Finite Group Databasebeta

← #27417936

Subgroups — #27417936

Order 13568.

1068 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810951C211yesmaximal, characteristic, complemented
2C2×#16810666C211yesmaximal, characteristic, complemented
3#20279032C211yesmaximal, characteristic
4#20276050C212yesmaximal, complemented
5#20275915C212yesmaximal, complemented
6#20275149C211yesmaximal, characteristic, complemented
7#20275013C211yesmaximal, characteristic
8#20274983C212yesmaximal, complemented
9#20274908C212yesmaximal, complemented
10#20274810C211yesmaximal, characteristic, complemented
11C2×C53×#552C211yesmaximal, characteristic
12C22×#11719917C2211yescharacteristic
13C22×C4⋊Dic53C2211yescharacteristic
14#16810951—24no—
15#16810666—24no—
16C2×#13133150C2212yes—
17C2×#13133148C2212yes—
18C2×#13133148C2212yes—
19C2×#13133141C2212yes—
20C2×#13133141C2212yes—
21C2×C22⋊#88322C2211yescharacteristic
22C2×C22⋊#88322C2212yes—
23#16810562—21no—
24#16810559—21no—
25#16810521C2212yes—
26#16810503—21no—
27#16810495—21no—
28#16810483C2211yescharacteristic
29#16810483C2212yes—
30#16810385C2212yes—
31#16810385C2212yes—
32#16810385C2212yes—
33#16810378C2212yes—
34C53×#552—24no—
35C53×#552—24no—
36C2×#126×C53—21no—
37C2×#125×C53—21no—
38C2×#125×C53C2211yescharacteristic
39C2×#123×C53—21no—
40C2×C22⋊Q8×C53C2211yescharacteristic
41C2×C22⋊Q8×C53C2211yescharacteristic
42C2×C4×Q8×C53C2211yescharacteristic
43C2×C4×D4×C53C2211yescharacteristic
44C2×#117×C53—21no—
45C2×#117×C53C2211yescharacteristic
46C22×C4⋊C4×C53C2211yescharacteristic
47C23×Dic53—21no—
48C2×#11719917—24no—
49C2×#11719917—22no—
50C22×#88322C2312yes—
51C22×#88322C2312yes—
52C2×C4⋊Dic53—21no—
53C2×C4⋊Dic53—22no—
54C2×C4⋊Dic53—22no—
55C2×C4⋊Dic53C2312yes—
56C2×C4⋊Dic53C2312yes—
57C2×#11719919—22no—
58C2×C22⋊Dic53—21no—
59C2×C22⋊Dic53—22no—
60#13133150—24no—
61#13133148—24no—
62#13133148—24no—
63#13133141—24no—
64#13133141—24no—
65C22⋊#88322—24no—
66C22⋊#88322—22no—
67#13133124—22no—
68#13133124—21no—
69#13133124—22no—
70C22×Q8×C53C2311yescharacteristic
71C22×D4×C53D411yescharacteristic
72C23×C4×C53—21no—
73C23×C4×C53C2311yescharacteristic
74C23×C4×C53C2311yescharacteristic
75#126×C53—28no—
76#125×C53—28no—
77#125×C53—22no—
78#125×C53—22no—
79#123×C53—28no—
80C22⋊Q8×C53—22no—
81C22⋊Q8×C53—22no—
82C22⋊Q8×C53—22no—
83C22⋊Q8×C53D414yes—
84C4×Q8×C53—22no—
85C4×Q8×C53—22no—
86C4×D4×C53—22no—
87C4×D4×C53—22no—
88#117×C53—28no—
89#117×C53—22no—
90#117×C53D414yes—
91C2×C4⋊C4×C53—22no—
92C2×C4⋊C4×C53—22no—
93C2×C4⋊C4×C53—21no—
94C2×C4⋊C4×C53—21no—
95C2×C4⋊C4×C53—21no—
96C2×C4⋊C4×C53—21no—
97C2×C4⋊C4×C53—21no—
98C2×C4⋊C4×C53—21no—
99C2×C4⋊C4×C53D411yescharacteristic
100C2×C4⋊C4×C53C2311yescharacteristic
101C2×C4⋊C4×C53C2311yescharacteristic
102C2×C4⋊C4×C53C2311yescharacteristic
103C2×C22⋊C4×C53—21no—
104C2×C22⋊C4×C53—21no—
105C2×C22⋊C4×C53—21no—
106C2×C22⋊C4×C53D411yescharacteristic
107C2×C22⋊C4×C53D411yescharacteristic
108C2×C42×C53—21no—
109C2×C42×C53—21no—
110C2×C42×C53C2311yescharacteristic
111C24×C53C2×D411yescharacteristic
