Finite Group Databasebeta

← #27425543

Subgroups — #27425543

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×#16810904C211yesmaximal, characteristic, complemented
3C4×#13133242C211yesmaximal, characteristic
4#20276155C212yesmaximal, complemented
5#20276006C212yesmaximal
6#20275703C211yesmaximal, characteristic, complemented
7#20275500C212yesmaximal, complemented
8#20275491C211yesmaximal, characteristic, complemented
9#20275377C212yesmaximal, complemented
10#20274704C211yesmaximal, characteristic, complemented
11C53×#2925C211yesmaximal, characteristic, complemented
12C2×#13133242C2211yescharacteristic
13C2×#13133217C2211yescharacteristic
14C2×C4×D212C2211yescharacteristic
15#16810918—21no—
16#16810913—21no—
17#16810908—21no—
18#16810904—21no—
19#16810904—21no—
20#16810904—22no—
21#16810903—21no—
22#16810733—21no—
23#16810721—21no—
24C4×#11719924—21no—
25C4×#11719922—21no—
26C2×#13133153C2212yes—
27#16810666C2211yescharacteristic
28#16810666C2212yes—
29C2×#13133151C2212yes—
30#16810647C2211yescharacteristic
31#16810627C2211yescharacteristic
32#16810624C2212yes—
33#16810624C2211yescharacteristic
34C2×#13133139C2212yes—
35C2×#13133133C2212yes—
36C4×#11719917C2211yescharacteristic
37#16810391C2212yes—
38#16810391C2212yes—
39#16810391C2212yes—
40#16810388C2212yes—
41C4×#41×C53C2211yescharacteristic
42C53×#433—22no—
43C53×#432—21no—
44C53×#431—21no—
45C53×#423C2211yescharacteristic
46C2×C4⋊C8×C53C2211yescharacteristic
47C53×#408C2212yes—
48C53×#405C2212yes—
49C22×Dic106—21no—
50C22×D212C2311yescharacteristic
51C22×C4×D53—21no—
52#13133242—22no—
53#13133242—22no—
54C2×#11719924—21no—
55C2×#11719922—21no—
56C2×#11719917C2311yescharacteristic
57C2×#11719917C2312yes—
58C2×#11719917C2311yescharacteristic
59#13133217—21no—
60#13133217—21no—
61#13133217—21no—
62#13133217—21no—
63C4×D212—21no—
64C4×D212—21no—
65C4×D212—21no—
66C4×D212—21no—
67C2×C4⋊Dic53—21no—
68C2×C4⋊Dic53C2311yescharacteristic
69C2×C4⋊Dic53C2312yes—
70C2×#11719921—21no—
71#13133153—24no—
72#13133151—22no—
73#13133151—22no—
74#13133150—22no—
75#13133150—21no—
76#13133148—22no—
77#13133148—21no—
78C4⋊#88322—21no—
79C4⋊#88322—21no—
80#13133141—22no—
81#13133141—21no—
82#13133139—22no—
83#13133139—22no—
84#13133138—22no—
85#13133138—21no—
86#13133137—21no—
87#13133137—21no—
88#13133133—24no—
89#13133128—21no—
90C4×#88322—21no—
91C2×#41×C53C2311yescharacteristic
92C2×M16×C53C2312yes—
93C22×C8×C53C2312yes—
94C4×Q8×C53D411yescharacteristic
95C4×Q8×C53D411yescharacteristic
96C4×D4×C53—21no—
97C4×D4×C53—21no—
98C4×D4×C53D411yescharacteristic
99C4×D4×C53D411yescharacteristic
100#117×C53—21no—
101#117×C53C2311yescharacteristic
102C2×C42×C53—21no—
103C2×C42×C53C2311yescharacteristic
104C4⋊C8×C53—21no—
105C4⋊C8×C53—21no—
106C4⋊C8×C53D411yescharacteristic
107C4⋊C8×C53D411yescharacteristic
108C4⋊C8×C53D412yes—
109C22⋊C8×C53—22no—
110C22⋊C8×C53—22no—
111#97×C53—22no—
112C4×C8×C53—22no—
113C23×D53—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×C53C2×D411yescharacteristic
