Finite Group Databasebeta

← #27417662

Subgroups — #27417662

Order 13568.

1334 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810939C211yesmaximal, characteristic, complemented
2C22×#13133148C211yesmaximal, characteristic, complemented
3#20279032C211yesmaximal, characteristic, complemented
4#20276045C212yesmaximal, complemented
5#20275901C212yesmaximal, complemented
6#20275144C211yesmaximal, characteristic, complemented
7#20275008C211yesmaximal, characteristic, complemented
8#20274976C212yesmaximal, complemented
9#20274908C212yesmaximal, complemented
10#20274801C211yesmaximal, characteristic, complemented
11C2×C53×#547C211yesmaximal, characteristic
12C23×#88322C2211yescharacteristic
13C22×C4⋊Dic53C2211yescharacteristic
14#16810939—24no—
15C2×#13133148—24no—
16C2×#13133148C2212yes—
17C2×#13133148C2212yes—
18C2×#13133148C2212yes—
19C2×#13133138C2212yes—
20C2×#13133138C2212yes—
21C2×C22⋊#88322C2211yescharacteristic
22C2×C22⋊#88322C2212yes—
23#16810562—21no—
24#16810559—21no—
25#16810521C2212yes—
26#16810503—21no—
27#16810495—21no—
28#16810483C2211yescharacteristic
29#16810483C2212yes—
30#16810385C2212yes—
31#16810385C2212yes—
32#16810385C2212yes—
33#16810378C2212yes—
34C22×#41×C53—21no—
35C53×#547—24no—
36C53×#547—24no—
37C2×C4⋊Q8×C53C2211yescharacteristic
38C2×#123×C53—21no—
39C2×#123×C53—21no—
40C2×C22⋊Q8×C53—21no—
41C2×#121×C53C2211yescharacteristic
42C2×#121×C53C2211yescharacteristic
43C2×C4×D4×C53C2211yescharacteristic
44C2×C4×D4×C53C2211yescharacteristic
45C22×C4⋊C4×C53C2211yescharacteristic
46C22×C4⋊C4×C53C2211yescharacteristic
47C23×Dic53—21no—
48C22×#88322—24no—
49C22×#88322—22no—
50C22×#88322C2312yes—
51C22×#88322C2312yes—
52C2×C4⋊Dic53—21no—
53C2×C4⋊Dic53—22no—
54C2×C4⋊Dic53—22no—
55C2×C4⋊Dic53C2312yes—
56C2×C4⋊Dic53C2312yes—
57C2×#11719919—22no—
58C2×C22⋊Dic53—21no—
59C2×C22⋊Dic53—22no—
60#13133148—24no—
61#13133148—24no—
62#13133148—24no—
63#13133138—24no—
64#13133138—24no—
65C22⋊#88322—24no—
66C22⋊#88322—22no—
67#13133124—22no—
68#13133124—21no—
69#13133124—22no—
70C2×#41×C53—28no—
71C2×#41×C53—44no—
72C2×#41×C53—44no—
73C22×Q8×C53—21no—
74C22×D4×C53—21no—
75C22×D4×C53—21no—
76C22×D4×C53D411yescharacteristic
77C22×D4×C53C2311yescharacteristic
78C23×C4×C53—21no—
79C23×C4×C53—21no—
80C23×C4×C53—21no—
81C23×C4×C53C2311yescharacteristic
82C23×C4×C53C2311yescharacteristic
83C4⋊Q8×C53—22no—
84C4⋊Q8×C53—22no—
85#123×C53—28no—
86#123×C53—28no—
87C22⋊Q8×C53—28no—
88#121×C53—22no—
89#121×C53—22no—
90#121×C53—22no—
91#121×C53D414yes—
92C4×D4×C53—22no—
93C4×D4×C53—22no—
94C4×D4×C53—22no—
95C4×D4×C53—22no—
96C2×C4⋊C4×C53—22no—
97C2×C4⋊C4×C53—22no—
