Finite Group Databasebeta

← #27438801

Subgroups — #27438801

Order 13568.

1122 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×#16811033C211yesmaximal, characteristic, complemented
2C2×#16810913C211yesmaximal, characteristic, complemented
3#20279707C211yesmaximal, characteristic, complemented
4#20275901C212yesmaximal, complemented
5#20275894C212yesmaximal, complemented
6#20275673C211yesmaximal, characteristic, complemented
7#20275514C211yesmaximal, characteristic, complemented
8#20274914C212yesmaximal, complemented
9#20274908C212yesmaximal, complemented
10#20274540C211yesmaximal, characteristic, complemented
11C2×C53×#553C211yesmaximal, characteristic
12C2×#13133217C2211yescharacteristic
13C2×#13133204—21no—
14#16811033—24no—
15#16810913—24no—
16C2×#13133150C2212yes—
17C2×#13133148C2212yes—
18C2×#13133148C2212yes—
19C2×C4⋊#88322C2211yescharacteristic
20C2×#13133138C2212yes—
21C2×#13133138C2212yes—
22C2×C22⋊#88322C2212yes—
23C2×C22⋊#88322C2212yes—
24C2×#13133128C2211yescharacteristic
25#16810548—21no—
26#16810548—21no—
27#16810528—21no—
28#16810483C2212yes—
29#16810483C2212yes—
30C4×C4⋊Dic53C2211yescharacteristic
31#16810385C2212yes—
32#16810385C2212yes—
33#16810385C2212yes—
34C53×#553—24no—
35C53×#553—24no—
36C2×#126×C53—21no—
37C2×#125×C53C2211yescharacteristic
38C2×#123×C53—21no—
39C2×#121×C53—21no—
40C2×#121×C53C2211yescharacteristic
41C2×#121×C53C2211yescharacteristic
42C2×C4×D4×C53—21no—
43C2×C4×D4×C53C2211yescharacteristic
44C2×C4×D4×C53C2211yescharacteristic
45C2×#117×C53C2211yescharacteristic
46C22×C4⋊C4×C53C2211yescharacteristic
47C22×Dic106—21no—
48C22×#88322C2312yes—
49C22×#88322C2312yes—
50#13133217—22no—
51#13133217—22no—
52#13133204—44no—
53C2×C4⋊Dic53—21no—
54C2×C4⋊Dic53C2312yes—
55C2×C4⋊Dic53C2312yes—
56C2×C22⋊Dic53—22no—
57C2×C22⋊Dic53—22no—
58C2×C4×Dic53—21no—
59#13133150—24no—
60#13133148—24no—
61#13133148—24no—
62C4⋊#88322—22no—
63#13133138—24no—
64#13133138—24no—
65C22⋊#88322—24no—
66C22⋊#88322—24no—
67#13133128—22no—
68#13133124—22no—
69#13133124—22no—
70#13133124—21no—
71C22×D4×C53—21no—
72C22×D4×C53—21no—
73C22×D4×C53D411yescharacteristic
74C22×D4×C53C2311yescharacteristic
75C23×C4×C53—21no—
76C23×C4×C53—21no—
77C23×C4×C53C2311yescharacteristic
78C23×C4×C53C2311yescharacteristic
79#126×C53—28no—
80#125×C53—22no—
81#125×C53D414yes—
82#123×C53—28no—
83#121×C53—28no—
84#121×C53—22no—
85#121×C53—22no—
86#121×C53—22no—
87#121×C53—22no—
88C4×D4×C53—28no—
89C4×D4×C53—22no—
90C4×D4×C53—22no—
91C4×D4×C53—22no—
92C4×D4×C53D414yes—
93#117×C53—22no—
94#117×C53—22no—
95C2×C4⋊C4×C53—22no—
96C2×C4⋊C4×C53—22no—
97C2×C4⋊C4×C53—21no—
98C2×C4⋊C4×C53—21no—
99C2×C4⋊C4×C53—21no—
100C2×C4⋊C4×C53D411yescharacteristic
101C2×C4⋊C4×C53C2311yescharacteristic
102C2×C4⋊C4×C53C2311yescharacteristic
103C2×C4⋊C4×C53C2311yescharacteristic
104C2×C22⋊C4×C53—21no—
105C2×C22⋊C4×C53—21no—
106C2×C22⋊C4×C53—21no—
107C2×C22⋊C4×C53—21no—
108C2×C22⋊C4×C53D411yescharacteristic
109C2×C22⋊C4×C53D411yescharacteristic
110C2×C42×C53—21no—
111C2×C42×C53C2311yescharacteristic
112C24×C53—21no—
113C24×C53C2×D411yescharacteristic
