Finite Group Databasebeta

← #27354203

Subgroups — #27354203

Order 13568.

598 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20284780C211yesmaximal, characteristic
2#20283824C211yesmaximal, characteristic
3#20283810C211yesmaximal, characteristic, complemented
4#20273589C211yesmaximal, characteristic, complemented
5#20273575C211yesmaximal, characteristic, complemented
6#20273568C211yesmaximal, characteristic, complemented
7#20273554C211yesmaximal, characteristic, complemented
8#20273531C211yesmaximal, characteristic, complemented
9#20273530C211yesmaximal, characteristic, complemented
10#20273519C211yesmaximal, characteristic, complemented
11#20273512C211yesmaximal, characteristic, complemented
12#20273498C211yesmaximal, characteristic, complemented
13#20273150C211yesmaximal, characteristic
14#20273015C211yesmaximal, characteristic, complemented
15C53×#1481C211yesmaximal, characteristic, complemented
16C4×#11719945C2211yescharacteristic
17#16810961C411yescharacteristic
18#16810961C411yescharacteristic
19#16810905C411yescharacteristic
20#16810900C411yescharacteristic
21#16810830C411yescharacteristic
22#16810825C411yescharacteristic
23#16810783C411yescharacteristic
24#16810783C411yescharacteristic
25#16810754—21no—
26#16810754—21no—
27#16810754C2211yescharacteristic
28#16810754C2211yescharacteristic
29Q8×#88322C2211yescharacteristic
30Q8×#88322C2211yescharacteristic
31#16810747C2211yescharacteristic
32#16810747C2211yescharacteristic
33C4×#11719925C2211yescharacteristic
34C4×#11719925C2211yescharacteristic
35C4×#11719924C2211yescharacteristic
36C4×#11719924C2211yescharacteristic
37#16810688C2211yescharacteristic
38#16810688C2211yescharacteristic
39C8×C22⋊D53—21no—
40C8×C22⋊D53—21no—
41#16810684—21no—
42#16810684—21no—
43#16810680C2211yescharacteristic
44#16810676C2211yescharacteristic
45#16810676C2211yescharacteristic
46#16810675—21no—
47C4⋊C8×D53—21no—
48#16810375C2211yescharacteristic
49#16810375C2211yescharacteristic
50#16810373C2211yescharacteristic
51#16810373C2211yescharacteristic
52#16810355C2211yescharacteristic
53#16810355C2211yescharacteristic
54#16810352C2211yescharacteristic
55#16810352C2211yescharacteristic
56#16810345C2211yescharacteristic
57#16810345C2211yescharacteristic
58#16810344C2211yescharacteristic
59C8×#88322C2211yescharacteristic
60C53×#533C2211yescharacteristic
61C53×#431C2211yescharacteristic
62C2×C4⋊C8×C53C2211yescharacteristic
63C53×Q8⋊C8C2211yescharacteristic
64C53×Q8⋊C8C2211yescharacteristic
65C53×Q8⋊C8C2211yescharacteristic
66C53×Q8⋊C8C2211yescharacteristic
67C2×#11719945C2×C411yescharacteristic
68C2×#11719925C2×C411yescharacteristic
69C2×#11719925C2×C411yescharacteristic
70C2×#11719924C2×C411yescharacteristic
71C2×#11719924C2×C411yescharacteristic
72C2×C8×D53—21no—
73C2×C8×D53—21no—
74C4×Dic106C2×C411yescharacteristic
75C4×Dic106C2311yescharacteristic
76C4×D212C2×C411yescharacteristic
77C4×D212C2311yescharacteristic
78C4×C22⋊D53—21no—
79C4×C22⋊D53—21no—
80#13133176C2×C411yescharacteristic
81#13133174—21no—
82C42×D53—21no—
83#13133150C2×C411yescharacteristic
84#13133150C2×C411yescharacteristic
85#13133148C2×C411yescharacteristic
86#13133148C2×C411yescharacteristic
87C4⋊#88322—21no—
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#13133141C2×C411yescharacteristic
100#13133141C2×C411yescharacteristic
101#13133141C2×C411yescharacteristic
