Finite Group Databasebeta

← #27416524

Subgroups — #27416524

Order 13568.

1228 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810778C211yesmaximal, characteristic, complemented
2C2×#16810631C211yesmaximal, characteristic, complemented
3#20278080C211yesmaximal, characteristic
4#20275863C212yesmaximal, complemented
5#20275723C212yesmaximal, complemented
6#20275149C211yesmaximal, characteristic, complemented
7#20275056C212yesmaximal, complemented
8#20275022C212yesmaximal, complemented
9#20275013C211yesmaximal, characteristic
10#20274810C211yesmaximal, characteristic, complemented
11C2×C53×#552C211yesmaximal, characteristic
12C23×Dic106C2211yescharacteristic
13C22×#11719925C2212yes—
14C22×#11719917C2211yescharacteristic
15C2×#13133196C2212yes—
16#16810778—24no—
17#16810631—24no—
18C2×#13133141C2212yes—
19C2×#13133140C2212yes—
20C2×#13133140C2212yes—
21C2×#13133140C2212yes—
22C2×C22⋊#88322C2211yescharacteristic
23C2×C22⋊#88322C2212yes—
24#16810543—21no—
25#16810535—21no—
26#16810529C2212yes—
27#16810503—21no—
28#16810495—21no—
29#16810483C2211yescharacteristic
30#16810385C2212yes—
31#16810385C2212yes—
32#16810385C2212yes—
33#16810378C2212yes—
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×Dic106—22no—
48C22×Dic106—24no—
49C22×Dic106—22no—
50C22×Dic106—21no—
51C22×Dic106—42no—
52C22×Dic106C2312yes—
53C23×Dic53—21no—
54C2×#11719925—24no—
55C2×#11719925—24no—
56C2×#11719925—42no—
57C2×#11719917—22no—
58C2×#11719917—42no—
59C22×#88322C2312yes—
60C22×#88322C2312yes—
61#13133196—24no—
62#13133196—42no—
63C2×C4⋊Dic53C2312yes—
64C2×#11719919—22no—
65C2×C22⋊Dic53—22no—
66C2×C22⋊Dic53—21no—
67#13133141—42no—
68#13133140—24no—
69#13133140—24no—
70#13133140—24no—
71C22⋊#88322—22no—
72C22⋊#88322—42no—
73#13133124—22no—
74#13133124—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×C53—22no—
85#123×C53—28no—
86C22⋊Q8×C53—22no—
87C22⋊Q8×C53—22no—
88C22⋊Q8×C53—22no—
89C22⋊Q8×C53D414yes—
90C4×Q8×C53—22no—
91C4×Q8×C53—22no—
92C4×D4×C53—22no—
93C4×D4×C53—22no—
94#117×C53—28no—
95#117×C53—22no—
96#117×C53D414yes—
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—22no—
127C22×C4×C53—22no—
128C22×C4×C53—24no—
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—42no—
138C22×C4×C53—42no—
139C22×C4×C53C2×D411yescharacteristic
140C22×C4×C53C2×D411yescharacteristic
141C22×C4×C53C2×D411yescharacteristic
142C22×C4×C53C2×D411yescharacteristic
143C22×C4×C53C2×D412yes—
144C22×C4×C53C2×D412yes—
145C22×C4×C53C2411yescharacteristic
146C22×Dic53—22no—
147C22×Dic53—22no—
148C22×Dic53—44no—
149C22×Dic53—42no—
150C2×Dic106—28no—
151C2×Dic106—22no—
152C2×Dic106—22no—
153C2×Dic106—22no—
154C2×Dic106—22no—
155C2×Dic106—22no—
156C2×Dic106—48no—
157C2×Dic106—44no—
158C2×Dic106—44no—
159C2×Dic106—42no—
160C2×Dic106#4114yes—
161C4⋊C4×C53—24no—
162C4⋊C4×C53—24no—
