Finite Group Databasebeta

← #27429441

Subgroups — #27429441

Order 13568.

700 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20279633C211yesmaximal, characteristic
2#20276282C211yesmaximal, characteristic, complemented
3#20276272C211yesmaximal, characteristic, complemented
4#20276129C211yesmaximal, characteristic
5#20276119C211yesmaximal, characteristic
6#20275656C211yesmaximal, characteristic, complemented
7#20275478C211yesmaximal, characteristic, complemented
8#20275342C211yesmaximal, characteristic
9#20275342C211yesmaximal, characteristic
10#20275300C211yesmaximal, characteristic, complemented
11#20275293C211yesmaximal, characteristic, complemented
12#20274848C211yesmaximal, characteristic, complemented
13#20274722C211yesmaximal, characteristic
14#20274543C211yesmaximal, characteristic
15C53×#2355C211yesmaximal, characteristic, complemented
16C2×#13133193—21no—
17C2×C4×D212C2211yescharacteristic
18C2×#13133148C2211yescharacteristic
19C2×#13133148C2211yescharacteristic
20C2×C4⋊#88322C2211yescharacteristic
21C2×#13133137C2211yescharacteristic
22C2×#13133137C2211yescharacteristic
23C2×#13133137C2211yescharacteristic
24C2×#13133137C2211yescharacteristic
25C2×#13133128C2211yescharacteristic
26#16810545—21no—
27#16810545—21no—
28#16810525C2211yescharacteristic
29#16810525C2211yescharacteristic
30#16810521C2211yescharacteristic
31#16810521C2211yescharacteristic
32#16810493C2211yescharacteristic
33#16810480—21no—
34#16810385C2211yescharacteristic
35#16810385C2211yescharacteristic
36#16810385C2211yescharacteristic
37#16810385C2211yescharacteristic
38#16810385C2211yescharacteristic
39#16810385C2211yescharacteristic
40#16810385C2211yescharacteristic
41#16810385C2211yescharacteristic
42#16810385C2211yescharacteristic
43#16810385C2211yescharacteristic
44#16810378C2211yescharacteristic
45#16810378C2211yescharacteristic
46#16810378C2211yescharacteristic
47#16810378C2211yescharacteristic
48C2×C4×Q8×C53C2211yescharacteristic
49C53×#391—21no—
50C53×#387—21no—
51C53×#387—21no—
52C53×#384C2211yescharacteristic
53C53×#382C2211yescharacteristic
54C53×#382C2211yescharacteristic
55C4×C4⋊C4×C53—21no—
56C4×C4⋊C4×C53C2211yescharacteristic
57C4×C4⋊C4×C53C2211yescharacteristic
58C4×C4⋊C4×C53C2211yescharacteristic
59C22×D212C2311yescharacteristic
60C22×C4×D53—21no—
61C22×#88322C2311yescharacteristic
62C22×#88322C2311yescharacteristic
63C22×#88322C2311yescharacteristic
64C22×#88322C2311yescharacteristic
65#13133193—44no—
66C4×D212—22no—
67C4×D212—22no—
68C2×C4⋊Dic53—21no—
69C2×C4⋊Dic53—21no—
70C2×C4⋊Dic53C2311yescharacteristic
71C2×C4⋊Dic53C2311yescharacteristic
72C2×C4⋊Dic53C2311yescharacteristic
73C2×#11719919—21no—
74C2×#11719919—21no—
75C2×#11719921—21no—
76C2×#11719921—21no—
77C2×#11719921—21no—
