Finite Group Databasebeta

← #27420180

Subgroups — #27420180

Order 13568.

1294 conjugacy classes of subgroups.

The subgroup lattice has 574 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4⋊Q8×D53C211yesmaximal, characteristic
2C2×#16810908C211yesmaximal, characteristic, complemented
3C4×#13133193C211yesmaximal, characteristic
4#20276046C212yesmaximal, complemented
5#20275854C212yesmaximal, complemented
6#20275668C211yesmaximal, characteristic, complemented
7#20275666C211yesmaximal, characteristic, complemented
8#20275019C212yesmaximal
9#20274906C212yesmaximal
10#20274692C211yesmaximal, characteristic
11C2×C53×#475C211yesmaximal, characteristic, complemented
12C22×Q8×D53—21no—
13C2×#13133217C2211yescharacteristic
14C2×#13133216—21no—
15C2×#13133216C2212yes—
16C2×#13133215—21no—
17C4⋊Q8×D53—24no—
18C4⋊Q8×D53—24no—
19C2×#13133193C2211yescharacteristic
20C2×C4⋊C4×D53—21no—
21C2×C4⋊C4×D53C2212yes—
22C2×C42×D53C2211yescharacteristic
23#16810908—24no—
24C2×#13133148C2212yes—
25C2×C4⋊#88322C2211yescharacteristic
26C2×#13133141C2212yes—
27C2×#13133129C2212yes—
28#16810530C2211yescharacteristic
29#16810519C2211yescharacteristic
30#16810504—21no—
31#16810494—21no—
32#16810480C2211yescharacteristic
33C4×#11719919—21no—
34C4×C4⋊Dic53C2211yescharacteristic
35C4×#11719921—21no—
36#16810386C2212yes—
37#16810386C2212yes—
38#16810385C2212yes—
39#16810385C2212yes—
40#16810377C2212yes—
41C2×C4⋊Q8×C53C2211yescharacteristic
42C2×C4×Q8×C53C2211yescharacteristic
43C53×#475—24no—
44C2×#106×C53C2212yes—
45C2×C4⋊C8×C53C2211yescharacteristic
46C2×#103×C53C2212yes—
47C22×Dic106—21no—
48C22×Dic106—22no—
49C2×Q8×D53—28no—
50C2×Q8×D53—44no—
51C2×Q8×D53—44no—
52C22×C4×D53—22no—
53C22×C4×D53D411yescharacteristic
54C22×C4×D53D411yescharacteristic
55C22×C4×D53C2311yescharacteristic
56C22×#88322C2312yes—
57#13133217—22no—
58#13133217D414yes—
59#13133216—28no—
60#13133216—24no—
61#13133216—24no—
62#13133215—28no—
63#13133193—22no—
64#13133193Q814yescomplemented
65C2×C4⋊Dic53—21no—
66C2×C4⋊Dic53C2311yescharacteristic
67C2×C4⋊Dic53C2312yes—
68C2×C4⋊Dic53C2311yescharacteristic
69C2×#11719919—21no—
70C2×#11719919—22no—
71C2×#11719919—22no—
72C2×#11719919—21no—
73C4⋊C4×D53—28no—
74C4⋊C4×D53—24no—
75C4⋊C4×D53—24no—
76C2×#11719921—21no—
77C2×#11719921—21no—
78C2×C4×Dic53—21no—
79C2×C4×Dic53D411yescharacteristic
80C2×C4×Dic53D411yescharacteristic
81C2×C4×Dic53C2311yescharacteristic
82C42×D53—22no—
83C42×D53—22no—
84#13133148—24no—
85C4⋊#88322—22no—
86#13133141—24no—
87#13133129—24no—
88#13133124—21no—
89#13133124—21no—
90#13133124—21no—
91#13133124—21no—
92#13133124—21no—
93C22×Q8×C53—21no—
94C22×Q8×C53C2311yescharacteristic
