Finite Group Databasebeta

← #27441109

Subgroups — #27441109

Order 13568.

1176 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×#16810989C211yesmaximal, characteristic, complemented
2C2×#16810818C211yesmaximal, characteristic
3#20279714C211yesmaximal, characteristic, complemented
4#20275858C212yesmaximal, complemented
5#20275848C212yesmaximal, complemented
6#20275673C211yesmaximal, characteristic, complemented
7#20275656C211yesmaximal, characteristic, complemented
8#20275063C212yesmaximal, complemented
9#20275063C212yesmaximal, complemented
10#20274540C211yesmaximal, characteristic, complemented
11C2×C53×#554C211yesmaximal, characteristic, complemented
12C22×#11719922C2212yes—
13C2×#13133214C2211yescharacteristic
14C2×#13133206—21no—
15#16810989—24no—
16#16810818—24no—
17C2×C4⋊#88322C2211yescharacteristic
18C2×#13133138C2212yes—
19C2×#13133137C2212yes—
20C2×#13133137C2212yes—
21C2×#13133137C2212yes—
22C2×#13133128C2211yescharacteristic
23#16810540—21no—
24#16810540—21no—
25#16810525C2212yes—
26#16810525C2212yes—
27#16810524—21no—
28C4×C4⋊Dic53C2211yescharacteristic
29#16810385C2212yes—
30#16810385C2212yes—
31#16810385C2212yes—
32#16810378C2212yes—
33#16810378C2212yes—
34C53×#554—24no—
35C53×#554—24no—
36C2×C4⋊Q8×C53C2211yescharacteristic
37C2×#125×C53C2211yescharacteristic
38C2×#125×C53C2211yescharacteristic
39C2×C22⋊Q8×C53—21no—
40C2×C22⋊Q8×C53—21no—
41C2×C22⋊Q8×C53—21no—
42C2×C4×Q8×C53C2211yescharacteristic
43C2×C4×D4×C53C2211yescharacteristic
44C2×C4×D4×C53C2211yescharacteristic
45C2×C4×D4×C53C2211yescharacteristic
46C22×C4⋊C4×C53—21no—
47C22×D212—21no—
48C22×D212—21no—
49C22×D212C2312yes—
50C2×#11719922—24no—
51C2×#11719922—24no—
52C2×#11719922—24no—
53C22×#88322C2312yes—
54C22×#88322C2312yes—
55#13133214—22no—
56#13133214—22no—
57#13133206—44no—
58C2×C4⋊Dic53C2312yes—
59C2×#11719921—22no—
60C2×#11719921—22no—
61C2×#11719921—22no—
62C2×#11719921—22no—
63C2×C4×Dic53—21no—
64C4⋊#88322—22no—
65#13133138—24no—
66#13133137—24no—
67#13133137—24no—
68#13133137—24no—
69#13133128—22no—
70#13133124—21no—
71C22×Q8×C53—21no—
72C22×Q8×C53D411yescharacteristic
73C22×D4×C53C2311yescharacteristic
74C23×C4×C53—21no—
75C23×C4×C53—21no—
76C23×C4×C53—21no—
77C4⋊Q8×C53—22no—
78C4⋊Q8×C53—22no—
79#125×C53—22no—
80#125×C53—22no—
81#125×C53—22no—
82#125×C53D414yes—
83C22⋊Q8×C53—28no—
84C22⋊Q8×C53—28no—
85C22⋊Q8×C53—28no—
86C4×Q8×C53—22no—
87C4×Q8×C53—22no—
88C4×D4×C53—22no—
89C4×D4×C53—22no—
90C4×D4×C53—22no—
91C4×D4×C53—22no—
92C4×D4×C53—22no—
93C4×D4×C53D414yes—
94C2×C4⋊C4×C53—28no—
95C2×C4⋊C4×C53—21no—
96C2×C4⋊C4×C53—21no—
97C2×C4⋊C4×C53—21no—
98C2×C4⋊C4×C53—21no—
99C2×C4⋊C4×C53—21no—
100C2×C4⋊C4×C53D411yescharacteristic
101C2×C4⋊C4×C53D411yescharacteristic
102C2×C4⋊C4×C53D411yescharacteristic
103C2×C4⋊C4×C53C2311yescharacteristic
104C2×C4⋊C4×C53C2311yescharacteristic
105C2×C4⋊C4×C53C2311yescharacteristic
106C2×C22⋊C4×C53—21no—
107C2×C22⋊C4×C53—21no—
108C2×C22⋊C4×C53—21no—
109C2×C42×C53C2311yescharacteristic
110C2×C42×C53C2311yescharacteristic
111C2×C42×C53C2311yescharacteristic
112C24×C53—21no—
113C23×D53—22no—
114C2×Q8×C53—24no—
115C2×Q8×C53—22no—
116C2×Q8×C53—22no—
117C2×Q8×C53—41no—
