Finite Group Databasebeta

← #27417769

Subgroups — #27417769

Order 13568.

1138 conjugacy classes of subgroups.

The subgroup lattice has 548 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811330C211yesmaximal, characteristic
2C2×#16810953C211yesmaximal, characteristic, complemented
3C22×#13133148C211yesmaximal, characteristic, complemented
4#20276048C212yesmaximal, complemented
5#20275911C212yesmaximal, complemented
6#20275136C211yesmaximal, characteristic, complemented
7#20275008C211yesmaximal, characteristic
8#20274982C212yesmaximal, complemented
9#20274907C212yesmaximal, complemented
10#20274803C211yesmaximal, characteristic, complemented
11C53×#2486C211yesmaximal, characteristic
12C23×#88322C2211yescharacteristic
13#16811330—24no—
14#16811330—24no—
15C2×#13133217C2211yescharacteristic
16C2×#13133208—21no—
17C2×#13133196C2211yescharacteristic
18C2×#13133196C2211yescharacteristic
19C2×#13133194—21no—
20C2×#13133194—21no—
21C2×C4×Dic106C2211yescharacteristic
22C2×D4×Dic53—21no—
23C22×C4⋊Dic53C2211yescharacteristic
24C22×C4⋊Dic53C2211yescharacteristic
25#16810953—24no—
26C2×#13133148—24no—
27C2×#13133148C2212yes—
28C2×#13133148C2212yes—
29C2×#13133148C2212yes—
30C2×#13133140C2212yes—
31C2×#13133140C2212yes—
32C2×C22⋊#88322C2211yescharacteristic
33C2×C22⋊#88322C2212yes—
34#16810385C2212yes—
35#16810385C2212yes—
36#16810385C2212yes—
37#16810378C2212yes—
38C2×#123×C53—21no—
39C2×C4×D4×C53C2211yescharacteristic
40C22×C4⋊C4×C53C2211yescharacteristic
41C53×#398—21no—
42C53×#397—21no—
43C53×#389C2212yes—
44C53×#381—21no—
45C53×#380C2211yescharacteristic
46C53×#380C2212yes—
47C22×Dic106—21no—
48C22×Dic106C2311yescharacteristic
49C23×Dic53—21no—
50C23×Dic53—21no—
51C22×#88322—24no—
52C22×#88322—22no—
53C22×#88322C2312yes—
54C22×#88322C2312yes—
55#13133217—22no—
56#13133217—22no—
57#13133208—28no—
58#13133196—22no—
59#13133196—22no—
60#13133196—22no—
61#13133196D414yes—
62#13133194—28no—
63#13133194—28no—
64C4×Dic106—22no—
65C4×Dic106—22no—
66D4×Dic53—28no—
67C2×C4⋊Dic53—22no—
68C2×C4⋊Dic53—22no—
69C2×C4⋊Dic53—22no—
70C2×C4⋊Dic53—21no—
71C2×C4⋊Dic53—21no—
72C2×C4⋊Dic53—21no—
73C2×C4⋊Dic53D414yes—
74C2×C4⋊Dic53C2311yescharacteristic
75C2×C4⋊Dic53C2311yescharacteristic
76C2×C4⋊Dic53C2311yescharacteristic
77C2×#11719919—21no—
78C2×#11719919—21no—
79C2×#11719919—21no—
80C2×C22⋊Dic53—21no—
81C2×C22⋊Dic53—21no—
82C2×C4×Dic53—21no—
83#13133148—24no—
84#13133148—24no—
85#13133148—24no—
