Finite Group Databasebeta

← #27355291

Subgroups — #27355291

Order 13568.

600 conjugacy classes of subgroups.

The subgroup lattice has 566 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20284791C211yesmaximal, characteristic
2#20283875C211yesmaximal, characteristic
3#20283805C211yesmaximal, characteristic, complemented
4#20273609C211yesmaximal, characteristic, complemented
5#20273608C211yesmaximal, characteristic, complemented
6#20273596C211yesmaximal, characteristic, complemented
7#20273589C211yesmaximal, characteristic, complemented
8#20273583C211yesmaximal, characteristic, complemented
9#20273569C211yesmaximal, characteristic, complemented
10#20273519C211yesmaximal, characteristic, complemented
11#20273518C211yesmaximal, characteristic, complemented
12#20273501C211yesmaximal, characteristic, complemented
13C2×#16810355C211yesmaximal, characteristic, complemented
14#20272998C211yesmaximal, characteristic, complemented
15C53×#1494C211yesmaximal, characteristic
16C2×C4×Dic106C2211yescharacteristic
17#16810951C411yescharacteristic
18#16810934C411yescharacteristic
19#16810927C411yescharacteristic
20#16810927C411yescharacteristic
21#16810842C411yescharacteristic
22#16810842C411yescharacteristic
23#16810778C411yescharacteristic
24#16810761C411yescharacteristic
25#16810754C2211yescharacteristic
26#16810751—21no—
27#16810751—21no—
28#16810750C2211yescharacteristic
29#16810750C2211yescharacteristic
30Q8×#88322C2211yescharacteristic
31#16810747C2211yescharacteristic
32#16810747C2211yescharacteristic
33C4×#11719925C2211yescharacteristic
34C4×#11719925C2211yescharacteristic
35C4×#11719923C2211yescharacteristic
36C4×#11719923C2211yescharacteristic
37#16810701C2211yescharacteristic
38#16810695—21no—
39D4×#88322—21no—
40D4×#88322C2211yescharacteristic
41#16810684—21no—
42#16810677—21no—
43#16810676C2211yescharacteristic
44#16810676C2211yescharacteristic
45#16810673—21no—
46#16810650—21no—
47C2×C4⋊#88322C2211yescharacteristic
48#16810375C2211yescharacteristic
49#16810375C2211yescharacteristic
50#16810373C2211yescharacteristic
51#16810373C2211yescharacteristic
52#16810355C2211yescharacteristic
53#16810355C2211yescharacteristic
54#16810355C2211yescharacteristic
55#16810355C2211yescharacteristic
56#16810344C2211yescharacteristic
57#16810344C2211yescharacteristic
58C8×#88322C2211yescharacteristic
59C8×#88322C2211yescharacteristic
60C53×#529C2211yescharacteristic
61C53×#432C2211yescharacteristic
62C2×C4×C8×C53C2211yescharacteristic
63C53×Q8⋊C8C2211yescharacteristic
64C53×Q8⋊C8C2211yescharacteristic
65C53×D4⋊C8C2211yescharacteristic
66C53×D4⋊C8C2211yescharacteristic
67C22×Dic106C2×C411yescharacteristic
68C2×#11719925C2×C411yescharacteristic
69C2×#11719925C2×C411yescharacteristic
70C2×#11719923C2×C411yescharacteristic
71C2×#11719923C2×C411yescharacteristic
72C2×#11719917—21no—
73C22×#88322—21no—