98C2×C4⋊C4×C53—21no—
99C2×C4⋊C4×C53—21no—
100C2×C4⋊C4×C53—22no—
101C2×C4⋊C4×C53—21no—
102C2×C4⋊C4×C53D414yes—
103C2×C4⋊C4×C53D411yescharacteristic
104C2×C4⋊C4×C53C2311yescharacteristic
105C2×C4⋊C4×C53C2311yescharacteristic
106C2×C4⋊C4×C53C2311yescharacteristic
107C2×C22⋊C4×C53—21no—
108C2×C22⋊C4×C53—21no—
109C2×C22⋊C4×C53—21no—
110C2×C22⋊C4×C53D411yescharacteristic
111C2×C22⋊C4×C53D411yescharacteristic
112C2×C42×C53C2311yescharacteristic
113C24×C53—21no—
114C24×C53C2×D411yescharacteristic
115C24×C53C2×D411yescharacteristic
116#41×C53—432no—
117C2×Q8×C53—24no—
118C2×Q8×C53—42no—
119C2×Q8×C53—42no—
120C2×D4×C53—22no—
121C2×D4×C53—22no—
122C2×D4×C53—24no—
123C2×D4×C53—24no—
124C2×D4×C53—21no—
125C2×D4×C53—22no—
126C2×D4×C53—22no—
127C2×D4×C53—42no—
128C2×D4×C53—42no—
129C2×D4×C53—42no—
130C2×D4×C53—42no—
131C2×D4×C53SD1614yes—
132C2×D4×C53C2×D412yes—
133C22×C4×C53—22no—
134C22×C4×C53—22no—
135C22×C4×C53—24no—
136C22×C4×C53—24no—
137C22×C4×C53—24no—
138C22×C4×C53—21no—
139C22×C4×C53—21no—
140C22×C4×C53—21no—
141C22×C4×C53—21no—
142C22×C4×C53—21no—
143C22×C4×C53—22no—
144C22×C4×C53—22no—
145C22×C4×C53—42no—
146C22×C4×C53—42no—
147C22×C4×C53—42no—
148C22×C4×C53—42no—
149C22×C4×C53—42no—
150C22×C4×C53—42no—
151C22×C4×C53C2×D411yescharacteristic
152C22×C4×C53C2×D411yescharacteristic
153C22×C4×C53C2×D411yescharacteristic
154C22×C4×C53C2×D412yes—
155C22×C4×C53C2×D412yes—
156C22×C4×C53C2×D411yescharacteristic
157C22×C4×C53C2411yescharacteristic
158C22×Dic53—22no—
159C22×Dic53—22no—
160C22×Dic53—42no—
161C22×Dic53—82no—
162C4⋊C4×C53—24no—
163C4⋊C4×C53—24no—
164C4⋊C4×C53—24no—
165C4⋊C4×C53—22no—
166C4⋊C4×C53—21no—
167C4⋊C4×C53—21no—
168C4⋊C4×C53—21no—
169C4⋊C4×C53C2×D412yes—
170C4⋊C4×C53C2×D412yes—
171C4⋊C4×C53C2×D412yes—
172C22⋊C4×C53—24no—
173C22⋊C4×C53—24no—
174C22⋊C4×C53—22no—
175C22⋊C4×C53—24no—
176C22⋊C4×C53—22no—
177C42×C53—21no—
178C42×C53—21no—
179C22⋊Dic53—42no—
180C22⋊Dic53—82no—
181C4⋊Dic53—22no—
182C4⋊Dic53—22no—
183C4⋊Dic53—22no—
184C4⋊Dic53—22no—
185C4⋊Dic53—42no—
186#11719919—82no—
187C2×#88322—28no—
188C2×#88322—22no—
189C2×#88322—22no—
190C2×#88322—22no—
191C2×#88322—22no—
192C2×#88322—22no—
193C2×#88322#4114yescomplemented
194C23×C53—22no—
195C23×C53—22no—
196C23×C53—21no—
197C23×C53—22no—
198C23×C53—21no—
199C23×C53—22no—
200C23×C53—22no—
201C23×C53—42no—
202C23×C53—41no—
203C23×C53—41no—
204C23×C53—42no—
205C23×C53#12012yes—
206C23×C53C2×SD1612yes—
207C23×C53C2×SD1612yes—
208C23×C53AGL(1,ℤ/8)12yes—
209C23×C53C22×D411yescharacteristic
210Q8×C53—48no—
211D4×C53—28no—
212D4×C53—24no—
213D4×C53—48no—
214D4×C53—48no—
215D4×C53—41no—
216D4×C53—41no—
