Finite Group Databasebeta

← #27366942

Subgroups — #27366942

Order 13568.

766 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#20284742C211yesmaximal, characteristic, complemented
2C2×#16810820C211yesmaximal, characteristic, complemented
3M16×Dic106C211yesmaximal, characteristic, complemented
4#20274196C212yesmaximal, complemented
5#20274184C212yesmaximal, complemented
6#20274072C212yesmaximal, complemented
7#20274060C212yesmaximal, complemented
8#20273467C211yesmaximal, characteristic, complemented
9#20273392C211yesmaximal, characteristic, complemented
10#20273383C211yesmaximal, characteristic, complemented
11C53×#3151C211yesmaximal, characteristic
12C22×#11719923—21no—
13C2×C4×Dic106C2211yescharacteristic
14#16810939—21no—
15#16810905C2211yescharacteristic, complemented
16#16810905C2211yescharacteristic, complemented
17#16810820—21no—
18#16810820—21no—
19#16810820C2212yes—
20#16810820C2212yes—
21#16810820C2211yescharacteristic, complemented
22#16810820C2211yescharacteristic, complemented
23#16810761—21no—
24#16810753C2212yescomplemented
25#16810751—22no—
26C8×Dic106C2212yescomplemented
27C4×#11719923—21no—
28C4×#11719923—21no—
29#16810676C2211yescharacteristic
30#16810647C2211yescharacteristic
31#16810646—21no—
32C2×C4⋊#88322C2211yescharacteristic
33C2×#13133138—21no—
34M16×Dic53—21no—
35#16810376C2212yescomplemented
36#16810374C2212yescomplemented
37#16810369C2212yescomplemented
38#16810367C2212yes—
39#16810367C2212yes—
40#16810367C2212yescomplemented
41C2×AGL(1,ℤ/8)×C53—21no—
42C2×AGL(1,ℤ/8)×C53—21no—
43C53×#532C2211yescharacteristic
44C2×#127×C53C2211yescharacteristic
45C53×#496—22no—
46C53×#493C2212yes—
47C53×#492C2212yes—
48C4×M16×C53C2211yescharacteristic
49C22×Dic106D411yescharacteristic
50C2×#11719923—21no—
51C2×#11719923—21no—
52C2×#11719923—22no—
53C2×#11719923—22no—
54C2×#11719923—44no—
55C2×#11719917—21no—
56C22×#88322—21no—
57C4×Dic106—21no—
58C4×Dic106—21no—
59C4×Dic106D411yescharacteristic
60C4×Dic106D411yescharacteristic
61C4×Dic106C2311yescharacteristic
62C4×Dic106C2311yescharacteristic
63C2×C4⋊Dic53D411yescharacteristic
64C2×#11719919—21no—
65C2×C4×Dic53—21no—
66#13133148—21no—
67#13133148—21no—
68C4⋊#88322—21no—
69C4⋊#88322—21no—
70C4⋊#88322C2312yes—
71C4⋊#88322C2312yes—
72C4⋊#88322C2311yescharacteristic
73C4⋊#88322C2311yescharacteristic
74#13133146—22no—
75#13133146—22no—
76#13133140—21no—
77#13133140—21no—
78#13133138—22no—
79#13133138—22no—
80#13133138—21no—
81#13133138—21no—
82C22⋊#88322—21no—
83#13133127—22no—
84C4×#88322—21no—
85C8×Dic53—22no—
86C2×#41×C53—21no—
87C22×D4×C53—21no—
88C22×D4×C53—21no—
89C22×D4×C53—21no—
90AGL(1,ℤ/8)×C53—44no—
91AGL(1,ℤ/8)×C53—44no—
92C2×SD16×C53—22no—
93C2×SD16×C53—22no—
94C2×D8×C53—22no—
95C2×D8×C53—22no—
96C2×M16×C53D411yescharacteristic
97C2×M16×C53D411yescharacteristic
98C4⋊Q8×C53C2311yescharacteristic
99#127×C53—21no—
100#127×C53—21no—
101#127×C53—21no—
102#127×C53C2312yes—
103#127×C53C2311yescharacteristic
104#124×C53—21no—
105#121×C53—21no—
106#121×C53—21no—
107C4×D4×C53—21no—
108C4×D4×C53—21no—
109C2×C42×C53C2311yescharacteristic
110#97×C53D412yes—