74C22×#88322—21no—
75C4×Dic106—21no—
76C4×Dic106—21no—
77C4×Dic106C2×C411yescharacteristic
78C4×Dic106C2×C411yescharacteristic
79C4×Dic106C2311yescharacteristic
80C4×Dic106C2311yescharacteristic
81C2×C4⋊Dic53C2×C411yescharacteristic
82C2×#11719919—21no—
83C2×C4×Dic53—21no—
84#13133150C2×C411yescharacteristic
85#13133150C2×C411yescharacteristic
86#13133148C2×C411yescharacteristic
87#13133148C2×C411yescharacteristic
88C4⋊#88322—21no—
89C4⋊#88322—21no—
90C4⋊#88322—21no—
91C4⋊#88322C2×C411yescharacteristic
92C4⋊#88322C2×C411yescharacteristic
93C4⋊#88322D411yescharacteristic
94C4⋊#88322D411yescharacteristic
95C4⋊#88322C2311yescharacteristic
96C4⋊#88322C2311yescharacteristic
97#13133146—21no—
98#13133146—21no—
99#13133146—21no—
100#13133146—21no—
101#13133141C2×C411yescharacteristic
102#13133141C2×C411yescharacteristic
103#13133140C2×C411yescharacteristic
104#13133140C2×C411yescharacteristic
105#13133140C2×C411yescharacteristic
106#13133140C2×C411yescharacteristic
107#13133138C2×C411yescharacteristic
108#13133138C2×C411yescharacteristic
109C22⋊#88322—21no—
110C22⋊#88322—21no—
111C22⋊#88322C2×C411yescharacteristic
112C22⋊#88322C2×C411yescharacteristic
113C22⋊#88322D411yescharacteristic
114C22⋊#88322D411yescharacteristic
115#13133128—21no—
116#13133127—21no—
117#13133127—21no—
118C4×#88322—21no—
119C4×#88322C2311yescharacteristic
120C4×#88322C2311yescharacteristic
121C4×#88322C2311yescharacteristic
122C4×#88322C2311yescharacteristic
123C8×Dic53—21no—
124C8×Dic53—21no—
125C22×C8×C53—21no—
126#126×C53—21no—
127#125×C53C2×C411yescharacteristic
128#124×C53C2×C411yescharacteristic
129#123×C53—21no—
130C22⋊Q8×C53C2×C411yescharacteristic
131#121×C53C2×C411yescharacteristic
132C4×Q8×C53C2311yescharacteristic
133C4×D4×C53—21no—
134C4×D4×C53C2311yescharacteristic
135#117×C53—21no—
136C2×C42×C53C2311yescharacteristic
137C4⋊C8×C53C2311yescharacteristic
138C4⋊C8×C53C2311yescharacteristic
139C22⋊C8×C53—21no—
140#97×C53—21no—
141C4×C8×C53—21no—
142C4×C8×C53C2311yescharacteristic
143C4×C8×C53C2311yescharacteristic
144C2×Q8×C53C22×C411yescharacteristic
145C2×D4×C53—21no—
146C2×D4×C53C22×C411yescharacteristic
147C22×C4×C53—21no—
148C22×C4×C53C22×C411yescharacteristic
149C22×C4×C53C2×D411yescharacteristic
150C22×C4×C53#4111yescharacteristic
151C22×Dic53—21no—
152C22×Dic53—21no—
153C2×Dic106—22no—
154C2×Dic106—21no—
155C2×Dic106—41no—
156C2×Dic106C22×C411yescharacteristic
157C2×Dic106C22×C411yescharacteristic
158C2×C8×C53—24no—
159C2×C8×C53—21no—
160C2×C8×C53—21no—
161C2×C8×C53—21no—
162C2×C8×C53—21no—
163C4⋊C4×C53—21no—
164C4⋊C4×C53—21no—
165C4⋊C4×C53—21no—
166C4⋊C4×C53—21no—
167C4⋊C4×C53C22×C411yescharacteristic
168C4⋊C4×C53C22×C411yescharacteristic
169C22⋊C4×C53—21no—
170C22⋊C4×C53—21no—
171C22⋊C4×C53—21no—
172C22⋊C4×C53—21no—
173C22⋊C4×C53—21no—
174C42×C53—21no—
175C42×C53C22×C411yescharacteristic