217C2×C4×C53—21no—
218C2×C4×C53—22no—
219C2×C4×C53—21no—
220C2×C4×C53—22no—
221C2×C4×C53—22no—
222C2×C4×C53—24no—
223C2×C4×C53—22no—
224C2×C4×C53—22no—
225C2×C4×C53—22no—
226C2×C4×C53—21no—
227C2×C4×C53—21no—
228C2×C4×C53—22no—
229C2×C4×C53—22no—
230C2×C4×C53—48no—
231C2×C4×C53—48no—
232C2×C4×C53—41no—
233C2×C4×C53—41no—
234C2×C4×C53—41no—
235C2×C4×C53—41no—
236C2×C4×C53—41no—
237C2×C4×C53—41no—
238C2×C4×C53—48no—
239C2×C4×C53—41no—
240C2×C4×C53—41no—
241C2×C4×C53—42no—
242C2×C4×C53#12012yes—
243C2×C4×C53C2×SD1612yes—
244C2×C4×C53C22×D411yescharacteristic
245C2×C4×C53C22×D411yescharacteristic
246C2×C4×C53C2×#4112yes—
247C2×C4×C532−1+412yes—
248C2×Dic53—42no—
249C2×Dic53—42no—
250C2×Dic53—42no—
251C2×Dic53—42no—
252C2×Dic53—42no—
253C2×Dic53—42no—
254C2×Dic53—84no—
255#88322—24no—
256#88322—42no—
257#24089—531nomaximal, Sylow
258C22×C53—24no—
259C22×C53—24no—
260C22×C53—22no—
261C22×C53—22no—
262C22×C53—24no—
263C22×C53—22no—
264C22×C53—42no—
265C22×C53—41no—
266C22×C53—41no—
267C22×C53—42no—
268C22×C53—42no—
269C22×C53—42no—
270C22×C53—82no—
271C22×C53—82no—
272C22×C53#45014yes—
273C22×C53C2×#12011yescharacteristic
274C22×C53#54712yes—
275C22×C53C22×SD1611yescharacteristic
276C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
277C22×C53#57712yes—
278C4×C53—22no—
279C4×C53—22no—
280C4×C53—41no—
281C4×C53—42no—
282C4×C53—42no—
283C4×C53—42no—
284C4×C53—41no—
285C4×C53#52511yescharacteristic
286C4×C53#53611yescharacteristic
287C4×C53#54712yes—
288Dic53—82no—
289Dic53—82no—
290C2×#547D5311yescharacteristic
291C2×#481—531no—
292C22×#106—531no—
293#2487—531no—
294#2058—532no—
295#2034—532no—
296#1893—531no—
297#1869—531no—
298#1864—532no—
299#1853—532no—
300#1823—531no—
301C2×C53—24no—
302C2×C53—42no—
303C2×C53—82no—
304C2×C53#240611yescharacteristic
305C2×C53C2×#45011yescharacteristic
306C2×C53#301011yescharacteristic
307C2×C53#333512yes—
308C2×C53#334512yes—
309C22×#41—21no—
310C23×C8—531no—
311#547—24no—
312#547—24no—
313C2×C4⋊Q8C2×D5311yescharacteristic
314C2×#123—21no—
315C2×#123—21no—
316C2×C22⋊Q8—21no—
317C2×#121C2×D5311yescharacteristic
318C2×#121C2×D5311yescharacteristic
319C2×C4×D4C2×D5311yescharacteristic
320C2×C4×D4C2×D5311yescharacteristic
321C22×C4⋊C4—531no—
322C22×C4⋊C4C2×D5311yescharacteristic
323C22×C4⋊C4C2×D5311yescharacteristic
324#481—1064no—
325C2×#106—532no—
326C2×#106—532no—
327C2×#106—532no—
328C2×#106—1064no—
329C2×#102—532no—
330C2×#102—532no—
331C2×C22⋊C8—531no—
332C2×C22⋊C8—532no—
333#400—1061no—
334#399—1061no—
335#389—532no—
336#386—1061no—
337#385—1061no—
338#380—531no—
339#380—532no—
340#340—532no—
341#340—532no—
342#340—532no—
