Finite Group Databasebeta

← #27354197

Subgroups — #27354197

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#20284786C211yesmaximal, characteristic
2#20283809C211yesmaximal, characteristic, complemented
3#20273586C211yesmaximal, characteristic, complemented
4#20273572C211yesmaximal, characteristic, complemented
5#20273565C211yesmaximal, characteristic, complemented
6#20273551C211yesmaximal, characteristic, complemented
7#20273532C211yesmaximal, characteristic, complemented
8#20273527C211yesmaximal, characteristic, complemented
9C8×#11719925C211yesmaximal, characteristic, complemented
10C8×#11719923C211yesmaximal, characteristic, complemented
11#20273498C211yesmaximal, characteristic, complemented
12#20273149C211yesmaximal, characteristic
13#20273117C211yesmaximal, characteristic, complemented
14#20273015C211yesmaximal, characteristic, complemented
15C53×#2988C211yesmaximal, characteristic
16#16811180C2211yescharacteristic
17#16810964C411yescharacteristic
18#16810964C411yescharacteristic
19#16810908C411yescharacteristic
20#16810903C411yescharacteristic
21#16810833C411yescharacteristic
22#16810828C411yescharacteristic
23#16810785C411yescharacteristic
24#16810785C411yescharacteristic
25#16810753C2211yescharacteristic
26#16810753C2211yescharacteristic
27#16810752—21no—
28#16810752—21no—
29C8×Dic106C2211yescharacteristic
30C8×Dic106C2211yescharacteristic
31C4×#11719925C2211yescharacteristic
32C4×#11719925C2211yescharacteristic
33C4×#11719923C2211yescharacteristic
34C4×#11719923C2211yescharacteristic
35#16810681—21no—
36#16810681—21no—
37#16810680C2211yescharacteristic
38#16810676C2211yescharacteristic
39#16810676C2211yescharacteristic
40#16810375C2211yescharacteristic
41#16810375C2211yescharacteristic
42#16810373C2211yescharacteristic
43#16810373C2211yescharacteristic
44Q8⋊#88322C2211yescharacteristic
45Q8⋊#88322C2211yescharacteristic
46#16810353C2211yescharacteristic
47#16810353C2211yescharacteristic
48#16810353C2211yescharacteristic
49#16810353C2211yescharacteristic
50D4⋊#88322C2211yescharacteristic
51D4⋊#88322C2211yescharacteristic
52#16810345C2211yescharacteristic
53#16810345C2211yescharacteristic
54#16810344C2211yescharacteristic
55C8×#88322C2211yescharacteristic
56C4×#41×C53C2211yescharacteristic
57C8×Q8×C53C2211yescharacteristic
58C8×Q8×C53C2211yescharacteristic
59C8×D4×C53—21no—
60C8×D4×C53—21no—
61C8×D4×C53C2211yescharacteristic
62C8×D4×C53C2211yescharacteristic
63C53×#431—21no—
64C53×#431C2211yescharacteristic
65C2×C4⋊C8×C53—21no—
66C2×C4⋊C8×C53C2211yescharacteristic
67C2×#11719925C2×C411yescharacteristic
68C2×#11719925C2×C411yescharacteristic
69C2×#11719923C2×C411yescharacteristic
70C2×#11719923C2×C411yescharacteristic
71#13133217C2×C411yescharacteristic
72#13133217C2×C411yescharacteristic
73#13133209—21no—
74#13133196—21no—
75#13133196C2×C411yescharacteristic
76#13133196C2×C411yescharacteristic
77C4×Dic106—21no—
78C4×Dic106C2311yescharacteristic
79C4×Dic106C2311yescharacteristic
80#13133176—21no—
81#13133150C2×C411yescharacteristic
82#13133150C2×C411yescharacteristic
83#13133148C2×C411yescharacteristic
84#13133148C2×C411yescharacteristic
85C4⋊#88322—21no—
86C4⋊#88322C2×C411yescharacteristic
87C4⋊#88322C2×C411yescharacteristic
88C4⋊#88322C2311yescharacteristic
89C4⋊#88322C2311yescharacteristic
90#13133146—21no—
91#13133146—21no—
92#13133146—21no—
93#13133146—21no—
