Finite Group Databasebeta

← #27437451

Subgroups — #27437451

Order 13568.

1126 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810993C211yesmaximal, characteristic, complemented
2C2×#16810833C211yesmaximal, characteristic, complemented
3#20279668C211yesmaximal, characteristic
4#20275723C212yesmaximal, complemented
5#20275723C212yesmaximal, complemented
6#20275673C211yesmaximal, characteristic, complemented
7#20275514C211yesmaximal, characteristic
8#20275027C212yesmaximal, complemented
9#20275022C212yesmaximal, complemented
10#20274545C211yesmaximal, characteristic, complemented
11C2×C53×#552C211yesmaximal, characteristic
12C22×#11719925C2212yes—
13C2×#13133217C2211yescharacteristic
14C2×#13133196C2212yes—
15C2×#13133196C2212yes—
16#16810993—24no—
17#16810833—24no—
18C2×C4⋊#88322C2211yescharacteristic
19C2×#13133141C2212yes—
20C2×#13133140C2212yes—
21C2×#13133140C2212yes—
22C2×#13133140C2212yes—
23C2×C22⋊#88322C2212yes—
24C2×C22⋊#88322C2212yes—
25C2×#13133128C2211yescharacteristic
26#16810559—21no—
27#16810559—21no—
28#16810526—21no—
29#16810492—21no—
30C4×C4⋊Dic53C2211yescharacteristic
31#16810385C2212yes—
32#16810385C2212yes—
33#16810385C2212yes—
34C53×#552—24no—
35C53×#552—24no—
36C2×#126×C53—21no—
37C2×#125×C53—21no—
38C2×#125×C53C2211yescharacteristic
39C2×#123×C53—21no—
40C2×C22⋊Q8×C53C2211yescharacteristic
41C2×C22⋊Q8×C53C2211yescharacteristic
42C2×C4×Q8×C53C2211yescharacteristic
43C2×C4×D4×C53C2211yescharacteristic
44C2×#117×C53—21no—
45C2×#117×C53C2211yescharacteristic
46C22×C4⋊C4×C53C2211yescharacteristic
47C22×Dic106C2312yes—
48C2×#11719925—24no—
49C2×#11719925—42no—
50C2×#11719925—42no—
51C22×#88322C2312yes—
52C22×#88322C2312yes—
53#13133217—22no—
54#13133217—22no—
55#13133196—24no—
56#13133196—24no—
57#13133196—42no—
58#13133196—42no—
59C2×C4⋊Dic53—21no—
60C2×C4⋊Dic53—21no—
61C2×C4⋊Dic53C2312yes—
62C2×#11719919—22no—
63C2×#11719919—22no—
64C2×C22⋊Dic53—22no—
65C2×C22⋊Dic53—22no—
66C2×C4×Dic53—21no—
67C4⋊#88322—22no—
68#13133141—24no—
69#13133140—24no—
70#13133140—24no—
71#13133140—24no—
72C22⋊#88322—42no—
73C22⋊#88322—42no—
74#13133128—22no—
75#13133124—21no—
76C22×Q8×C53C2311yescharacteristic
77C22×D4×C53D411yescharacteristic
78C23×C4×C53—21no—
79C23×C4×C53C2311yescharacteristic
80C23×C4×C53C2311yescharacteristic
81#126×C53—28no—
82#125×C53—28no—
83#125×C53—22no—
84#125×C53D414yes—
85#123×C53—28no—
86C22⋊Q8×C53—22no—
87C22⋊Q8×C53—22no—
88C22⋊Q8×C53—22no—
89C22⋊Q8×C53—22no—
90C4×Q8×C53—22no—
91C4×Q8×C53D414yes—
92C4×D4×C53—22no—
