Finite Group Databasebeta

← #27366933

Subgroups — #27366933

Order 13568.

728 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285108C211yesmaximal, characteristic, complemented
2C2×#16810828C211yesmaximal, characteristic, complemented
3D4×#11719917C211yesmaximal, characteristic, complemented
4#20274274C212yesmaximal, complemented
5#20274260C212yesmaximal, complemented
6#20274150C212yesmaximal, complemented
7#20274136C212yesmaximal, complemented
8#20273468C211yesmaximal, characteristic, complemented
9#20273399C211yesmaximal, characteristic, complemented
10#20273392C211yesmaximal, characteristic, complemented
11C53×#3100C211yesmaximal, characteristic
12C22×#11719923—21no—
13C22×#11719917—21no—
14C2×#13133217C2211yescharacteristic
15#16810982—22no—
16#16810973C2212yescomplemented
17#16810969C2212yescomplemented
18#16810828—21no—
19#16810828—21no—
20#16810828C2212yes—
21#16810828C2212yes—
22#16810701C2212yes—
23#16810695—22no—
24D4×#88322C2212yes—
25#16810647C2211yescharacteristic
26C2×C4⋊#88322C2211yescharacteristic
27#16810624—21no—
28#16810624—21no—
29C2×#13133140—21no—
30#16810587—21no—
31C4×#11719917C2211yescharacteristic
32#16810375C2212yescomplemented
33#16810373C2212yescomplemented
34#16810370C2212yescomplemented
35#16810370C2212yescomplemented
36#16810369C2212yescomplemented
37#16810367C2212yescomplemented
38C53×#532C2211yescharacteristic
39C2×C4×D4×C53C2211yescharacteristic
40C53×#480—21no—
41C53×#474C2211yescharacteristic
42C53×#474C2211yescharacteristic
43C53×#459C2211yescharacteristic
44C53×#459C2211yescharacteristic
45C53×#447—21no—
46C4×D8×C53—21no—
47C4×D8×C53—21no—
48C53×#431C2211yescharacteristic
49C22×Dic106—21no—
50C2×#11719923—22no—
51C2×#11719923—22no—
52C2×#11719923—44no—
53C2×#11719917—22no—
54C2×#11719917—21no—
55C2×#11719917—44no—
56C2×#11719917D411yescharacteristic
57C2×#11719917D411yescharacteristic
58C22×#88322—22no—
59C22×#88322—21no—
60#13133217—21no—
61#13133217—21no—
62#13133217—21no—
63#13133217C2312yes—
64C2×C4⋊Dic53—21no—
65C2×C4⋊Dic53—21no—
66C4⋊#88322—21no—
67C4⋊#88322—21no—
68C4⋊#88322C2312yes—
69C4⋊#88322C2312yes—
70#13133141—22no—
71#13133141—22no—
72#13133140—22no—
73#13133140—22no—
74#13133138—22no—
75#13133138—22no—
76C22⋊#88322—22no—
77C22⋊#88322—22no—
78#13133128D412yes—
79C4×#88322C2312yes—
80C2×#41×C53—21no—
81C22×D4×C53D411yescharacteristic
82C23×C4×C53—21no—
83C2×D8×C53—21no—
84C2×D8×C53—21no—
85C4⋊Q8×C53C2311yescharacteristic
86#127×C53C2311yescharacteristic
87#124×C53—21no—
88#121×C53—21no—
89#121×C53—21no—
90C4×D4×C53—21no—
91C4×D4×C53—21no—
92C4×D4×C53—21no—
93C4×D4×C53—21no—
94C4×D4×C53D411yescharacteristic
95C4×D4×C53D411yescharacteristic
96C4×D4×C53C2311yescharacteristic
97C4×D4×C53C2311yescharacteristic
98C2×C4⋊C4×C53D411yescharacteristic
99C2×C22⋊C4×C53—21no—
100C2×C42×C53C2311yescharacteristic
101#107×C53—21no—
102#107×C53—21no—
103C4⋊C8×C53C2311yescharacteristic
104C4⋊C8×C53C2311yescharacteristic
105#102×C53—21no—
106#102×C53—21no—
107#102×C53—21no—
108#102×C53—21no—
109C22⋊C8×C53—21no—
110C4×C8×C53—21no—
