Finite Group Databasebeta

← #27355549

Subgroups — #27355549

Order 13568.

638 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20283667C211yesmaximal, characteristic
2Q8×#11719912C211yesmaximal, characteristic
3C4×Q16×D53C211yesmaximal, characteristic
4#20273610C211yesmaximal, characteristic, complemented
5#20273609C211yesmaximal, characteristic, complemented
6#20273567C211yesmaximal, characteristic, complemented
7#20273566C211yesmaximal, characteristic, complemented
8#20273564C211yesmaximal, characteristic, complemented
9C8×#11719925C211yesmaximal, characteristic, complemented
10#20273519C211yesmaximal, characteristic, complemented
11#20273472C211yesmaximal, characteristic
12#20273142C211yesmaximal, characteristic
13#20273142C211yesmaximal, characteristic
14#20272994C211yesmaximal, characteristic
15C53×#1584C211yesmaximal, characteristic, complemented
16C2×Q16×D53C411yescharacteristic
17C4×Q8×D53C2211yescharacteristic
18C4×Q8×D53C2211yescharacteristic
19#16810754C2211yescharacteristic
20#16810753—21no—
21#16810753C2211yescharacteristic
22#16810752—21no—
23#16810751—21no—
24#16810750—21no—
25#16810750C2211yescharacteristic
26Q8×#88322C2211yescharacteristic
27C8×Dic106C2211yescharacteristic
28#16810747C2211yescharacteristic
29C4×Dic212C2211yescharacteristic
30Q16×Dic53C411yescharacteristic
31#16810717C411yescharacteristic
32C4×#11719925C2211yescharacteristic
33C4×#11719925C2211yescharacteristic
34#16810715C411yescharacteristic
35#16810715C411yescharacteristic
36#16810674C2211yescharacteristic
37#107×D53C411yescharacteristic
38#16810645—21no—
39#16810645—21no—
40#16810645C2211yescharacteristic
41#16810645C2211yescharacteristic
42#103×D53C411yescharacteristic
43#103×D53C411yescharacteristic
44#16810575—21no—
45C4×#11719912—21no—
46C4×#11719912C2211yescharacteristic
47C4×C8×D53C2211yescharacteristic
48#16810376C2211yescharacteristic
49#16810375C2211yescharacteristic
50#16810375C2211yescharacteristic
51#16810355C2211yescharacteristic
52#16810355C2211yescharacteristic
53Q8⋊#88322C2211yescharacteristic
54Q8⋊#88322C2211yescharacteristic
55#16810353C2211yescharacteristic
56#16810353C2211yescharacteristic
57#16810345C2211yescharacteristic
58C8×#88322C2211yescharacteristic
59C8×#88322C2211yescharacteristic
60C53×#446C2211yescharacteristic
61C8×Q8×C53C2211yescharacteristic
62C4×Q16×C53C2211yescharacteristic
63C53×#335C2211yescharacteristic
64C53×Q8⋊C8C2211yescharacteristic
65C53×Q8⋊C8C2211yescharacteristic
66C53×#322C2211yescharacteristic
67C2×Q8×D53C2×C411yescharacteristic
68C2×Q8×D53C2×C411yescharacteristic
69C2×Dic212C2×C411yescharacteristic
70C2×#11719925C2×C411yescharacteristic
71C2×#11719925C2×C411yescharacteristic
72Q16×D53—24no—
73C2×#11719912—21no—
74C2×#11719912—21no—
75C2×#11719912D411yescharacteristic
76C2×#11719912D411yescharacteristic
77C2×C8×D53C2×C411yescharacteristic