118C2×Q8×C53—82no—
119C2×D4×C53—21no—
120C2×D4×C53—22no—
121C2×D4×C53C2×D412yes—
122C2×D4×C53#4114yes—
123C22×C4×C53—24no—
124C22×C4×C53—21no—
125C22×C4×C53—21no—
126C22×C4×C53—21no—
127C22×C4×C53—24no—
128C22×C4×C53—24no—
129C22×C4×C53—42no—
130C22×C4×C53—42no—
131C22×C4×C53—82no—
132C22×C4×C53—82no—
133C22×C4×C53C2×D411yescharacteristic
134C22×C4×C53C2×D411yescharacteristic
135C22×C4×C53C2×D411yescharacteristic
136C22×C4×C53C2×D411yescharacteristic
137C22×C4×C53C2×D411yescharacteristic
138C22×C4×C53C2×D411yescharacteristic
139C22×C4×C53C2411yescharacteristic
140C22×Dic53—22no—
141C2×D212—22no—
142C2×D212—22no—
143C2×D212—22no—
144C2×D212—22no—
145C2×D212—22no—
146C2×D212—22no—
147C2×D212—42no—
148C2×D212—42no—
149C2×D212—42no—
150C2×D212—42no—
151C2×D212—82no—
152C4⋊C4×C53—24no—
153C4⋊C4×C53—24no—
154C4⋊C4×C53—24no—
155C4⋊C4×C53—22no—
156C4⋊C4×C53—24no—
157C4⋊C4×C53—24no—
158C4⋊C4×C53—22no—
159C4⋊C4×C53—22no—
160C4⋊C4×C53—21no—
161C4⋊C4×C53—21no—
162C4⋊C4×C53—21no—
163C4⋊C4×C53C2×D412yes—
164C4⋊C4×C53C2×D412yes—
165C4⋊C4×C53C2×D412yes—
166C22⋊C4×C53—24no—
167C22⋊C4×C53—24no—
168C22⋊C4×C53—24no—
169C42×C53—21no—
170C42×C53—21no—
171C42×C53—21no—
172C42×C53—21no—
173C42×C53C2×D412yes—
174C42×C53C2×D412yes—
175#11719922—48no—
176#11719921—44no—
177#11719921—44no—
178#11719921—82no—
179#11719921—82no—
180C4⋊Dic53—22no—
181C4⋊Dic53—22no—
182C4×Dic53—42no—
183C2×#88322—22no—
184C2×#88322—22no—
185C2×#88322—22no—
186C2×#88322—22no—
187C2×#88322—22no—
188C2×#88322—22no—
189C23×C53—22no—
190C23×C53—24no—
191C23×C53—41no—
192C23×C53—41no—
193C23×C53—42no—
194C23×C53C22×D411yescharacteristic
195C22×D53—42no—
196C22×D53—42no—
197C22×D53—42no—
198C22×D53—42no—
199C22×D53—42no—
200C22×D53—42no—
201C22×D53—42no—
202Q8×C53—44no—
203Q8×C53—84no—
204D4×C53—22no—
205D4×C53—24no—
206D4×C53#12414yescomplemented
207C2×C4×C53—21no—
208C2×C4×C53—21no—
209C2×C4×C53—21no—
210C2×C4×C53—21no—
211C2×C4×C53—22no—
212C2×C4×C53—21no—
213C2×C4×C53—22no—
214C2×C4×C53—22no—
215C2×C4×C53—22no—
216C2×C4×C53—21no—
217C2×C4×C53—21no—
218C2×C4×C53—21no—
219C2×C4×C53—21no—
220C2×C4×C53—22no—
221C2×C4×C53—22no—
222C2×C4×C53—48no—
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×C53AGL(1,ℤ/8)12yes—
232C2×C4×C53C22×D411yescharacteristic
233C2×C4×C53C22×D411yescharacteristic
234C2×C4×C53C2×#4112yes—
235C2×C4×C532−1+412yes—
236C2×Dic53—42no—
237C2×Dic53—42no—
238C2×Dic53—42no—
239D212—42no—
240D212—42no—
241D212—42no—
242D212—42no—
243D212—84no—
244D212—84no—
245#88322—42no—
246#88322—42no—
247#25946—531nomaximal, Sylow
248C22×C53—24no—
249C22×C53—44no—
250C22×C53—44no—
251C22×C53—42no—
252C22×C53—42no—
253C22×C53C2×#12011yescharacteristic
254C22×C53#54812yes—
255C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
256C22×C53#57511yescharacteristic
257C22×C53#57612yes—
258C2×D53—82no—
259C2×D53—82no—
260C2×D53—82no—
261C2×D53—82no—
262C2×D53—82no—
263C2×D53—82no—
264C2×D53—82no—
265C4×C53—22no—
266C4×C53—22no—
267C4×C53—22no—
268C4×C53—41no—
269C4×C53—42no—
270C4×C53—41no—
271C4×C53—41no—
272C4×C53—42no—
273C4×C53—42no—
274C4×C53C2×#12411yescharacteristic
275C4×C53#53611yescharacteristic
276C4×C53#54812yes—
