Finite Group Databasebeta

← #27425420

Subgroups — #27425420

Order 13568.

828 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810918C211yesmaximal, characteristic, complemented
2C2×#16810679C211yesmaximal, characteristic
3#20283068C211yesmaximal, characteristic
4C4×#13133193C211yesmaximal, characteristic
5#20276142C212yesmaximal
6#20275909C212yesmaximal
7#20275662C211yesmaximal, characteristic, complemented
8#20275476C211yesmaximal, characteristic, complemented
9#20275457C212yesmaximal, complemented
10#20275407C212yesmaximal, complemented
11C53×#1803C211yesmaximal, characteristic, complemented
12C2×#13133217C2211yescharacteristic
13C2×#13133197—21no—
14C2×#13133193C2211yescharacteristic
15C2×C42×D53C2211yescharacteristic
16#16810918—24no—
17#16810679—24no—
18C2×#13133150C2212yes—
19C2×#13133148C2212yes—
20C2×C4⋊#88322C2211yescharacteristic
21C2×#13133141C2212yes—
22C2×#13133129C2212yes—
23C2×#13133127C2212yes—
24#16810530C2211yescharacteristic
25#16810528—21no—
26#16810522—21no—
27#16810522C2212yes—
28#16810519C2211yescharacteristic
29#16810504—21no—
30#16810494—21no—
31#16810480—21no—
32#16810480C2211yescharacteristic
33#16810480C2212yes—
34C4×#11719919—21no—
35C4×C4⋊Dic53C2211yescharacteristic
36C4×#11719921—21no—
37#16810386C2212yes—
38#16810386C2212yes—
39#16810385C2212yes—
40C2×C4×Q8×C53C2211yescharacteristic
41C2×C4⋊C8×C53C2211yescharacteristic
42C2×#103×C53C2212yes—
43C53×#384C2211yescharacteristic
44C53×#340C2212yes—
45C22×Dic106—21no—
46C22×C4×D53D411yescharacteristic
47C22×C4×D53D411yescharacteristic
48C22×C4×D53C2311yescharacteristic
49C22×#88322C2312yes—
50#13133217—22no—
51#13133217—22no—
52#13133197—44no—
53#13133193—22no—
54#13133193—22no—
55C2×C4⋊Dic53—21no—
56C2×C4⋊Dic53—22no—
57C2×C4⋊Dic53C2312yes—
58C2×C4⋊Dic53C2311yescharacteristic
59C2×C4⋊Dic53C2311yescharacteristic
60C2×#11719919—21no—
61C2×#11719919—21no—
62C2×#11719921—21no—
63C2×#11719921—22no—
64C2×#11719921—21no—
65C2×C4×Dic53—21no—
66C2×C4×Dic53D411yescharacteristic
67C2×C4×Dic53D411yescharacteristic
68C2×C4×Dic53C2311yescharacteristic
69C42×D53—22no—
70C42×D53—22no—
71#13133150—24no—
72#13133148—24no—
73C4⋊#88322D414yes—
74#13133141—24no—
75#13133129—24no—
76#13133127—24no—
77#13133124—21no—
78#13133124—21no—
79#13133124—21no—
80#13133124—21no—
81#13133124—22no—
82#13133124—22no—
83#13133124—21no—
84C22×Q8×C53C2311yescharacteristic
85C22×C8×C53C2312yes—
86C4×Q8×C53—22no—
87C4×Q8×C53—22no—
88C2×C4⋊C4×C53—21no—
89C2×C4⋊C4×C53—21no—
90C2×C4⋊C4×C53—21no—
91C2×C4⋊C4×C53C2311yescharacteristic
92C2×C4⋊C4×C53C2312yes—
93C2×C42×C53—21no—
94C2×C42×C53C2311yescharacteristic
95C4⋊C8×C53—22no—
96#103×C53—24no—
97C23×D53C2×D411yescharacteristic
98C2×Q8×C53—21no—
99C2×Q8×C53—21no—
100C2×Q8×C53—21no—
101C2×Q8×C53—21no—
102C2×Q8×C53—21no—
103C2×Q8×C53#4112yes—
104C22×C4×C53—21no—
105C22×C4×C53—21no—
106C22×C4×C53—22no—
107C22×C4×C53C2×D411yescharacteristic
108C22×C4×C53C2×D411yescharacteristic
109C22×C4×C53C2411yescharacteristic
110C22×Dic53—21no—
111C22×Dic53—21no—
112C22×Dic53—22no—
