Finite Group Databasebeta

← #27417497

Subgroups — #27417497

Order 13568.

1066 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810939C211yesmaximal, characteristic
2C2×#16810951C211yesmaximal, characteristic, complemented
3#20277335C211yesmaximal, characteristic, complemented
4#20276045C212yesmaximal, complemented
5#20275951C212yesmaximal, complemented
6#20275038C211yesmaximal, characteristic
7#20275028C211yesmaximal, characteristic, complemented
8#20274983C212yesmaximal, complemented
9#20274976C212yesmaximal, complemented
10#20274786C211yesmaximal, characteristic, complemented
11C2×C53×#542C211yesmaximal, characteristic
12C2×#13133176C2211yescharacteristic
13C22×C4⋊Dic53C2211yescharacteristic
14#16810951—24no—
15#16810939—24no—
16C2×#13133148C2212yes—
17C2×#13133148C2212yes—
18C2×#13133141C2212yes—
19C2×#13133141C2212yes—
20C2×#13133138C2212yes—
21C2×#13133138C2212yes—
22C2×C22⋊#88322C2211yescharacteristic
23C2×C22⋊#88322C2211yescharacteristic
24#16810521C2212yes—
25#16810521C2212yes—
26#16810495—21no—
27#16810495—21no—
28#16810492—21no—
29#16810488—21no—
30#16810385C2212yes—
31#16810385C2212yes—
32#16810378C2212yes—
33#16810378C2212yes—
34C53×#542—24no—
35C53×#542—24no—
36C2×C4⋊Q8×C53C2211yescharacteristic
37C2×#126×C53—21no—
38C2×#124×C53—21no—
39C2×#124×C53C2211yescharacteristic
40C2×#123×C53—21no—
41C2×C22⋊Q8×C53—21no—
42C2×C22⋊Q8×C53C2211yescharacteristic
43C2×#121×C53C2211yescharacteristic
44C2×C4×Q8×C53C2211yescharacteristic
45C2×C4×D4×C53C2211yescharacteristic
46C2×#117×C53C2211yescharacteristic
47C23×Dic53—21no—
48C22×#88322C2312yes—
49C22×#88322C2312yes—
50#13133176—24no—
51C2×C4⋊Dic53—21no—
52C2×C4⋊Dic53—22no—
53C2×C4⋊Dic53—22no—
54C2×C4⋊Dic53C2312yes—
55C2×C4⋊Dic53C2312yes—
56C2×#11719919—22no—
57C2×#11719919—22no—
58C2×C22⋊Dic53—21no—
59C2×C4×Dic53—21no—
60#13133148—24no—
61#13133148—24no—
62#13133141—24no—
63#13133141—24no—
64#13133138—24no—
65#13133138—24no—
66C22⋊#88322—22no—
67C22⋊#88322—22no—
68#13133124—22no—
69#13133124—22no—
70C22×Q8×C53—21no—
71C22×Q8×C53C2311yescharacteristic
72C22×D4×C53—21no—
73C22×D4×C53C2311yescharacteristic
74C23×C4×C53—21no—
75C23×C4×C53C2311yescharacteristic
76C4⋊Q8×C53—22no—
77C4⋊Q8×C53—22no—
78#126×C53—28no—
79#124×C53—28no—
80#124×C53—22no—
81#124×C53—22no—
82#123×C53—28no—
83C22⋊Q8×C53—28no—
84C22⋊Q8×C53—22no—
85C22⋊Q8×C53D414yes—
86#121×C53—22no—
87#121×C53D414yes—
88C4×Q8×C53—22no—
89C4×Q8×C53—22no—
90C4×D4×C53—22no—
91C4×D4×C53—22no—
92#117×C53—22no—
93#117×C53—22no—
94C2×C4⋊C4×C53—21no—
95C2×C4⋊C4×C53—21no—
96C2×C4⋊C4×C53—21no—
97C2×C4⋊C4×C53D411yescharacteristic
98C2×C4⋊C4×C53D411yescharacteristic
99C2×C4⋊C4×C53C2311yescharacteristic
100C2×C4⋊C4×C53C2311yescharacteristic
101C2×C22⋊C4×C53—21no—
102C2×C22⋊C4×C53—21no—
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×C53C2311yescharacteristic
110C2×C42×C53C2311yescharacteristic
111C24×C53—21no—
