Finite Group Databasebeta

← #27355227

Subgroups — #27355227

Order 13568.

592 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20284786C211yesmaximal, characteristic
2#20283735C211yesmaximal, characteristic
3#20283658C211yesmaximal, characteristic, complemented
4#20273614C211yesmaximal, characteristic, complemented
5#20273608C211yesmaximal, characteristic, complemented
6#20273596C211yesmaximal, characteristic, complemented
7#20273583C211yesmaximal, characteristic, complemented
8#20273562C211yesmaximal, characteristic, complemented
9#20273561C211yesmaximal, characteristic, complemented
10#20273531C211yesmaximal, characteristic, complemented
11#20273519C211yesmaximal, characteristic, complemented
12#20273498C211yesmaximal, characteristic, complemented
13#20273189C211yesmaximal, characteristic, complemented
14#20273010C211yesmaximal, characteristic, complemented
15C53×#1477C211yesmaximal, characteristic
16#16811180C2211yescharacteristic
17#16810964C411yescharacteristic
18#16810964C411yescharacteristic
19#16810908C411yescharacteristic
20#16810903C411yescharacteristic
21#16810833C411yescharacteristic
22#16810828C411yescharacteristic
23#16810785C411yescharacteristic
24#16810785C411yescharacteristic
25#16810752—21no—
26#16810752—21no—
27#16810750C2211yescharacteristic
28#16810750C2211yescharacteristic
29Q8×#88322C2211yescharacteristic
30Q8×#88322C2211yescharacteristic
31#16810747C2211yescharacteristic
32#16810747C2211yescharacteristic
33C4×#11719925C2211yescharacteristic
34C4×#11719925C2211yescharacteristic
35C4×#11719923C2211yescharacteristic
36C4×#11719923C2211yescharacteristic
37#16810701—21no—
38#16810701—21no—
39#16810701C2211yescharacteristic
40#16810701C2211yescharacteristic
41#16810680—21no—
42#16810680C2211yescharacteristic
43#16810676C2211yescharacteristic
44#16810649—21no—
45#16810578—21no—
46C4×#11719917—21no—
47C4×#11719917C2211yescharacteristic
48#16810375C2211yescharacteristic
49#16810375C2211yescharacteristic
50#16810373C2211yescharacteristic
51#16810373C2211yescharacteristic
52#16810355C2211yescharacteristic
53#16810355C2211yescharacteristic
54#16810355C2211yescharacteristic
55#16810355C2211yescharacteristic
56#16810345C2211yescharacteristic
57#16810345C2211yescharacteristic
58#16810344C2211yescharacteristic
59C8×#88322C2211yescharacteristic
60C4×#41×C53C2211yescharacteristic
61C53×#431C2211yescharacteristic
62C2×C4⋊C8×C53C2211yescharacteristic
63C53×Q8⋊C8C2211yescharacteristic
64C53×Q8⋊C8C2211yescharacteristic
65C53×D4⋊C8C2211yescharacteristic
66C53×D4⋊C8C2211yescharacteristic
67C2×#11719925C2×C411yescharacteristic
68C2×#11719925C2×C411yescharacteristic
69C2×#11719923C2×C411yescharacteristic
70C2×#11719923C2×C411yescharacteristic
71C2×#11719917—21no—
72C2×#11719917—21no—
73C2×#11719917D411yescharacteristic