112C24×C53C2×D411yescharacteristic
113C2×Q8×C53—21no—
114C2×Q8×C53—22no—
115C2×Q8×C53—22no—
116C2×Q8×C53C2×D412yes—
117C2×D4×C53—22no—
118C2×D4×C53—22no—
119C2×D4×C53—41no—
120C22×C4×C53—22no—
121C22×C4×C53—22no—
122C22×C4×C53—24no—
123C22×C4×C53—21no—
124C22×C4×C53—21no—
125C22×C4×C53—21no—
126C22×C4×C53—21no—
127C22×C4×C53—21no—
128C22×C4×C53—21no—
129C22×C4×C53—22no—
130C22×C4×C53—22no—
131C22×C4×C53—42no—
132C22×C4×C53—42no—
133C22×C4×C53C2×D411yescharacteristic
134C22×C4×C53C2×D411yescharacteristic
135C22×C4×C53C2×D411yescharacteristic
136C22×C4×C53C2×D411yescharacteristic
137C22×C4×C53C2×D412yes—
138C22×C4×C53C2×D412yes—
139C22×C4×C53C2411yescharacteristic
140C22×Dic53—22no—
141C22×Dic53—22no—
142C22×Dic53—42no—
143C22×Dic53—82no—
144C4⋊C4×C53—24no—
145C4⋊C4×C53—24no—
146C4⋊C4×C53—24no—
147C4⋊C4×C53—24no—
148C4⋊C4×C53—24no—
149C4⋊C4×C53—24no—
150C4⋊C4×C53—22no—
151C4⋊C4×C53—21no—
152C4⋊C4×C53—21no—
153C4⋊C4×C53—21no—
154C4⋊C4×C53C2×D412yes—
155C4⋊C4×C53C2×D412yes—
156C4⋊C4×C53C2×D412yes—
157C22⋊C4×C53—24no—
158C22⋊C4×C53—24no—
159C22⋊C4×C53—24no—
160C22⋊C4×C53—22no—
161C22⋊C4×C53—22no—
162C42×C53—24no—
163C42×C53—24no—
164C42×C53—21no—
165C42×C53—21no—
166C22⋊Dic53—42no—
167C22⋊Dic53—82no—
168C4⋊Dic53—22no—
169C4⋊Dic53—22no—
170C4⋊Dic53—22no—
171C4⋊Dic53—22no—
172C4⋊Dic53—42no—
173#11719919—82no—
174#11719917—28no—
175C2×#88322—22no—
176C2×#88322—22no—
177C2×#88322—22no—
178C2×#88322—22no—
179C2×#88322—22no—
180C2×#88322#4114yes—
181C23×C53—21no—
182C23×C53—22no—
183C23×C53—22no—
184C23×C53—21no—
185C23×C53—21no—
186C23×C53—21no—
187C23×C53—22no—
188C23×C53—22no—
189C23×C53#12012yes—
190C23×C53#13712yes—
191C23×C53C22×D411yescharacteristic
192Q8×C53—24no—
193Q8×C53—41no—
194Q8×C53—41no—
195D4×C53—44no—
196C2×C4×C53—22no—
197C2×C4×C53—22no—
198C2×C4×C53—22no—
199C2×C4×C53—24no—
200C2×C4×C53—22no—
201C2×C4×C53—22no—
202C2×C4×C53—21no—
203C2×C4×C53—22no—
204C2×C4×C53—22no—
205C2×C4×C53—21no—
206C2×C4×C53—21no—
207C2×C4×C53—22no—
208C2×C4×C53—21no—
209C2×C4×C53—21no—
210C2×C4×C53—22no—
211C2×C4×C53—48no—
212C2×C4×C53—42no—
213C2×C4×C53—41no—
214C2×C4×C53—41no—
215C2×C4×C53—41no—
216C2×C4×C53—41no—
217C2×C4×C53—42no—
218C2×C4×C53—41no—
219C2×C4×C53—41no—
220C2×C4×C53—42no—
221C2×C4×C53#12012yes—
222C2×C4×C53C22×D411yescharacteristic
223C2×C4×C53C22×D411yescharacteristic
224C2×C4×C53C2×#4112yes—
225C2×C4×C532−1+412yes—
226C2×Dic53—42no—
227C2×Dic53—42no—
228C2×Dic53—42no—
229C2×Dic53—42no—
230C2×Dic53—42no—
231C2×Dic53—42no—
232C2×Dic53—84no—
233#88322—24no—
234#88322—42no—
235#24111—531nomaximal, Sylow
236C22×C53—24no—
237C22×C53—22no—
238C22×C53—22no—
239C22×C53—22no—
240C22×C53—22no—
241C22×C53—42no—
242C22×C53—42no—
243C22×C53—42no—
244C22×C53—41no—
245C22×C53—41no—
246C22×C53C2×#12011yescharacteristic
247C22×C53#54712yes—
248C22×C53C2×#13711yescharacteristic
249C22×C53#57511yescharacteristic
250C22×C53#57712yes—
251C4×C53—22no—
252C4×C53—41no—
253C4×C53—42no—
254C4×C53—41no—
255C4×C53—42no—
256C4×C53—42no—
257C4×C53—41no—
258C4×C53—42no—
