Finite Group Databasebeta

← #27417888

Subgroups — #27417888

Order 13568.

1082 conjugacy classes of subgroups.

The subgroup lattice has 546 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810956C211yesmaximal, characteristic, complemented
2C2×#16810666C211yesmaximal, characteristic, complemented
3#20279023C211yesmaximal, characteristic
4#20276040C212yesmaximal, complemented
5#20275915C212yesmaximal, complemented
6#20275149C211yesmaximal, characteristic, complemented
7#20275013C211yesmaximal, characteristic
8#20274983C212yesmaximal, complemented
9#20274914C212yesmaximal, complemented
10#20274810C211yesmaximal, characteristic, complemented
11C2×C53×#556C211yesmaximal, characteristic
12C22×#11719917C2211yescharacteristic
13C2×#13133198—21no—
14C22×C4⋊Dic53C2211yescharacteristic
15#16810956—24no—
16#16810666—24no—
17C2×#13133150C2212yes—
18C2×#13133150C2212yes—
19C2×#13133148C2212yes—
20C2×#13133141C2212yes—
21C2×#13133141C2212yes—
22C2×C22⋊#88322C2211yescharacteristic
23C2×C22⋊#88322C2212yes—
24#16810553—21no—
25#16810548—21no—
26#16810521C2212yes—
27#16810484—21no—
28#16810483C2211yescharacteristic
29#16810483C2212yes—
30#16810385C2212yes—
31#16810385C2212yes—
32#16810385C2212yes—
33#16810378C2212yes—
34C53×#556—24no—
35C53×#556—24no—
36C2×C4⋊Q8×C53C2211yescharacteristic
37C2×#126×C53—21no—
38C2×#126×C53—21no—
39C2×#124×C53—21no—
40C2×C22⋊Q8×C53C2211yescharacteristic
41C2×C22⋊Q8×C53C2211yescharacteristic
42C2×C4×Q8×C53—21no—
43C2×C4×Q8×C53C2211yescharacteristic
44C2×C4×D4×C53C2211yescharacteristic
45C2×#117×C53C2211yescharacteristic
46C2×#117×C53C2211yescharacteristic
47C23×Dic53—21no—
48C2×#11719917—24no—
49C2×#11719917—22no—
50C22×#88322C2312yes—
51C22×#88322C2312yes—
52#13133198—44no—
53C2×C4⋊Dic53—21no—
54C2×C4⋊Dic53—22no—
55C2×C4⋊Dic53—22no—
56C2×C4⋊Dic53C2312yes—
57C2×C4⋊Dic53C2312yes—
58C2×#11719919—22no—
59C2×C22⋊Dic53—22no—
60C2×C22⋊Dic53—21no—
61#13133150—24no—
62#13133150—24no—
63#13133148—24no—
64#13133141—24no—
65#13133141—24no—
66C22⋊#88322—24no—
67C22⋊#88322—22no—
68#13133124—22no—
69#13133124—21no—
70#13133124—22no—
71C22×Q8×C53—21no—
72C22×Q8×C53C2311yescharacteristic
73C22×D4×C53D411yescharacteristic
74C23×C4×C53C2311yescharacteristic
75C23×C4×C53C2311yescharacteristic
76C4⋊Q8×C53—22no—
77C4⋊Q8×C53—22no—
78#126×C53—28no—
79#126×C53—28no—
80#124×C53—28no—
81C22⋊Q8×C53—22no—
82C22⋊Q8×C53—22no—
83C22⋊Q8×C53—22no—
84C22⋊Q8×C53D414yes—
85C4×Q8×C53—28no—
86C4×Q8×C53—22no—
87C4×Q8×C53—22no—
88C4×D4×C53—22no—
89C4×D4×C53—22no—
90#117×C53—22no—
91#117×C53—22no—
92#117×C53—22no—
93#117×C53D414yes—
94C2×C4⋊C4×C53—21no—
95C2×C4⋊C4×C53—21no—
96C2×C4⋊C4×C53—21no—
97C2×C4⋊C4×C53—21no—
98C2×C4⋊C4×C53D411yescharacteristic
99C2×C4⋊C4×C53C2311yescharacteristic
100C2×C4⋊C4×C53C2311yescharacteristic
101C2×C4⋊C4×C53C2311yescharacteristic
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×C53—21no—
110C2×C42×C53—21no—
111C2×C42×C53C2311yescharacteristic