176C42×C53C2×D411yescharacteristic
177C42×C53#4111yescharacteristic
178C42×C53C2411yescharacteristic
179#11719925—22no—
180#11719925—41no—
181#11719923—22no—
182#11719923—41no—
183C4⋊Dic53—21no—
184C4⋊Dic53C22×C411yescharacteristic
185C4⋊Dic53C22×C411yescharacteristic
186#11719919—21no—
187#11719919—21no—
188#11719919—21no—
189#11719919—21no—
190C4×Dic53—21no—
191C4×Dic53—21no—
192C4×Dic53—21no—
193C4×Dic53—21no—
194#11719917—81no—
195C2×#88322—21no—
196C2×#88322—21no—
197C2×#88322—81no—
198C2×#88322—81no—
199C2×#88322C22×C411yescharacteristic
200C2×#88322C22×C411yescharacteristic
201C2×#88322C22×C411yescharacteristic
202C2×#88322C22×C411yescharacteristic
203C2×#88322C2×D411yescharacteristic
204C2×#88322C2×D411yescharacteristic
205C2×#88322C2×D411yescharacteristic
206C2×#88322C2×D411yescharacteristic
207C23×C53—21no—
208C23×C53C4×D411yescharacteristic
209Q8×C53—21no—
210Q8×C53—21no—
211D4×C53—21no—
212D4×C53—21no—
213D4×C53—42no—
214C2×C4×C53—22no—
215C2×C4×C53—21no—
216C2×C4×C53—21no—
217C2×C4×C53—21no—
218C2×C4×C53—21no—
219C2×C4×C53—21no—
220C2×C4×C53—21no—
221C2×C4×C53—21no—
222C2×C4×C53—41no—
223C2×C4×C53—41no—
224C2×C4×C53C4×D411yescharacteristic
225C2×C4×C53C4×D411yescharacteristic
226C2×C4×C53C4×D411yescharacteristic
227C2×C4×C53C23×C411yescharacteristic
228C2×C4×C53C22×D411yescharacteristic
229C2×C4×C53C2×#4111yescharacteristic
230C2×Dic53—21no—
231C2×Dic53—21no—
232C2×Dic53—21no—
233C2×Dic53—21no—
234C2×Dic53—42no—
235C2×Dic53—42no—
236Dic106—21no—
237Dic106—21no—
238Dic106—21no—
239Dic106—42no—
240Dic106#13112yes—
241C8×C53—22no—
242C8×C53—22no—
243C8×C53—41no—
244C8×C53—41no—
245#88322—21no—
246#88322—21no—
247#88322—21no—
248#88322—21no—
249#88322—21no—
250#88322—21no—
251#88322—21no—
252#88322—21no—
253#88322—41no—
254#88322—41no—
255#18017—531nomaximal, Sylow
256C22×C53—21no—
257C22×C53—21no—
258C22×C53—41no—
259C22×C53—41no—
260C22×C53—41no—
261C22×C53#44312yes—
262C22×C53C2×C4×D411yescharacteristic
263C4×C53—21no—
264C4×C53—21no—
265C4×C53—21no—
266C4×C53—21no—
267C4×C53—41no—
268C4×C53—41no—
269C4×C53—41no—
270C4×C53C2×#13111yescharacteristic
271C4×C53#56811yescharacteristic
272C4×C53#57711yescharacteristic
273C4×C53#57811yescharacteristic
274Dic53—22no—
275Dic53—41no—
276Dic53—41no—
277Dic53—41no—
278#3108—531no—
279#2994—531no—
280#2984—531no—
281#1591—531no—
282#1590—531no—
283#1588—531no—
284#1587—531no—
285#1587—531no—
286#1585—531no—
287C8×Q16—531no—
288C8×SD16—531no—
289#1573—531no—
290#1494D5311yescharacteristic
291C2×Q8⋊C8—531no—
292#1450—531no—
293C2×C53—22no—
294C2×C53—81no—
295C2×C53#293311yescharacteristic
296C2×C53#294711yescharacteristic
297C2×C53C2×#44311yescharacteristic
298#529C2×D5311yescharacteristic
299C2×C4×Q8—531no—
300#483—531no—
301#480—531no—
302#479—531no—
303#479—531no—
304#464—531no—
305#464—531no—
306#451—531no—
307#448—531no—
308#446—531no—