259C4×C53#52511yescharacteristic
260C4×C53#53611yescharacteristic
261C4×C53#54712yes—
262Dic53—82no—
263Dic53—82no—
264C2×#552D5311yescharacteristic
265C2×#483—531no—
266C2×#428—531no—
267#2487—531no—
268#2059—532no—
269#2036—532no—
270#1894—531no—
271#1870—531no—
272#1865—532no—
273#1853—532no—
274#1825—531no—
275C2×C53—42no—
276C2×C53—42no—
277C2×C53#240611yescharacteristic
278C2×C53#300711yescharacteristic
279C2×C53#301011yescharacteristic
280C2×C53#334412yes—
281C2×C53#335112yes—
282C22×M16—531no—
283#552—24no—
284#552—24no—
285C2×#126—21no—
286C2×#125—21no—
287C2×#125C2×D5311yescharacteristic
288C2×#123—21no—
289C2×C22⋊Q8C2×D5311yescharacteristic
290C2×C22⋊Q8C2×D5311yescharacteristic
291C2×C4×Q8C2×D5311yescharacteristic
292C2×C4×D4C2×D5311yescharacteristic
293C2×#117—21no—
294C2×#117C2×D5311yescharacteristic
295C22×C4⋊C4—531no—
296C22×C4⋊C4C2×D5311yescharacteristic
297#483—1064no—
298#428—1064no—
299C2×#107—532no—
300C2×#106—532no—
301C2×#106—532no—
302C2×#103—532no—
303C2×#103—532no—
304C2×C22⋊C8—531no—
305C2×C22⋊C8—532no—
306#400—1061no—
307#399—1061no—
308#389—532no—
309#386—1061no—
310#385—1061no—
311#380—531no—
312#380—532no—
313#340—532no—
314#340—532no—
315#340—532no—
316#336—532no—
317C53#2411111yescharacteristic, complemented, Sylow
318C22×Q8C22×D5311yescharacteristic
319C22×D4C22⋊D5311yescharacteristic
320C23×C4—21no—
321C23×C4—1061no—
322C23×C4C22×D5311yescharacteristic
323C23×C4C22×D5311yescharacteristic
324C2×M16—1064no—
325C2×M16—1062no—
326C22×C8—532no—
327C22×C8—532no—
328#126—28no—
329#125—28no—
330#125—22no—
331#125—22no—
332#123—28no—
333C22⋊Q8—22no—
334C22⋊Q8—22no—
335C22⋊Q8—22no—
336C22⋊Q8C22⋊D5314yes—
337C4×Q8—22no—
338C4×Q8—22no—
339C4×D4—22no—
340C4×D4—22no—
341#117—28no—
342#117—22no—
343#117C22⋊D5314yes—
344C2×C4⋊C4—21no—
345C2×C4⋊C4—21no—
346C2×C4⋊C4—21no—
347C2×C4⋊C4—21no—
348C2×C4⋊C4—21no—
349C2×C4⋊C4—21no—
350C2×C4⋊C4—22no—
351C2×C4⋊C4—22no—
352C2×C4⋊C4—532no—
353C2×C4⋊C4—532no—
354C2×C4⋊C4—1062no—
355C2×C4⋊C4—1062no—
356C2×C4⋊C4—1062no—
357C2×C4⋊C4—1061no—
358C2×C4⋊C4C22⋊D5311yescharacteristic
359C2×C4⋊C4C22×D5311yescharacteristic
360C2×C4⋊C4C22×D5311yescharacteristic
361C2×C4⋊C4C22×D5311yescharacteristic
362C2×C22⋊C4—21no—
363C2×C22⋊C4—21no—
364C2×C22⋊C4—21no—
365C2×C22⋊C4—1062no—
366C2×C22⋊C4—1061no—
367C2×C22⋊C4C22⋊D5311yescharacteristic
368C2×C22⋊C4C22⋊D5311yescharacteristic
369C2×C42—21no—
370C2×C42—21no—
371C2×C42C22×D5311yescharacteristic
372#107—1064no—
373#106—1064no—
374#106—1064no—
375#103—1064no—
376#103—1064no—
377C22⋊C8—1064no—
378C22⋊C8—1062no—
379#95—1062no—
380#95—1061no—
381#95—1062no—
382C24C2×C22⋊D5311yescharacteristic
383C24C2×C22⋊D5311yescharacteristic
384C2×Q8—21no—
385C2×Q8—22no—
386C2×Q8—22no—
387C2×Q8C2×C22⋊D5312yes—
388C2×D4—22no—
389C2×D4—22no—
390C2×D4—41no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—24no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—1062no—
405C22×C4—1062no—
406C22×C4—4242no—
407C22×C4—2122no—
408C22×C4C2×C22⋊D5311yescharacteristic
409C22×C4C2×C22⋊D5311yescharacteristic