78C2×#11719921—21no—
79C2×#11719921—21no—
80#13133148—22no—
81#13133148—22no—
82C4⋊#88322—22no—
83#13133137—22no—
84#13133137—22no—
85#13133137—22no—
86#13133137—22no—
87#13133128—22no—
88#13133124—21no—
89#13133124—21no—
90C22×Q8×C53D411yescharacteristic
91C4×Q8×C53—22no—
92C4×Q8×C53—22no—
93C2×C4⋊C4×C53—21no—
94C2×C4⋊C4×C53—21no—
95C2×C4⋊C4×C53D411yescharacteristic
96C2×C4⋊C4×C53D411yescharacteristic
97C2×C4⋊C4×C53D411yescharacteristic
98C2×C4⋊C4×C53C2311yescharacteristic
99C2×C4⋊C4×C53C2311yescharacteristic
100C2×C4⋊C4×C53C2311yescharacteristic
101C2×C4⋊C4×C53C2311yescharacteristic
102C2×C42×C53—21no—
103C2×C42×C53—21no—
104C2×C42×C53—21no—
105C2×C42×C53C2311yescharacteristic
106C2×C42×C53C2311yescharacteristic
107C2×C42×C53C2311yescharacteristic
108#95×C53—21no—
109#95×C53—21no—
110#95×C53—21no—
111#95×C53—21no—
112#95×C53—21no—
113#95×C53—21no—
114#95×C53—21no—
115C23×D53—21no—
116C2×Q8×C53—22no—
117C2×Q8×C53—22no—
118C2×Q8×C53—41no—
119C22×C4×C53—21no—
120C22×C4×C53—21no—
121C22×C4×C53—21no—
122C22×C4×C53—21no—
123C22×C4×C53C2×D411yescharacteristic
124C22×C4×C53C2×D411yescharacteristic
125C22×C4×C53C2×D411yescharacteristic
126C22×C4×C53C2×D411yescharacteristic
127C22×C4×C53C2×D411yescharacteristic
128C22×C4×C53C2×D411yescharacteristic
129C22×C4×C53C2411yescharacteristic
130C22×Dic53—21no—
131C22×Dic53—21no—
132C22×Dic53—21no—
133C2×D212—21no—
134C2×D212—21no—
135C2×D212—21no—
136C2×D212—21no—
137C2×D212—21no—
138C2×D212#4112yes—
139C2×C4×D53—42no—
140C2×C4×D53—42no—
141C2×C4×D53—42no—
142C4⋊C4×C53—22no—
143C4⋊C4×C53—22no—
144C4⋊C4×C53—22no—
145C4⋊C4×C53—21no—
146C4⋊C4×C53—21no—
147C4⋊C4×C53—21no—
148C4⋊C4×C53—21no—
149C4⋊C4×C53—21no—
150C4⋊C4×C53—21no—
151C4⋊C4×C53—21no—
152C4⋊C4×C53—42no—
153C4⋊C4×C53—81no—
154C4⋊C4×C53#4112yes—
155C42×C53—21no—
156C42×C53—21no—
157C42×C53—21no—
158C42×C53—21no—
159C42×C53—42no—
160C42×C53—81no—
161C42×C53—81no—
162C42×C53#4112yes—
163C42×C53#4112yes—
164#11719921—42no—
165#11719921—42no—
166#11719921—42no—
167#11719921—81no—
168#11719921—81no—
169C4⋊Dic53—21no—
170C4⋊Dic53—21no—
171C4⋊Dic53—21no—
172C4⋊Dic53—21no—
173C4⋊Dic53—21no—
174C4⋊Dic53—21no—
175C4⋊Dic53—42no—
176C4⋊Dic53—81no—
177#11719919—42no—
178#11719919—42no—
179C2×#88322—21no—
180C2×#88322—21no—
181C2×#88322—21no—
182C2×#88322—21no—
183C2×#88322—21no—
184C2×#88322—21no—
185C2×#88322—21no—
186C2×#88322—21no—
187C2×#88322—21no—
188C2×#88322—21no—
189C2×#88322—21no—
190C2×#88322—21no—
191C23×C53C22×D411yescharacteristic
192C22×D53—22no—
193C22×D53—41no—
194C22×D53—41no—
195C22×D53—41no—
196C22×D53—41no—
197C22×D53—41no—
198C22×D53—41no—
199Q8×C53—44no—
200C2×C4×C53—21no—
201C2×C4×C53—21no—
202C2×C4×C53—21no—
203C2×C4×C53—21no—
204C2×C4×C53—21no—
205C2×C4×C53—21no—