111C24×C53—21no—
112#41×C53—42no—
113#41×C53—42no—
114C2×Q8×C53—21no—
115C2×D4×C53—22no—
116C2×D4×C53—21no—
117C2×D4×C53—21no—
118C2×D4×C53—21no—
119C2×D4×C53—21no—
120C2×D4×C53—41no—
121C2×D4×C53—41no—
122C2×D4×C53C2×D411yescharacteristic
123C2×D4×C53C2×D411yescharacteristic
124C22×C4×C53—21no—
125C22×C4×C53—21no—
126C22×C4×C53—21no—
127C22×C4×C53—21no—
128C22×C4×C53—21no—
129C22×C4×C53—42no—
130C22×C4×C53—42no—
131C22×C4×C53—41no—
132C22×C4×C53C2×D411yescharacteristic
133C22×C4×C53C2×D411yescharacteristic
134C22×C4×C53#4111yescharacteristic
135C22×Dic53—41no—
136C2×Dic106—22no—
137C2×Dic106—42no—
138C2×Dic106—42no—
139C2×Dic106—41no—
140D8×C53—42no—
141D8×C53—42no—
142C2×C8×C53—21no—
143C2×C8×C53—21no—
144C4⋊C4×C53—21no—
145C4⋊C4×C53—21no—
146C4⋊C4×C53—21no—
147C4⋊C4×C53C2×D411yescharacteristic
148C4⋊C4×C53C2×D411yescharacteristic
149C22⋊C4×C53—21no—
150C22⋊C4×C53—21no—
151C22⋊C4×C53—21no—
152C22⋊C4×C53—21no—
153C22⋊C4×C53—41no—
154C22⋊C4×C53—41no—
155C42×C53C2×D411yescharacteristic
156C42×C53C2×D411yescharacteristic
157C42×C53#4111yescharacteristic
158C42×C53C2411yescharacteristic
159#11719923—44no—
160#11719923—44no—
161C4⋊Dic53—22no—
162C4⋊Dic53—22no—
163C4⋊Dic53—41no—
164C4⋊Dic53—41no—
165#11719917—22no—
166#11719917—22no—
167#11719917—44no—
168#11719917—42no—
169C2×#88322—22no—
170C2×#88322—22no—
171C2×#88322—44no—
172C2×#88322—42no—
173C2×#88322C2×D412yes—
174C2×#88322C2×D412yes—
175C23×C53—21no—
176C23×C53—21no—
177C23×C53—42no—
178C23×C53—42no—
179C23×C53—41no—
180C23×C53—81no—
181C23×C53#12111yescharacteristic
182Q8×C53—41no—
183Q8×C53—41no—
184D4×C53—21no—
185D4×C53—21no—
186D4×C53—21no—
187D4×C53—42no—
188D4×C53—41no—
189D4×C53—41no—
190D4×C53—41no—
191D4×C53—41no—
192D4×C53—81no—
193D4×C53—82no—
194D4×C53AGL(1,ℤ/8)12yes—
195C2×C4×C53—21no—
196C2×C4×C53—21no—
197C2×C4×C53—21no—
198C2×C4×C53—21no—
199C2×C4×C53—41no—
200C2×C4×C53—42no—
201C2×C4×C53—41no—
202C2×C4×C53—41no—
203C2×C4×C53—41no—
204C2×C4×C53—41no—
205C2×C4×C53—42no—
206C2×C4×C53—42no—
207C2×C4×C53—41no—
208C2×C4×C53—82no—
209C2×C4×C53#12111yescharacteristic
210C2×C4×C53#12111yescharacteristic
211C2×C4×C53#12111yescharacteristic
212C2×C4×C53#13512yes—
213C2×C4×C53AGL(1,ℤ/8)12yes—
214C2×C4×C53C22×D411yescharacteristic
215C2×C4×C53C22×D411yescharacteristic
216C2×C4×C53C2×#4111yescharacteristic
217C2×Dic53—42no—
218C2×Dic53—82no—
219Dic106—42no—
220Dic106—42no—
221Dic106—82no—
222C8×C53—41no—
223C8×C53—41no—
224#88322—22no—
225#88322—22no—
226#88322—42no—
227#88322—42no—
228#19169—531nomaximal, Sylow
229C22×C53—21no—
230C22×C53—21no—
231C22×C53—22no—
232C22×C53—41no—
233C22×C53—41no—
234C22×C53—41no—
235C22×C53—41no—
236C22×C53—41no—
237C22×C53—84no—
238C22×C53—81no—
239C22×C53—81no—
240C22×C53—81no—
241C22×C53#46312yes—
242C22×C53C2×#12111yescharacteristic
243C4×C53—21no—
244C4×C53—21no—
245C4×C53—21no—
246C4×C53—41no—
247C4×C53—41no—
248C4×C53—81no—
249C4×C53#46312yes—
250C4×C53C2×#13511yescharacteristic
251C4×C53C2×AGL(1,ℤ/8)11yescharacteristic
252C4×C53C2×AGL(1,ℤ/8)11yescharacteristic