114C24×C53C2×D411yescharacteristic
115C2×D4×C53—24no—
116C2×D4×C53—22no—
117C2×D4×C53—24no—
118C2×D4×C53—22no—
119C2×D4×C53—21no—
120C2×D4×C53—22no—
121C2×D4×C53—22no—
122C2×D4×C53—41no—
123C2×D4×C53—42no—
124C2×D4×C53—42no—
125C2×D4×C53—82no—
126C2×D4×C53C2×D412yes—
127C22×C4×C53—21no—
128C22×C4×C53—21no—
129C22×C4×C53—21no—
130C22×C4×C53—24no—
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—24no—
140C22×C4×C53—82no—
141C22×C4×C53—82no—
142C22×C4×C53C2×D411yescharacteristic
143C22×C4×C53C2×D411yescharacteristic
144C22×C4×C53C2×D411yescharacteristic
145C22×C4×C53C2×D411yescharacteristic
146C22×C4×C53C2411yescharacteristic
147C22×Dic53—22no—
148C22×Dic53—22no—
149C2×Dic106—42no—
150C2×Dic106—42no—
151C2×Dic106—42no—
152C4⋊C4×C53—24no—
153C4⋊C4×C53—24no—
154C4⋊C4×C53—24no—
155C4⋊C4×C53—22no—
156C4⋊C4×C53—21no—
157C4⋊C4×C53—21no—
158C4⋊C4×C53—21no—
159C4⋊C4×C53C2×D412yes—
160C4⋊C4×C53C2×D412yes—
161C4⋊C4×C53C2×D412yes—
162C22⋊C4×C53—24no—
163C22⋊C4×C53—24no—
164C22⋊C4×C53—24no—
165C22⋊C4×C53—24no—
166C22⋊C4×C53—22no—
167C22⋊C4×C53—22no—
168C42×C53—24no—
169C42×C53C2×D412yes—
170C42×C53C2×D412yes—
171C22⋊Dic53—44no—
172C22⋊Dic53—44no—
173C4⋊Dic53—22no—
174C4⋊Dic53—22no—
175C4⋊Dic53—22no—
176C4⋊Dic53—22no—
177C4⋊Dic53—42no—
178C4×Dic53—42no—
179C2×#88322—22no—
180C2×#88322—22no—
181C2×#88322—22no—
182C2×#88322—22no—
183C2×#88322—22no—
184C2×#88322#4114yes—
185C23×C53—21no—
186C23×C53—22no—
187C23×C53—21no—
188C23×C53—22no—
189C23×C53—21no—
190C23×C53—22no—
191C23×C53—22no—
192C23×C53—21no—
193C23×C53—21no—
194C23×C53—22no—
195C23×C53—42no—
196C23×C53—41no—
197C23×C53—41no—
198C23×C53—42no—
199C23×C53AGL(1,ℤ/8)12yes—
200C23×C53C22×D411yescharacteristic
201D4×C53—22no—
202D4×C53—22no—
203D4×C53—44no—
204D4×C53—48no—
205D4×C53—42no—
206D4×C53—84no—
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—41no—
218C2×C4×C53—41no—
219C2×C4×C53—42no—
220C2×C4×C53—42no—
221C2×C4×C53—41no—
222C2×C4×C53—41no—
223C2×C4×C53—41no—
224C2×C4×C53—41no—
225C2×C4×C53—42no—
226C2×C4×C53—42no—
227C2×C4×C53—84no—
228C2×C4×C53—84no—
229C2×C4×C53#12012yes—
230C2×C4×C53#12012yes—
231C2×C4×C53C22×D411yescharacteristic
232C2×C4×C53C22×D411yescharacteristic
233C2×C4×C53C2×#4112yes—
234C2×C4×C532−1+412yes—
235C2×Dic53—42no—
236C2×Dic53—42no—
237C2×Dic53—42no—
238C2×Dic53—42no—
239C2×Dic53—42no—
240C2×Dic53—42no—
241Dic106—84no—
242#88322—24no—
243#88322—42no—
244#25714—531nomaximal, Sylow
245C22×C53—24no—
246C22×C53—22no—
247C22×C53—42no—
248C22×C53—41no—
249C22×C53—41no—
250C22×C53—42no—
251C22×C53—42no—
252C22×C53—41no—
253C22×C53—42no—
254C22×C53—41no—
255C22×C53—42no—
256C22×C53—42no—
257C22×C53—41no—
258C22×C53—82no—
259C22×C53—82no—
260C22×C53C2×#12011yescharacteristic
261C22×C53#54712yes—
262C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
263C22×C53#57511yescharacteristic
264C22×C53#57712yes—
265C4×C53—22no—
266C4×C53—22no—
267C4×C53—41no—
268C4×C53—41no—
269C4×C53—42no—
270C4×C53—42no—
271C4×C53—42no—
272C4×C53#53211yescharacteristic
273C4×C53#53611yescharacteristic
274C4×C53#54812yes—
275Dic53—82no—
276Dic53—82no—