206C2×C4×C53—21no—
207C2×C4×C53—21no—
208C2×C4×C53—21no—
209C2×C4×C53—22no—
210C2×C4×C53—21no—
211C2×C4×C53—21no—
212C2×C4×C53—21no—
213C2×C4×C53—21no—
214C2×C4×C53—21no—
215C2×C4×C53—21no—
216C2×C4×C53—41no—
217C2×C4×C53—41no—
218C2×C4×C53—41no—
219C2×C4×C53—41no—
220C2×C4×C53—41no—
221C2×C4×C53—41no—
222C2×C4×C53—41no—
223C2×C4×C53—41no—
224C2×C4×C53—41no—
225C2×C4×C53—41no—
226C2×C4×C53—41no—
227C2×C4×C53#12312yes—
228C2×C4×C53#12312yes—
229C2×C4×C53AGL(1,ℤ/8)12yes—
230C2×C4×C53C2×#4111yescharacteristic
231C2×C4×C53C2×#4111yescharacteristic
232C2×C4×C53C2×#4111yescharacteristic
233C2×C4×C53C2×#4111yescharacteristic
234C2×C4×C532−1+411yescharacteristic
235C2×C4×C532−1+411yescharacteristic
236C2×Dic53—41no—
237C2×Dic53—41no—
238C2×Dic53—41no—
239C2×Dic53—41no—
240C2×Dic53—41no—
241C2×Dic53—41no—
242C2×Dic53—41no—
243C2×Dic53—41no—
244C2×Dic53—41no—
245D212—22no—
246D212—22no—
247D212—41no—
248D212—41no—
249C4×D53—84no—
250#88322—41no—
251#88322—41no—
252#88322—41no—
253#88322—41no—
254#25006—531nomaximal, Sylow
255C22×C53C2×#12311yescharacteristic
256C22×C53#54811yescharacteristic
257C22×C53#54811yescharacteristic
258C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
259C22×C53#57511yescharacteristic
260C22×C53#57711yescharacteristic
261C22×C53#57711yescharacteristic
262C2×D53—42no—
263C2×D53—42no—
264C2×D53—42no—
265C2×D53—81no—
266C2×D53—81no—
267C2×D53—81no—
268C2×D53—81no—
269C4×C53—22no—
270C4×C53—22no—
271C4×C53—22no—
272C4×C53—41no—
273C4×C53—41no—
274C4×C53—41no—
275C4×C53—42no—
276C4×C53—81no—
277C4×C53—81no—
278C4×C53—81no—
279C4×C53#52911yescharacteristic
280C4×C53#54111yescharacteristic
281C4×C53#55211yescharacteristic
282C4×C53#55611yescharacteristic
283Dic53—81no—
284Dic53—81no—
285Dic53—81no—
286#2538—531no—
287#2355D5311yescharacteristic, complemented
288#2101—531no—
289#2099—531no—
290#2073—531no—
291#2071—531no—
292#1985—531no—
293#1948—531no—
294#1926—531no—
295#1926—531no—
296#1920—531no—
297#1919—531no—
298#1836—531no—
299#1804—531no—
300#1762—531no—
301C2×C53#242111yescharacteristic
302C2×C53#309411yescharacteristic
303C2×C53#309911yescharacteristic
304C2×C53#336811yescharacteristic
305C2×C53#337311yescharacteristic
306C2×C53#337611yescharacteristic
307C2×C53#337611yescharacteristic
308D53—82no—
309C2×C22⋊Q8—1061no—
310C2×C4×Q8C2×D5311yescharacteristic
311C2×C4×D4—531no—
312C2×#106—531no—
313C2×#106—531no—
314C2×C4⋊C8—531no—
315C2×#102—531no—
316C2×#102—531no—
317C2×#102—531no—
318C2×#102—531no—
319C2×#97—531no—
320#396—1061no—
321#396—1061no—
322#391—21no—
323#390—531no—
324#390—531no—
325#389—531no—
326#389—531no—
327#387—21no—
328#387—21no—
329#384—531no—
330#384C2×D5311yescharacteristic
331#382C2×D5311yescharacteristic
332#382C2×D5311yescharacteristic
