Finite Group Databasebeta

← #27417446

Subgroups — #27417446

Order 13568.

1130 conjugacy classes of subgroups.

The subgroup lattice has 558 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810939C211yesmaximal, characteristic, complemented
2C2×#16810934C211yesmaximal, characteristic, complemented
3#20277335C211yesmaximal, characteristic, complemented
4#20276050C212yesmaximal, complemented
5#20275946C212yesmaximal, complemented
6#20275038C211yesmaximal, characteristic, complemented
7#20275038C211yesmaximal, characteristic, complemented
8#20274976C212yesmaximal, complemented
9#20274962C212yesmaximal, complemented
10#20274786C211yesmaximal, characteristic, complemented
11C2×C53×#540C211yesmaximal, characteristic
12C2×#13133176C2211yescharacteristic
13C22×C4⋊Dic53C2211yescharacteristic
14#16810939—24no—
15#16810934—24no—
16C2×#13133150C2212yes—
17C2×#13133148C2212yes—
18C2×#13133138C2212yes—
19C2×#13133138C2212yes—
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×#540—24no—
35C53×#540—24no—
36C2×#127×C53C2211yescharacteristic
37C2×#125×C53C2211yescharacteristic
38C2×#123×C53—21no—
39C2×#123×C53—21no—
40C2×#121×C53—21no—
41C2×#121×C53—21no—
42C2×#121×C53C2211yescharacteristic
43C2×#121×C53C2211yescharacteristic
44C2×C4×D4×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#13133150—24no—
61#13133148—24no—
62#13133138—24no—
63#13133138—24no—
64#13133138—24no—
65#13133138—24no—
66C22⋊#88322—22no—
67C22⋊#88322—22no—
68#13133124—22no—
69#13133124—22no—
70C22×D4×C53—21no—
71C22×D4×C53—21no—
72C22×D4×C53—21no—
73C22×D4×C53—21no—
74C22×D4×C53C2311yescharacteristic
75C22×D4×C53C2311yescharacteristic
76C23×C4×C53—21no—
77C23×C4×C53—21no—
78C23×C4×C53C2311yescharacteristic
79#127×C53—22no—
80#127×C53—22no—
81#125×C53—22no—
82#125×C53—22no—
83#123×C53—28no—
84#123×C53—28no—
85#121×C53—28no—
86#121×C53—28no—
87#121×C53—22no—
88#121×C53—22no—
89#121×C53D414yes—
90#121×C53D414yes—
91C4×D4×C53—22no—
92C4×D4×C53—22no—
93C4×D4×C53—22no—
94C4×D4×C53—22no—
95#117×C53—22no—
96#117×C53—22no—
97C2×C4⋊C4×C53—21no—
98C2×C4⋊C4×C53—21no—
99C2×C4⋊C4×C53D411yescharacteristic
100C2×C4⋊C4×C53D411yescharacteristic
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×C53—21no—
107C2×C22⋊C4×C53D411yescharacteristic
108C2×C22⋊C4×C53D411yescharacteristic
109C2×C42×C53C2311yescharacteristic
110C2×C42×C53C2311yescharacteristic
111C24×C53—21no—
112C24×C53—21no—
113C24×C53C2×D411yescharacteristic
114C2×D4×C53—24no—
115C2×D4×C53—24no—
116C2×D4×C53—24no—
117C2×D4×C53—24no—
118C2×D4×C53—21no—
119C2×D4×C53—21no—
120C2×D4×C53—22no—
121C2×D4×C53—22no—
122C2×D4×C53—22no—
123C2×D4×C53—22no—
124C2×D4×C53—82no—
125C2×D4×C53—82no—
126C2×D4×C53—82no—
127C2×D4×C53—82no—
128C2×D4×C53C2×D412yes—
129C2×D4×C53C2×D412yes—
130C22×C4×C53—22no—
131C22×C4×C53—22no—
132C22×C4×C53—21no—
133C22×C4×C53—21no—
134C22×C4×C53—24no—
135C22×C4×C53—24no—
136C22×C4×C53—82no—
137C22×C4×C53—82no—
