Finite Group Databasebeta

← #27425560

Subgroups — #27425560

Order 13568.

706 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20284530C211yesmaximal, characteristic, complemented
2C2×#16810913C211yesmaximal, characteristic, complemented
3#20283381C211yesmaximal, characteristic, complemented
4#20276163C212yesmaximal, complemented
5#20276004C212yesmaximal
6#20275704C211yesmaximal, characteristic, complemented
7#20275492C212yesmaximal, complemented
8#20275490C211yesmaximal, characteristic, complemented
9#20275380C212yesmaximal, complemented
10#20274707C211yesmaximal, characteristic, complemented
11C4×AGL(1,ℤ/8)×C53C211yesmaximal, characteristic
12C2×#13133217C2211yescharacteristic
13#16810918—21no—
14#16810913—21no—
15#16810913—21no—
16#16810913—22no—
17#16810913—21no—
18#16810908—21no—
19#16810903—21no—
20#16810684—22no—
21#16810680—21no—
22#16810676—21no—
23C2×#13133152C2212yes—
24#16810666C2212yes—
25C2×#13133148C2212yes—
26#16810647C2211yescharacteristic
27#16810647C2211yescharacteristic
28C2×C4⋊#88322C2211yescharacteristic
29#16810624C2212yes—
30C2×#13133138C2212yes—
31C2×#13133134C2212yes—
32#16810590C2212yes—
33#16810579C2212yes—
34#16810392C2212yes—
35#16810391C2212yes—
36#16810391C2212yes—
37#16810387C2212yes—
38C2×AGL(1,ℤ/8)×C53C2211yescharacteristic
39C4×#41×C53C2211yescharacteristic
40C2×C4×D4×C53C2211yescharacteristic
41C53×#442—21no—
42C53×#440—21no—
43C4×SD16×C53—21no—
44C4×D8×C53—21no—
45C53×#428C2211yescharacteristic
46C53×#418C2211yescharacteristic
47C53×#417C2211yescharacteristic
48C4×M16×C53C2211yescharacteristic
49C22×Dic106—21no—
50C2×#11719917C2312yes—
51C2×#11719917C2312yes—
52C22×#88322C2312yes—
53#13133217—21no—
54#13133217—21no—
55#13133217—21no—
56#13133217—21no—
57C2×C4⋊Dic53—21no—
58C2×C4⋊Dic53C2312yes—
59#13133152—24no—
60#13133150—22no—
61#13133148—22no—
62#13133148—22no—
63#13133148—22no—
64C4⋊#88322—21no—
65C4⋊#88322—21no—
66C4⋊#88322—21no—
67C4⋊#88322—21no—
68C4⋊#88322D411yescharacteristic
69C4⋊#88322D411yescharacteristic
70C4⋊#88322D412yes—
71#13133141—22no—
72#13133138—22no—
73#13133138—22no—
74#13133138—22no—
75#13133134—24no—
76C22⋊#88322—22no—
77C22⋊#88322—22no—
78#13133128—22no—
79C4×#88322—22no—
80C2×#41×C53C2311yescharacteristic
81C22×D4×C53C2311yescharacteristic
82C23×C4×C53—21no—
83AGL(1,ℤ/8)×C53—22no—
84AGL(1,ℤ/8)×C53—22no—
85C2×SD16×C53—21no—
86C2×D8×C53—21no—
87C2×M16×C53C2311yescharacteristic
88C2×M16×C53C2311yescharacteristic
89C4×Q8×C53D411yescharacteristic
90C4×Q8×C53D411yescharacteristic
91C4×D4×C53—21no—
92C4×D4×C53—21no—
93C4×D4×C53—21no—
94C4×D4×C53—21no—
95C4×D4×C53—21no—
96C4×D4×C53—21no—
97C4×D4×C53D411yescharacteristic
98C4×D4×C53D411yescharacteristic
99#117×C53—21no—
100#117×C53C2311yescharacteristic
101C2×C4⋊C4×C53C2311yescharacteristic
102C2×C22⋊C4×C53—21no—
103C2×C42×C53—21no—
104C2×C42×C53C2311yescharacteristic
105#107×C53—21no—
106#106×C53—21no—
107#103×C53—21no—
108#102×C53—21no—
109#102×C53—21no—
110#102×C53—21no—
111#97×C53—21no—
112C4×C8×C53—21no—
113C24×C53—21no—
114#41×C53—21no—
115#41×C53—21no—
