Finite Group Databasebeta

← #27512579

Subgroups — #27512579

Order 13568.

698 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#20278417C211yesmaximal, characteristic
2#20276273C211yesmaximal, characteristic, complemented
3#20276272C211yesmaximal, characteristic, complemented
4#20276206C211yesmaximal, characteristic, complemented
5#20276141C211yesmaximal, characteristic
6#20276141C211yesmaximal, characteristic
7#20276043C211yesmaximal, characteristic, complemented
8#20275675C211yesmaximal, characteristic, complemented
9#20275473C211yesmaximal, characteristic, complemented
10#20275445C211yesmaximal, characteristic, complemented
11#20275323C211yesmaximal, characteristic
12#20275286C211yesmaximal, characteristic, complemented
13#20274902C211yesmaximal, characteristic
14#20274799C211yesmaximal, characteristic
15C53×#2496C211yesmaximal, characteristic
16C2×#13133217C2211yescharacteristic
17C2×#13133194—21no—
18C22×C4⋊Dic53C2211yescharacteristic
19C2×#13133150C2211yescharacteristic
20C2×#13133150C2211yescharacteristic
21C2×#13133150C2211yescharacteristic
22C2×#13133150C2211yescharacteristic
23C2×C4⋊#88322C2211yescharacteristic
24C2×C22⋊#88322C2211yescharacteristic
25C2×C22⋊#88322C2211yescharacteristic
26C2×C4×#88322C2211yescharacteristic
27#16810555—21no—
28#16810544—21no—
29#16810521C2211yescharacteristic
30#16810521C2211yescharacteristic
31#16810498—21no—
32#16810493C2211yescharacteristic
33#16810483C2211yescharacteristic
34#16810385C2211yescharacteristic
35#16810385C2211yescharacteristic
36#16810385C2211yescharacteristic
37#16810385C2211yescharacteristic
38#16810385C2211yescharacteristic
39#16810385C2211yescharacteristic
40#16810385C2211yescharacteristic
41#16810385C2211yescharacteristic
42#16810385C2211yescharacteristic
43#16810385C2211yescharacteristic
44#16810385C2211yescharacteristic
45#16810385C2211yescharacteristic
46#16810378C2211yescharacteristic
47#16810378C2211yescharacteristic
48C2×#126×C53—21no—
49C53×#400—21no—
50C53×#397—21no—
51C53×#389C2211yescharacteristic
52C53×#389C2211yescharacteristic
53C53×#387—21no—
54C53×#384C2211yescharacteristic
55C53×#380C2211yescharacteristic
56C53×#380C2211yescharacteristic
57C4×C4⋊C4×C53C2211yescharacteristic
58C4×C22⋊C4×C53C2211yescharacteristic
59C22×Dic106—21no—
60C23×Dic53—21no—
61C22×#88322C2311yescharacteristic
62C22×#88322C2311yescharacteristic
63C22×#88322C2311yescharacteristic
64C22×#88322C2311yescharacteristic
65#13133217—22no—
66#13133217—41no—
67#13133194—44no—
68C2×C4⋊Dic53—21no—
69C2×C4⋊Dic53—21no—
70C2×C4⋊Dic53—21no—
71C2×C4⋊Dic53—21no—
72C2×C4⋊Dic53—22no—
73C2×C4⋊Dic53—41no—
74C2×C4⋊Dic53C2311yescharacteristic
75C2×C4⋊Dic53C2311yescharacteristic
76C2×C4⋊Dic53C2311yescharacteristic
77C2×C4⋊Dic53C2311yescharacteristic
78C2×#11719919—21no—
79C2×#11719919—21no—
80C2×C22⋊Dic53—21no—
81#13133150—22no—
82#13133150—22no—
83#13133150—41no—
84#13133150—41no—
85C4⋊#88322—22no—
86C22⋊#88322—22no—
87C22⋊#88322—41no—
88C4×#88322—41no—
89#13133124—21no—
90#13133124—21no—
91#13133124—21no—
92C23×C4×C53C2311yescharacteristic
93#126×C53—44no—
94C2×C4⋊C4×C53—21no—