112C24×C53C2×D411yescharacteristic
113C24×C53C2×D411yescharacteristic
114C2×Q8×C53—24no—
115C2×Q8×C53—21no—
116C2×Q8×C53—22no—
117C2×Q8×C53—22no—
118C2×Q8×C53—42no—
119C2×Q8×C53—42no—
120C2×Q8×C53C2×D412yes—
121C2×D4×C53—22no—
122C2×D4×C53—22no—
123C2×D4×C53—41no—
124C22×C4×C53—22no—
125C22×C4×C53—22no—
126C22×C4×C53—21no—
127C22×C4×C53—21no—
128C22×C4×C53—21no—
129C22×C4×C53—21no—
130C22×C4×C53—21no—
131C22×C4×C53—21no—
132C22×C4×C53—22no—
133C22×C4×C53—22no—
134C22×C4×C53C2×D411yescharacteristic
135C22×C4×C53C2×D411yescharacteristic
136C22×C4×C53C2×D411yescharacteristic
137C22×C4×C53C2×D412yes—
138C22×C4×C53C2×D412yes—
139C22×C4×C53C2×D411yescharacteristic
140C22×C4×C53C2411yescharacteristic
141C22×Dic53—22no—
142C22×Dic53—22no—
143C22×Dic53—42no—
144C22×Dic53—42no—
145C22×Dic53—42no—
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—24no—
161C22⋊C4×C53—22no—
162C22⋊C4×C53—22no—
163C42×C53—24no—
164C42×C53—24no—
165C42×C53—24no—
166C42×C53—21no—
167C42×C53—21no—
168C22⋊Dic53—44no—
169C22⋊Dic53—42no—
170C4⋊Dic53—22no—
171C4⋊Dic53—22no—
172C4⋊Dic53—22no—
173C4⋊Dic53—22no—
174C4⋊Dic53—42no—
175#11719919—44no—
176#11719917—28no—
177C2×#88322—22no—
178C2×#88322—22no—
179C2×#88322—22no—
180C2×#88322—22no—
181C2×#88322—22no—
182C2×#88322#4114yes—
183C23×C53—22no—
184C23×C53—21no—
185C23×C53—21no—
186C23×C53—22no—
187C23×C53—21no—
188C23×C53—22no—
189C23×C53—21no—
190C23×C53—21no—
191C23×C53—22no—
192C23×C53#12012yes—
193C23×C53C22×D411yescharacteristic
194Q8×C53—24no—
195Q8×C53—48no—
196Q8×C53—41no—
197Q8×C53—41no—
198D4×C53—44no—
199C2×C4×C53—21no—
200C2×C4×C53—22no—
201C2×C4×C53—21no—
202C2×C4×C53—22no—
203C2×C4×C53—22no—
204C2×C4×C53—24no—
205C2×C4×C53—21no—
206C2×C4×C53—22no—
207C2×C4×C53—22no—
208C2×C4×C53—22no—
209C2×C4×C53—22no—
210C2×C4×C53—21no—
211C2×C4×C53—22no—
212C2×C4×C53—22no—
213C2×C4×C53—42no—
214C2×C4×C53—41no—
215C2×C4×C53—41no—
216C2×C4×C53—41no—
217C2×C4×C53—41no—
218C2×C4×C53—42no—
219C2×C4×C53—41no—
220C2×C4×C53—41no—
221C2×C4×C53—42no—
222C2×C4×C53#12012yes—
223C2×C4×C53AGL(1,ℤ/8)12yes—
224C2×C4×C53C22×D411yescharacteristic
225C2×C4×C53C22×D411yescharacteristic
226C2×C4×C53C2×#4112yes—
227C2×C4×C532−1+412yes—
228C2×Dic53—42no—
229C2×Dic53—42no—
230C2×Dic53—42no—
231C2×Dic53—42no—
232C2×Dic53—42no—
233C2×Dic53—42no—
234C2×Dic53—84no—
235#88322—24no—
236#88322—42no—
237#24107—531nomaximal, Sylow
238C22×C53—22no—
239C22×C53—22no—
240C22×C53—22no—
241C22×C53—42no—
242C22×C53—42no—
243C22×C53—42no—
244C22×C53—42no—
245C22×C53—41no—
246C22×C53—41no—
247C22×C53—41no—
248C22×C53C2×#12011yescharacteristic
249C22×C53#54712yes—
250C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
251C22×C53#57511yescharacteristic
252C22×C53#57812yes—
253C4×C53—22no—
254C4×C53—22no—
255C4×C53—41no—
256C4×C53—42no—
257C4×C53—42no—
258C4×C53—42no—
259C4×C53—41no—
260C4×C53—42no—
261C4×C53#52511yescharacteristic
262C4×C53#53611yescharacteristic
263C4×C53#54712yes—
264Dic53—82no—
265Dic53—82no—