116#41×C53—21no—
117#41×C53—21no—
118C2×Q8×C53C2×D411yescharacteristic
119C2×D4×C53—21no—
120C2×D4×C53—21no—
121C2×D4×C53—21no—
122C2×D4×C53—21no—
123C2×D4×C53—21no—
124C2×D4×C53—21no—
125C2×D4×C53C2×D411yescharacteristic
126C2×D4×C53#4112yes—
127C22×C4×C53—21no—
128C22×C4×C53—21no—
129C22×C4×C53—21no—
130C22×C4×C53—21no—
131C22×C4×C53—41no—
132C22×C4×C53—41no—
133C22×C4×C53—82no—
134C22×C4×C53#4111yescharacteristic
135C22×C4×C53#4111yescharacteristic
136C22×C4×C53C2411yescharacteristic
137C22×Dic53—22no—
138C2×Dic106—41no—
139C2×Dic106—41no—
140C2×Dic106—82no—
141SD16×C53—24no—
142D8×C53—24no—
143M16×C53—21no—
144M16×C53—21no—
145M16×C53—21no—
146M16×C53#4112yes—
147C2×C8×C53—21no—
148C2×C8×C53—21no—
149C4⋊C4×C53—21no—
150C4⋊C4×C53—21no—
151C4⋊C4×C53—21no—
152C4⋊C4×C53—21no—
153C4⋊C4×C53C2×D411yescharacteristic
154C4⋊C4×C53C2×D411yescharacteristic
155C22⋊C4×C53—21no—
156C22⋊C4×C53—21no—
157C22⋊C4×C53—21no—
158C22⋊C4×C53—41no—
159C22⋊C4×C53—41no—
160C42×C53—21no—
161C42×C53—21no—
162C42×C53—21no—
163C42×C53C2×D411yescharacteristic
164C42×C53C2×D411yescharacteristic
165C42×C53C2×D411yescharacteristic
166C42×C53C2×D411yescharacteristic
167C4⋊Dic53—22no—
168C4⋊Dic53—22no—
169C4⋊Dic53—41no—
170C4⋊Dic53—41no—
171#11719917—22no—
172#11719917—22no—
173#11719917—22no—
174#11719917—22no—
175C2×#88322—22no—
176C2×#88322—22no—
177C2×#88322—22no—
178C2×#88322—22no—
179C2×#88322C2×D412yes—
180C2×#88322C2×D412yes—
181C23×C53—21no—
182C23×C53—22no—
183C23×C53—41no—
184C23×C53—41no—
185C23×C53—81no—
186C23×C53—81no—
187C23×C53C2×#4111yescharacteristic
188Q8×C53—21no—
189Q8×C53—21no—
190D4×C53—22no—
191D4×C53—21no—
192D4×C53—21no—
193D4×C53—22no—
194D4×C53—41no—
195D4×C53—41no—
196D4×C53—41no—
197D4×C53—41no—
198C2×C4×C53—22no—
199C2×C4×C53—21no—
200C2×C4×C53—21no—
201C2×C4×C53—21no—
202C2×C4×C53—41no—
203C2×C4×C53—41no—
204C2×C4×C53—41no—
205C2×C4×C53—42no—
206C2×C4×C53—41no—
207C2×C4×C53—81no—
208C2×C4×C53—82no—
209C2×C4×C53—82no—
210C2×C4×C53—81no—
211C2×C4×C53#12111yescharacteristic
212C2×C4×C53#12111yescharacteristic
213C2×C4×C53#12111yescharacteristic
214C2×C4×C53#12111yescharacteristic
215C2×C4×C53C22×D411yescharacteristic
216C2×C4×C53C22×D411yescharacteristic
217C2×C4×C53C2×#4111yescharacteristic
218C2×C4×C53C2×#4111yescharacteristic
219C2×C4×C532−1+411yescharacteristic
220C2×C4×C532−1+411yescharacteristic
221C2×Dic53—42no—
222C2×Dic53—82no—
223Dic106—82no—
224Dic106—82no—
225C8×C53—22no—
226C8×C53—41no—
227#88322—22no—
228#88322—22no—
229#88322—42no—
230#88322—42no—
231#24721—531nomaximal, Sylow
232C22×C53—21no—
233C22×C53—21no—
234C22×C53—41no—
235C22×C53—42no—
236C22×C53—41no—
237C22×C53—41no—
238C22×C53—82no—
239C22×C53—82no—
240C22×C53—81no—
241C22×C53—81no—
242C22×C53C2×#12111yescharacteristic
243C22×C53#55211yescharacteristic
244C22×C53#55611yescharacteristic
245C4×C53—21no—
246C4×C53—21no—
247C4×C53—21no—
248C4×C53—41no—
249C4×C53—41no—
250C4×C53—41no—
251C4×C53—81no—
252C4×C53#47112yes—
253C4×C53#52511yescharacteristic
254C4×C53#53611yescharacteristic
255C4×C53#54711yescharacteristic