333#380—1061no—
334C4×C4⋊C4—21no—
335C4×C4⋊C4C2×D5311yescharacteristic
336C4×C4⋊C4C2×D5311yescharacteristic
337C4×C4⋊C4C2×D5311yescharacteristic
338#340—531no—
339#340—531no—
340#340—531no—
341#340—531no—
342#340—531no—
343#340—531no—
344#340—531no—
345#340—531no—
346#340—531no—
347#340—531no—
348#336—531no—
349#336—531no—
350#336—531no—
351#336—531no—
352C53#2500611yescharacteristic, complemented, Sylow
353C22×Q8C22⋊D5311yescharacteristic
354C22×D4—531no—
355C23×C4—1061no—
356C22×C8—531no—
357C22×C8—531no—
358C22×C8—531no—
359C22×C8—531no—
360C22⋊Q8—2124no—
361C4×Q8—22no—
362C4×Q8—22no—
363C4×D4—1062no—
364C4×D4—1062no—
365C2×C4⋊C4—21no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—531no—
368C2×C4⋊C4—531no—
369C2×C4⋊C4—531no—
370C2×C4⋊C4—1061no—
371C2×C4⋊C4—1061no—
372C2×C4⋊C4—1061no—
373C2×C4⋊C4—1061no—
374C2×C4⋊C4C22⋊D5311yescharacteristic
375C2×C4⋊C4C22⋊D5311yescharacteristic
376C2×C4⋊C4C22⋊D5311yescharacteristic
377C2×C4⋊C4C22×D5311yescharacteristic
378C2×C4⋊C4C22×D5311yescharacteristic
379C2×C4⋊C4C22×D5311yescharacteristic
380C2×C4⋊C4C22×D5311yescharacteristic
381C2×C22⋊C4—1061no—
382C2×C22⋊C4—1061no—
383C2×C22⋊C4—1061no—
384C2×C22⋊C4—1061no—
385C2×C22⋊C4—1061no—
386C2×C42—21no—
387C2×C42—21no—
388C2×C42—21no—
389C2×C42C22×D5311yescharacteristic
390C2×C42C22×D5311yescharacteristic
391C2×C42C22×D5311yescharacteristic
392#106—1062no—
393#106—1062no—
394C4⋊C8—1062no—
395#102—1062no—
396#102—1062no—
397#102—1062no—
398#102—1062no—
399#97—1062no—
400#95—21no—
401#95—21no—
402#95—21no—
403#95—21no—
404#95—21no—
405#95—21no—
406#95—21no—
407#95—1061no—
408#95—1061no—
409C24—1061no—
410C2×Q8—22no—
411C2×Q8—22no—
412C2×Q8—41no—
413C2×D4—532no—
414C2×D4—1061no—
415C2×D4—1061no—
416C2×D4—1061no—
417C2×D4—1061no—
418C2×D4—1061no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—1061no—
424C22×C4—1061no—
425C22×C4—1061no—
426C22×C4—2122no—
427C22×C4—2122no—
428C22×C4—2122no—
429C22×C4C2×C22⋊D5311yescharacteristic
430C22×C4C2×C22⋊D5311yescharacteristic
431C22×C4C2×C22⋊D5311yescharacteristic
432C22×C4C2×C22⋊D5311yescharacteristic
433C22×C4C2×C22⋊D5311yescharacteristic
434C22×C4C2×C22⋊D5311yescharacteristic
435C22×C4C23×D5311yescharacteristic
436C2×C8—1061no—
437C2×C8—1061no—
438C2×C8—1061no—
439C2×C8—1061no—
440C2×C8—1061no—
441C2×C8—1061no—
442C2×C8—1061no—
443C2×C8—1061no—
444C2×C8—1061no—
445C2×C8—1061no—
446C2×C8—1061no—
447C2×C8—1061no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—22no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—42no—
459C4⋊C4—81no—
460C4⋊C4—1061no—
461C4⋊C4—1061no—
462C4⋊C4—1061no—
463C4⋊C4—1061no—
464C4⋊C4—1061no—
465C4⋊C4—1061no—
466C4⋊C4—4241no—
467C4⋊C4—2122no—
468C4⋊C4—2122no—
469C4⋊C4—2122no—
470C4⋊C4#1171994912yes—
471C22⋊C4—4241no—