93C4×D4×C53—22no—
94#117×C53—28no—
95#117×C53—22no—
96#117×C53—22no—
97C2×C4⋊C4×C53—22no—
98C2×C4⋊C4×C53—22no—
99C2×C4⋊C4×C53—21no—
100C2×C4⋊C4×C53—21no—
101C2×C4⋊C4×C53—21no—
102C2×C4⋊C4×C53—21no—
103C2×C4⋊C4×C53—21no—
104C2×C4⋊C4×C53—21no—
105C2×C4⋊C4×C53D411yescharacteristic
106C2×C4⋊C4×C53C2311yescharacteristic
107C2×C4⋊C4×C53C2311yescharacteristic
108C2×C4⋊C4×C53C2311yescharacteristic
109C2×C22⋊C4×C53—21no—
110C2×C22⋊C4×C53—21no—
111C2×C22⋊C4×C53—21no—
112C2×C22⋊C4×C53D411yescharacteristic
113C2×C22⋊C4×C53D411yescharacteristic
114C2×C42×C53—21no—
115C2×C42×C53—21no—
116C2×C42×C53C2311yescharacteristic
117C24×C53C2×D411yescharacteristic
118C24×C53C2×D411yescharacteristic
119C2×Q8×C53—21no—
120C2×Q8×C53—22no—
121C2×Q8×C53—22no—
122C2×Q8×C53C2×D412yes—
123C2×D4×C53—22no—
124C2×D4×C53—22no—
125C2×D4×C53—41no—
126C22×C4×C53—24no—
127C22×C4×C53—21no—
128C22×C4×C53—21no—
129C22×C4×C53—21no—
130C22×C4×C53—21no—
131C22×C4×C53—21no—
132C22×C4×C53—21no—
133C22×C4×C53—21no—
134C22×C4×C53—21no—
135C22×C4×C53—22no—
136C22×C4×C53—22no—
137C22×C4×C53—22no—
138C22×C4×C53—22no—
139C22×C4×C53—42no—
140C22×C4×C53—42no—
141C22×C4×C53C2×D411yescharacteristic
142C22×C4×C53C2×D411yescharacteristic
143C22×C4×C53C2×D411yescharacteristic
144C22×C4×C53C2×D411yescharacteristic
145C22×C4×C53C2411yescharacteristic
146C22×Dic53—22no—
147C22×Dic53—22no—
148C2×Dic106—22no—
149C2×Dic106—22no—
150C2×Dic106—22no—
151C2×Dic106—22no—
152C2×Dic106—22no—
153C2×Dic106—22no—
154C4⋊C4×C53—24no—
155C4⋊C4×C53—24no—
156C4⋊C4×C53—24no—
157C4⋊C4×C53—24no—
158C4⋊C4×C53—24no—
159C4⋊C4×C53—24no—
160C4⋊C4×C53—22no—
161C4⋊C4×C53—21no—
162C4⋊C4×C53—21no—
163C4⋊C4×C53—21no—
164C4⋊C4×C53C2×D412yes—
165C4⋊C4×C53C2×D412yes—
166C4⋊C4×C53C2×D412yes—
167C22⋊C4×C53—24no—
168C22⋊C4×C53—24no—
169C22⋊C4×C53—24no—
170C22⋊C4×C53—22no—
171C22⋊C4×C53—22no—
172C42×C53—24no—
173C42×C53—24no—
174C42×C53C2×D412yes—
175C42×C53C2×D412yes—
176C22⋊Dic53—44no—
177C22⋊Dic53—44no—
178#11719925—48no—
179C4⋊Dic53—22no—
180C4⋊Dic53—42no—
181C4⋊Dic53—42no—
182C4⋊Dic53#4114yes—
183#11719919—44no—
184#11719919—44no—
185C4×Dic53—42no—
186C2×#88322—22no—
187C2×#88322—22no—
188C2×#88322—22no—
189C2×#88322—22no—
190C2×#88322—22no—
191C2×#88322—22no—
192C23×C53—21no—
193C23×C53—22no—
194C23×C53—21no—
195C23×C53—22no—
196C23×C53—21no—
197C23×C53—22no—
198C23×C53—21no—
199C23×C53—21no—
200C23×C53—22no—
201C23×C53#13712yes—
202C23×C53C22×D411yescharacteristic
203Q8×C53—22no—
204Q8×C53—22no—
205Q8×C53—42no—
206D4×C53—44no—