74C2×#11719917D411yescharacteristic
75#13133217C2×C411yescharacteristic
76#13133217C2×C411yescharacteristic
77#13133209—21no—
78#13133196—21no—
79#13133196C2×C411yescharacteristic
80#13133196C2×C411yescharacteristic
81C4×Dic106—21no—
82C4×Dic106C2311yescharacteristic
83C4×Dic106C2311yescharacteristic
84#13133176—21no—
85#13133150C2×C411yescharacteristic
86#13133150C2×C411yescharacteristic
87#13133148C2×C411yescharacteristic
88#13133148C2×C411yescharacteristic
89C4⋊#88322—21no—
90C4⋊#88322—21no—
91C4⋊#88322C2×C411yescharacteristic
92C4⋊#88322C2×C411yescharacteristic
93C4⋊#88322C2311yescharacteristic
94C4⋊#88322C2311yescharacteristic
95#13133146—21no—
96#13133146—21no—
97#13133146—21no—
98#13133146—21no—
99#13133141C2×C411yescharacteristic
100#13133141C2×C411yescharacteristic
101#13133140C2×C411yescharacteristic
102#13133140C2×C411yescharacteristic
103#13133140C2×C411yescharacteristic
104#13133140C2×C411yescharacteristic
105#13133138C2×C411yescharacteristic
106#13133138C2×C411yescharacteristic
107C22⋊#88322—21no—
108C22⋊#88322—21no—
109C22⋊#88322—21no—
110C22⋊#88322C2×C411yescharacteristic
111C22⋊#88322C2×C411yescharacteristic
112#13133128—21no—
113#13133128D411yescharacteristic
114#13133128D411yescharacteristic
115#13133127—21no—
116#13133127—21no—
117C4×#88322—21no—
118C4×#88322—21no—
119C4×#88322C2311yescharacteristic
120C4×#88322C2311yescharacteristic
121C4×#88322C2311yescharacteristic
122C4×#88322C2311yescharacteristic
123C8×Dic53—21no—
124C8×Dic53—21no—
125C2×#41×C53C2×C411yescharacteristic
126C22×C8×C53—21no—
127C4×Q8×C53C2×C411yescharacteristic
128C4×Q8×C53C2311yescharacteristic
129C4×D4×C53—21no—
130C4×D4×C53—21no—
131C4×D4×C53C2×C411yescharacteristic
132C4×D4×C53C2311yescharacteristic
133#117×C53—21no—
134#117×C53C2×C411yescharacteristic
135C2×C42×C53—21no—
136C2×C42×C53C2311yescharacteristic
137C4⋊C8×C53—21no—
138C4⋊C8×C53—21no—
139C4⋊C8×C53C2311yescharacteristic
140C4⋊C8×C53C2311yescharacteristic
141C22⋊C8×C53—21no—
142C4×C8×C53C2311yescharacteristic
143C4×C8×C53C2311yescharacteristic
144#41×C53—22no—
145#41×C53—41no—
146C2×Q8×C53C22×C411yescharacteristic
147C2×D4×C53—21no—
148C2×D4×C53C22×C411yescharacteristic
149C22×C4×C53—21no—
150C22×C4×C53—21no—
151C22×C4×C53—21no—
152C22×C4×C53C22×C411yescharacteristic
153C22×C4×C53C2×D411yescharacteristic
154C22×C4×C53#4111yescharacteristic
155C2×Dic106—21no—
156C2×Dic106C22×C411yescharacteristic
157C2×Dic106C22×C411yescharacteristic
158C2×C8×C53—21no—
159C2×C8×C53—21no—
160C2×C8×C53—21no—
161C2×C8×C53—21no—
162C2×C8×C53—42no—
163C4⋊C4×C53—21no—
164C4⋊C4×C53—21no—
165C4⋊C4×C53C22×C411yescharacteristic
166C4⋊C4×C53C22×C411yescharacteristic
167C22⋊C4×C53—21no—
168C22⋊C4×C53—21no—
169C22⋊C4×C53—21no—
170C42×C53—21no—
171C42×C53—21no—
172C42×C53—21no—
173C42×C53C22×C411yescharacteristic