343#336—532no—
344C53#2408911yescharacteristic, complemented, Sylow
345C2×#41—28no—
346C2×#41—44no—
347C2×#41—44no—
348C22×Q8—21no—
349C22×D4—21no—
350C22×D4—21no—
351C22×D4C22⋊D5311yescharacteristic
352C22×D4C22×D5311yescharacteristic
353C23×C4—21no—
354C23×C4—21no—
355C23×C4—21no—
356C23×C4—1061no—
357C23×C4C22×D5311yescharacteristic
358C23×C4C22×D5311yescharacteristic
359C22×C8—532no—
360C22×C8—532no—
361C22×C8—1064no—
362C22×C8—1062no—
363C4⋊Q8—22no—
364C4⋊Q8—22no—
365#123—28no—
366#123—28no—
367C22⋊Q8—28no—
368#121—22no—
369#121—22no—
370#121—22no—
371#121C22⋊D5314yes—
372C4×D4—22no—
373C4×D4—22no—
374C4×D4—22no—
375C4×D4—22no—
376C2×C4⋊C4—21no—
377C2×C4⋊C4—21no—
378C2×C4⋊C4—22no—
379C2×C4⋊C4—21no—
380C2×C4⋊C4—22no—
381C2×C4⋊C4—22no—
382C2×C4⋊C4—532no—
383C2×C4⋊C4—532no—
384C2×C4⋊C4—1062no—
385C2×C4⋊C4—1062no—
386C2×C4⋊C4—1062no—
387C2×C4⋊C4—1061no—
388C2×C4⋊C4C22⋊D5311yescharacteristic
389C2×C4⋊C4C22⋊D5314yes—
390C2×C4⋊C4C22×D5311yescharacteristic
391C2×C4⋊C4C22×D5311yescharacteristic
392C2×C4⋊C4C22×D5311yescharacteristic
393C2×C22⋊C4—21no—
394C2×C22⋊C4—21no—
395C2×C22⋊C4—21no—
396C2×C22⋊C4—1062no—
397C2×C22⋊C4—1061no—
398C2×C22⋊C4C22⋊D5311yescharacteristic
399C2×C22⋊C4C22⋊D5311yescharacteristic
400C2×C42C22×D5311yescharacteristic
401#106—1064no—
402#106—1064no—
403#106—1064no—
404#102—1064no—
405#102—1064no—
406C22⋊C8—1064no—
407C22⋊C8—1062no—
408#95—1062no—
409#95—1061no—
410#95—1062no—
411C24—21no—
412C24C2×C22⋊D5311yescharacteristic
413C24C2×C22⋊D5311yescharacteristic
414#41—432no—
415C2×Q8—24no—
416C2×Q8—42no—
417C2×Q8—42no—
418C2×D4—22no—
419C2×D4—22no—
420C2×D4—24no—
421C2×D4—24no—
422C2×D4—21no—
423C2×D4—22no—
424C2×D4—22no—
425C2×D4—42no—
426C2×D4—42no—
427C2×D4—42no—
428C2×D4—42no—
429C2×D4#1171992314yes—
430C2×D4C2×C22⋊D5312yes—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—1062no—
450C22×C4—1062no—
451C22×C4—4242no—
452C22×C4—2122no—
453C22×C4C2×C22⋊D5311yescharacteristic
454C22×C4C2×C22⋊D5311yescharacteristic
455C22×C4C2×C22⋊D5311yescharacteristic
456C22×C4C2×C22⋊D5312yes—
457C22×C4C2×C22⋊D5312yes—
458C22×C4C2×C22⋊D5311yescharacteristic
459C22×C4C23×D5311yescharacteristic
460C2×C8—534no—
461C2×C8—1068no—
462C2×C8—1062no—
463C2×C8—1062no—
464C2×C8—1062no—
465C2×C8—1062no—
466C2×C8—1062no—
467C4⋊C4—24no—
468C4⋊C4—24no—
469C4⋊C4—24no—
470C4⋊C4—22no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—1062no—
475C4⋊C4—1062no—
476C4⋊C4—1062no—
477C4⋊C4—1062no—
478C4⋊C4—4242no—
479C4⋊C4—2122no—
480C4⋊C4C2×C22⋊D5312yes—
481C4⋊C4C2×C22⋊D5312yes—
482C4⋊C4C2×C22⋊D5312yes—
483C22⋊C4—24no—
484C22⋊C4—24no—
485C22⋊C4—22no—
486C22⋊C4—24no—