121C22×C4×C53—21no—
122C22×C4×C53—21no—
123C22×C4×C53—21no—
124C22×C4×C53#4111yescharacteristic
125C22×C4×C53#4111yescharacteristic
126C22×C4×C53C2411yescharacteristic
127C22×Dic53—21no—
128C22×Dic53—22no—
129C2×D212—21no—
130C2×D212—21no—
131C2×D212—21no—
132C2×D212—21no—
133C2×D212—21no—
134C2×D212#4112yes—
135C2×C4×D53—41no—
136C2×C4×D53—41no—
137C2×C4×D53—82no—
138C2×Dic106—41no—
139C2×Dic106—41no—
140C2×Dic106—82no—
141M16×C53—22no—
142M16×C53—22no—
143C2×C8×C53—22no—
144C2×C8×C53—22no—
145C2×C8×C53—22no—
146C2×C8×C53C2×D412yes—
147C2×C8×C53C2×D412yes—
148C4⋊C4×C53—21no—
149C4⋊C4×C53—21no—
150C4⋊C4×C53C2×D411yescharacteristic
151C4⋊C4×C53C2×D411yescharacteristic
152C22⋊C4×C53—21no—
153C22⋊C4×C53—21no—
154C22⋊C4×C53—21no—
155C42×C53—21no—
156C42×C53—21no—
157C42×C53—21no—
158C42×C53C2×D411yescharacteristic
159C42×C53C2×D411yescharacteristic
160C42×C53C2×D411yescharacteristic
161C42×C53C2×D411yescharacteristic
162#11719924—24no—
163#11719922—24no—
164#11719921—41no—
165#11719921—41no—
166C4⋊Dic53—21no—
167C4⋊Dic53—22no—
168C4⋊Dic53—22no—
169C4⋊Dic53—21no—
170C4⋊Dic53—41no—
171C4⋊Dic53—41no—
172#11719917—21no—
173#11719917—22no—
174#11719917—22no—
175#11719917—21no—
176#11719917—21no—
177#11719917#4112yes—
178C2×#88322—21no—
179C2×#88322—21no—
180C2×#88322—22no—
181C23×C53—21no—
182C23×C53C2×#4111yescharacteristic
183C22×D53—22no—
184C22×D53—41no—
185C22×D53—41no—
186C22×D53—81no—
187C22×D53—81no—
188Q8×C53—21no—
189Q8×C53—21no—
190D4×C53—21no—
191D4×C53—21no—
192D4×C53—41no—
193D4×C53—41no—
194C2×C4×C53—22no—
195C2×C4×C53—21no—
196C2×C4×C53—21no—
197C2×C4×C53—21no—
198C2×C4×C53—41no—
199C2×C4×C53—41no—
200C2×C4×C53—41no—
201C2×C4×C53—42no—
202C2×C4×C53—81no—
203C2×C4×C53#12111yescharacteristic
204C2×C4×C53#12111yescharacteristic
205C2×C4×C53#12111yescharacteristic
206C2×C4×C53#12111yescharacteristic
207C2×C4×C53C22×D411yescharacteristic
208C2×C4×C53C22×D411yescharacteristic
209C2×C4×C53C2×#4111yescharacteristic
210C2×C4×C53C2×#4111yescharacteristic
211C2×C4×C532−1+411yescharacteristic
212C2×C4×C532−1+411yescharacteristic
213C2×Dic53—42no—
214C2×Dic53—41no—
215C2×Dic53—82no—
216C2×Dic53—81no—
217D212—22no—
218D212—22no—
219D212—41no—
220D212—41no—
221C4×D53—82no—
222C4×D53—82no—
223Dic106—82no—
224Dic106—82no—
225C8×C53—22no—
226C8×C53—22no—
227C8×C53—42no—
228#88322—22no—
229#88322—42no—
230#88322—41no—
231#24720—531nomaximal, Sylow
232C22×C53—21no—
233C22×C53—21no—
234C22×C53—41no—
235C22×C53—41no—
236C22×C53—41no—
237C22×C53C2×#12111yescharacteristic
238C22×C53#55211yescharacteristic
239C22×C53#55611yescharacteristic
240C2×D53—42no—
241C2×D53—82no—
242C2×D53—81no—
243C2×D53—81no—
244C2×D53—82no—
245C4×C53—21no—
246C4×C53—21no—
247C4×C53—21no—
248C4×C53—41no—
249C4×C53—41no—
250C4×C53—41no—
251C4×C53#47212yes—
252C4×C53#52511yescharacteristic
253C4×C53#53611yescharacteristic
254C4×C53#54711yescharacteristic
255C4×C53#54811yescharacteristic
256Dic53—82no—
257Dic53—81no—
258#3079—531no—
259C2×#474—531no—
260C4×AGL(1,ℤ/8)—531no—