163C4⋊C4×C53—24no—
164C4⋊C4×C53—24no—
165C4⋊C4×C53—24no—
166C4⋊C4×C53—24no—
167C4⋊C4×C53—22no—
168C4⋊C4×C53—21no—
169C4⋊C4×C53—21no—
170C4⋊C4×C53—21no—
171C4⋊C4×C53C2×D412yes—
172C4⋊C4×C53C2×D412yes—
173C4⋊C4×C53C2×D412yes—
174C22⋊C4×C53—24no—
175C22⋊C4×C53—24no—
176C22⋊C4×C53—24no—
177C22⋊C4×C53—22no—
178C22⋊C4×C53—22no—
179C42×C53—24no—
180C42×C53—24no—
181C42×C53—21no—
182C42×C53—21no—
183C22⋊Dic53—44no—
184C22⋊Dic53—42no—
185#11719925—48no—
186C4⋊Dic53—22no—
187C4⋊Dic53—22no—
188#11719919—44no—
189#11719917—44no—
190C2×#88322—22no—
191C2×#88322—22no—
192C2×#88322—22no—
193C2×#88322—22no—
194C2×#88322—22no—
195C2×#88322—22no—
196C23×C53—21no—
197C23×C53—22no—
198C23×C53—22no—
199C23×C53—21no—
200C23×C53—21no—
201C23×C53—21no—
202C23×C53—22no—
203C23×C53—22no—
204C23×C53#12012yes—
205C23×C53#13712yes—
206C23×C53C22×D411yescharacteristic
207Q8×C53—24no—
208Q8×C53—41no—
209Q8×C53—41no—
210D4×C53—44no—
211C2×C4×C53—22no—
212C2×C4×C53—22no—
213C2×C4×C53—22no—
214C2×C4×C53—24no—
215C2×C4×C53—22no—
216C2×C4×C53—22no—
217C2×C4×C53—21no—
218C2×C4×C53—22no—
219C2×C4×C53—22no—
220C2×C4×C53—21no—
221C2×C4×C53—21no—
222C2×C4×C53—22no—
223C2×C4×C53—21no—
224C2×C4×C53—21no—
225C2×C4×C53—22no—
226C2×C4×C53—48no—
227C2×C4×C53—42no—
228C2×C4×C53—41no—
229C2×C4×C53—41no—
230C2×C4×C53—41no—
231C2×C4×C53—41no—
232C2×C4×C53—42no—
233C2×C4×C53—41no—
234C2×C4×C53—41no—
235C2×C4×C53—42no—
236C2×C4×C53#12012yes—
237C2×C4×C53C22×D411yescharacteristic
238C2×C4×C53C22×D411yescharacteristic
239C2×C4×C53C2×#4112yes—
240C2×C4×C53H(5,𝔽2)12yes—
241C2×Dic53—24no—
242C2×Dic53—48no—
243C2×Dic53—42no—
244C2×Dic53—42no—
245C2×Dic53—42no—
246C2×Dic53—42no—
247C2×Dic53—42no—
248Dic106—24no—
249Dic106—24no—
250Dic106—48no—
251Dic106—42no—
252Dic106—42no—
253#88322—42no—
254#88322—42no—
255#23983—531nomaximal, Sylow
256C22×C53—22no—
257C22×C53—22no—
258C22×C53—22no—
259C22×C53—22no—
260C22×C53—24no—
261C22×C53—41no—
262C22×C53—41no—
263C22×C53—42no—
264C22×C53—42no—
265C22×C53—42no—
266C22×C53C2×#12011yescharacteristic
267C22×C53#54612yes—
268C22×C53C2×#13711yescharacteristic
269C22×C53#57511yescharacteristic
270C22×C53#57812yes—
271C4×C53—22no—
272C4×C53—41no—
273C4×C53—42no—
274C4×C53—41no—
275C4×C53—42no—
276C4×C53—42no—
277C4×C53—41no—
278C4×C53—42no—
279C4×C53#52511yescharacteristic
280C4×C53#53511yescharacteristic
281C4×C53#54612yes—
282Dic53—44no—
283Dic53—82no—
284C2×#552D5311yescharacteristic
285C2×#451—531no—
286C2×#419—531no—
287#2409—531no—
288#2026—532no—
289#2001—532no—
290#1894—531no—
291#1878—532no—
292#1872—532no—
293#1870—531no—
294#1825—531no—
295C2×C53—42no—
296C2×C53—42no—
297C2×C53#240411yescharacteristic
298C2×C53#300711yescharacteristic
299C2×C53#301111yescharacteristic
300C2×C53#331212yes—
301C2×C53#332712yes—
302C23×Q8—531no—