256C4×C53#54811yescharacteristic
257Dic53—82no—
258#3079—531no—
259C2×#476—531no—
260C4×AGL(1,ℤ/8)D5311yescharacteristic
261#2925—531no—
262#2079—532no—
263#2051—532no—
264#1997—531no—
265#1951—532no—
266#1950—531no—
267#1931—532no—
268#1800—531no—
269C2×C53—21no—
270C2×C53—21no—
271C2×C53—81no—
272C2×C53—82no—
273C2×C53#241511yescharacteristic
274C2×C53C2×#47111yescharacteristic
275C2×C53#306811yescharacteristic
276C2×AGL(1,ℤ/8)C2×D5311yescharacteristic
277C2×C4⋊Q8—531no—
278C4×#41C2×D5311yescharacteristic
279C2×C4×D4C2×D5311yescharacteristic
280#477—1061no—
281#476—1061no—
282#476—1061no—
283#476—1062no—
284#476—1061no—
285#475—1061no—
286#474—1061no—
287#442—21no—
288#440—21no—
289C4×SD16—21no—
290C4×D8—21no—
291#433—1062no—
292#432—1061no—
293#431—1061no—
294C2×#108—532no—
295#428—532no—
296#428C2×D5311yescharacteristic
297C2×#106—532no—
298#423—531no—
299#423—531no—
300C2×C4⋊C8—531no—
301#418C2×D5311yescharacteristic
302#417—532no—
303#417C2×D5311yescharacteristic
304C2×#102—532no—
305C2×#100—532no—
306#408—532no—
307#405—532no—
308C4×M16C2×D5311yescharacteristic
309#343—532no—
310#343—532no—
311#343—532no—
312#341—532no—
313C53#2472111yescharacteristic, complemented, Sylow
314C2×#41C22×D5311yescharacteristic
315C22×Q8—1061no—
316C22×D4C22×D5311yescharacteristic
317C23×C4—21no—
318AGL(1,ℤ/8)—22no—
319AGL(1,ℤ/8)—22no—
320C2×SD16—21no—
321C2×D8—21no—
322C2×M16—532no—
323C2×M16—532no—
324C2×M16C22×D5311yescharacteristic
325C2×M16C22×D5311yescharacteristic
326C22×C8—532no—
327C4⋊Q8—1061no—
328C4⋊Q8—1061no—
329C4⋊Q8—1061no—
330C4⋊Q8—1061no—
331C4×Q8C22⋊D5311yescharacteristic
332C4×Q8C22⋊D5311yescharacteristic
333C4×D4—21no—
334C4×D4—21no—
335C4×D4—21no—
336C4×D4—21no—
337C4×D4—21no—
338C4×D4—21no—
339C4×D4C22⋊D5311yescharacteristic
340C4×D4C22⋊D5311yescharacteristic
341#117—21no—
342#117C22×D5311yescharacteristic
343C2×C4⋊C4—532no—
344C2×C4⋊C4—1061no—
345C2×C4⋊C4C22×D5311yescharacteristic
346C2×C22⋊C4—21no—
347C2×C42—21no—
348C2×C42C22×D5311yescharacteristic
349#108—1064no—
350#107—21no—
351#107—1062no—
352#106—21no—
353#106—1062no—
354#106—1062no—
355#106—1062no—
356C4⋊C8—531no—
357C4⋊C8—531no—
358C4⋊C8—532no—
359C4⋊C8—1061no—
360C4⋊C8—1061no—
361C4⋊C8—1061no—
362C4⋊C8—1061no—
363#103—21no—
364#103—1062no—
365#102—21no—
366#102—21no—
367#102—21no—
368#102—1062no—
369#102—1062no—
370#102—1062no—
371#100—1064no—
372C22⋊C8—1062no—
373C22⋊C8—1062no—
374#97—21no—
375#97—1062no—
376C4×C8—21no—
377C4×C8—1062no—
378C24—21no—
379#41—21no—
380#41—21no—
381#41—21no—
382#41—21no—
383C2×Q8—4242no—
384C2×Q8—2121no—
385C2×Q8—2121no—
386C2×Q8C2×C22⋊D5311yescharacteristic
387C2×D4—21no—
388C2×D4—21no—
389C2×D4—21no—
390C2×D4—21no—
391C2×D4—21no—
392C2×D4—21no—
393C2×D4#1171994512yes—
394C2×D4C2×C22⋊D5311yescharacteristic
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—41no—
400C22×C4—41no—
401C22×C4—82no—
402C22×C4—1062no—
403C22×C4#1171994711yescharacteristic
404C22×C4#1171994711yescharacteristic
405C22×C4C23×D5311yescharacteristic
406SD16—24no—
407D8—24no—
408M16—21no—
409M16—21no—