94#13133141C2×C411yescharacteristic
95#13133141C2×C411yescharacteristic
96#13133140C2×C411yescharacteristic
97#13133140C2×C411yescharacteristic
98#13133140C2×C411yescharacteristic
99#13133140C2×C411yescharacteristic
100#13133138C2×C411yescharacteristic
101#13133138C2×C411yescharacteristic
102C22⋊#88322—21no—
103C22⋊#88322—21no—
104C22⋊#88322C2×C411yescharacteristic
105C22⋊#88322C2×C411yescharacteristic
106#13133128—21no—
107#13133127—21no—
108#13133127—21no—
109C4×#88322C2311yescharacteristic
110C4×#88322C2311yescharacteristic
111C4×#88322C2311yescharacteristic
112C4×#88322C2311yescharacteristic
113C8×Dic53—21no—
114C8×Dic53—21no—
115C2×#41×C53C2×C411yescharacteristic
116C22×C8×C53—21no—
117C22×C8×C53—21no—
118C22×C8×C53—21no—
119C4×Q8×C53C2×C411yescharacteristic
120C4×Q8×C53C2311yescharacteristic
121C4×D4×C53—21no—
122C4×D4×C53—21no—
123C4×D4×C53C2×C411yescharacteristic
124C4×D4×C53C2311yescharacteristic
125#117×C53—21no—
126#117×C53C2×C411yescharacteristic
127C2×C42×C53—21no—
128C2×C42×C53C2311yescharacteristic
129C4⋊C8×C53—21no—
130C4⋊C8×C53—21no—
131C4⋊C8×C53—21no—
132C4⋊C8×C53D411yescharacteristic
133C4⋊C8×C53D411yescharacteristic
134C4⋊C8×C53C2311yescharacteristic
135C4⋊C8×C53C2311yescharacteristic
136C22⋊C8×C53—21no—
137C22⋊C8×C53—21no—
138C22⋊C8×C53D411yescharacteristic
139C22⋊C8×C53D411yescharacteristic
140C4×C8×C53—21no—
141C4×C8×C53—21no—
142C4×C8×C53C2311yescharacteristic
143C4×C8×C53C2311yescharacteristic
144#41×C53—22no—
145#41×C53—22no—
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—81no—
161C2×C8×C53—81no—
162C2×C8×C53—81no—
163C2×C8×C53C2×D411yescharacteristic
164C2×C8×C53C2×D411yescharacteristic
165C2×C8×C53C2×D411yescharacteristic
166C2×C8×C53C2×D411yescharacteristic
167C4⋊C4×C53—21no—
168C4⋊C4×C53—21no—
169C4⋊C4×C53C22×C411yescharacteristic
170C4⋊C4×C53C22×C411yescharacteristic
171C22⋊C4×C53—21no—
172C22⋊C4×C53—21no—
173C22⋊C4×C53—21no—
174C42×C53—21no—
175C42×C53—21no—
176C42×C53—21no—
177C42×C53C22×C411yescharacteristic
178C42×C53C2×D411yescharacteristic
179C42×C53#4111yescharacteristic
180C42×C53C2411yescharacteristic
181C22⋊Dic53—21no—
182C22⋊Dic53—21no—
183#11719925—22no—
184#11719925—22no—
185#11719923—22no—
186#11719923—22no—
187C4⋊Dic53—21no—
188C4⋊Dic53—21no—
189C4⋊Dic53—21no—
190C4⋊Dic53C22×C411yescharacteristic
191C4⋊Dic53C22×C411yescharacteristic
192#11719919—21no—
193#11719919—21no—
194#11719919—21no—
195#11719919—21no—
196C4×Dic53—21no—
197C4×Dic53—21no—
198C2×#88322—21no—
199C2×#88322—21no—
200C2×#88322—21no—
201C2×#88322—21no—
202C2×#88322C22×C411yescharacteristic
203C2×#88322C22×C411yescharacteristic
204C2×#88322C22×C411yescharacteristic
205C2×#88322C22×C411yescharacteristic
206C23×C53—21no—
207C23×C53C4×D411yescharacteristic
208Q8×C53—21no—
209Q8×C53#13112yes—
210D4×C53—24no—
211D4×C53—21no—
212D4×C53#13112yes—
213C2×C4×C53—24no—
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×C53—41no—
223C2×C4×C53—41no—
224C2×C4×C53—81no—
225C2×C4×C53C4×D411yescharacteristic
226C2×C4×C53C4×D411yescharacteristic
227C2×C4×C53C4×D411yescharacteristic
228C2×C4×C53C23×C411yescharacteristic
229C2×C4×C53C22×D411yescharacteristic
230C2×C4×C53C2×#4111yescharacteristic
231C2×Dic53—21no—
232C2×Dic53—21no—
233C2×Dic53—21no—
234C2×Dic53—21no—
235Dic106—21no—
236Dic106—21no—
237Dic106—21no—
238Dic106—21no—
239Dic106—81no—
240C8×C53—21no—
241C8×C53—21no—
242C8×C53—21no—