138C22×C4×C53C2×D411yescharacteristic
139C22×C4×C53C2×D411yescharacteristic
140C22×C4×C53C2×D411yescharacteristic
141C22×C4×C53C2×D411yescharacteristic
142C22×C4×C53C2×D411yescharacteristic
143C22×C4×C53C2×D412yes—
144C22×C4×C53C2×D412yes—
145C22×C4×C53C2411yescharacteristic
146C22×Dic53—22no—
147C22×Dic53—22no—
148C22×Dic53—42no—
149C22×Dic53—82no—
150C4⋊C4×C53—24no—
151C4⋊C4×C53—24no—
152C4⋊C4×C53—22no—
153C4⋊C4×C53—22no—
154C4⋊C4×C53—21no—
155C4⋊C4×C53—21no—
156C4⋊C4×C53C2×D412yes—
157C4⋊C4×C53C2×D412yes—
158C22⋊C4×C53—24no—
159C22⋊C4×C53—24no—
160C22⋊C4×C53—22no—
161C22⋊C4×C53—22no—
162C22⋊C4×C53—24no—
163C22⋊C4×C53—24no—
164C42×C53—21no—
165C42×C53—21no—
166C42×C53—21no—
167C42×C53—21no—
168C22⋊Dic53—24no—
169C4⋊Dic53—22no—
170C4⋊Dic53—22no—
171C4⋊Dic53—22no—
172C4⋊Dic53—42no—
173C4⋊Dic53#4114yes—
174#11719919—82no—
175#11719919—82no—
176C4×Dic53—24no—
177C2×#88322—22no—
178C2×#88322—22no—
179C2×#88322—22no—
180C2×#88322—22no—
181C2×#88322—22no—
182C2×#88322—22no—
183C23×C53—21no—
184C23×C53—21no—
185C23×C53—22no—
186C23×C53—22no—
187C23×C53—22no—
188C23×C53—22no—
189C23×C53—41no—
190C23×C53—41no—
191C23×C53—41no—
192C23×C53—41no—
193C23×C53—42no—
194C23×C53—42no—
195C23×C53—42no—
196C23×C53—42no—
197C23×C53#12012yes—
198C23×C53C22×D411yescharacteristic
199D4×C53—24no—
200D4×C53—24no—
201D4×C53—42no—
202D4×C53—42no—
203D4×C53—84no—
204D4×C53—84no—
205D4×C53—84no—
206D4×C53—84no—
207C2×C4×C53—21no—
208C2×C4×C53—21no—
209C2×C4×C53—24no—
210C2×C4×C53—24no—
211C2×C4×C53—21no—
212C2×C4×C53—21no—
213C2×C4×C53—22no—
214C2×C4×C53—22no—
215C2×C4×C53—21no—
216C2×C4×C53—21no—
217C2×C4×C53—22no—
218C2×C4×C53—22no—
219C2×C4×C53—22no—
220C2×C4×C53—22no—
221C2×C4×C53—42no—
222C2×C4×C53—42no—
223C2×C4×C53—84no—
224C2×C4×C53—84no—
225C2×C4×C53#12012yes—
226C2×C4×C53C22×D411yescharacteristic
227C2×C4×C53C22×D411yescharacteristic
228C2×C4×C53C2×#4112yes—
229C2×C4×C532−1+412yes—
230C2×Dic53—24no—
231C2×Dic53—42no—
232C2×Dic53—42no—
233C2×Dic53—42no—
234C2×Dic53—42no—
235C2×Dic53—42no—
236C2×Dic53—84no—
237#88322—42no—
238#88322—42no—
239#24069—531nomaximal, Sylow
240C22×C53—22no—
241C22×C53—22no—
242C22×C53—22no—
243C22×C53—42no—
244C22×C53—42no—
245C22×C53—42no—
246C22×C53—42no—
247C22×C53—42no—
248C22×C53—42no—
249C22×C53—42no—
250C22×C53—42no—
251C22×C53—82no—
252C22×C53—82no—
253C22×C53—82no—
254C22×C53—82no—
255C22×C53C2×#12011yescharacteristic
256C22×C53#52511yescharacteristic
257C22×C53#53611yescharacteristic
258C22×C53#57612yes—
259C22×C53#57712yes—
260C4×C53—22no—
261C4×C53—41no—
262C4×C53—41no—
263C4×C53—42no—
264C4×C53—41no—
265C4×C53—41no—
266C4×C53—42no—
267C4×C53#54712yes—
268C4×C53#54712yes—
269Dic53—44no—
270Dic53—82no—
271C2×#540D5311yescharacteristic
272C2×#481—531no—
273C2×#480—531no—
274#2317—531no—
275#2059—532no—
276#2041—532no—
277#1875—531no—
278#1875—531no—
279#1864—532no—
280#1862—532no—
281#1818—531no—
282C2×C53—42no—
283C2×C53—82no—