78C2×C8×D53C2×C411yescharacteristic
79Q8×Dic53C2×C411yescharacteristic
80Q8×Dic53C2×C411yescharacteristic
81C4×Dic106—21no—
82C4×Dic106—21no—
83C4×Dic106C2311yescharacteristic
84C4×Dic106C2311yescharacteristic
85#13133182—21no—
86#13133182—21no—
87#13133182C2×C411yescharacteristic
88#13133182C2×C411yescharacteristic
89C4⋊C4×D53—21no—
90C4⋊C4×D53—21no—
91C4⋊C4×D53C2×C411yescharacteristic
92C4⋊C4×D53C2×C411yescharacteristic
93C42×D53—21no—
94C42×D53—21no—
95C42×D53C2311yescharacteristic
96#13133151C2×C411yescharacteristic
97#13133150C2×C411yescharacteristic
98#13133150C2×C411yescharacteristic
99C4⋊#88322—21no—
100C4⋊#88322—21no—
101C4⋊#88322C2311yescharacteristic
102C4⋊#88322C2311yescharacteristic
103#13133146—21no—
104#13133146—21no—
105#13133146—21no—
106#13133146—21no—
107#13133146—21no—
108#13133146—21no—
109#13133146—21no—
110#13133142C2×C411yescharacteristic
111#13133142C2×C411yescharacteristic
112#13133141C2×C411yescharacteristic
113#13133141C2×C411yescharacteristic
114#13133140C2×C411yescharacteristic
115#13133140C2×C411yescharacteristic
116#13133129—21no—
117#13133128—21no—
118#13133127—21no—
119#13133127—21no—
120#13133127D411yescharacteristic
121#13133127D411yescharacteristic
122C4×#88322—21no—
123C4×#88322C2311yescharacteristic
124C4×#88322C2311yescharacteristic
125C4×#88322C2311yescharacteristic
126C8×Dic53—21no—
127C8×Dic53—21no—
128C8×Dic53C2×C411yescharacteristic
129C8×Dic53C2×C411yescharacteristic
130C2×Q16×C53C2×C411yescharacteristic
131C4×Q8×C53C2311yescharacteristic
132C4×Q8×C53C2311yescharacteristic
133#107×C53C2×C411yescharacteristic
134C4⋊C8×C53—21no—
135C4⋊C8×C53—21no—
136C4⋊C8×C53C2311yescharacteristic
137C4⋊C8×C53C2311yescharacteristic
138#103×C53C2×C411yescharacteristic
139#103×C53C2×C411yescharacteristic
140#97×C53—21no—
141C4×C8×C53—21no—
142C4×C8×C53C2311yescharacteristic
143C4×C8×C53C2311yescharacteristic
144C4×C8×C53C2311yescharacteristic
145C2×Q8×C53C22×C411yescharacteristic
146C2×Q8×C53C22×C411yescharacteristic
147Q8×D53—22no—
148Q8×D53—22no—
149Q8×D53—41no—
150Q8×D53—41no—
151C2×C4×D53—21no—
152C2×C4×D53—21no—
153C2×C4×D53—21no—
154C2×C4×D53—21no—
155C2×C4×D53C22×C411yescharacteristic
156C2×C4×D53C2×D411yescharacteristic
157C2×C4×D53#4111yescharacteristic
158C2×Dic106—21no—
159C2×Dic106—21no—
160C2×Dic106C22×C411yescharacteristic
161C2×Dic106C22×C411yescharacteristic
162Q16×C53—22no—
163C2×C8×C53—21no—
164C2×C8×C53—21no—
165C2×C8×C53—21no—
166C2×C8×C53C22×C411yescharacteristic
167C2×C8×C53C22×C411yescharacteristic
168C2×C8×C53C2×D411yescharacteristic
169C2×C8×C53C2×D411yescharacteristic
170C4⋊C4×C53—21no—
171C4⋊C4×C53—21no—
172C4⋊C4×C53C22×C411yescharacteristic
173C4⋊C4×C53C22×C411yescharacteristic
174C42×C53—21no—
175C42×C53—21no—
176C42×C53C2411yescharacteristic
177#11719925—22no—
178#11719925M1614yescomplemented
179C4⋊Dic53—21no—
180C4⋊Dic53—21no—
181C4⋊Dic53C22×C411yescharacteristic