95C22×C8×C53C2312yes—
96C4⋊Q8×C53—22no—
97C4⋊Q8×C53D414yes—
98C4×Q8×C53—22no—
99C4×Q8×C53—22no—
100C2×C4⋊C4×C53—21no—
101C2×C4⋊C4×C53—21no—
102C2×C4⋊C4×C53C2311yescharacteristic
103C2×C4⋊C4×C53C2312yes—
104C2×C42×C53—21no—
105C2×C42×C53C2311yescharacteristic
106#106×C53—24no—
107C4⋊C8×C53—22no—
108#103×C53—24no—
109C23×D53C2×D411yescharacteristic
110C2×Q8×C53—24no—
111C2×Q8×C53—21no—
112C2×Q8×C53—21no—
113C2×Q8×C53—21no—
114C2×Q8×C53—21no—
115C2×Q8×C53—21no—
116C2×Q8×C53—42no—
117C2×Q8×C53—42no—
118C2×Q8×C53C2×Q812yescomplemented
119C22×C4×C53—21no—
120C22×C4×C53—22no—
121C22×C4×C53—21no—
122C22×C4×C53C2×D411yescharacteristic
123C22×C4×C53C2×D411yescharacteristic
124C22×C4×C53C2411yescharacteristic
125C22×Dic53—21no—
126C22×Dic53—22no—
127C22×Dic53—21no—
128C22×Dic53C2×D411yescharacteristic
129C22×Dic53C2×D411yescharacteristic
130C22×Dic53C2×D411yescharacteristic
131Q8×D53—432no—
132C2×C4×D53—22no—
133C2×C4×D53—22no—
134C2×C4×D53—22no—
135C2×C4×D53—22no—
136C2×C4×D53—21no—
137C2×C4×D53—21no—
138C2×C4×D53—21no—
139C2×C4×D53—21no—
140C2×C4×D53—22no—
141C2×C4×D53—28no—
142C2×C4×D53—44no—
143C2×C4×D53—44no—
144C2×C4×D53SD1614yes—
145C2×C4×D53C2×Q812yes—
146C2×C4×D53#4112yes—
147C2×Dic106—28no—
148C2×Dic106—24no—
149C2×Dic106—44no—
150C2×Dic106—42no—
151C2×Dic106—42no—
152C2×Dic106—44no—
153C2×C8×C53—22no—
154C2×C8×C53—22no—
155C2×C8×C53—22no—
156C4⋊C4×C53—24no—
157C4⋊C4×C53—22no—
158C4⋊C4×C53—21no—
159C4⋊C4×C53—21no—
160C4⋊C4×C53—42no—
161C4⋊C4×C53C2×D414yes—
162C42×C53—42no—
163C42×C53C2×D412yes—
164C42×C53C2×D412yes—
165#11719921—24no—
166#11719921—42no—
167C4⋊Dic53—24no—
168C4⋊Dic53—22no—
169C4⋊Dic53—21no—
170C4⋊Dic53—21no—
171C4⋊Dic53—21no—
172C4⋊Dic53C2×D414yes—
173C4⋊Dic53C2×Q812yescomplemented
174#11719919—28no—
175#11719919—28no—
176#11719919—24no—
177#11719919—42no—
178C4×Dic53—22no—
179C4×Dic53—22no—
180C4×Dic53—21no—
181C4×Dic53—21no—
182C4×Dic53—42no—
183C2×#88322—22no—
184C2×#88322—22no—
185C2×#88322—22no—
186C23×C53C22×D411yescharacteristic
187C22×D53—21no—
188C22×D53—21no—
189C22×D53—21no—
190C22×D53—21no—
191C22×D53C22⋊Q812yes—
192C22×D53C2×SD1612yes—
193C22×D53#13712yes—
194Q8×C53—22no—
195Q8×C53—22no—
196Q8×C53—41no—
197Q8×C53—41no—
198Q8×C53—48no—
199C2×C4×C53—21no—
200C2×C4×C53—21no—
201C2×C4×C53—24no—
202C2×C4×C53—22no—
203C2×C4×C53—41no—
204C2×C4×C53—41no—
205C2×C4×C53—42no—
206C2×C4×C53—42no—
207C2×C4×C53—41no—
208C2×C4×C53—41no—
209C2×C4×C53—41no—
210C2×C4×C53#12012yes—
211C2×C4×C53#12012yes—
212C2×C4×C53C2×SD1612yes—
213C2×C4×C53AGL(1,ℤ/8)12yes—
214C2×C4×C53C22×D411yescharacteristic
215C2×C4×C53C22×D411yescharacteristic
216C2×C4×C53C22×Q811yescharacteristic
217C2×C4×C53C2×#4111yescharacteristic
