Finite Group Databasebeta

← #27366939

Subgroups — #27366939

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#20285111C211yesmaximal, characteristic, complemented
2#20284715C211yesmaximal, characteristic
3D4×#11719917C211yesmaximal, characteristic, complemented
4#20274278C212yesmaximal, complemented
5#20274264C212yesmaximal, complemented
6#20274154C212yesmaximal, complemented
7#20274140C212yesmaximal, complemented
8#20273465C211yesmaximal, characteristic, complemented
9#20273400C211yesmaximal, characteristic, complemented
10#20273390C211yesmaximal, characteristic, complemented
11C2×C53×#461C211yesmaximal, characteristic, complemented
12C22×#11719917—21no—
13#16811173C2211yescharacteristic
14#16810985—22no—
15#16810975C2212yescomplemented
16#16810971C2212yescomplemented
17#16810936—21no—
18#16810903C2211yescharacteristic
19#16810903C2211yescharacteristic
20#16810818C2211yescharacteristic
21#16810818C2211yescharacteristic
22#16810757—21no—
23C4×#11719922—21no—
24C4×#11719922—21no—
25#16810701C2212yes—
26#16810695—22no—
27D4×#88322C2212yes—
28#16810676C2211yescharacteristic
29#16810647C2211yescharacteristic
30#16810622—21no—
31#16810622—21no—
32#16810587—21no—
33C4×#11719917C2211yescharacteristic
34#16810375C2212yescomplemented
35#16810373C2212yescomplemented
36#16810369C2212yescomplemented
37#16810369C2212yescomplemented
38#16810368C2212yescomplemented
39#16810366C2212yescomplemented
40C22×SD16×C53—21no—
41C2×C4⋊Q8×C53C2211yescharacteristic
42C2×C4×D4×C53C2211yescharacteristic
43C53×#461—21no—
44C53×#461—21no—
45C53×#461C2212yes—
46C53×#461C2212yes—
47C2×C4⋊C8×C53C2211yescharacteristic
48C2×#103×C53—21no—
49C2×#11719945—21no—
50C2×#11719922—21no—
51C2×#11719922—21no—
52C2×#11719917—22no—
53C2×#11719917—21no—
54C2×#11719917—44no—
55C2×#11719917D411yescharacteristic
56C2×#11719917D411yescharacteristic
57C22×#88322—22no—
58#13133217C2311yescharacteristic
59#13133214C2311yescharacteristic
60#13133205—21no—
61#13133191—21no—
62#13133191—21no—
63C4×D212—21no—
64C4×D212—21no—
65#13133150—21no—
66#13133150—21no—
67C4⋊#88322—21no—
68C4⋊#88322C2311yescharacteristic
69C4⋊#88322C2311yescharacteristic
70C4⋊#88322C2312yes—
71#13133140—22no—
72#13133140—22no—
73#13133138—21no—
74#13133138—21no—
75#13133137—21no—
76#13133137—21no—
77#13133137—22no—
78#13133137—22no—
79C22⋊#88322—22no—
80C22⋊#88322—22no—
81C22⋊#88322—21no—
82#13133128D412yes—
83C4×#88322—21no—
84C4×#88322C2312yes—
85C22×Q8×C53—21no—
86C22×D4×C53D411yescharacteristic
87C23×C4×C53—21no—
88C2×SD16×C53—22no—
89C2×SD16×C53—22no—
90C2×SD16×C53—44no—
91C22×C8×C53—21no—
92C4⋊Q8×C53—21no—
93C4⋊Q8×C53—21no—
94C4⋊Q8×C53—21no—
95C4⋊Q8×C53C2312yes—
96C4×D4×C53—21no—
97C4×D4×C53—21no—
98C4×D4×C53D411yescharacteristic
99C4×D4×C53D411yescharacteristic
100C4×D4×C53C2311yescharacteristic
101C4×D4×C53C2311yescharacteristic
102C2×C4⋊C4×C53—21no—