309#446—531no—
310#446—531no—
311#446—1061no—
312#446—1061no—
313C8×Q8—531no—
314C8×Q8—531no—
315C8×Q8—531no—
316C4×Q16—531no—
317C4×Q16—531no—
318C4×SD16—531no—
319C4×SD16—531no—
320#436—531no—
321#435—1061no—
322C8×D4—531no—
323C8×D4—1061no—
324#433—1061no—
325#432—1061no—
326#432C2×D5311yescharacteristic
327#431—531no—
328#431—531no—
329#431—1061no—
330#424—1061no—
331C2×C4⋊C8—531no—
332C2×C4×C8C2×D5311yescharacteristic
333#335—531no—
334#335—531no—
335#334—531no—
336#334—531no—
337Q8⋊C8—531no—
338Q8⋊C8—531no—
339Q8⋊C8—531no—
340Q8⋊C8—531no—
341Q8⋊C8C2×D5311yescharacteristic
342Q8⋊C8C2×D5311yescharacteristic
343D4⋊C8C2×D5311yescharacteristic
344D4⋊C8C2×D5311yescharacteristic
345#322—531no—
346#322—531no—
347C82—531no—
348C82—531no—
349C53#1801711yescharacteristic, complemented, Sylow
350C22×Q8—531no—
351C2×Q16—531no—
352C2×Q16—531no—
353C2×SD16—531no—
354C2×SD16—531no—
355C2×M16—1061no—
356C22×C8—21no—
357C22×C8—1061no—
358C22×C8—1061no—
359#126—21no—
360#125C4×D5311yescharacteristic
361#124C4×D5311yescharacteristic
362#123—21no—
363C22⋊Q8C4×D5311yescharacteristic
364#121C4×D5311yescharacteristic
365C4×Q8—531no—
366C4×Q8—531no—
367C4×Q8—531no—
368C4×Q8—531no—
369C4×Q8—1061no—
370C4×Q8—1061no—
371C4×Q8C22×D5311yescharacteristic
372C4×D4—21no—
373C4×D4C22×D5311yescharacteristic
374#117—21no—
375C2×C4⋊C4—531no—
376C2×C4⋊C4—1061no—
377C2×C42—1061no—
378C2×C42C22×D5311yescharacteristic
379#107—531no—
380#107—531no—
381#106—531no—
382#106—531no—
383C4⋊C8—531no—
384C4⋊C8—531no—
385C4⋊C8—531no—
386C4⋊C8—531no—
387C4⋊C8—531no—
388C4⋊C8—531no—
389C4⋊C8—1061no—
390C4⋊C8—1061no—
391C4⋊C8—1061no—
392C4⋊C8—1061no—
393C4⋊C8—1061no—
394C4⋊C8—1061no—
395C4⋊C8—1061no—
396C4⋊C8C22×D5311yescharacteristic
397C4⋊C8C22×D5311yescharacteristic
398#103—531no—
399#103—531no—
400#103—531no—
401#103—531no—
402#103—531no—
403#103—531no—
404#102—531no—
405#102—531no—
406C22⋊C8—21no—
407C22⋊C8—531no—
408C22⋊C8—531no—
409C22⋊C8—531no—
410C22⋊C8—531no—
411C22⋊C8—1061no—
412C22⋊C8—1061no—
413#97—21no—
414#97—1061no—
415#97—1061no—
416#97—1061no—
417C4×C8—21no—
418C4×C8—531no—
419C4×C8—531no—
420C4×C8—531no—
421C4×C8—531no—
422C4×C8—1061no—
423C4×C8—1061no—
424C4×C8—1061no—
425C4×C8C22×D5311yescharacteristic
426C4×C8C22×D5311yescharacteristic
427C2×Q8—531no—
428C2×Q8—531no—
429C2×Q8—1062no—
430C2×Q8—1061no—
431C2×Q8—2121no—
432C2×Q8C2×C4×D5311yescharacteristic
433C2×D4—21no—
434C2×D4C2×C4×D5311yescharacteristic
435C22×C4—21no—
436C22×C4—1061no—
437C22×C4—1061no—
438C22×C4C2×C4×D5311yescharacteristic
439C22×C4D4×D5311yescharacteristic
440C22×C4#1171994711yescharacteristic
441Q16—1062no—
442Q16—2121no—
443SD16—1062no—
444SD16—2121no—
445M16—4241no—
446C2×C8—24no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—21no—
450C2×C8—21no—
451C2×C8—531no—
452C2×C8—531no—
453C2×C8—531no—
454C2×C8—531no—
455C2×C8—531no—