284C2×C53—82no—
285C2×C53#240611yescharacteristic
286C2×C53#300811yescharacteristic
287C2×C53#300911yescharacteristic
288C2×C53#311912yes—
289C2×C53#324112yes—
290#540—24no—
291#540—24no—
292C2×#127C2×D5311yescharacteristic
293C2×#125C2×D5311yescharacteristic
294C2×#123—21no—
295C2×#123—21no—
296C2×#121—21no—
297C2×#121—21no—
298C2×#121C2×D5311yescharacteristic
299C2×#121C2×D5311yescharacteristic
300C2×C4×D4C2×D5311yescharacteristic
301C2×C4×D4C2×D5311yescharacteristic
302C2×#117—531no—
303C2×#117C2×D5311yescharacteristic
304C22×C4⋊C4—531no—
305#481—1064no—
306#480—1064no—
307C2×#107—532no—
308C2×#106—532no—
309C2×#102—532no—
310C2×#102—532no—
311C2×#102—532no—
312C2×#102—532no—
313C2×C22⋊C8—531no—
314C2×C22⋊C8—531no—
315#389—532no—
316#389—532no—
317#385—1061no—
318#385—1061no—
319#384—1061no—
320#382—1061no—
321#340—532no—
322#340—532no—
323#336—532no—
324#336—532no—
325C53#2406911yescharacteristic, complemented, Sylow
326C22×D4—21no—
327C22×D4—21no—
328C22×D4—21no—
329C22×D4—21no—
330C22×D4C22×D5311yescharacteristic
331C22×D4C22×D5311yescharacteristic
332C23×C4—21no—
333C23×C4—21no—
334C23×C4—1061no—
335C23×C4C22×D5311yescharacteristic
336C22×C8—532no—
337C22×C8—532no—
338#127—22no—
339#127—22no—
340#125—22no—
341#125—22no—
342#123—28no—
343#123—28no—
344#121—28no—
345#121—28no—
346#121—22no—
347#121—22no—
348#121C22⋊D5314yes—
349#121C22⋊D5314yes—
350C4×D4—22no—
351C4×D4—22no—
352C4×D4—22no—
353C4×D4—22no—
354#117—22no—
355#117—22no—
356#117—1064no—
357C2×C4⋊C4—21no—
358C2×C4⋊C4—21no—
359C2×C4⋊C4—532no—
360C2×C4⋊C4—532no—
361C2×C4⋊C4—1062no—
362C2×C4⋊C4—1062no—
363C2×C4⋊C4—1062no—
364C2×C4⋊C4—1062no—
365C2×C4⋊C4—1061no—
366C2×C4⋊C4C22⋊D5311yescharacteristic
367C2×C4⋊C4C22⋊D5311yescharacteristic
368C2×C4⋊C4C22×D5311yescharacteristic
369C2×C4⋊C4C22×D5311yescharacteristic
370C2×C22⋊C4—21no—
371C2×C22⋊C4—21no—
372C2×C22⋊C4—21no—
373C2×C22⋊C4—21no—
374C2×C22⋊C4—1061no—
375C2×C22⋊C4C22⋊D5311yescharacteristic
376C2×C22⋊C4C22⋊D5311yescharacteristic
377C2×C42—1061no—
378C2×C42C22×D5311yescharacteristic
379C2×C42C22×D5311yescharacteristic
380#107—1064no—
381#106—1064no—
382#102—1064no—
383#102—1064no—
384#102—1064no—
385#102—1064no—
386C22⋊C8—1062no—
387C22⋊C8—1062no—
388#95—1062no—
389#95—1062no—
390C24—21no—
391C24—21no—
392C24C2×C22⋊D5311yescharacteristic
393C2×D4—24no—
394C2×D4—24no—
395C2×D4—24no—
396C2×D4—24no—
397C2×D4—21no—
398C2×D4—21no—
399C2×D4—22no—
400C2×D4—22no—
401C2×D4—22no—
402C2×D4—22no—
403C2×D4—82no—
404C2×D4—82no—
405C2×D4—82no—
406C2×D4—82no—
407C2×D4C2×C22⋊D5312yes—
408C2×D4C2×C22⋊D5312yes—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—82no—
416C22×C4—82no—
417C22×C4—1062no—
418C22×C4—1062no—
419C22×C4—4242no—
420C22×C4—2122no—
421C22×C4C2×C22⋊D5311yescharacteristic
422C22×C4C2×C22⋊D5311yescharacteristic
423C22×C4C2×C22⋊D5311yescharacteristic
424C22×C4C2×C22⋊D5311yescharacteristic
425C22×C4C2×C22⋊D5311yescharacteristic