174C42×C53C2×D411yescharacteristic
175C42×C53#4111yescharacteristic
176C42×C53C2411yescharacteristic
177C22⋊Dic53—21no—
178C22⋊Dic53—21no—
179#11719925—22no—
180#11719925—22no—
181#11719923—41no—
182#11719923—41no—
183C4⋊Dic53—21no—
184C4⋊Dic53—21no—
185C4⋊Dic53—21no—
186C4⋊Dic53C22×C411yescharacteristic
187C4⋊Dic53C22×C411yescharacteristic
188#11719919—21no—
189#11719919—21no—
190#11719919—21no—
191#11719919—21no—
192C4×Dic53—21no—
193C4×Dic53—21no—
194#11719917—22no—
195#11719917—41no—
196#11719917—42no—
197#11719917—42no—
198C2×#88322—21no—
199C2×#88322—21no—
200C2×#88322C22×C411yescharacteristic
201C2×#88322C22×C411yescharacteristic
202C2×#88322C22×C411yescharacteristic
203C2×#88322C22×C411yescharacteristic
204C2×#88322C2×D411yescharacteristic
205C2×#88322C2×D411yescharacteristic
206C2×#88322C2×D411yescharacteristic
207C2×#88322C2×D411yescharacteristic
208C23×C53—21no—
209C23×C53C4×D411yescharacteristic
210Q8×C53—21no—
211Q8×C53#13112yes—
212D4×C53—21no—
213D4×C53—21no—
214D4×C53—42no—
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—42no—
224C2×C4×C53—41no—
225C2×C4×C53—41no—
226C2×C4×C53—81no—
227C2×C4×C53C4×D411yescharacteristic
228C2×C4×C53C4×D411yescharacteristic
229C2×C4×C53C4×D411yescharacteristic
230C2×C4×C53C23×C411yescharacteristic
231C2×C4×C53C22×D411yescharacteristic
232C2×C4×C53C2×#4111yescharacteristic
233C2×Dic53—21no—
234C2×Dic53—21no—
235C2×Dic53—21no—
236C2×Dic53—21no—
237Dic106—21no—
238Dic106—21no—
239Dic106—21no—
240Dic106—21no—
241Dic106—81no—
242C8×C53—41no—
243C8×C53—41no—
244C8×C53—41no—
245C8×C53—41no—
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#88322#13712yes—
256#18013—531nomaximal, Sylow
257C22×C53—21no—
258C22×C53—21no—
259C22×C53—21no—
260C22×C53—41no—
261C22×C53—41no—
262C22×C53—41no—
263C22×C53C2×C4×D411yescharacteristic
264C4×C53—21no—
265C4×C53—22no—
266C4×C53—21no—
267C4×C53—21no—
268C4×C53—21no—
269C4×C53—41no—
270C4×C53—41no—
271C4×C53C2×#13111yescharacteristic
272C4×C53#56811yescharacteristic
273C4×C53C2×#13711yescharacteristic
274C4×C53#57511yescharacteristic
275Dic53—41no—
276Dic53—41no—
277Dic53—41no—
278Dic53—41no—
279#3107—531no—
280#2971—531no—
281#2962—531no—
282#1592—531no—
283#1590—531no—
284#1588—531no—
285#1587—531no—
286#1584—531no—
287#1583—531no—
288#1579—531no—
289C8×Q16—531no—
290#1573—531no—
291#1497—531no—
292#1477D5311yescharacteristic
293#1453—531no—
294C2×C53—41no—
295C2×C53—81no—
296C2×C53#293311yescharacteristic
297C2×C53C4×#13711yescharacteristic
298C2×C53#295711yescharacteristic
299#533—531no—
300C4×#41C2×D5311yescharacteristic
301#485—531no—
302#485—531no—
303#475—531no—
304#474—531no—
305#462—531no—
306#461—531no—
307#452—531no—
308#452—531no—
309#446—531no—
310#446—531no—
311#446—1061no—
312#446—1061no—
313C8×Q8—531no—