410C22×C4C2×C22⋊D5311yescharacteristic
411C22×C4C2×C22⋊D5311yescharacteristic
412C22×C4C2×C22⋊D5312yes—
413C22×C4C2×C22⋊D5312yes—
414C22×C4C23×D5311yescharacteristic
415M16—1068no—
416C2×C8—534no—
417C2×C8—1062no—
418C2×C8—1062no—
419C2×C8—1062no—
420C2×C8—1062no—
421C2×C8—1062no—
422C4⋊C4—24no—
423C4⋊C4—24no—
424C4⋊C4—24no—
425C4⋊C4—24no—
426C4⋊C4—24no—
427C4⋊C4—24no—
428C4⋊C4—22no—
429C4⋊C4—21no—
430C4⋊C4—21no—
431C4⋊C4—21no—
432C4⋊C4—1062no—
433C4⋊C4—1062no—
434C4⋊C4—1062no—
435C4⋊C4—1062no—
436C4⋊C4—4242no—
437C4⋊C4—2122no—
438C4⋊C4C2×C22⋊D5312yes—
439C4⋊C4C2×C22⋊D5312yes—
440C4⋊C4C2×C22⋊D5312yes—
441C22⋊C4—24no—
442C22⋊C4—24no—
443C22⋊C4—24no—
444C22⋊C4—22no—
445C22⋊C4—22no—
446C22⋊C4—4242no—
447C22⋊C4—2122no—
448C42—24no—
449C42—24no—
450C42—21no—
451C42—21no—
452C23—22no—
453C23—22no—
454C23—21no—
455C23—21no—
456C23—21no—
457C23—21no—
458C23—22no—
459C23—22no—
460C23#1313318612yes—
461C23#1313324712yes—
462C23C22×C22⋊D5311yescharacteristic
463Q8—24no—
464Q8—41no—
465Q8—41no—
466D4—44no—
467C2×C4—22no—
468C2×C4—24no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—21no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—22no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—22no—
482C2×C4—48no—
483C2×C4—42no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—4244no—
493C2×C4—2122no—
494C2×C4—2122no—
495C2×C4—2122no—
496C2×C4—2122no—
497C2×C4—2122no—
498C2×C4—2122no—
499C2×C4#1313318612yes—
500C2×C4C22×C22⋊D5311yescharacteristic
501C2×C4C22×C22⋊D5311yescharacteristic
502C2×C4#41×D5312yes—
503C2×C4#1313326412yes—
504C8—1064no—
505C8—2122no—
506C22—22no—
507C22—22no—
508C22—22no—
509C22—22no—
510C22—24no—
511C22—41no—
512C22—42no—
513C22—42no—
514C22—41no—
515C22—42no—
516C22C2×#1313318611yescharacteristic
517C22#1681127212yes—
518C22C2×#1313324711yescharacteristic
519C22#1681142111yescharacteristic
520C22#1681143112yes—
521C4—22no—
522C4—41no—
523C4—42no—
524C4—41no—
525C4—42no—
526C4—42no—
527C4—41no—
528C4—42no—
529C4—4242no—
530C4—4242no—
531C4#1681113011yescharacteristic
532C4#1681119411yescharacteristic
533C4#1681127212yes—
534C2—42no—
535C2—42no—
536C2#2027804611yescharacteristic
537C2#2028397511yescharacteristic
538C2#2028400111yescharacteristic
539C2#2028717012yes—
540C2#2028727512yes—
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC2×C53×#552C2
socleC23×C53C22×D4
prime radicalC22×D4×C53D4
hypercenterC2×#552D53
hypocenterC53#24111
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#552

Complements

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

NH ≅ G/Ntype
1. C2×#16810951C2semidirect
2. C2×#16810666C2semidirect
4. #20276050C2semidirect
5. #20275915C2semidirect
6. #20275149C2semidirect
8. #20274983C2semidirect
9. #20274908C2semidirect
10. #20274810C2semidirect
317. C53#24111semidirect

Maximal subgroups

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

subgroupindex
C2×#168109512
C2×#168106662
#202790322
#202760502
#202759152
#202751492
#202750132
#202749832
#202749082
#202748102
C2×C53×#5522
#2411153