243C8×C53—21no—
244C8×C53—41no—
245C8×C53—41no—
246#88322—21no—
247#88322—21no—
248#88322—21no—
249#88322—21no—
250#88322—41no—
251#88322—41no—
252#88322—41no—
253#88322—41no—
254#17921—531nomaximal, Sylow
255C22×C53—21no—
256C22×C53—22no—
257C22×C53—21no—
258C22×C53—21no—
259C22×C53—41no—
260C22×C53—41no—
261C22×C53C2×C4×D411yescharacteristic
262C4×C53—21no—
263C4×C53—22no—
264C4×C53—21no—
265C4×C53—21no—
266C4×C53—21no—
267C4×C53—41no—
268C4×C53—41no—
269C4×C53C2×#13111yescharacteristic
270C4×C53C2×#13111yescharacteristic
271C4×C53#57711yescharacteristic
272C4×C53#57811yescharacteristic
273Dic53—41no—
274Dic53—41no—
275Dic53—41no—
276Dic53—41no—
277#3107—531no—
278#2988D5311yescharacteristic
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#1481—531no—
291#1454—531no—
292C2×C53—41no—
293C2×C53—42no—
294C2×C53#292911yescharacteristic
295C2×C53#294711yescharacteristic
296C2×C53#295711yescharacteristic
297#533—531no—
298C4×#41C2×D5311yescharacteristic
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×Q8C2×D5311yescharacteristic
314C8×Q8C2×D5311yescharacteristic
315C4×Q16—531no—
316C4×Q16—531no—
317C4×SD16—531no—
318C4×SD16—531no—
319C8×D4—21no—
320C8×D4—21no—
321C8×D4C2×D5311yescharacteristic
322C8×D4C2×D5311yescharacteristic
323#433—1061no—
324#433—1061no—
325#432—531no—
326#431—21no—
327#431—531no—
328#431—531no—
329#431C2×D5311yescharacteristic
330C2×C4⋊C8—21no—
331C2×C4⋊C8C2×D5311yescharacteristic
332#335—531no—
333#335—531no—
334#334—531no—
335#334—531no—
336Q8⋊C8—531no—
337Q8⋊C8—531no—
338Q8⋊C8—531no—
339Q8⋊C8—531no—
340Q8⋊C8—531no—
341Q8⋊C8—531no—
342D4⋊C8—531no—
343D4⋊C8—531no—
344#322—531no—
345#322—531no—
346#322—531no—
347C82—531no—
348C53#1792111yescharacteristic, complemented, Sylow
349C2×#41C4×D5311yescharacteristic
350C2×Q16—531no—
351C2×Q16—531no—
352C2×SD16—531no—
353C2×SD16—531no—
354C22×C8—21no—
355C22×C8—21no—
356C22×C8—21no—
357C4⋊Q8—531no—
358C4⋊Q8—531no—
359#125—1061no—
360C22⋊Q8—531no—
361C22⋊Q8—531no—
362C22⋊Q8—1061no—
363C4×Q8—531no—
364C4×Q8—531no—
365C4×Q8—1061no—
366C4×Q8C4×D5311yescharacteristic
367C4×Q8C22×D5311yescharacteristic
368C4×D4—21no—
369C4×D4—21no—
370C4×D4C4×D5311yescharacteristic
371C4×D4C22×D5311yescharacteristic
372#117—21no—
373#117—1061no—
374#117C4×D5311yescharacteristic
375C2×C42—21no—
376C2×C42C22×D5311yescharacteristic
377#107—531no—
378#107—531no—
379#106—531no—
380#106—531no—
381C4⋊C8—21no—
382C4⋊C8—21no—
383C4⋊C8—21no—
384C4⋊C8—531no—
385C4⋊C8—531no—
386C4⋊C8—531no—
387C4⋊C8—531no—
388C4⋊C8—1061no—
389C4⋊C8—1061no—
390C4⋊C8—1061no—
391C4⋊C8—1061no—
392C4⋊C8—1061no—
393C4⋊C8C22⋊D5311yescharacteristic
394C4⋊C8C22⋊D5311yescharacteristic
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—21no—
407C22⋊C8—531no—
408C22⋊C8—531no—
409C22⋊C8—1061no—
410C22⋊C8—1061no—
411C22⋊C8C22⋊D5311yescharacteristic
412C22⋊C8C22⋊D5311yescharacteristic
413#97—1061no—
414#97—1061no—
415#97—1061no—
416C4×C8—21no—
417C4×C8—21no—
418C4×C8—531no—
419C4×C8—531no—
420C4×C8—531no—
421C4×C8—531no—
422C4×C8—1061no—
423C4×C8—1061no—
424C4×C8C22×D5311yescharacteristic
425C4×C8C22×D5311yescharacteristic
426#41—22no—
427#41—22no—
428C2×Q8—531no—
429C2×Q8—531no—
430C2×Q8—1061no—
431C2×Q8C2×C4×D5311yescharacteristic
432C2×D4—21no—
433C2×D4C2×C4×D5311yescharacteristic
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4C2×C4×D5311yescharacteristic