456C2×C8—531no—
457C2×C8—531no—
458C2×C8—531no—
459C2×C8—1061no—
460C2×C8—1061no—
461C2×C8—4241no—
462C2×C8—4241no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—531no—
468C4⋊C4—531no—
469C4⋊C4—1061no—
470C4⋊C4—1061no—
471C4⋊C4—1061no—
472C4⋊C4—1061no—
473C4⋊C4—1061no—
474C4⋊C4C2×C4×D5311yescharacteristic
475C4⋊C4C2×C4×D5311yescharacteristic
476C22⋊C4—21no—
477C22⋊C4—21no—
478C22⋊C4—21no—
479C22⋊C4—21no—
480C22⋊C4—21no—
481C42—21no—
482C42—1061no—
483C42—1061no—
484C42—1061no—
485C42—1061no—
486C42C2×C4×D5311yescharacteristic
487C42D4×D5311yescharacteristic
488C42#1171994711yescharacteristic
489C42C23×D5311yescharacteristic
490C23—21no—
491C23#1313317711yescharacteristic
492Q8—21no—
493Q8—21no—
494Q8—532no—
495Q8—1061no—
496Q8—1061no—
497Q8—1061no—
498Q8—2122no—
499D4—21no—
500D4—21no—
501D4—42no—
502C2×C4—22no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—1061no—
513C2×C4—1061no—
514C2×C4—1061no—
515C2×C4—1061no—
516C2×C4—2122no—
517C2×C4—2122no—
518C2×C4#1313317711yescharacteristic
519C2×C4#1313317711yescharacteristic
520C2×C4#1313317711yescharacteristic
521C2×C4C22×C4×D5311yescharacteristic
522C2×C4C2×D4×D5311yescharacteristic
523C2×C4C2×#1171994711yescharacteristic
524C8—22no—
525C8—22no—
526C8—41no—
527C8—41no—
528C8—1061no—
529C8—1061no—
530C8—1061no—
531C8—1061no—
532C8—1061no—
533C8—1061no—
534C8—1061no—
535C8—1061no—
536C8—2121no—
537C8—2121no—
538C22—21no—
539C22—21no—
540C22—41no—
541C22—41no—
542C22—41no—
543C22#1681073712yes—
544C22C2×#1313317711yescharacteristic
545C4—21no—
546C4—21no—
547C4—21no—
548C4—21no—
549C4—41no—
550C4—41no—
551C4—41no—
552C4—1062no—
553C4—2121no—
554C4—2121no—
555C4—2121no—
556C4#131×D5311yescharacteristic
557C4#1681138811yescharacteristic
558C4#1681143011yescharacteristic
559C4#1681143711yescharacteristic
560C2—22no—
561C2—81no—
562C2#2028344511yescharacteristic
563C2#2028356111yescharacteristic
564C2C2×#1681073711yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC42C23×D53
FittingC53×#1494C2
socleC22×C53C2×C4×D4
prime radical#121×C53C2×C4
cosocleC42×C53C24
hypercenter#1494D53
hypocenterC53#18017
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1494

Complements

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

NH ≅ G/Ntype
3. #20283805C2semidirect
4. #20273609C2semidirect
5. #20273608C2semidirect
6. #20273596C2semidirect
7. #20273589C2semidirect
8. #20273583C2semidirect
9. #20273569C2semidirect
10. #20273519C2semidirect
11. #20273518C2semidirect
12. #20273501C2semidirect
13. C2×#16810355C2semidirect
14. #20272998C2semidirect
349. C53#18017semidirect

Maximal subgroups

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

subgroupindex
#202847912
#202838752
#202838052
#202736092
#202736082
#202735962
#202735892
#202735832
#202735692
#202735192
#202735182
#202735012
C2×#168103552
#202729982
C53×#14942
#1801753