111C4×C8×C53C2312yes—
112C24×C53—41no—
113#41×C53—42no—
114#41×C53—42no—
115C2×Q8×C53—21no—
116C2×D4×C53—21no—
117C2×D4×C53—21no—
118C2×D4×C53—22no—
119C2×D4×C53—22no—
120C2×D4×C53—21no—
121C2×D4×C53—22no—
122C2×D4×C53—42no—
123C2×D4×C53—42no—
124C2×D4×C53—41no—
125C2×D4×C53—41no—
126C2×D4×C53—41no—
127C2×D4×C53—41no—
128C2×D4×C53—42no—
129C2×D4×C53—42no—
130C2×D4×C53—82no—
131C22×C4×C53—41no—
132C22×C4×C53C2×D411yescharacteristic
133C22×C4×C53C2×D411yescharacteristic
134C22×C4×C53#4111yescharacteristic
135C22×Dic53—21no—
136C22×Dic53—21no—
137C2×Dic106—22no—
138C2×Dic106—21no—
139C2×Dic106—41no—
140C2×Dic106C2×D411yescharacteristic
141C2×Dic106C2×D411yescharacteristic
142SD16×C53—44no—
143SD16×C53—44no—
144D8×C53—44no—
145D8×C53—44no—
146M16×C53—22no—
147M16×C53SD1614yescomplemented
148C2×C8×C53C2×D412yes—
149C2×C8×C53C2×D412yes—
150C4⋊C4×C53—21no—
151C4⋊C4×C53—21no—
152C22⋊C4×C53—41no—
153C22⋊C4×C53—41no—
154C42×C53C2×D411yescharacteristic
155C42×C53C2×D411yescharacteristic
156C42×C53#4111yescharacteristic
157C42×C53C2411yescharacteristic
158#11719923—42no—
159#11719923—44no—
160#11719923—42no—
161#11719923—44no—
162C4⋊Dic53—21no—
163C4⋊Dic53C2×D411yescharacteristic
164C4⋊Dic53C2×D411yescharacteristic
165#11719919—21no—
166#11719919—21no—
167#11719919—21no—
168#11719919—21no—
169C4×Dic53—21no—
170C4×Dic53—21no—
171C4×Dic53—21no—
172C4×Dic53—21no—
173#11719917—42no—
174C2×#88322—22no—
175C2×#88322—21no—
176C2×#88322—21no—
177C2×#88322—22no—
178C2×#88322—42no—
179C23×C53—41no—
180C23×C53—42no—
181C23×C53—82no—
182C23×C53—82no—
183C23×C53—82no—
184C23×C53#12111yescharacteristic
185Q8×C53—41no—
186Q8×C53—41no—
187D4×C53—42no—
188D4×C53—41no—
189D4×C53—42no—
190D4×C53—42no—
191D4×C53—41no—
192D4×C53—41no—
193D4×C53—42no—
194D4×C53—41no—
195D4×C53—81no—
196D4×C53—82no—
197D4×C53—82no—
198D4×C53—82no—
199D4×C53—82no—
200D4×C53—82no—
201C2×C4×C53—21no—
202C2×C4×C53—21no—
203C2×C4×C53—41no—
204C2×C4×C53—41no—
205C2×C4×C53—82no—
206C2×C4×C53#12111yescharacteristic
207C2×C4×C53#12111yescharacteristic
208C2×C4×C53#12111yescharacteristic
209C2×C4×C53C2×SD1612yes—
210C2×C4×C53AGL(1,ℤ/8)12yes—
211C2×C4×C53C22×D411yescharacteristic
212C2×C4×C53C22×D411yescharacteristic
213C2×C4×C53C2×#4111yescharacteristic
214C2×Dic53—21no—
215C2×Dic53—21no—
216C2×Dic53—21no—
217C2×Dic53—21no—
218C2×Dic53—42no—
219C2×Dic53—81no—
220Dic106—21no—
221Dic106—21no—
222Dic106—21no—
223Dic106—42no—
224Dic106AGL(1,ℤ/8)12yescomplemented
225C8×C53—22no—
226C8×C53C2×SD1614yescomplemented
227#88322—42no—
228#88322—42no—
229#88322—41no—
230#88322—41no—
231#19170—531nomaximal, Sylow
232C22×C53—21no—
233C22×C53—21no—
234C22×C53—81no—
235C22×C53—82no—
236C22×C53—82no—
237C22×C53—81no—
238C22×C53—82no—
239C22×C53—81no—
240C22×C53—164no—
241C22×C53#46012yes—
242C22×C53C2×#12111yescharacteristic
243C4×C53—21no—
244C4×C53—21no—
245C4×C53—21no—
246C4×C53—81no—
247C4×C53#46012yes—
248C4×C53C22×SD1611yescharacteristic
249C4×C53C2×AGL(1,ℤ/8)11yescharacteristic
250C4×C53C2×AGL(1,ℤ/8)11yescharacteristic
251C4×C53#57511yescharacteristic
252Dic53—22no—
253Dic53—41no—
254Dic53—41no—
255Dic53—41no—