95C2×C4⋊C4×C53—21no—
96C2×C4⋊C4×C53—21no—
97C2×C4⋊C4×C53—21no—
98C2×C4⋊C4×C53C2311yescharacteristic
99C2×C4⋊C4×C53C2311yescharacteristic
100C2×C4⋊C4×C53C2311yescharacteristic
101C2×C4⋊C4×C53C2311yescharacteristic
102C2×C22⋊C4×C53—21no—
103C2×C22⋊C4×C53—21no—
104C2×C22⋊C4×C53D411yescharacteristic
105C2×C22⋊C4×C53D411yescharacteristic
106C2×C42×C53—21no—
107C2×C42×C53C2311yescharacteristic
108C2×C42×C53C2311yescharacteristic
109#95×C53—21no—
110#95×C53—21no—
111#95×C53—21no—
112#95×C53—21no—
113#95×C53—21no—
114#95×C53D411yescharacteristic
115#95×C53D411yescharacteristic
116C24×C53C2×D411yescharacteristic
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—21no—
124C22×C4×C53—21no—
125C22×C4×C53—21no—
126C22×C4×C53—21no—
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—21no—
134C22×Dic53—21no—
135C22×Dic53—21no—
136C22×Dic53—21no—
137C22×Dic53—42no—
138C22×Dic53—42no—
139C22×Dic53—81no—
140C2×Dic106—42no—
141C2×Dic106—42no—
142C2×Dic106—81no—
143C4⋊C4×C53—21no—
144C4⋊C4×C53—21no—
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—42no—
152C4⋊C4×C53—42no—
153C4⋊C4×C53—42no—
154C4⋊C4×C53—42no—
155C22⋊C4×C53—22no—
156C22⋊C4×C53—42no—
157C22⋊C4×C53—42no—
158C22⋊C4×C53D814yes—
159C42×C53—21no—
160C42×C53—21no—
161C42×C53—21no—
162C42×C53—21no—
163C42×C53—42no—
164C22⋊Dic53—42no—
165C4⋊Dic53—21no—
166C4⋊Dic53—21no—
167C4⋊Dic53—21no—
168C4⋊Dic53—21no—
169C4⋊Dic53—21no—
170C4⋊Dic53—21no—
171C4⋊Dic53—21no—
172C4⋊Dic53—42no—
173C4⋊Dic53—42no—
174C4⋊Dic53—42no—
175C4⋊Dic53—81no—
176C4⋊Dic53#4112yes—
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#4112yes—
191C23×C53—21no—
192C23×C53—21no—
193C23×C53—21no—
194C23×C53—21no—
195C23×C53—21no—
196C23×C53—21no—
197C23×C53C2×D812yes—
198C23×C53C22×D411yescharacteristic
199C2×C4×C53—21no—
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—21no—
210C2×C4×C53—21no—
211C2×C4×C53—41no—
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—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×C53C2×D812yes—
228C2×C4×C53C2×D812yes—
229C2×C4×C53#13712yes—
230C2×C4×C53C2×#4111yescharacteristic
231C2×C4×C53C2×#4111yescharacteristic
232C2×C4×C53H(5,𝔽2)11yescharacteristic
233C2×C4×C532−1+411yescharacteristic
234C2×C4×C532−1+411yescharacteristic
235C2×C4×C532−1+411yescharacteristic
236C2×Dic53—22no—
237C2×Dic53—41no—
238C2×Dic53—41no—
239C2×Dic53—41no—
240C2×Dic53—41no—
241C2×Dic53—41no—
242C2×Dic53—41no—
243C2×Dic53—41no—
244C2×Dic53—41no—
245C2×Dic53—41no—
246C2×Dic53—41no—
247C2×Dic53—41no—
248C2×Dic53—84no—
249Dic106—84no—
250#88322—22no—
251#88322—41no—
252#88322—41no—
253#88322—41no—
254#31422—531nomaximal, Sylow
255C22×C53—22no—
256C22×C53—22no—
257C22×C53—22no—
258C22×C53—41no—
259C22×C53—41no—
260C22×C53—41no—
261C22×C53—41no—
262C22×C53#53611yescharacteristic
263C22×C53#54611yescharacteristic
264C22×C53#54711yescharacteristic
265C22×C53C22×D811yescharacteristic
266C22×C53C2×#13711yescharacteristic
267C22×C53#57711yescharacteristic
268C22×C53#57811yescharacteristic
269C4×C53—22no—
270C4×C53—22no—
271C4×C53—22no—
272C4×C53—41no—
273C4×C53—41no—
274C4×C53—81no—