277Dic53—82no—
278C2×#554D5311yescharacteristic, complemented
279C2×#490—531no—
280C2×#459—531no—
281#2548—531no—
282#2025—532no—
283#2023—532no—
284#1989—531no—
285#1985—531no—
286#1879—532no—
287#1879—532no—
288#1762—531no—
289C2×C53—44no—
290C2×C53#243611yescharacteristic
291C2×C53#300611yescharacteristic
292C2×C53#300811yescharacteristic
293C2×C53#336612yes—
294C2×C53#337012yes—
295D53—162no—
296C22×D8—532no—
297#554—24no—
298#554—24no—
299C2×C4⋊Q8C2×D5311yescharacteristic
300C2×#127—531no—
301C2×#125C2×D5311yescharacteristic
302C2×#125C2×D5311yescharacteristic
303C2×#124—1061no—
304C2×C22⋊Q8—21no—
305C2×C22⋊Q8—21no—
306C2×C22⋊Q8—21no—
307C2×C4×Q8C2×D5311yescharacteristic
308C2×C4×D4C2×D5311yescharacteristic
309C2×C4×D4C2×D5311yescharacteristic
310C2×C4×D4C2×D5311yescharacteristic
311C22×C4⋊C4—21no—
312#490—1064no—
313#459—1064no—
314C2×C4⋊C8—531no—
315C2×#102—532no—
316C2×#102—532no—
317C2×#102—532no—
318C2×#102—532no—
319C2×#97—531no—
320#394—1061no—
321#394—1061no—
322#390—532no—
323#390—532no—
324#390—1061no—
325C4×C4⋊C4—531no—
326#340—532no—
327#340—532no—
328#340—532no—
329#336—532no—
330#336—532no—
331C53#2594611yescharacteristic, complemented, Sylow
332C22×Q8—21no—
333C22×Q8C22⋊D5311yescharacteristic
334C22×D4—532no—
335C22×D4—1061no—
336C22×D4—1061no—
337C22×D4C22×D5311yescharacteristic
338C23×C4—21no—
339C23×C4—21no—
340C23×C4—21no—
341C2×D8—1064no—
342C2×D8—1064no—
343C2×D8—1064no—
344C22×C8—532no—
345C22×C8—532no—
346C4⋊Q8—22no—
347C4⋊Q8—22no—
348#127—1062no—
349#127—1062no—
350#125—22no—
351#125—22no—
352#125—22no—
353#125C22⋊D5314yes—
354#124—2124no—
355C22⋊Q8—28no—
356C22⋊Q8—28no—
357C22⋊Q8—28no—
358C4×Q8—22no—
359C4×Q8—22no—
360C4×D4—22no—
361C4×D4—22no—
362C4×D4—22no—
363C4×D4—22no—
364C4×D4—22no—
365C4×D4C22⋊D5314yes—
366C2×C4⋊C4—28no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—21no—
370C2×C4⋊C4—21no—
371C2×C4⋊C4—21no—
372C2×C4⋊C4—532no—
373C2×C4⋊C4C22⋊D5311yescharacteristic
374C2×C4⋊C4C22⋊D5311yescharacteristic
375C2×C4⋊C4C22⋊D5311yescharacteristic
376C2×C4⋊C4C22×D5311yescharacteristic
377C2×C4⋊C4C22×D5311yescharacteristic
378C2×C4⋊C4C22×D5311yescharacteristic
379C2×C22⋊C4—21no—
380C2×C22⋊C4—21no—
381C2×C22⋊C4—21no—
382C2×C22⋊C4—1062no—
383C2×C22⋊C4—1062no—
384C2×C22⋊C4—1062no—
385C2×C22⋊C4—1062no—
386C2×C42—1061no—
387C2×C42C22×D5311yescharacteristic
388C2×C42C22×D5311yescharacteristic
389C2×C42C22×D5311yescharacteristic
390C4⋊C8—1062no—
391#102—1064no—
392#102—1064no—
393#102—1064no—
394#102—1064no—
395#97—1062no—
396#95—1061no—
397C24—21no—
398C24—1062no—
399C2×Q8—24no—
400C2×Q8—22no—
401C2×Q8—22no—
402C2×Q8—41no—
403C2×Q8—82no—
404C2×D4—21no—
405C2×D4—22no—
406C2×D4—1062no—
407C2×D4—1062no—
408C2×D4—1062no—
409C2×D4—1062no—
410C2×D4—1062no—
411C2×D4—1062no—
412C2×D4—4242no—
413C2×D4—2122no—
414C2×D4—2122no—
415C2×D4—2122no—
416C2×D4—2122no—
417C2×D4#1171994914yes—
418C2×D4C2×C22⋊D5312yes—
419C22×C4—24no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—82no—
428C22×C4—82no—
429C22×C4—1062no—
430C22×C4C2×C22⋊D5311yescharacteristic
431C22×C4C2×C22⋊D5311yescharacteristic
432C22×C4C2×C22⋊D5311yescharacteristic
433C22×C4C2×C22⋊D5311yescharacteristic