256#3151D5311yescharacteristic
257#3103—531no—
258C2×#460—531no—
259Q8×M16—531no—
260#1688—532no—
261#1686—532no—
262#1668—532no—
263#1666—532no—
264#1565—531no—
265#1547—531no—
266#1545—531no—
267C2×C53—22no—
268C2×C53—161no—
269C2×C53—162no—
270C2×C53C2×#46011yescharacteristic
271C2×C53#305511yescharacteristic
272C2×C53#306811yescharacteristic
273C2×AGL(1,ℤ/8)—21no—
274C2×AGL(1,ℤ/8)—21no—
275C22×SD16—1061no—
276#532C2×D5311yescharacteristic
277C2×#127C2×D5311yescharacteristic
278C2×C4×Q8—531no—
279#496—22no—
280#493C2×D5312yes—
281#492C2×D5312yes—
282#481—1061no—
283#475—531no—
284#475—531no—
285#460—531no—
286#460—531no—
287#460—532no—
288#460—532no—
289#460—1061no—
290#460—1061no—
291#448—1061no—
292#446—532no—
293#446—1062no—
294C8×Q8—532no—
295C4×SD16—1061no—
296C4×SD16—1061no—
297#431—531no—
298#423—531no—
299#423—1061no—
300C2×C4⋊C8—531no—
301C2×#102—1061no—
302C4×M16—1061no—
303C4×M16C2×D5311yescharacteristic
304#335—532no—
305#334—532no—
306#332—532no—
307#331—532no—
308#331—532no—
309#331—532no—
310C53#1917011yescharacteristic, complemented, Sylow
311C2×#41—21no—
312C22×Q8—531no—
313C22×D4—21no—
314C22×D4—21no—
315C22×D4—21no—
316AGL(1,ℤ/8)—44no—
317AGL(1,ℤ/8)—44no—
318C2×SD16—22no—
319C2×SD16—22no—
320C2×SD16—1062no—
321C2×SD16—1062no—
322C2×SD16—1061no—
323C2×SD16—1061no—
324C2×SD16—2124no—
325C2×D8—22no—
326C2×D8—22no—
327C2×M16—1061no—
328C2×M16C22⋊D5311yescharacteristic
329C2×M16C22⋊D5311yescharacteristic
330C22×C8—1061no—
331C4⋊Q8C22×D5311yescharacteristic
332#127—21no—
333#127—21no—
334#127—21no—
335#127C22×D5312yes—
336#127C22×D5311yescharacteristic
337#124—21no—
338#121—21no—
339#121—21no—
340C4×Q8—531no—
341C4×Q8—531no—
342C4×Q8—531no—
343C4×Q8—531no—
344C4×Q8—1061no—
345C4×Q8—1061no—
346C4×D4—21no—
347C4×D4—21no—
348C2×C4⋊C4—531no—
349C2×C4⋊C4—1061no—
350C2×C42—1061no—
351C2×C42C22×D5311yescharacteristic
352#106—1061no—
353#106—1061no—
354C4⋊C8—532no—
355C4⋊C8—531no—
356C4⋊C8—531no—
357C4⋊C8—532no—
358C4⋊C8—1062no—
359C4⋊C8—1062no—
360C4⋊C8—1061no—
361C4⋊C8—1061no—
362#103—1061no—
363#103—1061no—
364#102—1062no—
365#102—1062no—
366#102—1061no—
367#102—1061no—
368C22⋊C8—1061no—
369#97—1062no—
370#97C22⋊D5312yes—
371C4×C8—1062no—
372C4×C8—1061no—
373C4×C8C22×D5312yes—
374C24—41no—
375#41—42no—
376#41—42no—
377C2×Q8—21no—
378C2×Q8—531no—
379C2×Q8—531no—
380C2×Q8—1062no—
381C2×Q8—1061no—
382C2×Q8—2121no—
383C2×D4—21no—
384C2×D4—22no—
385C2×D4—21no—
386C2×D4—22no—
387C2×D4—21no—
388C2×D4—22no—
389C2×D4—42no—
390C2×D4—42no—
391C2×D4—41no—
392C2×D4—41no—
393C2×D4—41no—
394C2×D4—41no—
395C2×D4—42no—
396C2×D4—42no—
397C2×D4—82no—
398C22×C4—41no—
399C22×C4—1061no—
400C22×C4—1061no—
401C22×C4D4×D5311yescharacteristic
402C22×C4#1171994711yescharacteristic
403C22×C4C2×C22⋊D5311yescharacteristic
404SD16—44no—
405SD16—44no—
406SD16—2124no—
407SD16—2122no—
408SD16—2124no—
409SD16—2122no—
410D8—44no—
411D8—44no—
412M16—22no—
413M16—2122no—
414M16#1171992314yescomplemented
415C2×C8—1061no—
416C2×C8—1061no—
417C2×C8—1062no—
418C2×C8—1062no—
419C2×C8—2122no—
420C2×C8C2×C22⋊D5312yes—
421C2×C8C2×C22⋊D5312yes—
422C4⋊C4—21no—
423C4⋊C4—21no—