275C4×C53—81no—
276C4×C53—81no—
277C4×C53—81no—
278C4×C53#53911yescharacteristic
279C4×C53#54111yescharacteristic
280C4×C53#55611yescharacteristic
281C4×C53#56311yescharacteristic
282Dic53—42no—
283Dic53—81no—
284Dic53—81no—
285Dic53—81no—
286#2496D5311yescharacteristic
287#2440—531no—
288#2099—531no—
289#2099—531no—
290#2086—531no—
291#2075—531no—
292#2075—531no—
293#2057—531no—
294#1990—531no—
295#1947—531no—
296#1942—531no—
297#1923—531no—
298#1918—531no—
299#1852—531no—
300#1823—531no—
301C2×C53—42no—
302C2×C53#268911yescharacteristic
303C2×C53#322611yescharacteristic
304C2×C53#324211yescharacteristic
305C2×C53#329411yescharacteristic
306C2×C53#331211yescharacteristic
307C2×C53#333411yescharacteristic
308C2×C53#335611yescharacteristic
309C2×C4⋊Q8—531no—
310C2×#126—21no—
311C2×C22⋊Q8—1061no—
312C22×C4⋊C4—531no—
313C2×#107—531no—
314C2×#107—531no—
315C2×#107—531no—
316C2×#107—531no—
317C2×C4⋊C8—531no—
318C2×C22⋊C8—531no—
319C2×C22⋊C8—531no—
320C2×C4×C8—531no—
321#400—21no—
322#398—1061no—
323#397—21no—
324#395—1061no—
325#389—531no—
326#389—531no—
327#389C2×D5311yescharacteristic
328#389C2×D5311yescharacteristic
329#387—21no—
330#385—1061no—
331#384—531no—
332#384C2×D5311yescharacteristic
333#380—531no—
334#380C2×D5311yescharacteristic
335#380C2×D5311yescharacteristic
336C4×C4⋊C4C2×D5311yescharacteristic
337C4×C22⋊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#340—531no—
349#340—531no—
350#336—531no—
351#336—531no—
352C53#3142211yescharacteristic, complemented, Sylow
353C22×Q8—1061no—
354C23×C4—1061no—
355C23×C4C22×D5311yescharacteristic
356C22×C8—531no—
357C22×C8—531no—
358C22×C8—531no—
359C22×C8—531no—
360C4⋊Q8—1062no—
361C4⋊Q8—2121no—
362#126—44no—
363C22⋊Q8—2124no—
364C2×C4⋊C4—21no—
365C2×C4⋊C4—21no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—531no—
369C2×C4⋊C4—531no—
370C2×C4⋊C4—531no—
371C2×C4⋊C4—531no—
372C2×C4⋊C4—1062no—
373C2×C4⋊C4—1061no—
374C2×C4⋊C4—1061no—
375C2×C4⋊C4—1061no—
376C2×C4⋊C4—1061no—
377C2×C4⋊C4—1061no—
378C2×C4⋊C4—1061no—
379C2×C4⋊C4—2121no—
380C2×C4⋊C4C22×D5311yescharacteristic
381C2×C4⋊C4C22×D5311yescharacteristic
382C2×C4⋊C4C22×D5311yescharacteristic
383C2×C4⋊C4C22×D5311yescharacteristic
384C2×C22⋊C4—21no—
385C2×C22⋊C4—21no—
386C2×C22⋊C4—1061no—
387C2×C22⋊C4C22⋊D5311yescharacteristic
388C2×C22⋊C4C22⋊D5311yescharacteristic
389C2×C42—21no—
390C2×C42C22×D5311yescharacteristic
391C2×C42C22×D5311yescharacteristic
392#107—1062no—
393#107—1062no—
394#107—2121no—
395#107—2121no—
396C4⋊C8—1062no—
397C22⋊C8—1062no—
398C22⋊C8—2121no—
399C4×C8—2121no—
400#95—21no—
401#95—21no—
402#95—21no—
403#95—21no—
404#95—21no—
405#95—1061no—
406#95—1061no—
407#95—1061no—
408#95C22⋊D5311yescharacteristic
409#95C22⋊D5311yescharacteristic
410C24C2×C22⋊D5311yescharacteristic
411C2×Q8—4241no—
412C2×Q8—2122no—
413C2×Q8—2122no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—1061no—
425C22×C4—1061no—
426C22×C4—1061no—
427C22×C4—1061no—
428C22×C4—4241no—
429C22×C4—2122no—
430C22×C4—2122no—
431C22×C4C2×C22⋊D5311yescharacteristic
432C22×C4C2×C22⋊D5311yescharacteristic
433C22×C4C2×C22⋊D5311yescharacteristic