112C24×C53C2×D411yescharacteristic
113C2×Q8×C53—24no—
114C2×Q8×C53—21no—
115C2×Q8×C53—22no—
116C2×Q8×C53—22no—
117C2×Q8×C53—82no—
118C2×Q8×C53C2×D412yes—
119C2×D4×C53—24no—
120C2×D4×C53—21no—
121C2×D4×C53—22no—
122C2×D4×C53—22no—
123C2×D4×C53—82no—
124C2×D4×C53C2×D412yes—
125C22×C4×C53—22no—
126C22×C4×C53—22no—
127C22×C4×C53—21no—
128C22×C4×C53—21no—
129C22×C4×C53—21no—
130C22×C4×C53—24no—
131C22×C4×C53—82no—
132C22×C4×C53C2×D411yescharacteristic
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—22no—
148C4⋊C4×C53—22no—
149C4⋊C4×C53—21no—
150C4⋊C4×C53—21no—
151C4⋊C4×C53C2×D412yes—
152C4⋊C4×C53C2×D412yes—
153C22⋊C4×C53—24no—
154C22⋊C4×C53—24no—
155C22⋊C4×C53—24no—
156C22⋊C4×C53—24no—
157C22⋊C4×C53—22no—
158C22⋊C4×C53—22no—
159C22⋊C4×C53—24no—
160C42×C53—24no—
161C42×C53—21no—
162C42×C53—21no—
163C42×C53—21no—
164C42×C53—21no—
165C22⋊Dic53—24no—
166C4⋊Dic53—22no—
167C4⋊Dic53—22no—
168C4⋊Dic53—22no—
169C4⋊Dic53—42no—
170C4⋊Dic53#4114yes—
171#11719919—82no—
172#11719919—82no—
173C4×Dic53—24no—
174C2×#88322—22no—
175C2×#88322—22no—
176C2×#88322—22no—
177C2×#88322—22no—
178C2×#88322—22no—
179C2×#88322—22no—
180C23×C53—21no—
181C23×C53—21no—
182C23×C53—22no—
183C23×C53—22no—
184C23×C53—22no—
185C23×C53—41no—
186C23×C53—41no—
187C23×C53—42no—
188C23×C53—42no—
189C23×C53#12012yes—
190C23×C53C22×D411yescharacteristic
191Q8×C53—24no—
192Q8×C53—42no—
193Q8×C53—84no—
194D4×C53—24no—
195D4×C53—42no—
196D4×C53—84no—
197C2×C4×C53—21no—
198C2×C4×C53—21no—
199C2×C4×C53—24no—
200C2×C4×C53—24no—
201C2×C4×C53—21no—
202C2×C4×C53—22no—
203C2×C4×C53—22no—
204C2×C4×C53—22no—
205C2×C4×C53—21no—
206C2×C4×C53—21no—
207C2×C4×C53—21no—
208C2×C4×C53—22no—
209C2×C4×C53—22no—
210C2×C4×C53—22no—
211C2×C4×C53—22no—
212C2×C4×C53—41no—
213C2×C4×C53—41no—
214C2×C4×C53—42no—
215C2×C4×C53—42no—
216C2×C4×C53—84no—
217C2×C4×C53#12012yes—
218C2×C4×C53C22×D411yescharacteristic
219C2×C4×C53C22×D411yescharacteristic
220C2×C4×C53C2×#4112yes—
221C2×C4×C532−1+412yes—
222C2×Dic53—24no—
223C2×Dic53—42no—
224C2×Dic53—42no—
225C2×Dic53—42no—
226C2×Dic53—42no—
227C2×Dic53—42no—
228C2×Dic53—84no—
229#88322—42no—
230#88322—42no—
231#24074—531nomaximal, Sylow
232C22×C53—22no—
233C22×C53—22no—
234C22×C53—22no—
235C22×C53—42no—
236C22×C53—42no—
237C22×C53—42no—
238C22×C53—42no—
239C22×C53—42no—
240C22×C53—82no—
241C22×C53—82no—
242C22×C53C2×#12011yescharacteristic
243C22×C53#52511yescharacteristic
244C22×C53#53611yescharacteristic
245C22×C53#57712yes—
246C22×C53#57712yes—
247C4×C53—22no—
248C4×C53—42no—
249C4×C53—41no—
250C4×C53—41no—
251C4×C53—42no—
252C4×C53—42no—
253C4×C53—41no—
254C4×C53—41no—
255C4×C53#54712yes—
256C4×C53#54712yes—
257Dic53—44no—
258Dic53—82no—
259C2×#542D5311yescharacteristic
260C2×#481—531no—
261C2×#483—531no—
262#2317—531no—
263#2058—532no—
264#2042—532no—
265#1875—531no—
266#1873—531no—
267#1865—532no—
268#1864—532no—
269#1818—531no—