182C4⋊Dic53C22×C411yescharacteristic
183#11719919—21no—
184#11719919—21no—
185#11719919—21no—
186#11719919—21no—
187#11719919—21no—
188#11719919—21no—
189C4×Dic53—21no—
190C4×Dic53—21no—
191C4×Dic53—21no—
192C4×Dic53—21no—
193C4×Dic53—21no—
194C4×Dic53—21no—
195C4×Dic53C22×C411yescharacteristic
196C4×Dic53C2×D411yescharacteristic
197C4×Dic53#4111yescharacteristic
198C2×#88322—21no—
199C2×#88322—21no—
200C2×#88322—21no—
201C2×#88322C22×C411yescharacteristic
202C2×#88322C22×C411yescharacteristic
203C2×#88322C2×D411yescharacteristic
204C2×#88322C2×D411yescharacteristic
205Dic212—22no—
206#11719912—22no—
207#11719912—42no—
208#11719912—42no—
209#11719912Q1614yescomplemented
210C8×D53—22no—
211C8×D53—41no—
212C22×D53C4×D411yescharacteristic
213Q8×C53—21no—
214Q8×C53—21no—
215Q8×C53—21no—
216Q8×C53C2×M1612yes—
217C2×C4×C53—21no—
218C2×C4×C53—21no—
219C2×C4×C53—21no—
220C2×C4×C53—21no—
221C2×C4×C53C23×C411yescharacteristic
222C2×C4×C53C22×D411yescharacteristic
223C2×C4×C53C2×#4111yescharacteristic
224C2×Dic53—21no—
225C2×Dic53—21no—
226C2×Dic53—21no—
227C2×Dic53—21no—
228C2×Dic53C4×D411yescharacteristic
229C2×Dic53C4×D411yescharacteristic
230C2×Dic53C4×D411yescharacteristic
231C4×D53—21no—
232C4×D53—21no—
233C4×D53—41no—
234C4×D53—42no—
235C4×D53—42no—
236C4×D53—81no—
237C4×D53—81no—
238C4×D53C2×Q1612yes—
239C4×D53#13512yes—
240Dic106—21no—
241Dic106—21no—
242Dic106—21no—
243Dic106—42no—
244Dic106—42no—
245Dic106C2×M1612yes—
246C8×C53—21no—
247C8×C53—21no—
248C8×C53—21no—
249C8×C53—41no—
250C8×C53—41no—
251C8×C53—41no—
252C8×C53C2×Q1612yescomplemented
253#88322—21no—
254#88322—21no—
255#88322—41no—
256#88322—41no—
257#88322—41no—
258#88322C2×M1612yes—
259#88322C2×Q1612yescomplemented
260M16×Q16—531nomaximal, Sylow
261C22×C53C2×C4×D411yescharacteristic
262C2×D53—21no—
263C2×D53—21no—
264C2×D53C4×Q1612yes—
265C4×C53—22no—
266C4×C53—21no—
267C4×C53—41no—
268C4×C53—41no—
269C4×C53—41no—
270C4×C53C22×M1611yescharacteristic
271C4×C53#56811yescharacteristic
272C4×C53C22×Q1611yescharacteristic
273C4×C53C2×#13511yescharacteristic
274Dic53—22no—
275Dic53—21no—
276Dic53—21no—
277Dic53—41no—
278Dic53—41no—
279Dic53—41no—
280Dic53C4×Q1612yes—
281Q8×M16—531no—
282Q8×M16—531no—
283C2×C4×Q16—531no—
284#1591—531no—
285#1591—531no—
286#1584—531no—
287#1584—531no—
288#1584—531no—
289#1584D5311yescharacteristic, complemented
290C8×Q16—531no—
291C8×Q16—531no—
292#1567—531no—
293#1487—531no—
294#1487—531no—
295C8×M16—531no—
296C2×C53D4×M1611yescharacteristic
297C2×C53C2×C4×Q1611yescharacteristic
298C2×C53#295711yescharacteristic
299D53—22no—
300C22×Q16—531no—
301C2×C4×Q8—531no—
302C2×C4×Q8—531no—
303#446—531no—
304#446—531no—
305#446—531no—
306#446—1061no—
307#446—1061no—
308#446—1061no—
309#446—1061no—
310#446C2×D5311yescharacteristic
311C8×Q8—531no—
312C8×Q8—531no—
313C8×Q8—531no—
314C8×Q8C2×D5311yescharacteristic
