×C8—1062no—
423C2×C8—1062no—
424C2×C8—1062no—
425C2×C8—1062no—
426C2×C8—1062no—
427C4⋊C4—21no—
428C4⋊C4—21no—
429C4⋊C4—21no—
430C4⋊C4—21no—
431C4⋊C4—21no—
432C4⋊C4—42no—
433C4⋊C4—1062no—
434C4⋊C4—1062no—
435C4⋊C4—1062no—
436C4⋊C4—1062no—
437C4⋊C4—2124no—
438C4⋊C4—2122no—
439C4⋊C4—2122no—
440C4⋊C4—2122no—
441C4⋊C4#1171994512yes—
442C22⋊C4—24no—
443C22⋊C4—42no—
444C22⋊C4—42no—
445C22⋊C4—42no—
446C22⋊C4—42no—
447C22⋊C4—2124no—
448C42—22no—
449C42—22no—
450C42—21no—
451C42—21no—
452C42—42no—
453C42D4×D5312yes—
454C42#1171994712yes—
455C23—21no—
456C23—21no—
457C23—21no—
458C23—21no—
459C23—21no—
460C23—21no—
461C23—22no—
462C23—41no—
463C23—41no—
464C23—41no—
465C23—41no—
466C23—41no—
467C23—41no—
468C23#1313324712yes—
469C23C22×C22⋊D5311yescharacteristic
470Q8—4244no—
471D4—22no—
472D4—22no—
473D4—48no—
474D4—41no—
475D4—41no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—22no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—44no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—44no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—84no—
503C2×C4—2122no—
504C2×C4—2122no—
505C2×C4—2122no—
506C2×C4—2122no—
507C2×C4—2122no—
508C2×C4—2122no—
509C2×C4#1313318912yes—
510C2×C4#1313318912yes—
511C2×C4#1313324212yes—
512C2×C4#1313324312yes—
513C2×C4C2×D4×D5311yescharacteristic
514C2×C4C2×#1171994711yescharacteristic
515C2×C4C2×#1171994511yescharacteristic
516C2×C4C2×#1171994511yescharacteristic
517C2×C4#1313326411yescharacteristic
518C2×C4#1313326411yescharacteristic
519C8—1064no—
520C8—2122no—
521C22—22no—
522C22—22no—
523C22—22no—
524C22—41no—
525C22—41no—
526C22—41no—
527C22—41no—
528C22—42no—
529C22—42no—
530C22—42no—
531C22—81no—
532C22—81no—
533C22—81no—
534C22—81no—
535C22C2×#1313318911yescharacteristic
536C22#1681113411yescharacteristic
537C22#1681119411yescharacteristic
538C22C2×#1313324211yescharacteristic
539C22C2×#1313324311yescharacteristic
540C22C2×#1313324711yescharacteristic
541C22#1681142111yescharacteristic
542C4—22no—
543C4—41no—
544C4—42no—
545C4—41no—
546C4—41no—
547C4—81no—
548C4—81no—
549C4—4242no—
550C4—4242no—
551C4#1681087414yes—
552C4#1681127811yescharacteristic
553C4#1681128211yescharacteristic
554C4#1681131311yescharacteristic
555C4#1681134311yescharacteristic
556C2—42no—
557C2—82no—
558C2#2027814411yescharacteristic
559C2C2×#1681087411yescharacteristic
560C2#2028434711yescharacteristic
561C2#2028481411yescharacteristic
562C2#2028486611yescharacteristic
563C2#2028590911yescharacteristic
564C2#2028594211yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC4×#121×C53C2
socleC23×C53C22×D4
prime radicalC2×#121×C53C22
hypercenterC4×#121D53
hypocenterC53#24709
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C22×C42 ◁ C4×#121

Complements

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

NH ≅ G/Ntype
1. C2×#16810903C2semidirect
2. C2×#16810680C2semidirect
3. #20283067C2semidirect
4. #20276146C2semidirect
6. #20275667C2semidirect
7. #20275478C2semidirect
8. #20275455C2semidirect
9. #20275415C2semidirect
10. #20274715C2semidirect
61. C4⋊#88322D4semidirect
327. C53#24709semidirect

Maximal subgroups

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

subgroupindex
C2×#168109032
C2×#168106802
#202830672
#202761462
#202759012
#202756672
#202754782
#202754552
#202754152
#202747152
C4×#121×C532
#2470953