103C2×C4⋊C4×C53—21no—
104C2×C4⋊C4×C53D411yescharacteristic
105C2×C22⋊C4×C53—21no—
106C2×C42×C53C2311yescharacteristic
107C4⋊C8×C53—21no—
108C4⋊C8×C53C2312yes—
109#103×C53—22no—
110#103×C53—22no—
111C24×C53—21no—
112C2×Q8×C53—22no—
113C2×Q8×C53—42no—
114C2×Q8×C53—42no—
115C2×Q8×C53—41no—
116C2×D4×C53—22no—
117C2×D4×C53—21no—
118C2×D4×C53—41no—
119C2×D4×C53C2×D411yescharacteristic
120C2×D4×C53C2×D411yescharacteristic
121C22×C4×C53—21no—
122C22×C4×C53—21no—
123C22×C4×C53—21no—
124C22×C4×C53—21no—
125C22×C4×C53—21no—
126C22×C4×C53—42no—
127C22×C4×C53—42no—
128C22×C4×C53—41no—
129C22×C4×C53C2×D411yescharacteristic
130C22×C4×C53C2×D411yescharacteristic
131C22×C4×C53#4111yescharacteristic
132C2×C22⋊D53—41no—
133#11719945—42no—
134#11719945—42no—
135C2×D212—21no—
136C2×D212—21no—
137C2×D212—21no—
138C2×C4×D53—41no—
139C2×Dic106—21no—
140SD16×C53—44no—
141SD16×C53—44no—
142C2×C8×C53—22no—
143C2×C8×C53—42no—
144C4⋊C4×C53—21no—
145C4⋊C4×C53—22no—
146C4⋊C4×C53—22no—
147C4⋊C4×C53—41no—
148C4⋊C4×C53—41no—
149C4⋊C4×C53C2×D411yescharacteristic
150C4⋊C4×C53C2×D411yescharacteristic
151C22⋊C4×C53—21no—
152C22⋊C4×C53—21no—
153C22⋊C4×C53—21no—
154C22⋊C4×C53—21no—
155C42×C53C2×D411yescharacteristic
156C42×C53C2×D411yescharacteristic
157C42×C53#4111yescharacteristic
158C42×C53C2411yescharacteristic
159#11719922—42no—
160#11719922—42no—
161#11719921—41no—
162#11719921—41no—
163C4⋊Dic53—21no—
164C4⋊Dic53—21no—
165#11719917—22no—
166#11719917—22no—
167#11719917—44no—
168#11719917—42no—
169C2×#88322—21no—
170C2×#88322—21no—
171C2×#88322—22no—
172C2×#88322—44no—
173C2×#88322C2×D412yes—
174C2×#88322C2×D412yes—
175C23×C53—21no—
176C23×C53—21no—
177C23×C53—42no—
178C23×C53—42no—
179C23×C53—81no—
180C23×C53#12111yescharacteristic
181C22×D53—41no—
182Q8×C53—42no—
183Q8×C53—42no—
184Q8×C53—82no—
185D4×C53—21no—
186D4×C53—21no—
187D4×C53—21no—
188D4×C53—42no—
189D4×C53AGL(1,ℤ/8)12yes—
190C2×C4×C53—21no—
191C2×C4×C53—21no—
192C2×C4×C53—21no—
193C2×C4×C53—21no—
194C2×C4×C53—41no—
195C2×C4×C53—42no—
196C2×C4×C53—41no—
197C2×C4×C53—41no—
198C2×C4×C53—41no—
199C2×C4×C53—41no—
200C2×C4×C53—42no—
201C2×C4×C53—42no—
202C2×C4×C53—42no—
203C2×C4×C53—82no—
204C2×C4×C53#12111yescharacteristic
205C2×C4×C53#12111yescharacteristic
206C2×C4×C53#12111yescharacteristic
207C2×C4×C53#13512yes—
208C2×C4×C53AGL(1,ℤ/8)12yes—
209C2×C4×C53C22×D411yescharacteristic
210C2×C4×C53C22×D411yescharacteristic
211C2×C4×C53C2×#4111yescharacteristic
212C22⋊D53—82no—
213C2×Dic53—41no—
214D212—41no—
215D212—41no—
216D212—41no—
217D212—41no—
218D212—81no—
219C4×D53—82no—
220Dic106—41no—
221Dic106—41no—
222C8×C53—42no—
223#88322—22no—
224#88322—22no—
225#88322—42no—
226#88322—41no—
227#88322—41no—
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#46312yes—
239C22×C53C2×#12111yescharacteristic
240C2×D53—81no—
241C2×D53—81no—
242C2×D53—81no—
243C4×C53—21no—
244C4×C53—21no—