102#13133141C2×C411yescharacteristic
103#13133140C2×C411yescharacteristic
104#13133140C2×C411yescharacteristic
105#13133137C2×C411yescharacteristic
106#13133137C2×C411yescharacteristic
107C22⋊#88322—21no—
108C22⋊#88322C2×C411yescharacteristic
109C22⋊#88322C2×C411yescharacteristic
110C22⋊#88322D411yescharacteristic
111C22⋊#88322D411yescharacteristic
112#13133129—21no—
113#13133129—21no—
114#13133128—21no—
115#13133128—21no—
116#13133128—21no—
117C4×#88322—21no—
118C4×#88322—21no—
119C4×#88322C2311yescharacteristic
120C4×#88322C2311yescharacteristic
121C4×#88322C2311yescharacteristic
122C4×#88322C2311yescharacteristic
123C8×Dic53—21no—
124C8×Dic53—21no—
125C22×C8×C53—21no—
126C4⋊Q8×C53C2×C411yescharacteristic
127C4⋊Q8×C53C2×C411yescharacteristic
128#125×C53—21no—
129C22⋊Q8×C53—21no—
130C22⋊Q8×C53C2×C411yescharacteristic
131C22⋊Q8×C53C2×C411yescharacteristic
132C4×Q8×C53—21no—
133C4×Q8×C53C2311yescharacteristic
134C4×Q8×C53C2311yescharacteristic
135#117×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
144C2×Q8×C53—21no—
145C2×Q8×C53C22×C411yescharacteristic
146C2×Q8×C53C22×C411yescharacteristic
147C22×C4×C53C22×C411yescharacteristic
148C22×C4×C53C2×D411yescharacteristic
149C22×C4×C53#4111yescharacteristic
150C2×C22⋊D53—21no—
151#11719945—22no—
152#11719945—22no—
153C2×D212C22×C411yescharacteristic
154C2×C4×D53—21no—
155C2×C4×D53—21no—
156C2×C4×D53—21no—
157C2×Dic106C22×C411yescharacteristic
158C2×C8×C53—21no—
159C2×C8×C53—21no—
160C2×C8×C53—21no—
161C2×C8×C53—21no—
162C2×C8×C53—81no—
163C4⋊C4×C53—21no—
164C4⋊C4×C53—21no—
165C4⋊C4×C53—21no—
166C4⋊C4×C53—21no—
167C4⋊C4×C53—21no—
168C4⋊C4×C53—21no—
169C4⋊C4×C53—21no—
170C4⋊C4×C53C22×C411yescharacteristic
171C4⋊C4×C53C22×C411yescharacteristic
172C22⋊C4×C53—21no—
173C22⋊C4×C53—21no—
174C42×C53—21no—
175C42×C53—21no—
176C42×C53C22×C411yescharacteristic
177C42×C53C2×D411yescharacteristic
178C42×C53#4111yescharacteristic
179C42×C53C2411yescharacteristic
180C22⋊Dic53—21no—
181#11719925—22no—
182#11719925—22no—
183#11719924—22no—
184#11719924—22no—
185#11719921—21no—
186#11719921—21no—
187C4⋊Dic53C22×C411yescharacteristic
188C4⋊Dic53C22×C411yescharacteristic
189#11719919—21no—
190#11719919—21no—
191C4×Dic53—21no—
192C4×Dic53—21no—
193C4×Dic53—21no—
194C2×#88322—21no—
195C2×#88322—21no—
196C2×#88322C22×C411yescharacteristic
197C2×#88322C22×C411yescharacteristic
198C2×#88322C22×C411yescharacteristic
199C2×#88322C22×C411yescharacteristic
200C2×#88322C2×D411yescharacteristic
201C2×#88322C2×D411yescharacteristic
202C2×#88322C2×D411yescharacteristic
203C2×#88322C2×D411yescharacteristic
204C8×D53—81no—
205C8×D53—81no—
206C23×C53C4×D411yescharacteristic
207C22×D53—21no—
208Q8×C53—21no—
209Q8×C53—21no—
210Q8×C53—21no—
211Q8×C53—21no—
212Q8×C53—81no—
213C2×C4×C53—21no—
214C2×C4×C53—21no—
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—41no—
222C2×C4×C53C4×D411yescharacteristic
223C2×C4×C53C4×D411yescharacteristic
224C2×C4×C53C4×D411yescharacteristic
225C2×C4×C53C23×C411yescharacteristic
226C2×C4×C53C22×D411yescharacteristic
227C2×C4×C53C2×#4111yescharacteristic
228C22⋊D53—24no—
229C2×Dic53—21no—
230C2×Dic53—21no—
231C2×Dic53—21no—
232D212—21no—
233D212#13112yes—
234C4×D53—24no—
235C4×D53—41no—
236C4×D53—41no—
237C4×D53—81no—
238Dic106—21no—
239Dic106#13112yes—
240C8×C53—41no—
241C8×C53—41no—
242C8×C53—41no—
243C8×C53—41no—
244#88322—21no—