270C2×C53—42no—
271C2×C53—82no—
272C2×C53#240611yescharacteristic
273C2×C53#300911yescharacteristic
274C2×C53#301011yescharacteristic
275C2×C53#312512yes—
276C2×C53#324412yes—
277#542—24no—
278#542—24no—
279C2×C4⋊Q8C2×D5311yescharacteristic
280C2×#126—21no—
281C2×#124—21no—
282C2×#124C2×D5311yescharacteristic
283C2×#123—21no—
284C2×C22⋊Q8—21no—
285C2×C22⋊Q8C2×D5311yescharacteristic
286C2×#121C2×D5311yescharacteristic
287C2×C4×Q8C2×D5311yescharacteristic
288C2×C4×D4C2×D5311yescharacteristic
289C2×#117—531no—
290C2×#117C2×D5311yescharacteristic
291C22×C4⋊C4—531no—
292#483—1064no—
293#481—1064no—
294C2×#106—532no—
295C2×#106—532no—
296C2×#103—532no—
297C2×#103—532no—
298C2×#102—532no—
299C2×#102—532no—
300C2×C22⋊C8—531no—
301C2×C22⋊C8—531no—
302#389—532no—
303#389—532no—
304#385—1061no—
305#385—1061no—
306#384—1061no—
307#382—1061no—
308#340—532no—
309#340—532no—
310#336—532no—
311#336—532no—
312C53#2407411yescharacteristic, complemented, Sylow
313C22×Q8—21no—
314C22×Q8C22×D5311yescharacteristic
315C22×D4—21no—
316C22×D4C22×D5311yescharacteristic
317C23×C4—21no—
318C23×C4—1061no—
319C23×C4C22×D5311yescharacteristic
320C22×C8—532no—
321C22×C8—532no—
322C4⋊Q8—22no—
323C4⋊Q8—22no—
324#126—28no—
325#124—28no—
326#124—22no—
327#124—22no—
328#123—28no—
329C22⋊Q8—28no—
330C22⋊Q8—22no—
331C22⋊Q8C22⋊D5314yes—
332#121—22no—
333#121C22⋊D5314yes—
334C4×Q8—22no—
335C4×Q8—22no—
336C4×D4—22no—
337C4×D4—22no—
338#117—22no—
339#117—22no—
340#117—1064no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—21no—
343C2×C4⋊C4—21no—
344C2×C4⋊C4—532no—
345C2×C4⋊C4—532no—
346C2×C4⋊C4—1062no—
347C2×C4⋊C4—1062no—
348C2×C4⋊C4—1062no—
349C2×C4⋊C4—1062no—
350C2×C4⋊C4—1061no—
351C2×C4⋊C4C22⋊D5311yescharacteristic
352C2×C4⋊C4C22⋊D5311yescharacteristic
353C2×C4⋊C4C22×D5311yescharacteristic
354C2×C4⋊C4C22×D5311yescharacteristic
355C2×C22⋊C4—21no—
356C2×C22⋊C4—21no—
357C2×C22⋊C4—21no—
358C2×C22⋊C4—21no—
359C2×C22⋊C4—21no—
360C2×C22⋊C4—1061no—
361C2×C22⋊C4C22⋊D5311yescharacteristic
362C2×C22⋊C4C22⋊D5311yescharacteristic
363C2×C42—21no—
364C2×C42—1061no—
365C2×C42C22×D5311yescharacteristic
366C2×C42C22×D5311yescharacteristic
367#106—1064no—
368#106—1064no—
369#103—1064no—
370#103—1064no—
371#102—1064no—
372#102—1064no—
373C22⋊C8—1062no—
374C22⋊C8—1062no—
375#95—1062no—
376#95—1062no—
377C24—21no—
378C24C2×C22⋊D5311yescharacteristic
379C2×Q8—24no—
380C2×Q8—21no—
381C2×Q8—22no—
382C2×Q8—22no—
383C2×Q8—82no—
384C2×Q8C2×C22⋊D5312yes—
385C2×D4—24no—
386C2×D4—21no—
387C2×D4—22no—
388C2×D4—22no—
389C2×D4—82no—
390C2×D4C2×C22⋊D5312yes—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—24no—
397C22×C4—82no—
398C22×C4—1062no—
399C22×C4—1062no—
400C22×C4—4242no—
401C22×C4—2122no—
402C22×C4C2×C22⋊D5311yescharacteristic
403C22×C4C2×C22⋊D5311yescharacteristic
404C22×C4C2×C22⋊D5311yescharacteristic
405C22×C4C2×C22⋊D5311yescharacteristic
406C22×C4C2×C22⋊D5311yescharacteristic