113C22×Dic53C2×D411yescharacteristic
114C22×Dic53C2×D411yescharacteristic
115C22×Dic53C2×D411yescharacteristic
116C2×C4×D53—22no—
117C2×C4×D53—22no—
118C2×C4×D53—22no—
119C2×C4×D53—22no—
120C2×C4×D53—22no—
121C2×C4×D53—21no—
122C2×C4×D53—21no—
123C2×C4×D53—21no—
124C2×C4×D53—21no—
125C2×C4×D53—21no—
126C2×C4×D53—21no—
127C2×C4×D53—41no—
128C2×Dic106—42no—
129C2×Dic106—42no—
130C2×Dic106—42no—
131C2×C8×C53—22no—
132C2×C8×C53—22no—
133C2×C8×C53—22no—
134C4⋊C4×C53—21no—
135C4⋊C4×C53—22no—
136C4⋊C4×C53—21no—
137C4⋊C4×C53—22no—
138C4⋊C4×C53—42no—
139C4⋊C4×C53—42no—
140C4⋊C4×C53—42no—
141C42×C53—42no—
142C42×C53C2×D412yes—
143C42×C53#4112yes—
144#11719921—24no—
145#11719921—42no—
146#11719921—44no—
147C4⋊Dic53—21no—
148C4⋊Dic53—22no—
149C4⋊Dic53—22no—
150C4⋊Dic53—21no—
151C4⋊Dic53—21no—
152C4⋊Dic53—44no—
153C4⋊Dic53—42no—
154C4⋊Dic53#4112yes—
155#11719919—24no—
156#11719919—42no—
157C4×Dic53—22no—
158C4×Dic53—22no—
159C4×Dic53—21no—
160C4×Dic53—21no—
161C4×Dic53—42no—
162C2×#88322—22no—
163C2×#88322—22no—
164C2×#88322C2×D414yes—
165C23×C53C22×D411yescharacteristic
166C22×D53—21no—
167C22×D53—21no—
168C22×D53—21no—
169C22×D53—21no—
170C22×D53—21no—
171C22×D53—21no—
172C22×D53AGL(1,ℤ/8)12yes—
173Q8×C53—22no—
174Q8×C53—22no—
175Q8×C53—41no—
176Q8×C53—41no—
177C2×C4×C53—21no—
178C2×C4×C53—21no—
179C2×C4×C53—22no—
180C2×C4×C53—41no—
181C2×C4×C53—41no—
182C2×C4×C53—41no—
183C2×C4×C53—42no—
184C2×C4×C53—42no—
185C2×C4×C53—41no—
186C2×C4×C53—42no—
187C2×C4×C53—41no—
188C2×C4×C53#12112yes—
189C2×C4×C53#12112yes—
190C2×C4×C53#13712yes—
191C2×C4×C53#13712yes—
192C2×C4×C53C22×D411yescharacteristic
193C2×C4×C53C2×#4111yescharacteristic
194C2×C4×C53C2×#4111yescharacteristic
195C2×C4×C53C2×#4111yescharacteristic
196C2×C4×C532−1+411yescharacteristic
197C2×C4×C532−1+411yescharacteristic
198C2×Dic53—21no—
199C2×Dic53—21no—
200C2×Dic53—21no—
201C2×Dic53—21no—
202C2×Dic53—22no—
203C2×Dic53—21no—
204C2×Dic53—21no—
205C2×Dic53—21no—
206C2×Dic53—21no—
207C2×Dic53—21no—
208C2×Dic53—41no—
209C2×Dic53—41no—
210C2×Dic53—42no—
211C2×Dic53—41no—
212C2×Dic53—42no—
213C2×Dic53—41no—
214C2×Dic53—41no—
215C2×Dic53—42no—
216C4×D53—44no—
217C4×D53—41no—
218C4×D53—41no—
219C4×D53—41no—
220C4×D53—41no—
221C4×D53—44no—
222Dic106—84no—
223C8×C53—42no—
224#88322—24no—
225#24711—531nomaximal, Sylow
226C22×C53C2×#12111yescharacteristic
227C22×C53#52511yescharacteristic
228C22×C53#53611yescharacteristic
229C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
230C22×C53C2×#13711yescharacteristic
231C22×C53C2×#13711yescharacteristic
232C22×C53#57511yescharacteristic
233C2×D53—22no—
234C2×D53—22no—
235C2×D53—22no—
236C2×D53—41no—
237C2×D53—41no—
238C2×D53—41no—
239C2×D53—41no—
240C4×C53—22no—
241C4×C53—42no—
242C4×C53—82no—
243C4×C53—81no—
244C4×C53#47014yes—
245C4×C53#54711yescharacteristic
246C4×C53#54811yescharacteristic
247C4×C53#55211yescharacteristic
248C4×C53#55611yescharacteristic
249Dic53—42no—
250Dic53—41no—
251Dic53—41no—
252Dic53—41no—
253Dic53—81no—
254Dic53—82no—
255C2×#477—531no—