245#88322—21no—
246#88322—21no—
247#88322—21no—
248#88322—21no—
249#88322—21no—
250#88322—21no—
251#88322—21no—
252#88322—41no—
253#88322—41no—
254#17921—531nomaximal, Sylow
255C22×C53—21no—
256C22×C53—21no—
257C22×C53—21no—
258C22×C53C2×C4×D411yescharacteristic
259C2×D53—22no—
260C2×D53—41no—
261C2×D53—41no—
262C4×C53—21no—
263C4×C53—21no—
264C4×C53—21no—
265C4×C53—21no—
266C4×C53—41no—
267C4×C53—41no—
268C4×C53—41no—
269C4×C53—41no—
270C4×C53C2×#13111yescharacteristic
271C4×C53C2×#13111yescharacteristic
272C4×C53#57711yescharacteristic
273C4×C53#57811yescharacteristic
274Dic53—22no—
275Dic53—41no—
276Dic53—41no—
277#3107—531no—
278#2988—531no—
279#2985—531no—
280#1587—531no—
281#1585—531no—
282#1584—531no—
283#1582—531no—
284#1579—531no—
285SD16⋊C8—531no—
286C8×Q16—531no—
287C8×SD16—531no—
288#1573—531no—
289#1488—531no—
290#1481D5311yescharacteristic, complemented
291#1454—531no—
292C2×C53—41no—
293C2×C53#292911yescharacteristic
294C2×C53#294711yescharacteristic
295C2×C53#295711yescharacteristic
296D53—42no—
297#533C2×D5311yescharacteristic
298C4×#41—531no—
299#485—531no—
300#485—531no—
301#475—531no—
302#474—531no—
303#462—531no—
304#461—531no—
305#452—531no—
306#452—531no—
307#446—531no—
308#446—531no—
309#446—1061no—
310#446—1061no—
311C8×Q8—531no—
312C8×Q8—531no—
313C8×Q8—531no—
314C8×Q8—531no—
315C4×Q16—531no—
316C4×Q16—531no—
317C4×SD16—531no—
318C4×SD16—531no—
319C8×D4—531no—
320C8×D4—531no—
321C8×D4—1061no—
322C8×D4—1061no—
323#433—1061no—
324#433—1061no—
325#432—531no—
326#431—531no—
327#431—531no—
328#431—1061no—
329#431C2×D5311yescharacteristic
330C2×C4⋊C8—1061no—
331C2×C4⋊C8C2×D5311yescharacteristic
332#335—531no—
333#335—531no—
334#334—531no—
335#334—531no—
336Q8⋊C8—531no—
337Q8⋊C8—531no—
338Q8⋊C8C2×D5311yescharacteristic
339Q8⋊C8C2×D5311yescharacteristic
340Q8⋊C8C2×D5311yescharacteristic
341Q8⋊C8C2×D5311yescharacteristic
342D4⋊C8—531no—
343D4⋊C8—531no—
344#322—531no—
345#322—531no—
346#322—531no—
347C82—531no—
348C53#1792111yescharacteristic, complemented, Sylow
349C2×#41—531no—
350C2×Q16—531no—
351C2×Q16—531no—
352C2×SD16—531no—
353C2×SD16—531no—
354C22×C8—21no—
355C22×C8—1061no—
356C22×C8—1061no—
357C4⋊Q8C4×D5311yescharacteristic
358C4⋊Q8C4×D5311yescharacteristic
359#125—21no—
360C22⋊Q8—21no—
361C22⋊Q8C4×D5311yescharacteristic
362C22⋊Q8C4×D5311yescharacteristic
363C4×Q8—21no—
364C4×Q8—531no—
365C4×Q8—531no—
366C4×Q8C22×D5311yescharacteristic
367C4×Q8C22×D5311yescharacteristic
368C4×D4—531no—
369C4×D4—531no—
370C4×D4—1061no—
371C4×D4—1061no—
372#117—21no—
373#117—531no—
374#117—1061no—
375C2×C42—1061no—
376C2×C42C22×D5311yescharacteristic
377#107—531no—
378#107—531no—
379#106—531no—
380#106—531no—
381C4⋊C8—21no—
382C4⋊C8—21no—
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⋊C8C22×D5311yescharacteristic
396C4⋊C8C22×D5311yescharacteristic
397#103—531no—
398#103—531no—
399#103—531no—
400#103—531no—
401#103—531no—
402#103—531no—
403#102—531no—
404#102—531no—
405C22⋊C8—21no—
406C22⋊C8—531no—
407C22⋊C8—531no—
408C22⋊C8—531no—
409C22⋊C8—531no—
410C22⋊C8—1061no—
411C22⋊C8—1061no—
412C22⋊C8—1061no—
413#97—1061no—
414#97—1061no—
415#97—1061no—
416C4×C8—531no—
417C4×C8—531no—
418C4×C8—531no—
419C4×C8—531no—
420C4×C8—1061no—
421C4×C8—1061no—
422C4×C8—1061no—
423C4×C8—1061no—
424C4×C8C22×D5311yescharacteristic
425C4×C8C22×D5311yescharacteristic
426#41—1062no—