86#13133140—24no—
87#13133140—24no—
88C22⋊#88322—24no—
89C22⋊#88322—22no—
90C22×D4×C53D411yescharacteristic
91C23×C4×C53—21no—
92C23×C4×C53C2311yescharacteristic
93C23×C4×C53C2311yescharacteristic
94#123×C53—44no—
95C4×D4×C53—22no—
96C4×D4×C53—22no—
97C2×C4⋊C4×C53—21no—
98C2×C4⋊C4×C53—22no—
99C2×C4⋊C4×C53—22no—
100C2×C4⋊C4×C53—22no—
101C2×C4⋊C4×C53D411yescharacteristic
102C2×C4⋊C4×C53C2312yes—
103C2×C4⋊C4×C53C2312yes—
104C2×C22⋊C4×C53—21no—
105C2×C22⋊C4×C53—22no—
106C2×C22⋊C4×C53D411yescharacteristic
107C2×C22⋊C4×C53D411yescharacteristic
108C2×C42×C53C2311yescharacteristic
109#95×C53—22no—
110#95×C53—21no—
111#95×C53—22no—
112C24×C53C2×D411yescharacteristic
113C24×C53C2×D411yescharacteristic
114C2×D4×C53—22no—
115C2×D4×C53—22no—
116C2×D4×C53SD1614yes—
117C22×C4×C53—22no—
118C22×C4×C53—22no—
119C22×C4×C53—21no—
120C22×C4×C53—21no—
121C22×C4×C53—22no—
122C22×C4×C53—22no—
123C22×C4×C53—22no—
124C22×C4×C53—22no—
125C22×C4×C53—42no—
126C22×C4×C53—42no—
127C22×C4×C53—42no—
128C22×C4×C53C2×D411yescharacteristic
129C22×C4×C53C2×D411yescharacteristic
130C22×C4×C53C2×D411yescharacteristic
131C22×C4×C53C2×D412yes—
132C22×C4×C53C2×D412yes—
133C22×C4×C53C2×D411yescharacteristic
134C22×C4×C53C2411yescharacteristic
135C22×Dic53—21no—
136C22×Dic53—21no—
137C22×Dic53—21no—
138C22×Dic53—21no—
139C22×Dic53—24no—
140C22×Dic53—24no—
141C22×Dic53—82no—
142C22×Dic53—82no—
143C2×Dic106—22no—
144C2×Dic106—22no—
145C2×Dic106—21no—
146C2×Dic106—24no—
147C2×Dic106—82no—
148C2×Dic106C2×D412yes—
149C4⋊C4×C53—22no—
150C4⋊C4×C53—22no—
151C4⋊C4×C53—22no—
152C4⋊C4×C53—22no—
153C4⋊C4×C53—22no—
154C4⋊C4×C53—42no—
155C4⋊C4×C53—44no—
156C22⋊C4×C53—22no—
157C22⋊C4×C53—22no—
158C22⋊C4×C53—44no—
159C22⋊C4×C53—42no—
160C42×C53—21no—
161C42×C53—21no—
162C22⋊Dic53—24no—
163C22⋊Dic53—24no—
164C4⋊Dic53—24no—
165C4⋊Dic53—24no—
166C4⋊Dic53—24no—
167C4⋊Dic53—21no—
168C4⋊Dic53—21no—
169C4⋊Dic53—21no—
170C4⋊Dic53C2×D412yes—
171C4⋊Dic53C2×D412yes—
172C4⋊Dic53C2×D412yes—
173#11719919—24no—
174#11719919—24no—
175#11719919—24no—
176C4×Dic53—24no—
177C2×#88322—28no—
178C2×#88322—22no—
179C2×#88322—22no—
180C2×#88322—22no—
181C2×#88322—22no—
182C2×#88322—22no—
183C2×#88322#4114yes—
184C23×C53—22no—
185C23×C53—22no—
186C23×C53—21no—
187C23×C53—22no—
188C23×C53—21no—
189C23×C53—22no—
190C23×C53#12012yes—
191C23×C53C2×SD1612yes—
192C23×C53C2×SD1612yes—
193C23×C53#13712yes—
194C23×C53C22×D411yescharacteristic
195D4×C53—28no—