256C2×#432—531no—
257#2857—531no—
258C4×C22⋊Q8—531no—
259#2075—532no—
260#2035—532no—
261#1987—531no—
262#1948—531no—
263#1944—532no—
264#1935—532no—
265#1803D5311yescharacteristic, complemented
266C2×C53#241511yescharacteristic
267C2×C53C2×#47011yescharacteristic
268C2×C53#305511yescharacteristic
269C2×C53#311211yescharacteristic
270C2×C53#311611yescharacteristic
271C2×C53#323711yescharacteristic
272C2×C53#323811yescharacteristic
273D53—42no—
274C2×C4⋊Q8—531no—
275C2×C22⋊Q8—531no—
276C2×C22⋊Q8—1061no—
277C2×C4×Q8C2×D5311yescharacteristic
278C22×C42—531no—
279#477—1064no—
280#432—1064no—
281C2×#107—532no—
282C2×#106—532no—
283C2×C4⋊C8—531no—
284C2×C4⋊C8C2×D5311yescharacteristic
285C2×#103—532no—
286C2×#103C2×D5312yes—
287C2×C22⋊C8—532no—
288C2×#97—532no—
289#391—531no—
290#391—1061no—
291#389—531no—
292#389—532no—
293#389—1061no—
294#387—1061no—
295#385—1061no—
296#384C2×D5311yescharacteristic
297#380—531no—
298#380—532no—
299#380—1061no—
300C4×C4⋊C4—531no—
301C4×C4⋊C4—1061no—
302C4×C22⋊C4—1061no—
303#340—532no—
304#340—532no—
305#340—532no—
306#340C2×D5312yes—
307C53#2471111yescharacteristic, complemented, Sylow
308C22×Q8—1061no—
309C22×Q8C22×D5311yescharacteristic
310C23×C4—531no—
311C23×C4—531no—
312C23×C4—531no—
313C22×C8—532no—
314C22×C8C22×D5312yes—
315C4⋊Q8—1062no—
316C4⋊Q8—1062no—
317C22⋊Q8—1062no—
318C22⋊Q8—1062no—
319C22⋊Q8—2124no—
320C4×Q8—22no—
321C4×Q8—22no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—531no—
326C2×C4⋊C4—532no—
327C2×C4⋊C4—531no—
328C2×C4⋊C4—1061no—
329C2×C4⋊C4—1062no—
330C2×C4⋊C4—1061no—
331C2×C4⋊C4—1061no—
332C2×C4⋊C4C22×D5311yescharacteristic
333C2×C4⋊C4C22×D5312yes—
334C2×C22⋊C4—1061no—
335C2×C22⋊C4—1062no—
336C2×C22⋊C4—1061no—
337C2×C42—21no—
338C2×C42—531no—
339C2×C42—531no—
340C2×C42—531no—
341C2×C42—1062no—
342C2×C42—1062no—
343C2×C42—1061no—
344C2×C42C22×D5311yescharacteristic
345#107—1064no—
346#106—1064no—
347C4⋊C8—22no—
348C4⋊C8—534no—
349#103—24no—
350#103—1064no—
351C22⋊C8—1064no—
352#97—1064no—
353#95—1061no—
354#95—1061no—
355#95—1061no—
356#95—1061no—
357#95—1062no—
358#95—1062no—
359#95—1061no—
360C24—531no—
361C2×Q8—21no—
362C2×Q8—21no—
363C2×Q8—21no—
364C2×Q8—21no—
365C2×Q8—21no—
366C2×Q8—2122no—
367C2×Q8—2122no—
368C2×Q8—2122no—
369C2×Q8#1171994712yes—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—22no—
373C22×C4—531no—
374C22×C4—531no—
375C22×C4—531no—
376C22×C4—1062no—
377C22×C4—1061no—
378C22×C4—1062no—
379C22×C4—1062no—
380C22×C4—1062no—
381C22×C4—1062no—
382C22×C4—1061no—
383C22×C4—1061no—
384C22×C4—1061no—
385C22×C4—1061no—
386C22×C4—1061no—
387C22×C4—1061no—
388C22×C4—1061no—
389C22×C4—1062no—
390C22×C4—2121no—
391C22×C4D4×D5311yescharacteristic
392C22×C4D4×D5311yescharacteristic
393C22×C4C23×D5311yescharacteristic
394C2×C8—22no—
395C2×C8—22no—
396C2×C8—22no—
397C2×C8—534no—
398C2×C8—1062no—
399C2×C8—1062no—
400C4⋊C4—21no—
401C4⋊C4—22no—
402C4⋊C4—22no—
403C4⋊C4—21no—
404C4⋊C4—42no—
405C4⋊C4—42no—
406C4⋊C4—42no—
407C4⋊C4—532no—
408C4⋊C4—1064no—
409C4⋊C4—1061no—
410C4⋊C4—1062no—