245C4×C53—21no—
246C4×C53—41no—
247C4×C53—41no—
248C4×C53—82no—
249C4×C53#46312yes—
250C4×C53C2×#13511yescharacteristic
251C4×C53C2×AGL(1,ℤ/8)11yescharacteristic
252C4×C53C2×AGL(1,ℤ/8)11yescharacteristic
253C4×C53#57511yescharacteristic
254Dic53—81no—
255#3141—531no—
256#3100—531no—
257C2×#461D5311yescharacteristic, complemented
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×C53C2×#46311yescharacteristic
270C2×C53#305511yescharacteristic
271C2×C53#306911yescharacteristic
272D53—161no—
273C22×SD16—21no—
274C22×M16—1061no—
275#532—531no—
276C2×C4⋊Q8C2×D5311yescharacteristic
277C2×C4×D4C2×D5311yescharacteristic
278#489—1062no—
279#487—532no—
280#486—532no—
281#480—1061no—
282#474—531no—
283#474—531no—
284#461—21no—
285#461—21no—
286#461C2×D5312yes—
287#461C2×D5312yes—
288#459—531no—
289#459—531no—
290#447—1061no—
291C4×D8—1061no—
292C4×D8—1061no—
293#436—532no—
294#435—1062no—
295C8×D4—532no—
296#431—531no—
297#423—531no—
298C2×C4⋊C8C2×D5311yescharacteristic
299#417—1061no—
300#417—1061no—
301C2×#103—21no—
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—1061no—
312C22×Q8—21no—
313C22×D4C22⋊D5311yescharacteristic
314C23×C4—21no—
315C2×SD16—22no—
316C2×SD16—22no—
317C2×SD16—44no—
318C2×D8—1061no—
319C2×D8—1061no—
320C2×M16—531no—
321C2×M16—531no—
322C2×M16—1062no—
323C2×M16—1061no—
324C2×M16—2124no—
325C22×C8—21no—
326C22×C8—1062no—
327C4⋊Q8—21no—
328C4⋊Q8—21no—
329C4⋊Q8—21no—
330C4⋊Q8—531no—
331C4⋊Q8C22×D5312yes—
332#127—531no—
333#124—1061no—
334#121—1061no—
335#121—1061no—
336C4×D4—21no—
337C4×D4—21no—
338C4×D4—1061no—
339C4×D4—1061no—
340C4×D4C22⋊D5311yescharacteristic
341C4×D4C22⋊D5311yescharacteristic
342C4×D4C22×D5311yescharacteristic
343C4×D4C22×D5311yescharacteristic
344C2×C4⋊C4—21no—
345C2×C4⋊C4—21no—
346C2×C4⋊C4C22⋊D5311yescharacteristic
347C2×C22⋊C4—21no—
348C2×C42C22×D5311yescharacteristic
349#107—1061no—
350#107—1061no—
351C4⋊C8—21no—
352C4⋊C8—532no—
353C4⋊C8—531no—
354C4⋊C8—531no—
355C4⋊C8—1061no—
356C4⋊C8C22×D5312yes—
357#103—22no—
358#103—22no—
359#103—1062no—
360#103—1062no—
361#102—1062no—
362#102—1062no—
363#102—1061no—
364#102—1061no—
365#102—1061no—
366#102—1061no—
367C22⋊C8—1062no—
368C22⋊C8—1062no—
369C22⋊C8—1061no—
370#97—532no—
371C4×C8—532no—
372C4×C8—1061no—
373C24—21no—
374#41—2122no—
375#41—2122no—
376C2×Q8—22no—
377C2×Q8—42no—
378C2×Q8—42no—
379C2×Q8—41no—
380C2×Q8—1061no—
381C2×D4—22no—
382C2×D4—21no—
383C2×D4—41no—
384C2×D4—1061no—
385C2×D4—1061no—
386C2×D4—1061no—
387C2×D4—2121no—
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—44no—
403SD16—44no—
404D8—2122no—
405D8—2122no—
406M16—1062no—
407M16—1062no—
408M16—2124no—
409M16—2122no—
410C2×C8—22no—
411C2×C8—42no—
412C2×C8—532no—
413C2×C8—532no—
414C2×C8—1062no—
415C2×C8—1061no—
416C2×C8—1061no—
417C2×C8—2124no—
418C4⋊C4—21no—
419C4⋊C4—22no—
420C4⋊C4—22no—
421C4⋊C4—41no—
422C4⋊C4—41no—