196C2×C4×C53—21no—
197C2×C4×C53—22no—
198C2×C4×C53—21no—
199C2×C4×C53—22no—
200C2×C4×C53—22no—
201C2×C4×C53—24no—
202C2×C4×C53—21no—
203C2×C4×C53—21no—
204C2×C4×C53—22no—
205C2×C4×C53—22no—
206C2×C4×C53—41no—
207C2×C4×C53—41no—
208C2×C4×C53—42no—
209C2×C4×C53—42no—
210C2×C4×C53—42no—
211C2×C4×C53—42no—
212C2×C4×C53—42no—
213C2×C4×C53—42no—
214C2×C4×C53—42no—
215C2×C4×C53—84no—
216C2×C4×C53#12012yes—
217C2×C4×C53C2×SD1612yes—
218C2×C4×C53C22×D411yescharacteristic
219C2×C4×C53C22×D411yescharacteristic
220C2×C4×C53C2×#4112yes—
221C2×C4×C532−1+412yes—
222C2×Dic53—22no—
223C2×Dic53—22no—
224C2×Dic53—22no—
225C2×Dic53—22no—
226C2×Dic53—42no—
227C2×Dic53—42no—
228C2×Dic53—42no—
229C2×Dic53—42no—
230C2×Dic53—84no—
231C2×Dic53—84no—
232Dic106—24no—
233Dic106—41no—
234Dic106—41no—
235Dic106—84no—
236#88322—24no—
237#88322—42no—
238#24097—531nomaximal, Sylow
239C22×C53—24no—
240C22×C53—24no—
241C22×C53—22no—
242C22×C53—22no—
243C22×C53—24no—
244C22×C53—22no—
245C22×C53—42no—
246C22×C53—41no—
247C22×C53—41no—
248C22×C53#44814yes—
249C22×C53C2×#12011yescharacteristic
250C22×C53#54712yes—
251C22×C53C22×SD1611yescharacteristic
252C22×C53C2×#13711yescharacteristic
253C22×C53#57712yes—
254C4×C53—22no—
255C4×C53—22no—
256C4×C53—41no—
257C4×C53—41no—
258C4×C53—82no—
259C4×C53—82no—
260C4×C53#52511yescharacteristic
261C4×C53#53611yescharacteristic
262C4×C53#54712yes—
263Dic53—42no—
264Dic53—42no—
265Dic53—42no—
266Dic53—42no—
267C2×#554—531no—
268C2×#483—531no—
269C22×#106—531no—
270#2486D5311yescharacteristic
271#2058—532no—
272#2035—532no—
273#1892—531no—
274#1869—531no—
275#1865—532no—
276#1853—532no—
277#1824—531no—
278C2×C53—24no—
279C2×C53—42no—
280C2×C53#240611yescharacteristic
281C2×C53C2×#44811yescharacteristic
282C2×C53#300911yescharacteristic
283C2×C53#333512yes—
284C2×C53#334412yes—
285C23×C8—531no—
286#554—1064no—
287#554—1064no—
288C2×C4⋊Q8—531no—
289C2×#125—1061no—
290C2×#123—21no—
291C2×C22⋊Q8—531no—
292C2×C22⋊Q8—531no—
293C2×C22⋊Q8—1061no—
294C2×C22⋊Q8—1061no—
295C2×C4×Q8—531no—
296C2×C4×D4—1061no—
297C2×C4×D4C2×D5311yescharacteristic
298C22×C4⋊C4—531no—
299C22×C4⋊C4—531no—
300C22×C4⋊C4C2×D5311yescharacteristic
301#483—1064no—
302C2×#106—532no—
303C2×#106—532no—
304C2×#106—532no—
305C2×#106—1064no—
306C2×#103—532no—
307C2×#103—532no—
308C2×C22⋊C8—531no—
309C2×C22⋊C8—532no—
310#398—21no—
311#397—21no—
312#389C2×D5312yes—
313#381—21no—
314#380C2×D5311yescharacteristic
315#380C2×D5312yes—