427#41—1062no—
428C2×Q8—21no—
429C2×Q8—531no—
430C2×Q8C2×C4×D5311yescharacteristic
431C2×Q8C2×C4×D5311yescharacteristic
432C2×D4—531no—
433C2×D4—1061no—
434C22×C4—1061no—
435C22×C4—1061no—
436C22×C4—1061no—
437C22×C4C2×C4×D5311yescharacteristic
438C22×C4D4×D5311yescharacteristic
439C22×C4#1171994711yescharacteristic
440Q16—1062no—
441Q16—1062no—
442SD16—1062no—
443SD16—1062no—
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—81no—
449C2×C8—531no—
450C2×C8—531no—
451C2×C8—531no—
452C2×C8—531no—
453C2×C8—531no—
454C2×C8—531no—
455C2×C8—531no—
456C2×C8—531no—
457C2×C8—1061no—
458C2×C8—1061no—
459C2×C8—4241no—
460C2×C8—4241no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—531no—
469C4⋊C4—531no—
470C4⋊C4—1061no—
471C4⋊C4—1061no—
472C4⋊C4C2×C4×D5311yescharacteristic
473C4⋊C4C2×C4×D5311yescharacteristic
474C22⋊C4—21no—
475C22⋊C4—21no—
476C22⋊C4—1061no—
477C22⋊C4—1061no—
478C22⋊C4—1061no—
479C42—21no—
480C42—21no—
481C42—1061no—
482C42—1061no—
483C42—1061no—
484C42C2×C4×D5311yescharacteristic
485C42D4×D5311yescharacteristic
486C42#1171994711yescharacteristic
487C42C23×D5311yescharacteristic
488C23—1061no—
489C23#1313317711yescharacteristic
490Q8—21no—
491Q8—21no—
492Q8—21no—
493Q8—21no—
494Q8—81no—
495Q8—532no—
496Q8—1061no—
497D4—532no—
498D4—1064no—
499D4—1061no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—41no—
509C2×C4—1064no—
510C2×C4—1061no—
511C2×C4—1061no—
512C2×C4—1061no—
513C2×C4—4241no—
514C2×C4—2121no—
515C2×C4—2121no—
516C2×C4#1313317711yescharacteristic
517C2×C4#1313317711yescharacteristic
518C2×C4#1313317711yescharacteristic
519C2×C4C22×C4×D5311yescharacteristic
520C2×C4C2×D4×D5311yescharacteristic
521C2×C4C2×#1171994711yescharacteristic
522C8—41no—
523C8—41no—
524C8—41no—
525C8—41no—
526C8—1061no—
527C8—1061no—
528C8—1061no—
529C8—1061no—
530C8—1061no—
531C8—1061no—
532C8—1061no—
533C8—1061no—
534C8—2121no—
535C8—2121no—
536C22—21no—
537C22—21no—
538C22—21no—
539C22—1062no—
540C22—2121no—
541C22—2121no—
542C22C2×#1313317711yescharacteristic
543C4—21no—
544C4—21no—
545C4—21no—
546C4—21no—
547C4—41no—
548C4—41no—
549C4—41no—
550C4—41no—
551C4—1062no—
552C4—2121no—
553C4—2121no—
554C4#131×D5311yescharacteristic
555C4#131×D5311yescharacteristic
556C4#1681142911yescharacteristic
557C4#1681143611yescharacteristic
558C2—41no—
559C2—2122no—
560C2#2028341211yescharacteristic
561C2#2028355211yescharacteristic
562C2#2028362811yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC42C23×D53
FittingC53×#1481C2
socleC22×C53C2×C4×D4
prime radicalC2×#11719945C2×C4
cosocleC42×C53C24
hypercenter#1481D53
hypocenterC53#17921
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1481

Complements

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

NH ≅ G/Ntype
3. #20283810C2semidirect
4. #20273589C2semidirect
5. #20273575C2semidirect
6. #20273568C2semidirect
7. #20273554C2semidirect
8. #20273531C2semidirect
9. #20273530C2semidirect
10. #20273519C2semidirect
11. #20273512C2semidirect
12. #20273498C2semidirect
14. #20273015C2semidirect
15. C53×#1481C2semidirect
290. #1481D53semidirect
348. C53#17921semidirect

Maximal subgroups

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

subgroupindex
#202847802
#202838242
#202838102
#202735892
#202735752
#202735682
#202735542
#202735312
#202735302
#202735192
#202735122
#202734982
#202731502
#202730152
C53×#14812
#1792153