472C22⋊C4—4241no—
473C22⋊C4—2122no—
474C22⋊C4—2122no—
475C22⋊C4—2122no—
476C42—21no—
477C42—21no—
478C42—21no—
479C42—21no—
480C42—42no—
481C42—81no—
482C42—81no—
483C42#1171994512yes—
484C42#1171994512yes—
485C23—1062no—
486C23—2121no—
487C23—2121no—
488C23—2121no—
489C23—2121no—
490C23—2121no—
491C23—2121no—
492C23C22×C22⋊D5311yescharacteristic
493Q8—44no—
494D4—1062no—
495D4—1062no—
496D4—2121no—
497D4—2121no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—22no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—4244no—
526C2×C4—2121no—
527C2×C4—2121no—
528C2×C4—2121no—
529C2×C4—2121no—
530C2×C4—2121no—
531C2×C4—2121no—
532C2×C4—2121no—
533C2×C4—2121no—
534C2×C4—2121no—
535C2×C4#1313320012yes—
536C2×C4#1313320012yes—
537C2×C4#1313324212yes—
538C2×C4#41×D5311yescharacteristic
539C2×C4C2×#1171994511yescharacteristic
540C2×C4C2×#1171994511yescharacteristic
541C2×C4C2×#1171994911yescharacteristic
542C2×C4#1313326411yescharacteristic
543C2×C4#1313326511yescharacteristic
544C8—2121no—
545C8—2121no—
546C8—2121no—
547C8—2121no—
548C22—4241no—
549C22—4241no—
550C22—4241no—
551C22—4241no—
552C22—2122no—
553C22—2122no—
554C22—2122no—
555C22C2×#1313320011yescharacteristic
556C22#1681128011yescharacteristic
557C22#1681128411yescharacteristic
558C22C2×#1313324211yescharacteristic
559C22#1681142111yescharacteristic
560C22#1681143111yescharacteristic
561C22#1681143411yescharacteristic
562C4—22no—
563C4—22no—
564C4—22no—
565C4—41no—
566C4—41no—
567C4—42no—
568C4—41no—
569C4—81no—
570C4—81no—
571C4—81no—
572C4—4241no—
573C4—4241no—
574C4—4241no—
575C4#1681116011yescharacteristic
576C4#1681123111yescharacteristic
577C4#1681131411yescharacteristic
578C4#1681134611yescharacteristic
579C2—4242no—
580C2#2027821211yescharacteristic
581C2#2028465511yescharacteristic
582C2#2028470411yescharacteristic
583C2#2028744411yescharacteristic
584C2#2028750311yescharacteristic
585C2#2028753611yescharacteristic
586C2#2028754211yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2355C2
socleC23×C53C22×D4
prime radicalC22×D212C23
hypercenter#2355D53
hypocenterC53#25006
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ #2355

Complements

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

NH ≅ G/Ntype
2. #20276282C2semidirect
3. #20276272C2semidirect
6. #20275656C2semidirect
7. #20275478C2semidirect
10. #20275300C2semidirect
11. #20275293C2semidirect
12. #20274848C2semidirect
15. C53×#2355C2semidirect
287. #2355D53semidirect
352. C53#25006semidirect

Maximal subgroups

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

subgroupindex
#202796332
#202762822
#202762722
#202761292
#202761192
#202756562
#202754782
#202753422
#202753002
#202752932
#202748482
#202747222
#202745432
C53×#23552
#2500653