411C4⋊C4—1062no—
412C4⋊C4—1061no—
413C4⋊C4—1061no—
414C4⋊C4—2124no—
415C4⋊C4—2122no—
416C4⋊C4—2122no—
417C22⋊C4—1064no—
418C22⋊C4—2124no—
419C22⋊C4—2122no—
420C42—42no—
421C42—1062no—
422C42—1062no—
423C42—1061no—
424C42—1061no—
425C42—2122no—
426C42#1171994512yes—
427C42D4×D5312yes—
428C23—532no—
429C23—1061no—
430C23—1061no—
431C23—1061no—
432C23—1061no—
433C23—1061no—
434C23—1061no—
435C23C2×D4×D5311yescharacteristic
436Q8—22no—
437Q8—22no—
438Q8—41no—
439Q8—41no—
440Q8—4244no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—22no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—41no—
450C2×C4—42no—
451C2×C4—41no—
452C2×C4—1061no—
453C2×C4—1061no—
454C2×C4—1061no—
455C2×C4—1062no—
456C2×C4—1061no—
457C2×C4—1061no—
458C2×C4—1061no—
459C2×C4—1061no—
460C2×C4—1061no—
461C2×C4—1061no—
462C2×C4—2124no—
463C2×C4—2122no—
464C2×C4—2122no—
465C2×C4—2121no—
466C2×C4—2121no—
467C2×C4—2122no—
468C2×C4—2121no—
469C2×C4—2121no—
470C2×C4—2121no—
471C2×C4—2121no—
472C2×C4—2121no—
473C2×C4—2121no—
474C2×C4—2121no—
475C2×C4—2124no—
476C2×C4#1313319012yes—
477C2×C4#1313319012yes—
478C2×C4#1313324612yes—
479C2×C4#1313324912yes—
480C2×C4C2×D4×D5311yescharacteristic
481C2×C4#41×D5311yescharacteristic
482C2×C4C2×#1171994711yescharacteristic
483C2×C4C2×#1171994511yescharacteristic
484C2×C4#1313326411yescharacteristic
485C2×C4#1313326511yescharacteristic
486C8—42no—
487C8—1064no—
488C22—1062no—
489C22—1062no—
490C22—1062no—
491C22—2121no—
492C22—2121no—
493C22—2121no—
494C22—2121no—
495C22C2×#1313319011yescharacteristic
496C22#1681113311yescharacteristic
497C22#1681119611yescharacteristic
498C22AGL(1,ℤ/8)×D5311yescharacteristic
499C22C2×#1313324611yescharacteristic
500C22C2×#1313324911yescharacteristic
501C22#1681142211yescharacteristic
502C4—22no—
503C4—42no—
504C4—82no—
505C4—81no—
506C4—4242no—
507C4—4241no—
508C4—2121no—
509C4—2121no—
510C4—2121no—
511C4—2122no—
512C4#1681088014yes—
513C4#1681127511yescharacteristic
514C4#1681127911yescharacteristic
515C4#1681130811yescharacteristic
516C4#1681134111yescharacteristic
517C2—2122no—
518C2#2027814611yescharacteristic
519C2C2×#1681088011yescharacteristic
520C2#2028433711yescharacteristic
521C2#2028483611yescharacteristic
522C2#2028487611yescharacteristic
523C2#2028592811yescharacteristic
524C2#2028594011yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#1803C2
socleC23×C53C22×D4
prime radicalC23×D53C2×D4
hypercenter#1803D53
hypocenterC53#24711
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1803

Complements

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

NH ≅ G/Ntype
1. C2×#16810918C2semidirect
7. #20275662C2semidirect
8. #20275476C2semidirect
9. #20275457C2semidirect
10. #20275407C2semidirect
11. C53×#1803C2semidirect
265. #1803D53semidirect
307. C53#24711semidirect

Maximal subgroups

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

subgroupindex
C2×#168109182
C2×#168106792
#202830682
C4×#131331932
#202761422
#202759092
#202756622
#202754762
#202754572
#202754072
C53×#18032
#2471153