261#2925D5311yescharacteristic, complemented
262#2078—532no—
263#2051—532no—
264#1997—531no—
265#1952—532no—
266#1950—531no—
267#1931—532no—
268#1800—531no—
269C2×C53—21no—
270C2×C53—21no—
271C2×C53—81no—
272C2×C53#241511yescharacteristic
273C2×C53C2×#47211yescharacteristic
274C2×C53#306911yescharacteristic
275D53—82no—
276C2×AGL(1,ℤ/8)—531no—
277C2×C4⋊Q8—531no—
278C4×#41C2×D5311yescharacteristic
279C2×C4×D4—531no—
280#477—1061no—
281#476—1061no—
282#475—1061no—
283#474—1061no—
284#474—1061no—
285#474—1061no—
286#474—1062no—
287#442—1061no—
288#440—1061no—
289C4×SD16—1061no—
290C4×D8—1061no—
291#433—22no—
292#432—21no—
293#431—21no—
294C2×#108—532no—
295#428—532no—
296#428—531no—
297C2×#107—532no—
298#423—531no—
299#423C2×D5311yescharacteristic
300C2×C4⋊C8C2×D5311yescharacteristic
301#418—531no—
302#417—532no—
303#417—531no—
304C2×#102—532no—
305C2×#100—532no—
306#408C2×D5312yes—
307#405C2×D5312yes—
308C4×M16—531no—
309#343—532no—
310#343—532no—
311#343—532no—
312#341—532no—
313C53#2472011yescharacteristic, complemented, Sylow
314C2×#41C22×D5311yescharacteristic
315C22×Q8—1061no—
316C22×D4—531no—
317C23×C4—1061no—
318AGL(1,ℤ/8)—1062no—
319AGL(1,ℤ/8)—1062no—
320C2×SD16—1061no—
321C2×D8—1061no—
322C2×M16—531no—
323C2×M16—531no—
324C2×M16—532no—
325C2×M16C22×D5312yes—
326C22×C8C22×D5312yes—
327C4⋊Q8—1061no—
328C4⋊Q8—1061no—
329C4⋊Q8—1061no—
330C4⋊Q8—1061no—
331C4×Q8D21211yescharacteristic
332C4×Q8D21211yescharacteristic
333C4×D4—21no—
334C4×D4—21no—
335C4×D4—1061no—
336C4×D4—1061no—
337C4×D4—1061no—
338C4×D4—1061no—
339C4×D4D21211yescharacteristic
340C4×D4D21211yescharacteristic
341#117—21no—
342#117C22×D5311yescharacteristic
343C2×C4⋊C4—531no—
344C2×C4⋊C4—532no—
345C2×C4⋊C4—1061no—
346C2×C22⋊C4—1061no—
347C2×C42—21no—
348C2×C42C22×D5311yescharacteristic
349#108—1064no—
350#107—1062no—
351#107—1062no—
352#107—1061no—
353#107—1062no—
354#106—1061no—
355#106—1062no—
356C4⋊C8—21no—
357C4⋊C8—21no—
358C4⋊C8—1061no—
359C4⋊C8—1061no—
360C4⋊C8C22⋊D5311yescharacteristic
361C4⋊C8C22⋊D5311yescharacteristic
362C4⋊C8C22⋊D5312yes—
363#103—1061no—
364#103—1062no—
365#102—1061no—
366#102—1062no—
367#102—1062no—
368#102—1062no—
369#102—1061no—
370#102—1061no—
371#100—1064no—
372C22⋊C8—22no—
373C22⋊C8—22no—
374#97—22no—
375#97—1061no—
376C4×C8—22no—
377C4×C8—1061no—
378C24—1061no—
379#41—21no—
380#41—21no—
381#41—21no—
382#41—21no—
383C2×Q8—4242no—
384C2×Q8—2121no—
385C2×Q8—2121no—
386C2×Q8C2×D21211yescharacteristic
387C2×D4—21no—
388C2×D4—532no—
389C2×D4—1061no—
390C2×D4—1061no—
391C2×D4—1061no—
392C2×D4—1061no—
393C2×D4—1061no—
394C2×D4C2×D21211yescharacteristic
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—1061no—
399C22×C4—1062no—
400C22×C4—4242no—
401C22×C4—2121no—
402C22×C4—2121no—
403C22×C4#1171994511yescharacteristic
404C22×C4#1171994511yescharacteristic
405C22×C4C23×D5311yescharacteristic
406SD16—1064no—
407D8—1064no—
408M16—22no—
409M16—22no—
410M16—532no—
411M16—1061no—
412M16—1062no—
413M16—1062no—
414M16—1061no—
415M16—1061no—