407C22×C4C2×C22⋊D5312yes—
408C22×C4C2×C22⋊D5312yes—
409C22×C4C23×D5311yescharacteristic
410C2×C8—1062no—
411C2×C8—1062no—
412C2×C8—1062no—
413C2×C8—1062no—
414C2×C8—1062no—
415C2×C8—1062no—
416C4⋊C4—24no—
417C4⋊C4—24no—
418C4⋊C4—24no—
419C4⋊C4—22no—
420C4⋊C4—22no—
421C4⋊C4—21no—
422C4⋊C4—21no—
423C4⋊C4—534no—
424C4⋊C4—1062no—
425C4⋊C4—1062no—
426C4⋊C4—1062no—
427C4⋊C4—4242no—
428C4⋊C4—4242no—
429C4⋊C4—2122no—
430C4⋊C4C2×C22⋊D5312yes—
431C4⋊C4C2×C22⋊D5312yes—
432C22⋊C4—24no—
433C22⋊C4—24no—
434C22⋊C4—24no—
435C22⋊C4—24no—
436C22⋊C4—22no—
437C22⋊C4—22no—
438C22⋊C4—24no—
439C22⋊C4—1064no—
440C42—24no—
441C42—21no—
442C42—21no—
443C42—21no—
444C42—21no—
445C42—1064no—
446C23—21no—
447C23—22no—
448C23—22no—
449C23—21no—
450C23—22no—
451C23—41no—
452C23—42no—
453C23—42no—
454C23—41no—
455C23#1313318612yes—
456C23C22×C22⋊D5311yescharacteristic
457Q8—24no—
458Q8—42no—
459Q8—84no—
460D4—24no—
461D4—42no—
462D4—84no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—24no—
466C2×C4—24no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—22no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—41no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—41no—
482C2×C4—84no—
483C2×C4—1064no—
484C2×C4—4244no—
485C2×C4—2122no—
486C2×C4—2122no—
487C2×C4—2122no—
488C2×C4—2122no—
489C2×C4—2122no—
490C2×C4#1313318612yes—
491C2×C4C22×C22⋊D5311yescharacteristic
492C2×C4C22×C22⋊D5311yescharacteristic
493C2×C4#41×D5312yes—
494C2×C4#1313326412yes—
495C8—2122no—
496C8—2122no—
497C22—22no—
498C22—22no—
499C22—22no—
500C22—42no—
501C22—42no—
502C22—42no—
503C22—42no—
504C22—42no—
505C22—82no—
506C22—82no—
507C22C2×#1313318611yescharacteristic
508C22#1681113011yescharacteristic
509C22#1681119411yescharacteristic
510C22#1681143112yes—
511C22#1681143412yes—
512C4—22no—
513C4—42no—
514C4—41no—
515C4—41no—
516C4—42no—
517C4—42no—
518C4—41no—
519C4—41no—
520C4—4242no—
521C4—2124no—
522C4#1681127212yes—
523C4#1681127212yes—
524C2—42no—
525C2—82no—
526C2#2027804611yescharacteristic
527C2#2028399211yescharacteristic
528C2#2028400111yescharacteristic
529C2#2028498612yes—
530C2#2028600712yes—
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC2×C53×#542C2
socleC23×C53C22×D4
prime radicalC2×#121×C53C22
hypercenterC2×#542D53
hypocenterC53#24074
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#542

Complements

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

NH ≅ G/Ntype
2. C2×#16810951C2semidirect
3. #20277335C2semidirect
4. #20276045C2semidirect
5. #20275951C2semidirect
7. #20275028C2semidirect
8. #20274983C2semidirect
9. #20274976C2semidirect
10. #20274786C2semidirect
312. C53#24074semidirect

Maximal subgroups

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

subgroupindex
C2×#168109392
C2×#168109512
#202773352
#202760452
#202759512
#202750382
#202750282
#202749832
#202749762
#202747862
C2×C53×#5422
#2407453