277C2×#553D5311yescharacteristic
278C2×#501—531no—
279C2×#476—531no—
280#2547—531no—
281#2034—532no—
282#2033—532no—
283#1989—531no—
284#1955—531no—
285#1854—532no—
286#1853—532no—
287#1762—531no—
288C2×C53—42no—
289C2×C53—81no—
290C2×C53—82no—
291C2×C53#243211yescharacteristic
292C2×C53#300611yescharacteristic
293C2×C53#300911yescharacteristic
294C2×C53#334512yes—
295C2×C53#335012yes—
296#553—24no—
297#553—24no—
298C2×C4⋊Q8—531no—
299C2×#126—21no—
300C2×#125C2×D5311yescharacteristic
301C2×#124—1061no—
302C2×#123—21no—
303C2×#121—21no—
304C2×#121C2×D5311yescharacteristic
305C2×#121C2×D5311yescharacteristic
306C2×C4×D4—21no—
307C2×C4×D4C2×D5311yescharacteristic
308C2×C4×D4C2×D5311yescharacteristic
309C2×#117C2×D5311yescharacteristic
310C22×C4⋊C4C2×D5311yescharacteristic
311#501—1064no—
312#476—1064no—
313C2×#107—532no—
314C2×#106—532no—
315C2×#106—532no—
316C2×C4⋊C8—531no—
317C2×#102—532no—
318C2×#102—532no—
319C2×C22⋊C8—532no—
320C2×C22⋊C8—532no—
321C2×#97—531no—
322#397—1061no—
323#397—1061no—
324#391—1061no—
325#380—532no—
326#380—532no—
327C4×C4⋊C4—531no—
328#340—532no—
329#340—532no—
330#340—532no—
331C53#2571411yescharacteristic, complemented, Sylow
332C22×Q8—1061no—
333C22×D4—21no—
334C22×D4—21no—
335C22×D4C22⋊D5311yescharacteristic
336C22×D4C22×D5311yescharacteristic
337C23×C4—21no—
338C23×C4—21no—
339C23×C4C22×D5311yescharacteristic
340C23×C4C22×D5311yescharacteristic
341C22×C8—532no—
342C22×C8—532no—
343C4⋊Q8—1062no—
344C4⋊Q8—1062no—
345#126—28no—
346#125—22no—
347#125C22⋊D5314yes—
348#124—2124no—
349#123—28no—
350#121—28no—
351#121—22no—
352#121—22no—
353#121—22no—
354#121—22no—
355C4×D4—28no—
356C4×D4—22no—
357C4×D4—22no—
358C4×D4—22no—
359C4×D4C22⋊D5314yes—
360#117—22no—
361#117—22no—
362C2×C4⋊C4—21no—
363C2×C4⋊C4—21no—
364C2×C4⋊C4—21no—
365C2×C4⋊C4—22no—
366C2×C4⋊C4—22no—
367C2×C4⋊C4—532no—
368C2×C4⋊C4—532no—
369C2×C4⋊C4—1061no—
370C2×C4⋊C4C22⋊D5311yescharacteristic
371C2×C4⋊C4C22×D5311yescharacteristic
372C2×C4⋊C4C22×D5311yescharacteristic
373C2×C4⋊C4C22×D5311yescharacteristic
374C2×C22⋊C4—21no—
375C2×C22⋊C4—21no—
376C2×C22⋊C4—21no—
377C2×C22⋊C4—21no—
378C2×C22⋊C4—1062no—
379C2×C22⋊C4—1062no—
380C2×C22⋊C4C22⋊D5311yescharacteristic
381C2×C22⋊C4C22⋊D5311yescharacteristic
382C2×C42—21no—
383C2×C42—1061no—
384C2×C42C22×D5311yescharacteristic
385#107—1064no—
386#106—1064no—
387#106—1064no—
388C4⋊C8—1062no—
389#102—1064no—
390#102—1064no—
391C22⋊C8—1064no—
392C22⋊C8—1064no—
393#97—1062no—
394#95—1061no—
395#95—1062no—
396#95—1062no—
397C24—21no—
398C24C2×C22⋊D5311yescharacteristic
399C24C2×C22⋊D5311yescharacteristic
400C2×Q8—2122no—
401C2×Q8—2122no—
402C2×Q8—2122no—
403C2×D4—24no—
404C2×D4—22no—
405C2×D4—24no—
406C2×D4—22no—
407C2×D4—21no—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—41no—
411C2×D4—42no—
412C2×D4—42no—
413C2×D4—82no—
414C2×D4C2×C22⋊D5312yes—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—24no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—82no—
429C22×C4—82no—
430C22×C4—1062no—
431C22×C4—1062no—
432C22×C4C2×C22⋊D5311yescharacteristic
433C22×C4C2×C22⋊D5311yescharacteristic