315C4×Q16—531no—
316C4×Q16—531no—
317C4×Q16—531no—
318C4×Q16—531no—
319C4×Q16—531no—
320C4×Q16—531no—
321C4×Q16—531no—
322C4×Q16C2×D5311yescharacteristic
323#431—531no—
324C2×#107—531no—
325#423—531no—
326#423—531no—
327#423—1061no—
328#423—1061no—
329C2×#103—531no—
330C2×#103—531no—
331C4×M16—531no—
332C4×M16—1061no—
333C4×M16—1061no—
334C2×C4×C8—531no—
335#335—531no—
336#335—531no—
337#335—531no—
338#335C2×D5311yescharacteristic
339Q8⋊C8—531no—
340Q8⋊C8—531no—
341Q8⋊C8—531no—
342Q8⋊C8—531no—
343Q8⋊C8—531no—
344Q8⋊C8—531no—
345Q8⋊C8C2×D5311yescharacteristic
346Q8⋊C8C2×D5311yescharacteristic
347#322—531no—
348#322C2×D5311yescharacteristic
349C82—531no—
350C82—531no—
351C53M16×Q1611yescharacteristic, complemented, Sylow
352C22×Q8—531no—
353C22×Q8—531no—
354C2×Q16—531no—
355C2×Q16—531no—
356C2×Q16—531no—
357C2×Q16—1064no—
358C2×Q16C4×D5311yescharacteristic
359C2×M16—531no—
360C2×M16—531no—
361C2×M16—1061no—
362C2×M16—1061no—
363C22×C8—531no—
364C22×C8—531no—
365C4×Q8—531no—
366C4×Q8—531no—
367C4×Q8—531no—
368C4×Q8—531no—
369C4×Q8—531no—
370C4×Q8—531no—
371C4×Q8—1061no—
372C4×Q8—1061no—
373C4×Q8—1061no—
374C4×Q8—1061no—
375C4×Q8C22×D5311yescharacteristic
376C4×Q8C22×D5311yescharacteristic
377C2×C4⋊C4—531no—
378C2×C4⋊C4—531no—
379C2×C4⋊C4—1061no—
380C2×C4⋊C4—1061no—
381C2×C42—531no—
382C2×C42—1061no—
383C2×C42—1061no—
384#107—531no—
385#107—531no—
386#107—531no—
387#107C4×D5311yescharacteristic
388C4⋊C8—21no—
389C4⋊C8—21no—
390C4⋊C8—531no—
391C4⋊C8—531no—
392C4⋊C8—1061no—
393C4⋊C8—1061no—
394C4⋊C8—1061no—
395C4⋊C8—1061no—
396C4⋊C8—1061no—
397C4⋊C8—1061no—
398C4⋊C8—1061no—
399C4⋊C8—1061no—
400C4⋊C8—1061no—
401C4⋊C8C22×D5311yescharacteristic
402C4⋊C8C22×D5311yescharacteristic
403#103—531no—
404#103—531no—
405#103—531no—
406#103—531no—
407#103—531no—
408#103—531no—
409#103C4×D5311yescharacteristic
410#103C4×D5311yescharacteristic
411C22⋊C8—1061no—
412#97—21no—
413#97—531no—
414#97—531no—
415#97—1061no—
416#97—1061no—
417#97—1061no—
418C4×C8—21no—
419C4×C8—531no—
420C4×C8—531no—
421C4×C8—531no—
422C4×C8—531no—
423C4×C8—531no—
424C4×C8—1061no—
425C4×C8—1061no—
426C4×C8—1061no—
427C4×C8C22×D5311yescharacteristic
428C4×C8C22×D5311yescharacteristic
429C4×C8C22×D5311yescharacteristic
430C2×Q8—531no—
431C2×Q8—531no—
432C2×Q8—1061no—
433C2×Q8—1061no—
434C2×Q8—1062no—
435C2×Q8—1062no—
436C2×Q8—2121no—
437C2×Q8—2121no—
438C2×Q8C2×C4×D5311yescharacteristic
439C2×Q8C2×C4×D5311yescharacteristic
440C22×C4—531no—
441C22×C4—531no—
442C22×C4—531no—
443C22×C4—1061no—
444C22×C4—1061no—
445C22×C4—1061no—
446C22×C4—1061no—
447Q16—22no—
448Q16—534no—
449Q16—1062no—
450Q16—1062no—
451M16—534no—
452M16—1062no—
453M16—2122no—
454M16—2122no—
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—531no—
459C2×C8—531no—
460C2×C8—531no—
461C2×C8—531no—
462C2×C8—1062no—
463C2×C8—1061no—
464C2×C8—1061no—
465C2×C8—1061no—
466C2×C8—2121no—