207C2×C4×C53—21no—
208C2×C4×C53—21no—
209C2×C4×C53—22no—
210C2×C4×C53—22no—
211C2×C4×C53—22no—
212C2×C4×C53—21no—
213C2×C4×C53—21no—
214C2×C4×C53—22no—
215C2×C4×C53—22no—
216C2×C4×C53—22no—
217C2×C4×C53—22no—
218C2×C4×C53—48no—
219C2×C4×C53—42no—
220C2×C4×C53—41no—
221C2×C4×C53—41no—
222C2×C4×C53—41no—
223C2×C4×C53—41no—
224C2×C4×C53—42no—
225C2×C4×C53—41no—
226C2×C4×C53—41no—
227C2×C4×C53—41no—
228C2×C4×C53—41no—
229C2×C4×C53—42no—
230C2×C4×C53—42no—
231C2×C4×C53#12012yes—
232C2×C4×C53#12012yes—
233C2×C4×C53C22×D411yescharacteristic
234C2×C4×C53C22×D411yescharacteristic
235C2×C4×C53C2×#4112yes—
236C2×C4×C53H(5,𝔽2)12yes—
237C2×Dic53—24no—
238C2×Dic53—42no—
239C2×Dic53—42no—
240C2×Dic53—42no—
241C2×Dic53—42no—
242C2×Dic53—42no—
243Dic106—42no—
244Dic106—42no—
245Dic106—42no—
246Dic106—42no—
247#88322—42no—
248#88322—42no—
249#25584—531nomaximal, Sylow
250C22×C53—22no—
251C22×C53—24no—
252C22×C53—41no—
253C22×C53—41no—
254C22×C53—42no—
255C22×C53—41no—
256C22×C53—41no—
257C22×C53—41no—
258C22×C53—42no—
259C22×C53—42no—
260C22×C53C2×#12011yescharacteristic
261C22×C53#54612yes—
262C22×C53C2×#13711yescharacteristic
263C22×C53#57511yescharacteristic
264C22×C53#57812yes—
265C4×C53—22no—
266C4×C53—22no—
267C4×C53—41no—
268C4×C53—41no—
269C4×C53—42no—
270C4×C53—42no—
271C4×C53—42no—
272C4×C53—42no—
273C4×C53#53211yescharacteristic
274C4×C53#53511yescharacteristic
275C4×C53#55012yes—
276Dic53—44no—
277Dic53—82no—
278C2×#552D5311yescharacteristic
279C2×#491—531no—
280C2×#462—531no—
281#2542—531no—
282#2001—532no—
283#2001—532no—
284#1989—531no—
285#1955—531no—
286#1873—532no—
287#1872—532no—
288#1763—531no—
289C2×C53—42no—
290C2×C53—81no—
291C2×C53#243111yescharacteristic
292C2×C53#300711yescharacteristic
293C2×C53#301111yescharacteristic
294C2×C53#331212yes—
295C2×C53#331512yes—
296C22×Q16—532no—
297#552—24no—
298#552—24no—
299C2×C4⋊Q8—531no—
300C2×#126—21no—
301C2×#125—21no—
302C2×#125C2×D5311yescharacteristic
303C2×#123—21no—
304C2×C22⋊Q8—532no—
305C2×C22⋊Q8—532no—
306C2×C22⋊Q8C2×D5311yescharacteristic
307C2×C22⋊Q8C2×D5311yescharacteristic
308C2×C4×Q8C2×D5311yescharacteristic
309C2×C4×D4C2×D5311yescharacteristic
310C2×#117—21no—
311C2×#117C2×D5311yescharacteristic
312C22×C4⋊C4C2×D5311yescharacteristic
313#491—1064no—
314#462—1064no—
315C2×C4⋊C8—531no—
316C2×#103—532no—
317C2×#103—532no—
318C2×#103—532no—
319C2×#103—532no—
320C2×C22⋊C8—532no—
321C2×C22⋊C8—532no—
322C2×#97—531no—
323#399—1061no—
324#399—1061no—
325#390—1061no—
326#384—1061no—
327C4×C4⋊C4—531no—