426C22×C4C2×C22⋊D5312yes—
427C22×C4C2×C22⋊D5312yes—
428C22×C4C23×D5311yescharacteristic
429C2×C8—1062no—
430C2×C8—1062no—
431C2×C8—1062no—
432C2×C8—1062no—
433C2×C8—1062no—
434C2×C8—1062no—
435C4⋊C4—24no—
436C4⋊C4—24no—
437C4⋊C4—22no—
438C4⋊C4—22no—
439C4⋊C4—21no—
440C4⋊C4—21no—
441C4⋊C4—534no—
442C4⋊C4—1062no—
443C4⋊C4—1062no—
444C4⋊C4—1062no—
445C4⋊C4—4242no—
446C4⋊C4—4242no—
447C4⋊C4—2122no—
448C4⋊C4C2×C22⋊D5312yes—
449C4⋊C4C2×C22⋊D5312yes—
450C22⋊C4—24no—
451C22⋊C4—24no—
452C22⋊C4—22no—
453C22⋊C4—22no—
454C22⋊C4—24no—
455C22⋊C4—24no—
456C22⋊C4—1064no—
457C42—21no—
458C42—21no—
459C42—21no—
460C42—21no—
461C42—1064no—
462C23—21no—
463C23—22no—
464C23—22no—
465C23—21no—
466C23—22no—
467C23—22no—
468C23—41no—
469C23—41no—
470C23—42no—
471C23—42no—
472C23—42no—
473C23—42no—
474C23—41no—
475C23—41no—
476C23#1313318612yes—
477C23C22×C22⋊D5311yescharacteristic
478D4—24no—
479D4—24no—
480D4—42no—
481D4—42no—
482D4—84no—
483D4—84no—
484D4—84no—
485D4—84no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—24no—
489C2×C4—24no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—84no—
503C2×C4—84no—
504C2×C4—1064no—
505C2×C4—4244no—
506C2×C4—2122no—
507C2×C4—2122no—
508C2×C4—2122no—
509C2×C4—2122no—
510C2×C4—2122no—
511C2×C4#1313318612yes—
512C2×C4C22×C22⋊D5311yescharacteristic
513C2×C4C22×C22⋊D5311yescharacteristic
514C2×C4#41×D5312yes—
515C2×C4#1313326412yes—
516C8—2122no—
517C8—2122no—
518C22—22no—
519C22—22no—
520C22—22no—
521C22—42no—
522C22—42no—
523C22—42no—
524C22—42no—
525C22—42no—
526C22—42no—
527C22—42no—
528C22—42no—
529C22—82no—
530C22—82no—
531C22—82no—
532C22—82no—
533C22C2×#1313318611yescharacteristic
534C22#1681113011yescharacteristic
535C22#1681119411yescharacteristic
536C22#1681142612yes—
537C22#1681143412yes—
538C4—22no—
539C4—41no—
540C4—41no—
541C4—42no—
542C4—41no—
543C4—41no—
544C4—42no—
545C4—4242no—
546C4—2124no—
547C4#1681127212yes—
548C4#1681127212yes—
549C2—42no—
550C2—82no—
551C2—82no—
552C2#2027804611yescharacteristic
553C2#2028398311yescharacteristic
554C2#2028399211yescharacteristic
555C2#2028490812yes—
556C2#2028596212yes—
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
Fitting, prime radicalC2×C53×#540C2
socleC23×C53C22×D4
hypercenterC2×#540D53
hypocenterC53#24069
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#540

Complements

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

NH ≅ G/Ntype
1. C2×#16810939C2semidirect
2. C2×#16810934C2semidirect
3. #20277335C2semidirect
4. #20276050C2semidirect
5. #20275946C2semidirect
6. #20275038C2semidirect
7. #20275038C2semidirect
8. #20274976C2semidirect
9. #20274962C2semidirect
10. #20274786C2semidirect
325. C53#24069semidirect

Maximal subgroups

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

subgroupindex
C2×#168109392
C2×#168109342
#202773352
#202760502
#202759462
#202750382
#202749762
#202749622
#202747862
C2×C53×#5402
#2406953