253C4×C53#57511yescharacteristic
254Dic53—82no—
255#3141—531no—
256#3100D5311yescharacteristic
257C2×#461—531no—
258D4×M16—531no—
259#1701—532no—
260#1699—532no—
261#1681—532no—
262#1679—532no—
263#1565—531no—
264#1549—531no—
265#1547—531no—
266C2×C53—22no—
267C2×C53—42no—
268C2×C53—81no—
269C2×C53—161no—
270C2×C53C2×#46311yescharacteristic
271C2×C53#305511yescharacteristic
272C2×C53#306911yescharacteristic
273C22×SD16—1061no—
274C22×M16—1061no—
275#532C2×D5311yescharacteristic
276C2×C4⋊Q8—531no—
277C2×C4×D4C2×D5311yescharacteristic
278#489—1062no—
279#487—532no—
280#486—532no—
281#480—21no—
282#474C2×D5311yescharacteristic
283#474C2×D5311yescharacteristic
284#461—532no—
285#461—532no—
286#461—1061no—
287#461—1061no—
288#459C2×D5311yescharacteristic
289#459C2×D5311yescharacteristic
290#447—21no—
291C4×D8—21no—
292C4×D8—21no—
293#436—532no—
294#435—1062no—
295C8×D4—532no—
296#431C2×D5311yescharacteristic
297#423—531no—
298C2×C4⋊C8—531no—
299#417—1061no—
300#417—1061no—
301C2×#103—1061no—
302#407—1061no—
303C4×M16—531no—
304#335—532no—
305#334—532no—
306#332—532no—
307#332—532no—
308#332—532no—
309#331—532no—
310C53#1916911yescharacteristic, complemented, Sylow
311C2×#41—21no—
312C22×Q8—1061no—
313C22×D4C22⋊D5311yescharacteristic
314C23×C4—21no—
315C2×SD16—1062no—
316C2×SD16—1062no—
317C2×SD16—2124no—
318C2×D8—21no—
319C2×D8—21no—
320C2×M16—531no—
321C2×M16—531no—
322C2×M16—1062no—
323C2×M16—1061no—
324C2×M16—2124no—
325C22×C8—1062no—
326C22×C8—1061no—
327C4⋊Q8—532no—
328C4⋊Q8—1061no—
329C4⋊Q8—1061no—
330C4⋊Q8—1061no—
331C4⋊Q8C22×D5311yescharacteristic
332#127C22×D5311yescharacteristic
333#124—21no—
334#121—21no—
335#121—21no—
336C4×D4—21no—
337C4×D4—21no—
338C4×D4—21no—
339C4×D4—21no—
340C4×D4C22⋊D5311yescharacteristic
341C4×D4C22⋊D5311yescharacteristic
342C4×D4C22×D5311yescharacteristic
343C4×D4C22×D5311yescharacteristic
344C2×C4⋊C4—1061no—
345C2×C4⋊C4—1061no—
346C2×C4⋊C4C22⋊D5311yescharacteristic
347C2×C22⋊C4—21no—
348C2×C42C22×D5311yescharacteristic
349#107—21no—
350#107—21no—
351C4⋊C8—532no—
352C4⋊C8—532no—
353C4⋊C8—1061no—
354C4⋊C8—1061no—
355C4⋊C8C22×D5311yescharacteristic
356C4⋊C8C22×D5311yescharacteristic
357#103—1062no—
358#103—1062no—
359#103—1062no—
360#103—1062no—
361#102—21no—
362#102—21no—
363#102—21no—
364#102—21no—
365#102—1062no—
366#102—1062no—
367C22⋊C8—21no—
368C22⋊C8—1062no—
369C22⋊C8—1062no—
370#97—532no—
371C4×C8—21no—
372C4×C8—532no—
373C24—21no—
374#41—42no—
375#41—42no—
376C2×Q8—21no—
377C2×Q8—1062no—
378C2×Q8—2122no—
379C2×Q8—2122no—
380C2×Q8—2121no—
381C2×D4—22no—
382C2×D4—21no—
383C2×D4—21no—
384C2×D4—21no—
385C2×D4—21no—
386C2×D4—41no—
387C2×D4—41no—
388C2×D4C2×C22⋊D5311yescharacteristic
389C2×D4C2×C22⋊D5311yescharacteristic
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—41no—
398C22×C4—2121no—
399C22×C4D4×D5311yescharacteristic
400C22×C4#1171994711yescharacteristic
401C22×C4C2×C22⋊D5311yescharacteristic
402SD16—2124no—
403SD16—2124no—
404D8—42no—
405D8—42no—
406M16—1062no—
407M16—1062no—
408M16—2124no—
409M16—2122no—
410C2×C8—21no—