328#340—532no—
329#340—532no—
330#340—532no—
331C53#2558411yescharacteristic, complemented, Sylow
332C22×Q8—532no—
333C22×Q8C22×D5311yescharacteristic
334C22×D4C22⋊D5311yescharacteristic
335C23×C4—21no—
336C23×C4C22×D5311yescharacteristic
337C23×C4C22×D5311yescharacteristic
338C2×Q16—1064no—
339C2×Q16—2122no—
340C2×Q16—2122no—
341C22×C8—532no—
342C22×C8—532no—
343C4⋊Q8—1062no—
344C4⋊Q8—1062no—
345#126—28no—
346#125—28no—
347#125—22no—
348#125C22⋊D5314yes—
349#123—28no—
350C22⋊Q8—22no—
351C22⋊Q8—22no—
352C22⋊Q8—22no—
353C22⋊Q8—22no—
354C22⋊Q8—1064no—
355C22⋊Q8—1064no—
356C22⋊Q8—2122no—
357C22⋊Q8—2122no—
358C4×Q8—22no—
359C4×Q8C22⋊D5314yes—
360C4×D4—22no—
361C4×D4—22no—
362#117—28no—
363#117—22no—
364#117—22no—
365C2×C4⋊C4—21no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—21no—
370C2×C4⋊C4—21no—
371C2×C4⋊C4—22no—
372C2×C4⋊C4—22no—
373C2×C4⋊C4—532no—
374C2×C4⋊C4—1062no—
375C2×C4⋊C4—1062no—
376C2×C4⋊C4—1061no—
377C2×C4⋊C4—1061no—
378C2×C4⋊C4C22⋊D5311yescharacteristic
379C2×C4⋊C4C22×D5311yescharacteristic
380C2×C4⋊C4C22×D5311yescharacteristic
381C2×C4⋊C4C22×D5311yescharacteristic
382C2×C22⋊C4—21no—
383C2×C22⋊C4—21no—
384C2×C22⋊C4—21no—
385C2×C22⋊C4—1062no—
386C2×C22⋊C4—1062no—
387C2×C22⋊C4C22⋊D5311yescharacteristic
388C2×C22⋊C4C22⋊D5311yescharacteristic
389C2×C42—21no—
390C2×C42—21no—
391C2×C42—1061no—
392C2×C42C22×D5311yescharacteristic
393C4⋊C8—1062no—
394#103—1064no—
395#103—1064no—
396#103—1064no—
397#103—1064no—
398C22⋊C8—2122no—
399C22⋊C8—2122no—
400#97—1062no—
401#95—1061no—
402C24C2×C22⋊D5311yescharacteristic
403C24C2×C22⋊D5311yescharacteristic
404C2×Q8—21no—
405C2×Q8—22no—
406C2×Q8—22no—
407C2×Q8—1062no—
408C2×Q8—1062no—
409C2×Q8—1062no—
410C2×Q8—1062no—
411C2×Q8—1062no—
412C2×Q8—1062no—
413C2×Q8C2×C22⋊D5312yes—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—41no—
417C22×C4—24no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—1062no—
433C22×C4—1062no—
434C22×C4C2×C22⋊D5311yescharacteristic
435C22×C4C2×C22⋊D5311yescharacteristic
436C22×C4C2×C22⋊D5311yescharacteristic
437C22×C4C2×C22⋊D5311yescharacteristic
438C22×C4C23×D5311yescharacteristic
439Q16—2128no—
440C2×C8—1062no—
441C2×C8—1062no—
442C2×C8—1062no—
443C2×C8—1062no—
444C2×C8—1062no—
445C2×C8—1062no—
446C4⋊C4—24no—
447C4⋊C4—24no—
448C4⋊C4—24no—
449C4⋊C4—24no—
450C4⋊C4—24no—
451C4⋊C4—24no—
452C4⋊C4—22no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—534no—
457C4⋊C4—1062no—
458C4⋊C4—2124no—
459C4⋊C4—2124no—
460C4⋊C4—2122no—
461C4⋊C4—2122no—
462C4⋊C4C2×C22⋊D5312yes—