218C2×C4×C532−1+411yescharacteristic
219C2×C4×C532−1+411yescharacteristic
220C2×Dic53—21no—
221C2×Dic53—21no—
222C2×Dic53—21no—
223C2×Dic53—21no—
224C2×Dic53—21no—
225C2×Dic53—21no—
226C2×Dic53—24no—
227C2×Dic53—22no—
228C2×Dic53—42no—
229C2×Dic53—42no—
230C2×Dic53—41no—
231C2×Dic53—41no—
232C2×Dic53—41no—
233C2×Dic53—41no—
234C2×Dic53—41no—
235C2×Dic53C22⋊Q812yes—
236C2×Dic53C2×SD1612yes—
237C2×Dic53#13712yes—
238C4×D53—28no—
239C4×D53—22no—
240C4×D53—22no—
241C4×D53—22no—
242C4×D53—22no—
243C4×D53—28no—
244C4×D53—416no—
245Dic106—416no—
246Dic106—48no—
247C8×C53—42no—
248#88322—42no—
249#24301—531nomaximal, Sylow
250C22×C53C2×#12011yescharacteristic
251C22×C53C2×C22⋊Q811yescharacteristic
252C22×C53#53611yescharacteristic
253C22×C53C22×SD1611yescharacteristic
254C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
255C22×C53C2×#13711yescharacteristic
256C22×C53C2×#13711yescharacteristic
257C2×D53—22no—
258C2×D53—22no—
259C2×D53—22no—
260C2×D53—22no—
261C2×D53—22no—
262C2×D53—22no—
263C2×D53#47514yescomplemented
264C4×C53—22no—
265C4×C53—42no—
266C4×C53—44no—
267C4×C53—81no—
268C4×C53#45014yes—
269C4×C53#54811yescharacteristic
270C4×C53#54811yescharacteristic
271C4×C53D4×Q811yescharacteristic
272C4×C53D4×Q811yescharacteristic
273Dic53—22no—
274Dic53—22no—
275Dic53—22no—
276Dic53—42no—
277Dic53—44no—
278Dic53—81no—
279C22×C4⋊Q8—531no—
280C2×#475—531no—
281C2×#475D5311yescharacteristic, complemented
282C4×C22⋊Q8—531no—
283#2058—532no—
284#2024—532no—
285#1988—531no—
286#1988—531no—
287#1872—532no—
288#1853—532no—
289#1797—531no—
290C2×C53#240911yescharacteristic
291C2×C53C2×#45011yescharacteristic
292C2×C53#300411yescharacteristic
293C2×C53C2×#47511yescharacteristic
294C2×C53#307911yescharacteristic
295C2×C53#322911yescharacteristic
296C2×C53#323911yescharacteristic
297D53—24no—
298C23×Q8—1061no—
299C2×C4⋊Q8—532no—
300C2×C4⋊Q8—531no—
301C2×C4⋊Q8—1061no—
302C2×C4⋊Q8—1061no—
303C2×C4⋊Q8—1064no—
304C2×C4⋊Q8—1064no—
305C2×C4⋊Q8C2×D5311yescharacteristic
306C2×C22⋊Q8—531no—
307C2×C4×Q8C2×D5311yescharacteristic
308C22×C4⋊C4—532no—
309C22×C4⋊C4—1061no—
310C22×C42—531no—
311#475—24no—
312#475—1064no—
313C2×#106—532no—
314C2×#106C2×D5312yes—
315C2×C4⋊C8—531no—
316C2×C4⋊C8C2×D5311yescharacteristic
317C2×#103—532no—
318C2×#103C2×D5312yes—
319C2×C22⋊C8—532no—
320#391—531no—
321#389—531no—
322#387—1061no—
323#385—1061no—
324#380—531no—
325C4×C4⋊C4—531no—
326C4×C4⋊C4—1061no—
327C4×C22⋊C4—1061no—
328#340—532no—
329#340—532no—
330#340—532no—
331#340—532no—
332#336—532no—
333C53#2430111yescharacteristic, complemented, Sylow
334C22×Q8—21no—
335C22×Q8—1062no—
336C22×Q8—1068no—
337C22×Q8—1061no—