410M16—21no—
411M16—1062no—
412M16—1062no—
413M16—1062no—
414M16—1062no—
415M16#1171994512yes—
416C2×C8—21no—
417C2×C8—21no—
418C2×C8—532no—
419C2×C8—532no—
420C2×C8—1062no—
421C2×C8—1062no—
422C2×C8—1062no—
423C2×C8—1062no—
424C4⋊C4—21no—
425C4⋊C4—21no—
426C4⋊C4—21no—
427C4⋊C4—21no—
428C4⋊C4—1062no—
429C4⋊C4—1062no—
430C4⋊C4—2121no—
431C4⋊C4—2121no—
432C4⋊C4C2×C22⋊D5311yescharacteristic
433C4⋊C4C2×C22⋊D5311yescharacteristic
434C22⋊C4—21no—
435C22⋊C4—21no—
436C22⋊C4—21no—
437C22⋊C4—41no—
438C22⋊C4—41no—
439C42—21no—
440C42—21no—
441C42—21no—
442C42D4×D5311yescharacteristic
443C42D4×D5311yescharacteristic
444C42C2×C22⋊D5311yescharacteristic
445C42C2×C22⋊D5311yescharacteristic
446C23—21no—
447C23—22no—
448C23—41no—
449C23—41no—
450C23—81no—
451C23—81no—
452C23C2×#1171994711yescharacteristic
453Q8—21no—
454Q8—21no—
455Q8—4242no—
456Q8—4242no—
457D4—22no—
458D4—21no—
459D4—22no—
460D4—21no—
461D4—41no—
462D4—41no—
463D4—41no—
464D4—41no—
465C2×C4—22no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—42no—
473C2×C4—41no—
474C2×C4—81no—
475C2×C4—82no—
476C2×C4—82no—
477C2×C4—81no—
478C2×C4—4242no—
479C2×C4—2122no—
480C2×C4#1313318911yescharacteristic
481C2×C4#1313318911yescharacteristic
482C2×C4#1313318911yescharacteristic
483C2×C4#1313318911yescharacteristic
484C2×C4C2×D4×D5311yescharacteristic
485C2×C4C22×C22⋊D5311yescharacteristic
486C2×C4C2×#1171994511yescharacteristic
487C2×C4C2×#1171994511yescharacteristic
488C2×C4#1313326411yescharacteristic
489C2×C4#1313326411yescharacteristic
490C8—22no—
491C8—41no—
492C8—1062no—
493C8—1062no—
494C8—2122no—
495C8—2122no—
496C22—21no—
497C22—21no—
498C22—41no—
499C22—41no—
500C22—42no—
501C22—41no—
502C22—82no—
503C22—82no—
504C22—81no—
505C22—81no—
506C22C2×#1313318911yescharacteristic
507C22#1681131311yescharacteristic
508C22#1681134311yescharacteristic
509C4—21no—
510C4—21no—
511C4—21no—
512C4—41no—
513C4—41no—
514C4—41no—
515C4—81no—
516C4—4242no—
517C4#1681088412yes—
518C4#1681113411yescharacteristic
519C4#1681119411yescharacteristic
520C4#1681127811yescharacteristic
521C4#1681128211yescharacteristic
522C2—21no—
523C2—21no—
524C2—81no—
525C2—82no—
526C2#2027814411yescharacteristic
527C2C2×#1681088411yescharacteristic
528C2#2028444511yescharacteristic
subgroupG/H
centerC22C2×#13133189
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC4×AGL(1,ℤ/8)×C53C2
socleC22×C53C2×#121
prime radicalC2×AGL(1,ℤ/8)×C53C22
hypercenterC4×AGL(1,ℤ/8)D53
hypocenterC53#24721
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ C4×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. #20284530C2semidirect
2. C2×#16810913C2semidirect
3. #20283381C2semidirect
4. #20276163C2semidirect
6. #20275704C2semidirect
7. #20275492C2semidirect
8. #20275490C2semidirect
9. #20275380C2semidirect
10. #20274707C2semidirect
313. C53#24721semidirect

Maximal subgroups

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

subgroupindex
#202845302
C2×#168109132
#202833812
#202761632
#202760042
#202757042
#202754922
#202754902
#202753802
#202747072
C4×AGL(1,ℤ/8)×C532
#2472153