467C2×C8C2×C4×D5311yescharacteristic
468C2×C8C2×C4×D5311yescharacteristic
469C2×C8D4×D5311yescharacteristic
470C2×C8D4×D5311yescharacteristic
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—531no—
474C4⋊C4—531no—
475C4⋊C4—1061no—
476C4⋊C4—1061no—
477C4⋊C4—1061no—
478C4⋊C4—1061no—
479C4⋊C4—1061no—
480C4⋊C4—1061no—
481C4⋊C4—1061no—
482C4⋊C4—1061no—
483C4⋊C4C2×C4×D5311yescharacteristic
484C4⋊C4C2×C4×D5311yescharacteristic
485C42—21no—
486C42—21no—
487C42—531no—
488C42—531no—
489C42—531no—
490C42—1061no—
491C42—1061no—
492C42—1061no—
493C42—1061no—
494C42—1061no—
495C42—1061no—
496C42C23×D5311yescharacteristic
497C23—531no—
498Q8—21no—
499Q8—21no—
500Q8—21no—
501Q8—532no—
502Q8—1061no—
503Q8—1061no—
504Q8—1061no—
505Q8—2122no—
506Q8—2122no—
507Q8M16×D5312yes—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—531no—
513C2×C4—532no—
514C2×C4—532no—
515C2×C4—531no—
516C2×C4—531no—
517C2×C4—1061no—
518C2×C4—1061no—
519C2×C4—1061no—
520C2×C4—1061no—
521C2×C4—1061no—
522C2×C4—1061no—
523C2×C4—4241no—
524C2×C4—4241no—
525C2×C4—2121no—
526C2×C4—2122no—
527C2×C4—2122no—
528C2×C4C22×C4×D5311yescharacteristic
529C2×C4C2×D4×D5311yescharacteristic
530C2×C4#41×D5311yescharacteristic
531C8—21no—
532C8—21no—
533C8—21no—
534C8—41no—
535C8—41no—
536C8—41no—
537C8—532no—
538C8—532no—
539C8—1061no—
540C8—1061no—
541C8—2121no—
542C8—2121no—
543C8—2121no—
544C8Q16×D5312yescomplemented
545C22—532no—
546C22—1061no—
547C22—1061no—
548C22C4×D4×D5311yescharacteristic
549C4—22no—
550C4—21no—
551C4—41no—
552C4—41no—
553C4—41no—
554C4—532no—
555C4—1061no—
556C4—1062no—
557C4—1061no—
558C4—2121no—
559C4—2121no—
560C4—2121no—
561C4C2×M16×D5311yescharacteristic
562C4#1681138811yescharacteristic
563C4C2×Q16×D5311yescharacteristic
564C4#135×D5311yescharacteristic
565C2—1062no—
566C2#2028344811yescharacteristic
567C2C4×Q16×D5311yescharacteristic
568C2#2028362911yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC42C23×D53
FittingC53×#1584C2
socleC22×C53C2×C4×D4
prime radicalC22×D53C4×D4
cosocleC42×C53C24
hypercenter#1584D53
hypocenterC53M16×Q16
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ #1584

Complements

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

NH ≅ G/Ntype
4. #20273610C2semidirect
5. #20273609C2semidirect
6. #20273567C2semidirect
7. #20273566C2semidirect
8. #20273564C2semidirect
9. C8×#11719925C2semidirect
10. #20273519C2semidirect
15. C53×#1584C2semidirect
178. #11719925M16semidirect
209. #11719912Q16semidirect
252. C8×C53C2×Q16semidirect
259. #88322C2×Q16semidirect
289. #1584D53semidirect
351. C53M16×Q16semidirect
544. C8Q16×D53semidirect

Maximal subgroups

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

subgroupindex
#202836672
Q8×#117199122
C4×Q16×D532
#202736102
#202736092
#202735672
#202735662
#202735642
C8×#117199252
#202735192
#202734722
#202731422
#202729942
C53×#15842
M16×Q1653