266C2×#556D5311yescharacteristic
267C2×#484—531no—
268C2×#428—531no—
269#2486—531no—
270#2057—532no—
271#2036—532no—
272#1894—531no—
273#1870—531no—
274#1865—532no—
275#1854—532no—
276#1825—531no—
277C2×C53—42no—
278C2×C53—81no—
279C2×C53#240611yescharacteristic
280C2×C53#300611yescharacteristic
281C2×C53#301111yescharacteristic
282C2×C53#334912yes—
283C2×C53#335312yes—
284C22×M16—531no—
285#556—24no—
286#556—24no—
287C2×C4⋊Q8C2×D5311yescharacteristic
288C2×#126—21no—
289C2×#126—21no—
290C2×#124—21no—
291C2×#123—1061no—
292C2×C22⋊Q8C2×D5311yescharacteristic
293C2×C22⋊Q8C2×D5311yescharacteristic
294C2×C4×Q8—21no—
295C2×C4×Q8C2×D5311yescharacteristic
296C2×C4×D4C2×D5311yescharacteristic
297C2×#117C2×D5311yescharacteristic
298C2×#117C2×D5311yescharacteristic
299C22×C4⋊C4—531no—
300#484—1064no—
301#428—1064no—
302C2×#107—532no—
303C2×#107—532no—
304C2×#106—532no—
305C2×#103—532no—
306C2×#103—532no—
307C2×C22⋊C8—531no—
308C2×C22⋊C8—532no—
309#398—1061no—
310#397—1061no—
311#389—532no—
312#381—1061no—
313#380—531no—
314#380—532no—
315#340—532no—
316#340—532no—
317#340—532no—
318#336—532no—
319C53#2410711yescharacteristic, complemented, Sylow
320C22×Q8—21no—
321C22×Q8C22×D5311yescharacteristic
322C22×D4C22⋊D5311yescharacteristic
323C23×C4—1061no—
324C23×C4C22×D5311yescharacteristic
325C23×C4C22×D5311yescharacteristic
326C2×M16—1064no—
327C2×M16—1062no—
328C22×C8—532no—
329C22×C8—532no—
330C4⋊Q8—22no—
331C4⋊Q8—22no—
332#126—28no—
333#126—28no—
334#124—28no—
335#123—2124no—
336C22⋊Q8—22no—
337C22⋊Q8—22no—
338C22⋊Q8—22no—
339C22⋊Q8C22⋊D5314yes—
340C4×Q8—28no—
341C4×Q8—22no—
342C4×Q8—22no—
343C4×D4—22no—
344C4×D4—22no—
345#117—22no—
346#117—22no—
347#117—22no—
348#117C22⋊D5314yes—
349C2×C4⋊C4—21no—
350C2×C4⋊C4—21no—
351C2×C4⋊C4—21no—
352C2×C4⋊C4—21no—
353C2×C4⋊C4—532no—
354C2×C4⋊C4—532no—
355C2×C4⋊C4—1062no—
356C2×C4⋊C4—1062no—
357C2×C4⋊C4—1061no—
358C2×C4⋊C4—1062no—
359C2×C4⋊C4C22⋊D5311yescharacteristic
360C2×C4⋊C4C22×D5311yescharacteristic
361C2×C4⋊C4C22×D5311yescharacteristic
362C2×C4⋊C4C22×D5311yescharacteristic
363C2×C22⋊C4—21no—
364C2×C22⋊C4—21no—
365C2×C22⋊C4—21no—
366C2×C22⋊C4—21no—
367C2×C22⋊C4—1061no—
368C2×C22⋊C4—1062no—
369C2×C22⋊C4C22⋊D5311yescharacteristic
370C2×C22⋊C4C22⋊D5311yescharacteristic
371C2×C42—21no—
372C2×C42—21no—
373C2×C42—21no—
374C2×C42C22×D5311yescharacteristic
375#107—1064no—
376#107—1064no—
377#106—1064no—
378#103—1064no—
379#103—1064no—
380C22⋊C8—1064no—
381C22⋊C8—1062no—
382#95—1062no—
383#95—1061no—
384#95—1062no—
385C24C2×C22⋊D5311yescharacteristic
386C24C2×C22⋊D5311yescharacteristic
387C2×Q8—24no—
388C2×Q8—21no—
389C2×Q8—22no—
390C2×Q8—22no—
391C2×Q8—42no—
392C2×Q8—42no—
393C2×Q8C2×C22⋊D5312yes—
394C2×D4—22no—
395C2×D4—22no—
396C2×D4—41no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—1062no—
408C22×C4—1062no—
409C22×C4—2122no—
410C22×C4—2122no—
411C22×C4—2122no—
412C22×C4C2×C22⋊D5311yescharacteristic
413C22×C4C2×C22⋊D5311yescharacteristic