338C22×Q8—2124no—
339C22×Q8—2124no—
340C22×Q8C22×D5311yescharacteristic
341C23×C4—531no—
342C23×C4—531no—
343C23×C4—531no—
344C23×C4—1062no—
345C22×C8—532no—
346C22×C8C22×D5312yes—
347C4⋊Q8—22no—
348C4⋊Q8—534no—
349C4⋊Q8—1064no—
350C4⋊Q8—1068no—
351C4⋊Q8—1068no—
352C4⋊Q8—1064no—
353C4⋊Q8—1062no—
354C4⋊Q8C22⋊D5314yes—
355C22⋊Q8—534no—
356C22⋊Q8—1062no—
357C4×Q8—22no—
358C4×Q8—22no—
359C2×C4⋊C4—21no—
360C2×C4⋊C4—21no—
361C2×C4⋊C4—532no—
362C2×C4⋊C4—531no—
363C2×C4⋊C4—531no—
364C2×C4⋊C4—1064no—
365C2×C4⋊C4—1068no—
366C2×C4⋊C4—1062no—
367C2×C4⋊C4—1062no—
368C2×C4⋊C4—1061no—
369C2×C4⋊C4—1061no—
370C2×C4⋊C4—1064no—
371C2×C4⋊C4—1061no—
372C2×C4⋊C4C22×D5311yescharacteristic
373C2×C4⋊C4C22×D5312yes—
374C2×C22⋊C4—1061no—
375C2×C22⋊C4—1061no—
376C2×C42—21no—
377C2×C42—531no—
378C2×C42—531no—
379C2×C42—531no—
380C2×C42—1062no—
381C2×C42—1062no—
382C2×C42—1061no—
383C2×C42C22×D5311yescharacteristic
384#106—24no—
385#106—1064no—
386C4⋊C8—22no—
387C4⋊C8—1062no—
388#103—24no—
389#103—1064no—
390C22⋊C8—1064no—
391#95—1061no—
392#95—1061no—
393#95—1061no—
394#95—1061no—
395#95—1061no—
396C24—531no—
397C2×Q8—24no—
398C2×Q8—21no—
399C2×Q8—21no—
400C2×Q8—21no—
401C2×Q8—21no—
402C2×Q8—21no—
403C2×Q8—42no—
404C2×Q8—42no—
405C2×Q8—1068no—
406C2×Q8—1064no—
407C2×Q8—2124no—
408C2×Q8—2124no—
409C2×Q8—21232no—
410C2×Q8—2122no—
411C2×Q8—2122no—
412C2×Q8Q8×D5312yescomplemented
413C22×C4—21no—
414C22×C4—22no—
415C22×C4—21no—
416C22×C4—531no—
417C22×C4—531no—
418C22×C4—531no—
419C22×C4—534no—
420C22×C4—532no—
421C22×C4—532no—
422C22×C4—1062no—
423C22×C4—1062no—
424C22×C4—1068no—
425C22×C4—1062no—
426C22×C4—1062no—
427C22×C4—1062no—
428C22×C4—1061no—
429C22×C4—1061no—
430C22×C4—1061no—
431C22×C4—1061no—
432C22×C4—1061no—
433C22×C4—1062no—
434C22×C4—1061no—
435C22×C4—2124no—
436C22×C4—2124no—
437C22×C4D4×D5311yescharacteristic
438C22×C4D4×D5311yescharacteristic
439C22×C4C23×D5311yescharacteristic
440C2×C8—22no—
441C2×C8—22no—
442C2×C8—22no—
443C2×C8—1062no—
444C2×C8—1062no—
445C2×C8—1062no—
446C4⋊C4—24no—
447C4⋊C4—22no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—42no—
451C4⋊C4—534no—
452C4⋊C4—532no—
453C4⋊C4—1064no—
454C4⋊C4—1068no—
455C4⋊C4—1068no—
456C4⋊C4—1064no—
457C4⋊C4—1062no—
458C4⋊C4—1061no—
459C4⋊C4—1061no—
460C4⋊C4—1061no—
461C4⋊C4—2122no—
462C4⋊C4C2×C22⋊D5314yes—
463C22⋊C4—1064no—
464C22⋊C4—2122no—
465C42—42no—
466C42—1062no—
467C42—1062no—
468C42—1061no—
469C42—1061no—
470C42—2122no—
471C42D4×D5312yes—
472C42C2×C22⋊D5312yes—
473C23—532no—
474C23—532no—
475C23—532no—
476C23—1061no—
477C23—1061no—
478C23—1061no—
479C23—1061no—
480C23C2×D4×D5311yescharacteristic
481Q8—22no—
482Q8—22no—
483Q8—41no—
484Q8—41no—