314C8×Q8—531no—
315C8×Q8—531no—
316C8×Q8—531no—
317C4×Q16—531no—
318C4×Q16—531no—
319C4×SD16—531no—
320C4×SD16—531no—
321#436—531no—
322#436—531no—
323#436—1061no—
324#436—1061no—
325#432—531no—
326#432—1061no—
327#431—531no—
328#431C2×D5311yescharacteristic
329#424—1061no—
330C2×C4⋊C8C2×D5311yescharacteristic
331#405—1061no—
332C4×M16—531no—
333C4×M16—1061no—
334#335—531no—
335#335—531no—
336#334—531no—
337#334—531no—
338Q8⋊C8—531no—
339Q8⋊C8—531no—
340Q8⋊C8—531no—
341Q8⋊C8—531no—
342Q8⋊C8C2×D5311yescharacteristic
343Q8⋊C8C2×D5311yescharacteristic
344D4⋊C8C2×D5311yescharacteristic
345D4⋊C8C2×D5311yescharacteristic
346#322—531no—
347#322—531no—
348#322—531no—
349C82—531no—
350C53#1801311yescharacteristic, complemented, Sylow
351C2×#41C4×D5311yescharacteristic
352C2×Q16—531no—
353C2×Q16—531no—
354C2×SD16—531no—
355C2×SD16—531no—
356C2×M16—531no—
357C2×M16—531no—
358C2×M16—1061no—
359C2×M16—1061no—
360C22×C8—21no—
361C4⋊Q8—531no—
362C4⋊Q8—531no—
363#125—1061no—
364C22⋊Q8—531no—
365C22⋊Q8—531no—
366C22⋊Q8—1061no—
367C4×Q8—531no—
368C4×Q8—531no—
369C4×Q8—1061no—
370C4×Q8C4×D5311yescharacteristic
371C4×Q8C22×D5311yescharacteristic
372C4×D4—21no—
373C4×D4—21no—
374C4×D4C4×D5311yescharacteristic
375C4×D4C22×D5311yescharacteristic
376#117—21no—
377#117—1061no—
378#117C4×D5311yescharacteristic
379C2×C42—21no—
380C2×C42C22×D5311yescharacteristic
381#107—531no—
382#107—531no—
383#106—531no—
384#106—531no—
385C4⋊C8—21no—
386C4⋊C8—21no—
387C4⋊C8—531no—
388C4⋊C8—531no—
389C4⋊C8—531no—
390C4⋊C8—531no—
391C4⋊C8—1061no—
392C4⋊C8—1061no—
393C4⋊C8—1061no—
394C4⋊C8—1061no—
395C4⋊C8—1061no—
396C4⋊C8—1061no—
397C4⋊C8C22×D5311yescharacteristic
398C4⋊C8C22×D5311yescharacteristic
399#103—531no—
400#103—531no—
401#103—531no—
402#103—531no—
403#103—531no—
404#103—531no—
405#102—531no—
406#102—531no—
407C22⋊C8—21no—
408C22⋊C8—531no—
409C22⋊C8—531no—
410C22⋊C8—1061no—
411C22⋊C8—1061no—
412C22⋊C8—1061no—
413#97—531no—
414#97—531no—
415#97—1061no—
416#97—1061no—
417#97—1061no—
418C4×C8—531no—
419C4×C8—531no—
420C4×C8—531no—
421C4×C8—531no—
422C4×C8—1061no—
423C4×C8—1061no—
424C4×C8—1061no—
425C4×C8—1061no—
426C4×C8C22×D5311yescharacteristic
427C4×C8C22×D5311yescharacteristic
428#41—22no—
429#41—41no—
430C2×Q8—531no—
431C2×Q8—531no—
432C2×Q8—1061no—
433C2×Q8C2×C4×D5311yescharacteristic
434C2×D4—21no—
435C2×D4C2×C4×D5311yescharacteristic
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4C2×C4×D5311yescharacteristic
440C22×C4D4×D5311yescharacteristic
441C22×C4#1171994711yescharacteristic
442Q16—1062no—
443Q16—1062no—
444SD16—2121no—
445SD16—2121no—
446M16—1062no—
447M16—2122no—
448M16—2121no—
449M16—2122no—
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—21no—
453C2×C8—21no—
454C2×C8—42no—
455C2×C8—531no—
456C2×C8—531no—
457C2×C8—531no—
458C2×C8—531no—