434C22×C4C2×C22⋊D5311yescharacteristic
435C22×C4C2×C22⋊D5311yescharacteristic
436C22×C4C23×D5311yescharacteristic
437C2×C8—534no—
438C2×C8—1062no—
439C2×C8—1062no—
440C2×C8—1062no—
441C2×C8—1062no—
442C2×C8—1062no—
443C4⋊C4—24no—
444C4⋊C4—24no—
445C4⋊C4—24no—
446C4⋊C4—22no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—1062no—
451C4⋊C4—1062no—
452C4⋊C4—1062no—
453C4⋊C4—1062no—
454C4⋊C4—2122no—
455C4⋊C4C2×C22⋊D5312yes—
456C4⋊C4C2×C22⋊D5312yes—
457C4⋊C4C2×C22⋊D5312yes—
458C22⋊C4—24no—
459C22⋊C4—24no—
460C22⋊C4—24no—
461C22⋊C4—24no—
462C22⋊C4—22no—
463C22⋊C4—22no—
464C22⋊C4—2124no—
465C22⋊C4—2124no—
466C42—24no—
467C42—2122no—
468C42C2×C22⋊D5312yes—
469C42C2×C22⋊D5312yes—
470C23—22no—
471C23—21no—
472C23—21no—
473C23—22no—
474C23—22no—
475C23—21no—
476C23—22no—
477C23—21no—
478C23—21no—
479C23—22no—
480C23—42no—
481C23—41no—
482C23—41no—
483C23—42no—
484C23#1313324212yes—
485C23C22×C22⋊D5311yescharacteristic
486Q8—4244no—
487D4—22no—
488D4—22no—
489D4—44no—
490D4—48no—
491D4—42no—
492D4—84no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—21no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—21no—
502C2×C4—22no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—84no—
514C2×C4—84no—
515C2×C4—2122no—
516C2×C4—2122no—
517C2×C4—2122no—
518C2×C4—2122no—
519C2×C4—2122no—
520C2×C4—2122no—
521C2×C4#1313318612yes—
522C2×C4#1313318612yes—
523C2×C4C22×C22⋊D5311yescharacteristic
524C2×C4C22×C22⋊D5311yescharacteristic
525C2×C4#41×D5312yes—
526C2×C4#1313326412yes—
527C8—1064no—
528C8—2122no—
529C22—22no—
530C22—24no—
531C22—42no—
532C22—41no—
533C22—41no—
534C22—42no—
535C22—41no—
536C22—42no—
537C22—41no—
538C22—42no—
539C22—42no—
540C22—41no—
541C22—42no—
542C22—82no—
543C22—82no—
544C22C2×#1313318611yescharacteristic
545C22#1681127212yes—
546C22C2×#1313324211yescharacteristic
547C22#1681142111yescharacteristic
548C22#1681143412yes—
549C4—22no—
550C4—22no—
551C4—42no—
552C4—41no—
553C4—41no—
554C4—42no—
555C4—42no—
556C4—4242no—
557C4—4242no—
558C4#1681116911yescharacteristic
559C4#1681119411yescharacteristic
560C4#1681128012yes—
561C2—42no—
562C2—81no—
563C2—82no—
564C2#2027833911yescharacteristic
565C2#2028396711yescharacteristic
566C2#2028399211yescharacteristic
567C2#2028717912yes—
568C2#2028725412yes—
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
Fitting, prime radicalC2×C53×#553C2
socleC23×C53C22×D4
hypercenterC2×#553D53
hypocenterC53#25714
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#553

Complements

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

NH ≅ G/Ntype
1. C2×#16811033C2semidirect
2. C2×#16810913C2semidirect
3. #20279707C2semidirect
4. #20275901C2semidirect
5. #20275894C2semidirect
6. #20275673C2semidirect
7. #20275514C2semidirect
8. #20274914C2semidirect
9. #20274908C2semidirect
10. #20274540C2semidirect
331. C53#25714semidirect

Maximal subgroups

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

subgroupindex
C2×#168110332
C2×#168109132
#202797072
#202759012
#202758942
#202756732
#202755142
#202749142
#202749082
#202745402
C2×C53×#5532
#2571453