487C22⋊C4—22no—
488C22⋊C4—4242no—
489C22⋊C4—2122no—
490C42—21no—
491C42—21no—
492C23—22no—
493C23—22no—
494C23—22no—
495C23—21no—
496C23—21no—
497C23—22no—
498C23—22no—
499C23—42no—
500C23—41no—
501C23—41no—
502C23—42no—
503C23#1313318612yes—
504C23C2×#1171992312yes—
505C23C2×#1171992312yes—
506C23#1313324312yes—
507C23C22×C22⋊D5311yescharacteristic
508Q8—48no—
509D4—28no—
510D4—24no—
511D4—48no—
512D4—48no—
513D4—41no—
514D4—41no—
515C2×C4—22no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—24no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—48no—
529C2×C4—48no—
530C2×C4—48no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—42no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—4244no—
541C2×C4—2122no—
542C2×C4—2122no—
543C2×C4—2122no—
544C2×C4—2122no—
545C2×C4—2122no—
546C2×C4—2122no—
547C2×C4#1313318612yes—
548C2×C4C2×#1171992312yes—
549C2×C4C22×C22⋊D5311yescharacteristic
550C2×C4C22×C22⋊D5311yescharacteristic
551C2×C4#41×D5312yes—
552C2×C4#1313326412yes—
553C8—1064no—
554C8—2122no—
555C22—22no—
556C22—24no—
557C22—22no—
558C22—24no—
559C22—24no—
560C22—22no—
561C22—41no—
562C22—41no—
563C22—42no—
564C22—42no—
565C22—42no—
566C22—42no—
567C22—82no—
568C22—82no—
569C22#1681077414yes—
570C22C2×#1313318611yescharacteristic
571C22#1681127212yes—
572C22C22×#1171992311yescharacteristic
573C22C2×#1313324311yescharacteristic
574C22#1681143112yes—
575C4—22no—
576C4—22no—
577C4—41no—
578C4—41no—
579C4—42no—
580C4—42no—
581C4—42no—
582C4—4242no—
583C4—4242no—
584C4#1681113011yescharacteristic
585C4#1681119411yescharacteristic
586C4#1681127212yes—
587C2—24no—
588C2—42no—
589C2—82no—
590C2#2027804611yescharacteristic
591C2C2×#1681077411yescharacteristic
592C2#2028400111yescharacteristic
593C2#2028707212yes—
594C2#2028718512yes—
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
Fitting, prime radicalC2×C53×#547C2
socleC23×C53C22×D4
hypercenterC2×#547D53
hypocenterC53#24089
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#547

Complements

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

NH ≅ G/Ntype
1. C2×#16810939C2semidirect
2. C22×#13133148C2semidirect
3. #20279032C2semidirect
4. #20276045C2semidirect
5. #20275901C2semidirect
6. #20275144C2semidirect
7. #20275008C2semidirect
8. #20274976C2semidirect
9. #20274908C2semidirect
10. #20274801C2semidirect
193. C2×#88322#41semidirect
344. C53#24089semidirect

Maximal subgroups

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

subgroupindex
C2×#168109392
C22×#131331482
#202790322
#202760452
#202759012
#202751442
#202750082
#202749762
#202749082
#202748012
C2×C53×#5472
#2408953