416C2×C8—22no—
417C2×C8—22no—
418C2×C8—22no—
419C2×C8—1061no—
420C2×C8—1061no—
421C2×C8—1062no—
422C2×C8C2×C22⋊D5312yes—
423C2×C8C2×C22⋊D5312yes—
424C4⋊C4—21no—
425C4⋊C4—21no—
426C4⋊C4—1062no—
427C4⋊C4—1061no—
428C4⋊C4—1061no—
429C4⋊C4—1062no—
430C4⋊C4—2121no—
431C4⋊C4—2121no—
432C4⋊C4C2×D21211yescharacteristic
433C4⋊C4C2×D21211yescharacteristic
434C22⋊C4—21no—
435C22⋊C4—21no—
436C22⋊C4—21no—
437C22⋊C4—2121no—
438C22⋊C4—2121no—
439C42—21no—
440C42—21no—
441C42—21no—
442C42C2×D21211yescharacteristic
443C42C2×D21211yescharacteristic
444C42C2×C22⋊D5311yescharacteristic
445C42C2×C22⋊D5311yescharacteristic
446C23—21no—
447C23—1062no—
448C23—4241no—
449C23—4241no—
450C23—2121no—
451C23—2121no—
452C23C2×#1171994511yescharacteristic
453Q8—21no—
454Q8—21no—
455Q8—4242no—
456Q8—4242no—
457D4—21no—
458D4—21no—
459D4—41no—
460D4—41no—
461D4—1062no—
462D4—1062no—
463D4—2121no—
464D4—2121no—
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—81no—
474C2×C4—4242no—
475C2×C4—4242no—
476C2×C4—4241no—
477C2×C4—4242no—
478C2×C4—2121no—
479C2×C4—2122no—
480C2×C4#1313319111yescharacteristic
481C2×C4#1313319111yescharacteristic
482C2×C4#1313319111yescharacteristic
483C2×C4#1313319111yescharacteristic
484C2×C4C22×C22⋊D5311yescharacteristic
485C2×C4C22×D21211yescharacteristic
486C2×C4#41×D5311yescharacteristic
487C2×C4#41×D5311yescharacteristic
488C2×C4#1313326411yescharacteristic
489C2×C4#1313326411yescharacteristic
490C8—22no—
491C8—22no—
492C8—42no—
493C8—1062no—
494C8—2122no—
495C8—2121no—
496C22—21no—
497C22—21no—
498C22—41no—
499C22—41no—
500C22—41no—
501C22—4242no—
502C22—4242no—
503C22—4241no—
504C22—4241no—
505C22—2122no—
506C22C2×#1313319111yescharacteristic
507C22#1681131211yescharacteristic
508C22#1681134111yescharacteristic
509C4—21no—
510C4—21no—
511C4—21no—
512C4—41no—
513C4—41no—
514C4—41no—
515C4—4241no—
516C4—4242no—
517C4#1681089312yes—
518C4#1681113511yescharacteristic
519C4#1681119911yescharacteristic
520C4#1681127211yescharacteristic
521C4#1681128011yescharacteristic
522C2—21no—
523C2—21no—
524C2—81no—
525C2—4242no—
526C2#2027813811yescharacteristic
527C2C2×#1681089311yescharacteristic
528C2#2028445211yescharacteristic
subgroupG/H
centerC22C2×#13133191
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2925C2
socleC22×C53C2×#121
prime radicalC2×#13133242C22
hypercenter#2925D53
hypocenterC53#24720
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2925

Complements

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

NH ≅ G/Ntype
1. #20284530C2semidirect
2. C2×#16810904C2semidirect
4. #20276155C2semidirect
6. #20275703C2semidirect
7. #20275500C2semidirect
8. #20275491C2semidirect
9. #20275377C2semidirect
10. #20274704C2semidirect
11. C53×#2925C2semidirect
261. #2925D53semidirect
313. C53#24720semidirect

Maximal subgroups

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

subgroupindex
#202845302
C2×#168109042
C4×#131332422
#202761552
#202760062
#202757032
#202755002
#202754912
#202753772
#202747042
C53×#29252
#2472053