303C22×Q16—532no—
304C22×M16—531no—
305#552—24no—
306#552—24no—
307C2×#126—21no—
308C2×#125—21no—
309C2×#125C2×D5311yescharacteristic
310C2×#123—21no—
311C2×C22⋊Q8—532no—
312C2×C22⋊Q8C2×D5311yescharacteristic
313C2×C22⋊Q8C2×D5311yescharacteristic
314C2×C4×Q8C2×D5311yescharacteristic
315C2×C4×D4C2×D5311yescharacteristic
316C2×#117—21no—
317C2×#117C2×D5311yescharacteristic
318C22×C4⋊C4C2×D5311yescharacteristic
319#451—1064no—
320#419—1064no—
321C2×#103—532no—
322C2×#103—532no—
323C2×#103—532no—
324C2×#103—532no—
325C2×C22⋊C8—531no—
326C2×C22⋊C8—532no—
327#395—1061no—
328#393—1061no—
329#391—532no—
330#386—1061no—
331#385—1061no—
332#380—531no—
333#340—532no—
334#340—532no—
335#340—532no—
336#336—532no—
337C53#2398311yescharacteristic, complemented, Sylow
338C22×Q8—532no—
339C22×Q8—1064no—
340C22×Q8—1061no—
341C22×Q8—1062no—
342C22×Q8—1062no—
343C22×Q8—2122no—
344C22×Q8C22×D5311yescharacteristic
345C22×D4C22⋊D5311yescharacteristic
346C23×C4—21no—
347C23×C4—1061no—
348C23×C4C22×D5311yescharacteristic
349C23×C4C22×D5311yescharacteristic
350C2×Q16—1064no—
351C2×Q16—1064no—
352C2×Q16—2122no—
353C2×M16—1062no—
354C2×M16—2122no—
355C22×C8—532no—
356C22×C8—532no—
357#126—28no—
358#125—28no—
359#125—22no—
360#125—22no—
361#123—28no—
362C22⋊Q8—22no—
363C22⋊Q8—22no—
364C22⋊Q8—22no—
365C22⋊Q8—1064no—
366C22⋊Q8—2122no—
367C22⋊Q8C22⋊D5314yes—
368C4×Q8—22no—
369C4×Q8—22no—
370C4×D4—22no—
371C4×D4—22no—
372#117—28no—
373#117—22no—
374#117C22⋊D5314yes—
375C2×C4⋊C4—21no—
376C2×C4⋊C4—21no—
377C2×C4⋊C4—21no—
378C2×C4⋊C4—21no—
379C2×C4⋊C4—21no—
380C2×C4⋊C4—21no—
381C2×C4⋊C4—22no—
382C2×C4⋊C4—22no—
383C2×C4⋊C4—532no—
384C2×C4⋊C4—1062no—
385C2×C4⋊C4C22⋊D5311yescharacteristic
386C2×C4⋊C4C22×D5311yescharacteristic
387C2×C4⋊C4C22×D5311yescharacteristic
388C2×C4⋊C4C22×D5311yescharacteristic
389C2×C22⋊C4—21no—
390C2×C22⋊C4—21no—
391C2×C22⋊C4—21no—
392C2×C22⋊C4—1062no—
393C2×C22⋊C4—1061no—
394C2×C22⋊C4C22⋊D5311yescharacteristic
395C2×C22⋊C4C22⋊D5311yescharacteristic
396C2×C42—21no—
397C2×C42—21no—
398C2×C42C22×D5311yescharacteristic
399#103—1064no—
400#103—1064no—
401#103—1064no—
402#103—2122no—
403C22⋊C8—1062no—
404C22⋊C8—2122no—
405#95—1061no—
406#95—1062no—
407#95—1062no—
408C24C2×C22⋊D5311yescharacteristic
409C24C2×C22⋊D5311yescharacteristic
410C2×Q8—21no—
411C2×Q8—22no—
412C2×Q8—22no—
413C2×Q8—534no—
414C2×Q8—1068no—
415C2×Q8—1062no—
416C2×Q8—1062no—
417C2×Q8—1062no—
418C2×Q8—1062no—
419C2×Q8—1062no—
420C2×Q8—2124no—
421C2×Q8—2128no—
422C2×Q8—2124no—
423C2×Q8—2122no—
424C2×Q8C2×C22⋊D5312yes—
425C2×D4—22no—
426C2×D4—22no—
427C2×D4—41no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—1062no—
442C22×C4—1062no—
443C22×C4—2124no—
444C22×C4—2122no—
445C22×C4C2×C22⋊D5311yescharacteristic
446C22×C4C2×C22⋊D5311yescharacteristic
447C22×C4C2×C22⋊D5311yescharacteristic