434C22×C4C2×C22⋊D5311yescharacteristic
435C22×C4C2×C22⋊D5311yescharacteristic
436C22×C4C23×D5311yescharacteristic
437D8—2128no—
438C2×C8—1062no—
439C2×C8—1062no—
440C2×C8—1062no—
441C2×C8—1062no—
442C2×C8—1062no—
443C2×C8—1062no—
444C4⋊C4—24no—
445C4⋊C4—24no—
446C4⋊C4—24no—
447C4⋊C4—22no—
448C4⋊C4—24no—
449C4⋊C4—24no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—1062no—
456C4⋊C4—1062no—
457C4⋊C4C2×C22⋊D5312yes—
458C4⋊C4C2×C22⋊D5312yes—
459C4⋊C4C2×C22⋊D5312yes—
460C22⋊C4—24no—
461C22⋊C4—24no—
462C22⋊C4—24no—
463C22⋊C4—4242no—
464C22⋊C4—4242no—
465C22⋊C4—2124no—
466C22⋊C4—2124no—
467C42—21no—
468C42—21no—
469C42—21no—
470C42—21no—
471C42—2122no—
472C42C2×C22⋊D5312yes—
473C42C2×C22⋊D5312yes—
474C23—24no—
475C23—22no—
476C23—41no—
477C23—41no—
478C23—42no—
479C23—2122no—
480C23—2122no—
481C23—2122no—
482C23—2122no—
483C23—2122no—
484C23—2122no—
485C23—2122no—
486C23C22×C22⋊D5311yescharacteristic
487Q8—44no—
488Q8—84no—
489D4—24no—
490D4—22no—
491D4—4244no—
492D4—4244no—
493D4—2122no—
494D4—2122no—
495D4—2122no—
496D4—2122no—
497D4#1313320614yescomplemented
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—22no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—48no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—84no—
519C2×C4—84no—
520C2×C4—2122no—
521C2×C4—2122no—
522C2×C4—2122no—
523C2×C4#1313318612yes—
524C2×C4#1313318612yes—
525C2×C4#1313324212yes—
526C2×C4C22×C22⋊D5311yescharacteristic
527C2×C4C22×C22⋊D5311yescharacteristic
528C2×C4C2×#1171994912yes—
529C2×C4#1313326512yes—
530C8—2122no—
531C8—2122no—
532C22—24no—
533C22—42no—
534C22—42no—
535C22—44no—
536C22—44no—
537C22—4242no—
538C22—4242no—
539C22—4242no—
540C22—4242no—
541C22—4242no—
542C22—4242no—
543C22—4242no—
544C22C2×#1313318611yescharacteristic
545C22#1681128412yes—
546C22C2×#1313324211yescharacteristic
547C22#1681142111yescharacteristic
548C22#1681142612yes—
549C4—22no—
550C4—22no—
551C4—22no—
552C4—41no—
553C4—42no—
554C4—41no—
555C4—41no—
556C4—42no—
557C4—42no—
558C4—4242no—
559C4C2×#1313320611yescharacteristic
560C4#1681119711yescharacteristic
561C4#1681128412yes—
562C2—44no—
563C2—8482no—
564C2#2027836911yescharacteristic
565C2#2028396711yescharacteristic
566C2#2028398311yescharacteristic
567C2#2028742212yes—
568C2#2028746812yes—
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC2×C53×#554C2
socleC23×C53C22×D4
prime radicalC2×#16810818C2
hypercenterC2×#554D53
hypocenterC53#25946
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×#554

Complements

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

NH ≅ G/Ntype
1. C2×#16810989C2semidirect
3. #20279714C2semidirect
4. #20275858C2semidirect
5. #20275848C2semidirect
6. #20275673C2semidirect
7. #20275656C2semidirect
8. #20275063C2semidirect
9. #20275063C2semidirect
10. #20274540C2semidirect
11. C2×C53×#554C2semidirect
206. D4×C53#124semidirect
278. C2×#554D53semidirect
331. C53#25946semidirect
497. D4#13133206semidirect

Maximal subgroups

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

subgroupindex
C2×#168109892
C2×#168108182
#202797142
#202758582
#202758482
#202756732
#202756562
#202750632
#202745402
C2×C53×#5542
#2594653