414C22×C4C2×C22⋊D5311yescharacteristic
415C22×C4C2×C22⋊D5312yes—
416C22×C4C2×C22⋊D5312yes—
417C22×C4C2×C22⋊D5311yescharacteristic
418C22×C4C23×D5311yescharacteristic
419M16—1068no—
420C2×C8—534no—
421C2×C8—1062no—
422C2×C8—1062no—
423C2×C8—1062no—
424C2×C8—1062no—
425C2×C8—1062no—
426C4⋊C4—24no—
427C4⋊C4—24no—
428C4⋊C4—24no—
429C4⋊C4—24no—
430C4⋊C4—22no—
431C4⋊C4—21no—
432C4⋊C4—21no—
433C4⋊C4—21no—
434C4⋊C4—1062no—
435C4⋊C4—1062no—
436C4⋊C4—1062no—
437C4⋊C4—1062no—
438C4⋊C4—2122no—
439C4⋊C4—2124no—
440C4⋊C4C2×C22⋊D5312yes—
441C4⋊C4C2×C22⋊D5312yes—
442C4⋊C4C2×C22⋊D5312yes—
443C22⋊C4—24no—
444C22⋊C4—24no—
445C22⋊C4—24no—
446C22⋊C4—24no—
447C22⋊C4—22no—
448C22⋊C4—22no—
449C22⋊C4—2122no—
450C22⋊C4—2124no—
451C42—24no—
452C42—24no—
453C42—24no—
454C42—21no—
455C42—21no—
456C23—22no—
457C23—21no—
458C23—21no—
459C23—22no—
460C23—22no—
461C23—21no—
462C23—21no—
463C23—21no—
464C23—22no—
465C23#1313318612yes—
466C23C22×C22⋊D5311yescharacteristic
467Q8—24no—
468Q8—48no—
469Q8—41no—
470Q8—41no—
471D4—44no—
472C2×C4—22no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—24no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—42no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—4244no—
496C2×C4—2122no—
497C2×C4—2122no—
498C2×C4—2122no—
499C2×C4—2122no—
500C2×C4—2122no—
501C2×C4—2122no—
502C2×C4#1313318612yes—
503C2×C4#1313324212yes—
504C2×C4C22×C22⋊D5311yescharacteristic
505C2×C4C22×C22⋊D5311yescharacteristic
506C2×C4#41×D5312yes—
507C2×C4#1313326412yes—
508C8—1064no—
509C8—2122no—
510C22—22no—
511C22—22no—
512C22—22no—
513C22—42no—
514C22—42no—
515C22—42no—
516C22—41no—
517C22—42no—
518C22—41no—
519C22—41no—
520C22C2×#1313318611yescharacteristic
521C22#1681127212yes—
522C22C2×#1313324211yescharacteristic
523C22#1681142111yescharacteristic
524C22#1681143812yes—
525C4—22no—
526C4—22no—
527C4—41no—
528C4—42no—
529C4—42no—
530C4—42no—
531C4—41no—
532C4—42no—
533C4—4242no—
534C4—4242no—
535C4#1681113011yescharacteristic
536C4#1681119411yescharacteristic
537C4#1681127212yes—
538C2—42no—
539C2—81no—
540C2#2027804611yescharacteristic
541C2#2028396711yescharacteristic
542C2#2028401011yescharacteristic
543C2#2028723912yes—
544C2#2028729312yes—
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC2×C53×#556C2
socleC23×C53C22×D4
prime radicalC22×D4×C53D4
hypercenterC2×#556D53
hypocenterC53#24107
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#556

Complements

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

NH ≅ G/Ntype
1. C2×#16810956C2semidirect
2. C2×#16810666C2semidirect
4. #20276040C2semidirect
5. #20275915C2semidirect
6. #20275149C2semidirect
8. #20274983C2semidirect
9. #20274914C2semidirect
10. #20274810C2semidirect
319. C53#24107semidirect

Maximal subgroups

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

subgroupindex
C2×#168109562
C2×#168106662
#202790232
#202760402
#202759152
#202751492
#202750132
#202749832
#202749142
#202748102
C2×C53×#5562
#2410753