424C4⋊C4—531no—
425C4⋊C4—531no—
426C4⋊C4—1061no—
427C4⋊C4—1061no—
428C4⋊C4—1061no—
429C4⋊C4—1061no—
430C4⋊C4—1061no—
431C22⋊C4—41no—
432C22⋊C4—41no—
433C42—1061no—
434C42—1061no—
435C42—1061no—
436C42—1061no—
437C42D4×D5311yescharacteristic
438C42#1171994711yescharacteristic
439C42C2×C22⋊D5311yescharacteristic
440C42C23×D5311yescharacteristic
441C23—41no—
442C23—42no—
443C23—82no—
444C23—82no—
445C23—82no—
446C23#1313318811yescharacteristic
447Q8—41no—
448Q8—41no—
449Q8—532no—
450Q8—1061no—
451Q8—1061no—
452Q8—1061no—
453Q8—2122no—
454D4—41no—
455D4—42no—
456D4—42no—
457D4—41no—
458D4—41no—
459D4—42no—
460D4—41no—
461D4—42no—
462D4—81no—
463D4—82no—
464D4—82no—
465D4—82no—
466D4—82no—
467D4—82no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—82no—
473C2×C4—1061no—
474C2×C4—1061no—
475C2×C4—1061no—
476C2×C4—1061no—
477C2×C4—4241no—
478C2×C4—2122no—
479C2×C4#1313318811yescharacteristic
480C2×C4#1313318811yescharacteristic
481C2×C4#1313318811yescharacteristic
482C2×C4C2×#1171992312yes—
483C2×C4#1313324312yes—
484C2×C4C2×D4×D5311yescharacteristic
485C2×C4C22×C22⋊D5311yescharacteristic
486C2×C4C2×#1171994711yescharacteristic
487C8—22no—
488C8—2122no—
489C8—2121no—
490C8—2122no—
491C8—2121no—
492C8C2×#1171992314yescomplemented
493C22—21no—
494C22—21no—
495C22—82no—
496C22—81no—
497C22—82no—
498C22—81no—
499C22—81no—
500C22—82no—
501C22—164no—
502C22#1681082012yes—
503C22C2×#1313318811yescharacteristic
504C4—21no—
505C4—21no—
506C4—21no—
507C4—81no—
508C4—1062no—
509C4—2121no—
510C4—2121no—
511C4—2121no—
512C4#1681082012yes—
513C4C22×#1171992311yescharacteristic
514C4AGL(1,ℤ/8)×D5311yescharacteristic
515C4C2×#1313324311yescharacteristic
516C4#1681141911yescharacteristic
517C2—22no—
518C2—162no—
519C2—161no—
520C2C2×#1681082011yescharacteristic
521C2#2028432811yescharacteristic
522C2#2028443911yescharacteristic
subgroupG/H
centerC22C2×#13133188
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
Fitting, prime radicalC53×#3151C2
socleC22×C53C2×#121
hypercenter#3151D53
hypocenterC53#19170
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3151

Complements

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

NH ≅ G/Ntype
1. #20284742C2semidirect
2. C2×#16810820C2semidirect
3. M16×Dic106C2semidirect
4. #20274196C2semidirect
5. #20274184C2semidirect
6. #20274072C2semidirect
7. #20274060C2semidirect
8. #20273467C2semidirect
9. #20273392C2semidirect
10. #20273383C2semidirect
15. #16810905C22semidirect
16. #16810905C22semidirect
21. #16810820C22semidirect
22. #16810820C22semidirect
24. #16810753C22semidirect
26. C8×Dic106C22semidirect
35. #16810376C22semidirect
36. #16810374C22semidirect
37. #16810369C22semidirect
40. #16810367C22semidirect
147. M16×C53SD16semidirect
224. Dic106AGL(1,ℤ/8)semidirect
226. C8×C53C2×SD16semidirect
310. C53#19170semidirect
414. M16#11719923semidirect
492. C8C2×#11719923semidirect

Maximal subgroups

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

subgroupindex
#202847422
C2×#168108202
M16×Dic1062
#202741962
#202741842
#202740722
#202740602
#202734672
#202733922
#202733832
C53×#31512
#1917053