448C22×C4C2×C22⋊D5311yescharacteristic
449C22×C4C2×C22⋊D5312yes—
450C22×C4C2×C22⋊D5312yes—
451C22×C4C23×D5311yescharacteristic
452Q16—2128no—
453M16—2124no—
454C2×C8—1062no—
455C2×C8—1062no—
456C2×C8—1062no—
457C2×C8—1062no—
458C2×C8—1062no—
459C2×C8—1062no—
460C4⋊C4—24no—
461C4⋊C4—24no—
462C4⋊C4—24no—
463C4⋊C4—24no—
464C4⋊C4—24no—
465C4⋊C4—24no—
466C4⋊C4—22no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—1062no—
471C4⋊C4—1062no—
472C4⋊C4—2124no—
473C4⋊C4C2×C22⋊D5312yes—
474C4⋊C4C2×C22⋊D5312yes—
475C4⋊C4C2×C22⋊D5312yes—
476C22⋊C4—24no—
477C22⋊C4—24no—
478C22⋊C4—24no—
479C22⋊C4—22no—
480C22⋊C4—22no—
481C22⋊C4—2124no—
482C22⋊C4—2122no—
483C42—24no—
484C42—24no—
485C42—21no—
486C42—21no—
487C23—21no—
488C23—21no—
489C23—21no—
490C23—21no—
491C23—22no—
492C23—22no—
493C23—22no—
494C23—22no—
495C23#1313318612yes—
496C23#1313324712yes—
497C23C22×C22⋊D5311yescharacteristic
498Q8—24no—
499Q8—41no—
500Q8—41no—
501Q8—1064no—
502Q8—1064no—
503Q8—2128no—
504Q8—2122no—
505Q8—2122no—
506D4—44no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—24no—
511C2×C4—21no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—22no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—48no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—42no—
532C2×C4—1064no—
533C2×C4—2122no—
534C2×C4—2122no—
535C2×C4—2122no—
536C2×C4—2128no—
537C2×C4—2122no—
538C2×C4—2122no—
539C2×C4#1313318612yes—
540C2×C4C22×C22⋊D5311yescharacteristic
541C2×C4C22×C22⋊D5311yescharacteristic
542C2×C4#41×D5312yes—
543C2×C4#1313326312yes—
544C8—2122no—
545C8—2122no—
546C22—22no—
547C22—24no—
548C22—22no—
549C22—22no—
550C22—22no—
551C22—42no—
552C22—42no—
553C22—41no—
554C22—42no—
555C22—41no—
556C22C2×#1313318611yescharacteristic
557C22#1681126112yes—
558C22C2×#1313324711yescharacteristic
559C22#1681142111yescharacteristic
560C22#1681143812yes—
561C4—22no—
562C4—41no—
563C4—42no—
564C4—41no—
565C4—42no—
566C4—42no—
567C4—41no—
568C4—42no—
569C4—4242no—
570C4—2124no—
571C4#1681113011yescharacteristic
572C4#1681118711yescharacteristic
573C4#1681126112yes—
574C2—42no—
575C2—42no—
576C2#2027803011yescharacteristic
577C2#2028397511yescharacteristic
578C2#2028401011yescharacteristic
579C2#2028676012yes—
580C2#2028697712yes—
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#23983
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×#16810778C2semidirect
2. C2×#16810631C2semidirect
4. #20275863C2semidirect
5. #20275723C2semidirect
6. #20275149C2semidirect
7. #20275056C2semidirect
8. #20275022C2semidirect
10. #20274810C2semidirect
337. C53#23983semidirect

Maximal subgroups

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

subgroupindex
C2×#168107782
C2×#168106312
#202780802
#202758632
#202757232
#202751492
#202750562
#202750222
#202750132
#202748102
C2×C53×#5522
#2398353