423C4⋊C4—1061no—
424C4⋊C4—1061no—
425C4⋊C4C2×C22⋊D5311yescharacteristic
426C4⋊C4C2×C22⋊D5311yescharacteristic
427C22⋊C4—21no—
428C22⋊C4—21no—
429C22⋊C4—21no—
430C22⋊C4—21no—
431C22⋊C4—2121no—
432C22⋊C4—2121no—
433C42D4×D5311yescharacteristic
434C42#1171994711yescharacteristic
435C42C2×C22⋊D5311yescharacteristic
436C42C23×D5311yescharacteristic
437C23—21no—
438C23—21no—
439C23—42no—
440C23—42no—
441C23—81no—
442C23—2121no—
443C23#1313318911yescharacteristic
444Q8—42no—
445Q8—42no—
446Q8—82no—
447Q8—2121no—
448Q8—2121no—
449D4—21no—
450D4—21no—
451D4—21no—
452D4—42no—
453D4—4241no—
454D4—4242no—
455D4—2121no—
456D4—2121no—
457D4—2121no—
458D4—2121no—
459D4#1313324312yes—
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—42no—
473C2×C4—82no—
474C2×C4—4242no—
475C2×C4—2121no—
476C2×C4#1313318911yescharacteristic
477C2×C4#1313318911yescharacteristic
478C2×C4#1313318911yescharacteristic
479C2×C4#1313323812yes—
480C2×C4#1313324412yes—
481C2×C4C2×D4×D5311yescharacteristic
482C2×C4C22×C22⋊D5311yescharacteristic
483C2×C4C2×#1171994711yescharacteristic
484C8—42no—
485C8—1062no—
486C8—1062no—
487C8—2121no—
488C8—2122no—
489C8—2121no—
490C22—21no—
491C22—21no—
492C22—22no—
493C22—41no—
494C22—41no—
495C22—41no—
496C22—41no—
497C22—41no—
498C22—84no—
499C22—4241no—
500C22—4241no—
501C22—4241no—
502C22#1681084012yes—
503C22C2×#1313318911yescharacteristic
504C4—21no—
505C4—21no—
506C4—21no—
507C4—41no—
508C4—41no—
509C4—82no—
510C4—4241no—
511C4#1681084012yes—
512C4C2×#1313323811yescharacteristic
513C4C2×#1313324311yescharacteristic
514C4C2×#1313324411yescharacteristic
515C4#1681142111yescharacteristic
516C2—22no—
517C2—42no—
518C2—81no—
519C2—8481no—
520C2C2×#1681084011yescharacteristic
521C2#2028433011yescharacteristic
522C2#2028444911yescharacteristic
subgroupG/H
centerC22C2×#13133189
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC2×C53×#461C2
socleC22×C53C2×#121
prime radical#20284715C2
hypercenterC2×#461D53
hypocenterC53#19169
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#461

Complements

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

NH ≅ G/Ntype
1. #20285111C2semidirect
3. D4×#11719917C2semidirect
4. #20274278C2semidirect
5. #20274264C2semidirect
6. #20274154C2semidirect
7. #20274140C2semidirect
8. #20273465C2semidirect
9. #20273400C2semidirect
10. #20273390C2semidirect
11. C2×C53×#461C2semidirect
15. #16810975C22semidirect
16. #16810971C22semidirect
34. #16810375C22semidirect
35. #16810373C22semidirect
36. #16810369C22semidirect
37. #16810369C22semidirect
38. #16810368C22semidirect
39. #16810366C22semidirect
257. C2×#461D53semidirect
310. C53#19169semidirect

Maximal subgroups

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

subgroupindex
#202851112
#202847152
D4×#117199172
#202742782
#202742642
#202741542
#202741402
#202734652
#202734002
#202733902
C2×C53×#4612
#1916953