459C2×C8—531no—
460C2×C8—531no—
461C2×C8—531no—
462C2×C8—531no—
463C2×C8—1061no—
464C2×C8—1061no—
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⋊C4—1061no—
475C4⋊C4—1061no—
476C4⋊C4C2×C4×D5311yescharacteristic
477C4⋊C4C2×C4×D5311yescharacteristic
478C22⋊C4—21no—
479C22⋊C4—21no—
480C22⋊C4—21no—
481C22⋊C4—1061no—
482C22⋊C4—1061no—
483C42—21no—
484C42—21no—
485C42—21no—
486C42—1061no—
487C42—1061no—
488C42C2×C4×D5311yescharacteristic
489C42D4×D5311yescharacteristic
490C42#1171994711yescharacteristic
491C42C23×D5311yescharacteristic
492C23—21no—
493C23#1313317711yescharacteristic
494Q8—21no—
495Q8—1061no—
496Q8—1061no—
497Q8—1061no—
498Q8—1061no—
499Q8—4241no—
500Q8#1313322312yes—
501D4—21no—
502D4—21no—
503D4—42no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—41no—
512C2×C4—42no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—81no—
516C2×C4—1061no—
517C2×C4—1061no—
518C2×C4—1061no—
519C2×C4—1061no—
520C2×C4#1313317711yescharacteristic
521C2×C4#1313317711yescharacteristic
522C2×C4#1313317711yescharacteristic
523C2×C4C22×C4×D5311yescharacteristic
524C2×C4C2×D4×D5311yescharacteristic
525C2×C4C2×#1171994711yescharacteristic
526C8—41no—
527C8—41no—
528C8—41no—
529C8—41no—
530C8—532no—
531C8—1061no—
532C8—1061no—
533C8—1061no—
534C8—1061no—
535C8—1061no—
536C8—1061no—
537C8—1061no—
538C8—2121no—
539C8—2121no—
540C22—21no—
541C22—21no—
542C22—21no—
543C22—41no—
544C22—41no—
545C22—41no—
546C22C2×#1313317711yescharacteristic
547C4—21no—
548C4—22no—
549C4—21no—
550C4—21no—
551C4—21no—
552C4—41no—
553C4—41no—
554C4—2121no—
555C4—2121no—
556C4—2121no—
557C4—2121no—
558C4C2×#1313322311yescharacteristic
559C4#1681138911yescharacteristic
560C4#137×D5311yescharacteristic
561C4#1681141911yescharacteristic
562C2—41no—
563C2—81no—
564C2#2028343611yescharacteristic
565C2#2028353911yescharacteristic
566C2#2028362811yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC42C23×D53
FittingC53×#1477C2
socleC22×C53C2×C4×D4
prime radicalC2×#41×C53C2×C4
cosocleC42×C53C24
hypercenter#1477D53
hypocenterC53#18013
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1477

Complements

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

NH ≅ G/Ntype
3. #20283658C2semidirect
4. #20273614C2semidirect
5. #20273608C2semidirect
6. #20273596C2semidirect
7. #20273583C2semidirect
8. #20273562C2semidirect
9. #20273561C2semidirect
10. #20273531C2semidirect
11. #20273519C2semidirect
12. #20273498C2semidirect
13. #20273189C2semidirect
14. #20273010C2semidirect
350. C53#18013semidirect

Maximal subgroups

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

subgroupindex
#202847862
#202837352
#202836582
#202736142
#202736082
#202735962
#202735832
#202735622
#202735612
#202735312
#202735192
#202734982
#202731892
#202730102
C53×#14772
#1801353