463C4⋊C4C2×C22⋊D5312yes—
464C4⋊C4C2×C22⋊D5312yes—
465C22⋊C4—24no—
466C22⋊C4—24no—
467C22⋊C4—24no—
468C22⋊C4—22no—
469C22⋊C4—22no—
470C22⋊C4—2124no—
471C22⋊C4—2124no—
472C42—24no—
473C42—24no—
474C42—2122no—
475C42C2×C22⋊D5312yes—
476C42C2×C22⋊D5312yes—
477C23—21no—
478C23—21no—
479C23—22no—
480C23—22no—
481C23—21no—
482C23—21no—
483C23—21no—
484C23—22no—
485C23—22no—
486C23#1313324712yes—
487C23C22×C22⋊D5311yescharacteristic
488Q8—22no—
489Q8—22no—
490Q8—42no—
491Q8—2122no—
492Q8—2122no—
493Q8—2122no—
494Q8—2122no—
495D4—44no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—22no—
499C2×C4—21no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—21no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—48no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—1064no—
521C2×C4—2122no—
522C2×C4—2122no—
523C2×C4—2122no—
524C2×C4—2122no—
525C2×C4—2122no—
526C2×C4#1313318612yes—
527C2×C4#1313318612yes—
528C2×C4C22×C22⋊D5311yescharacteristic
529C2×C4C22×C22⋊D5311yescharacteristic
530C2×C4#41×D5312yes—
531C2×C4#1313326312yes—
532C8—2122no—
533C8—2122no—
534C22—24no—
535C22—22no—
536C22—41no—
537C22—42no—
538C22—41no—
539C22—42no—
540C22—41no—
541C22—41no—
542C22—42no—
543C22—41no—
544C22C2×#1313318611yescharacteristic
545C22#1681126112yes—
546C22C2×#1313324711yescharacteristic
547C22#1681142111yescharacteristic
548C22#1681143812yes—
549C4—22no—
550C4—22no—
551C4—41no—
552C4—41no—
553C4—42no—
554C4—42no—
555C4—42no—
556C4—42no—
557C4—4242no—
558C4—2124no—
559C4#1681116911yescharacteristic
560C4#1681118711yescharacteristic
561C4#1681129212yes—
562C2—42no—
563C2—81no—
564C2#2027833011yescharacteristic
565C2#2028397511yescharacteristic
566C2#2028401011yescharacteristic
567C2#2028676012yes—
568C2#2028680512yes—
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC2×C53×#552C2
socleC23×C53C22×D4
prime radicalC22×D4×C53D4
hypercenterC2×#552D53
hypocenterC53#25584
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#552

Complements

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

NH ≅ G/Ntype
1. C2×#16810993C2semidirect
2. C2×#16810833C2semidirect
4. #20275723C2semidirect
5. #20275723C2semidirect
6. #20275673C2semidirect
8. #20275027C2semidirect
9. #20275022C2semidirect
10. #20274545C2semidirect
331. C53#25584semidirect

Maximal subgroups

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

subgroupindex
C2×#168109932
C2×#168108332
#202796682
#202757232
#202756732
#202755142
#202750272
#202750222
#202745452
C2×C53×#5522
#2558453