411C2×C8—21no—
412C2×C8—532no—
413C2×C8—532no—
414C2×C8—1062no—
415C2×C8—1062no—
416C2×C8—2124no—
417C2×C8—2122no—
418C4⋊C4—21no—
419C4⋊C4—21no—
420C4⋊C4—21no—
421C4⋊C4—1062no—
422C4⋊C4—1062no—
423C4⋊C4—2121no—
424C4⋊C4—2121no—
425C4⋊C4C2×C22⋊D5311yescharacteristic
426C4⋊C4C2×C22⋊D5311yescharacteristic
427C22⋊C4—21no—
428C22⋊C4—21no—
429C22⋊C4—21no—
430C22⋊C4—21no—
431C22⋊C4—41no—
432C22⋊C4—41no—
433C42D4×D5311yescharacteristic
434C42#1171994711yescharacteristic
435C42C2×C22⋊D5311yescharacteristic
436C42C23×D5311yescharacteristic
437C23—21no—
438C23—21no—
439C23—42no—
440C23—42no—
441C23—41no—
442C23—81no—
443C23#1313318911yescharacteristic
444Q8—41no—
445Q8—41no—
446Q8—4242no—
447Q8—2122no—
448Q8—2122no—
449D4—21no—
450D4—21no—
451D4—21no—
452D4—42no—
453D4—41no—
454D4—41no—
455D4—41no—
456D4—41no—
457D4—81no—
458D4—82no—
459D4#1313324212yes—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—41no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—82no—
474C2×C4—4242no—
475C2×C4—2122no—
476C2×C4#1313318911yescharacteristic
477C2×C4#1313318911yescharacteristic
478C2×C4#1313318911yescharacteristic
479C2×C4#1313323612yes—
480C2×C4#1313324112yes—
481C2×C4C2×D4×D5311yescharacteristic
482C2×C4C22×C22⋊D5311yescharacteristic
483C2×C4C2×#1171994711yescharacteristic
484C8—41no—
485C8—41no—
486C8—1062no—
487C8—1062no—
488C8—2122no—
489C8—2122no—
490C22—21no—
491C22—21no—
492C22—22no—
493C22—41no—
494C22—41no—
495C22—41no—
496C22—41no—
497C22—41no—
498C22—84no—
499C22—81no—
500C22—81no—
501C22—81no—
502C22#1681083612yes—
503C22C2×#1313318911yescharacteristic
504C4—21no—
505C4—21no—
506C4—21no—
507C4—41no—
508C4—41no—
509C4—81no—
510C4—4242no—
511C4#1681083612yes—
512C4C2×#1313323611yescharacteristic
513C4C2×#1313324111yescharacteristic
514C4C2×#1313324211yescharacteristic
515C4#1681142111yescharacteristic
516C2—22no—
517C2—42no—
518C2—81no—
519C2—161no—
520C2C2×#1681083611yescharacteristic
521C2#2028433611yescharacteristic
522C2#2028445411yescharacteristic
subgroupG/H
centerC22C2×#13133189
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
Fitting, prime radicalC53×#3100C2
socleC22×C53C2×#121
hypercenter#3100D53
hypocenterC53#19169
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3100

Complements

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

NH ≅ G/Ntype
1. #20285108C2semidirect
2. C2×#16810828C2semidirect
3. D4×#11719917C2semidirect
4. #20274274C2semidirect
5. #20274260C2semidirect
6. #20274150C2semidirect
7. #20274136C2semidirect
8. #20273468C2semidirect
9. #20273399C2semidirect
10. #20273392C2semidirect
16. #16810973C22semidirect
17. #16810969C22semidirect
32. #16810375C22semidirect
33. #16810373C22semidirect
34. #16810370C22semidirect
35. #16810370C22semidirect
36. #16810369C22semidirect
37. #16810367C22semidirect
310. C53#19169semidirect

Maximal subgroups

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

subgroupindex
#202851082
C2×#168108282
D4×#117199172
#202742742
#202742602
#202741502
#202741362
#202734682
#202733992
#202733922
C53×#31002
#1916953