438C22×C4#1171994511yescharacteristic
439C22×C4C2×C22⋊D5311yescharacteristic
440Q16—1062no—
441Q16—1062no—
442SD16—1062no—
443SD16—1062no—
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—81no—
447C2×C8—81no—
448C2×C8—81no—
449C2×C8—531no—
450C2×C8—531no—
451C2×C8—531no—
452C2×C8—531no—
453C2×C8—1061no—
454C2×C8—1061no—
455C2×C8—1061no—
456C2×C8—1061no—
457C2×C8C2×C22⋊D5311yescharacteristic
458C2×C8C2×C22⋊D5311yescharacteristic
459C2×C8C2×C22⋊D5311yescharacteristic
460C2×C8C2×C22⋊D5311yescharacteristic
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—531no—
464C4⋊C4—531no—
465C4⋊C4—1061no—
466C4⋊C4—1061no—
467C4⋊C4—1061no—
468C4⋊C4—1061no—
469C4⋊C4—1061no—
470C4⋊C4—1061no—
471C4⋊C4—1061no—
472C4⋊C4C2×C4×D5311yescharacteristic
473C4⋊C4C2×C4×D5311yescharacteristic
474C22⋊C4—21no—
475C22⋊C4—21no—
476C22⋊C4—21no—
477C22⋊C4—1061no—
478C22⋊C4—1061no—
479C42—21no—
480C42—21no—
481C42—21no—
482C42—1061no—
483C42—1061no—
484C42C2×C4×D5311yescharacteristic
485C42#1171994511yescharacteristic
486C42C2×C22⋊D5311yescharacteristic
487C42C23×D5311yescharacteristic
488C23—21no—
489C23C4×C22⋊D5311yescharacteristic
490Q8—21no—
491Q8—1061no—
492Q8—1061no—
493Q8—1061no—
494Q8—1061no—
495Q8—4241no—
496Q8#1313322312yes—
497D4—24no—
498D4—21no—
499D4#1313322312yes—
500C2×C4—24no—
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—41no—
510C2×C4—41no—
511C2×C4—81no—
512C2×C4—1061no—
513C2×C4—1061no—
514C2×C4—1061no—
515C2×C4—1061no—
516C2×C4C4×C22⋊D5311yescharacteristic
517C2×C4C4×C22⋊D5311yescharacteristic
518C2×C4C4×C22⋊D5311yescharacteristic
519C2×C4C22×C4×D5311yescharacteristic
520C2×C4C22×C22⋊D5311yescharacteristic
521C2×C4C2×#1171994511yescharacteristic
522C8—21no—
523C8—21no—
524C8—21no—
525C8—21no—
526C8—41no—
527C8—41no—
528C8—1061no—
529C8—1061no—
530C8—1061no—
531C8—1061no—
532C8—2121no—
533C8—2121no—
534C8—2121no—
535C8—2121no—
536C22—21no—
537C22—22no—
538C22—21no—
539C22—21no—
540C22—41no—
541C22—41no—
542C22C2×C4×C22⋊D5311yescharacteristic
543C4—21no—
544C4—22no—
545C4—21no—
546C4—21no—
547C4—21no—
548C4—41no—
549C4—41no—
550C4—2121no—
551C4—2121no—
552C4—2121no—
553C4—2121no—
554C4C2×#1313322311yescharacteristic
555C4C2×#1313322311yescharacteristic
556C4#1681143411yescharacteristic
557C4#1681143811yescharacteristic
558C2—41no—
559C2—42no—
560C2#2028340911yescharacteristic
561C2#2028355411yescharacteristic
562C2#2028363011yescharacteristic
subgroupG/H
centerC2×C4C22×C22⋊D53
commutatorC2×C4×C53C23×C4
FrattiniC42C23×D53
FittingC53×#2988C2
socleC22×C53C2×C4×D4
prime radicalC2×#41×C53C2×C4
cosocleC42×C53C24
hypercenter#2988D53
hypocenterC53#17921
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ #431 ◁ #2988

Complements

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

NH ≅ G/Ntype
2. #20283809C2semidirect
3. #20273586C2semidirect
4. #20273572C2semidirect
5. #20273565C2semidirect
6. #20273551C2semidirect
7. #20273532C2semidirect
8. #20273527C2semidirect
9. C8×#11719925C2semidirect
10. C8×#11719923C2semidirect
11. #20273498C2semidirect
13. #20273117C2semidirect
14. #20273015C2semidirect
348. C53#17921semidirect

Maximal subgroups

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

subgroupindex
#202847862
#202838092
#202735862
#202735722
#202735652
#202735512
#202735322
#202735272
C8×#117199252
C8×#117199232
#202734982
#202731492
#202731172
#202730152
C53×#29882
#1792153