485Q8—48no—
486Q8—21216no—
487Q8—2128no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—24no—
491C2×C4—22no—
492C2×C4—41no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—532no—
500C2×C4—532no—
501C2×C4—532no—
502C2×C4—1068no—
503C2×C4—1061no—
504C2×C4—1061no—
505C2×C4—1061no—
506C2×C4—1061no—
507C2×C4—1061no—
508C2×C4—1061no—
509C2×C4—1062no—
510C2×C4—1062no—
511C2×C4—1068no—
512C2×C4—1062no—
513C2×C4—1062no—
514C2×C4—1064no—
515C2×C4—1062no—
516C2×C4—21216no—
517C2×C4—2121no—
518C2×C4—2121no—
519C2×C4—2121no—
520C2×C4—2122no—
521C2×C4—2121no—
522C2×C4—2122no—
523C2×C4—2121no—
524C2×C4#1313318512yes—
525C2×C4#1313318512yes—
526C2×C4SD16×D5312yes—
527C2×C4#1313324412yes—
528C2×C4C2×D4×D5311yescharacteristic
529C2×C4C22×C22⋊D5311yescharacteristic
530C2×C4C2×Q8×D5311yescharacteristic
531C2×C4#41×D5311yescharacteristic
532C2×C4#1313326411yescharacteristic
533C2×C4#1313326511yescharacteristic
534C8—42no—
535C8—2122no—
536C22—534no—
537C22—1062no—
538C22—1062no—
539C22—1062no—
540C22—1062no—
541C22—1062no—
542C22—1062no—
543C22C2×#1313318511yescharacteristic
544C22C22⋊Q8×D5311yescharacteristic
545C22#1681119611yescharacteristic
546C22C2×SD16×D5311yescharacteristic
547C22C2×#1313324411yescharacteristic
548C22#137×D5311yescharacteristic
549C22#137×D5311yescharacteristic
550C4—22no—
551C4—42no—
552C4—44no—
553C4—81no—
554C4—1062no—
555C4—1062no—
556C4—1062no—
557C4—4241no—
558C4—2122no—
559C4—2124no—
560C4#1681077214yes—
561C4#1681128011yescharacteristic
562C4#1681128311yescharacteristic
563C4#1681128611yescharacteristic
564C4Q8×C22⋊D5311yescharacteristic
565C2—1064no—
566C2#2027807611yescharacteristic
567C2C2×#1681077211yescharacteristic
568C2#2028394711yescharacteristic
569C2#475×D5311yescharacteristic
570C2#2028452511yescharacteristic
571C2#2028585811yescharacteristic
572C2#2028594911yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC2×C53×#475C2
socleC23×C53C22×D4
prime radicalC23×D53C2×D4
hypercenterC2×#475D53
hypocenterC53#24301
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#475

Complements

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

NH ≅ G/Ntype
2. C2×#16810908C2semidirect
4. #20276046C2semidirect
5. #20275854C2semidirect
6. #20275668C2semidirect
7. #20275666C2semidirect
11. C2×C53×#475C2semidirect
64. #13133193Q8semidirect
118. C2×Q8×C53C2×Q8semidirect
173. C4⋊Dic53C2×Q8semidirect
263. C2×D53#475semidirect
281. C2×#475D53semidirect
333. C53#24301semidirect
412. C2×Q8Q8×D53semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊Q8×D532
C2×#168109082
C4×#131331932
#202760462
#202758542
#202756682
#202756662
#202750192
#202749062
#202746922
C2×C53×#4752
#2430153