434C22×C4C2×C22⋊D5311yescharacteristic
435C22×C4C2×C22⋊D5311yescharacteristic
436C22×C4C23×D5311yescharacteristic
437C2×C8—532no—
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—
448C2×C8—1061no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—42no—
458C4⋊C4—42no—
459C4⋊C4—42no—
460C4⋊C4—42no—
461C4⋊C4—532no—
462C4⋊C4—1061no—
463C4⋊C4—1061no—
464C4⋊C4—1061no—
465C4⋊C4—1061no—
466C4⋊C4—1061no—
467C4⋊C4—1061no—
468C4⋊C4—1061no—
469C4⋊C4—4241no—
470C4⋊C4—2122no—
471C4⋊C4—2122no—
472C4⋊C4—2122no—
473C4⋊C4—2122no—
474C4⋊C4—2122no—
475C22⋊C4—22no—
476C22⋊C4—42no—
477C22⋊C4—42no—
478C22⋊C4—2122no—
479C22⋊C4#1171992214yes—
480C42—21no—
481C42—21no—
482C42—21no—
483C42—21no—
484C42—42no—
485C23—21no—
486C23—21no—
487C23—21no—
488C23—21no—
489C23—21no—
490C23—21no—
491C23C2×#1171992212yes—
492C23C22×C22⋊D5311yescharacteristic
493Q8—4244no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
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—1062no—
523C2×C4—4244no—
524C2×C4—2121no—
525C2×C4—2121no—
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×C4C2×#1171992212yes—
536C2×C4C2×#1171992212yes—
537C2×C4#1313324712yes—
538C2×C4#41×D5311yescharacteristic
539C2×C4#41×D5311yescharacteristic
540C2×C4#1313326311yescharacteristic
541C2×C4#1313326411yescharacteristic
542C2×C4#1313326411yescharacteristic
543C2×C4#1313326411yescharacteristic
544C8—1062no—
545C8—2121no—
546C8—2121no—
547C8—2121no—
548C22—22no—
549C22—22no—
550C22—22no—
551C22—41no—
552C22—41no—
553C22—41no—
554C22—41no—
555C22#1681119411yescharacteristic
556C22#1681126111yescharacteristic
557C22#1681127211yescharacteristic
558C22C22×#1171992211yescharacteristic
559C22C2×#1313324711yescharacteristic
560C22#1681143111yescharacteristic
561C22#1681143811yescharacteristic
562C4—22no—
563C4—22no—
564C4—22no—
565C4—41no—
566C4—41no—
567C4—81no—
568C4—81no—
569C4—81no—
570C4—81no—
571C4—4241no—
572C4—4241no—
573C4—4241no—
574C4—2122no—
575C4#1681121411yescharacteristic
576C4#1681123011yescharacteristic
577C4#1681134011yescharacteristic
578C4#1681137511yescharacteristic
579C2—42no—
580C2#2028116011yescharacteristic
581C2#2028583311yescharacteristic
582C2#2028597711yescharacteristic
583C2#2028652211yescharacteristic
584C2#2028676011yescharacteristic
585C2#2028705811yescharacteristic
586C2#2028732411yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2496C2
socleC23×C53C22×D4
prime radicalC24×C53C2×D4
hypercenter#2496D53
hypocenterC53#31422
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2496

Complements

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

NH ≅ G/Ntype
2. #20276273C2semidirect
3. #20276272C2semidirect
4. #20276206C2semidirect
7. #20276043C2semidirect
8. #20275675C2semidirect
9. #20275473C2semidirect
10. #20275445C2semidirect
12. #20275286C2semidirect
352. C53#31422semidirect

Maximal subgroups

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

subgroupindex
#202784172
#202762732
#202762722
#202762062
#202761412
#202760432
#202756752
#202754732
#202754452
#202753232
#202752862
#202749022
#202747992
C53×#24962
#3142253