316#340—532no—
317#340—532no—
318#340—532no—
319#336—532no—
320C53#2409711yescharacteristic, complemented, Sylow
321C22×Q8—531no—
322C22×Q8—1061no—
323C22×D4C22⋊D5311yescharacteristic
324C23×C4—21no—
325C23×C4—1061no—
326C23×C4—1061no—
327C23×C4C22×D5311yescharacteristic
328C23×C4C22×D5311yescharacteristic
329C22×C8—532no—
330C22×C8—532no—
331C22×C8—1064no—
332C22×C8—1062no—
333C4⋊Q8—1062no—
334C4⋊Q8—1062no—
335#125—1068no—
336#123—44no—
337C22⋊Q8—534no—
338C22⋊Q8—1068no—
339C22⋊Q8—1068no—
340C22⋊Q8—1062no—
341C22⋊Q8—1062no—
342C22⋊Q8—1062no—
343C4×Q8—1062no—
344C4×Q8—1062no—
345C4×D4—22no—
346C4×D4—22no—
347C4×D4—1068no—
348C2×C4⋊C4—22no—
349C2×C4⋊C4—22no—
350C2×C4⋊C4—21no—
351C2×C4⋊C4—22no—
352C2×C4⋊C4—534no—
353C2×C4⋊C4—531no—
354C2×C4⋊C4—531no—
355C2×C4⋊C4—531no—
356C2×C4⋊C4—1062no—
357C2×C4⋊C4—1062no—
358C2×C4⋊C4—1062no—
359C2×C4⋊C4—1061no—
360C2×C4⋊C4—1061no—
361C2×C4⋊C4—1061no—
362C2×C4⋊C4—1061no—
363C2×C4⋊C4—1061no—
364C2×C4⋊C4—1061no—
365C2×C4⋊C4C22⋊D5311yescharacteristic
366C2×C4⋊C4C22×D5312yes—
367C2×C4⋊C4C22×D5312yes—
368C2×C22⋊C4—21no—
369C2×C22⋊C4—22no—
370C2×C22⋊C4—1061no—
371C2×C22⋊C4—1061no—
372C2×C22⋊C4C22⋊D5311yescharacteristic
373C2×C22⋊C4C22⋊D5311yescharacteristic
374C2×C42—1061no—
375C2×C42C22×D5311yescharacteristic
376#106—1064no—
377#106—1064no—
378#106—1064no—
379#103—1064no—
380#103—1064no—
381C22⋊C8—1064no—
382C22⋊C8—1062no—
383#95—22no—
384#95—21no—
385#95—22no—
386C24C2×C22⋊D5311yescharacteristic
387C24C2×C22⋊D5311yescharacteristic
388C2×Q8—532no—
389C2×Q8—1064no—
390C2×Q8—1061no—
391C2×Q8—1062no—
392C2×Q8—1062no—
393C2×Q8—4242no—
394C2×D4—22no—
395C2×D4—22no—
396C2×D4#1171992314yes—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—1064no—
409C22×C4—1064no—
410C22×C4—1061no—
411C22×C4—1061no—
412C22×C4—1061no—
413C22×C4—1061no—
414C22×C4—4242no—
415C22×C4—4242no—
416C22×C4D4×D5312yes—
417C22×C4D4×D5312yes—
418C22×C4C2×C22⋊D5311yescharacteristic
419C22×C4C2×C22⋊D5311yescharacteristic
420C22×C4C2×C22⋊D5311yescharacteristic
421C22×C4C2×C22⋊D5311yescharacteristic
422C22×C4C23×D5311yescharacteristic
423C2×C8—534no—
424C2×C8—1068no—
425C2×C8—1062no—
426C2×C8—1062no—
427C2×C8—1062no—
428C2×C8—1062no—
429C2×C8—1062no—
430C4⋊C4—22no—
431C4⋊C4—22no—
432C4⋊C4—22no—
433C4⋊C4—22no—
434C4⋊C4—22no—
435C4⋊C4—42no—
436C4⋊C4—44no—
437C4⋊C4—532no—
438C4⋊C4—532no—
439C4⋊C4—532no—
440C4⋊C4—1064no—
441C4⋊C4—1064no—
442C4⋊C4—1064no—
443C4⋊C4—1064no—
444C4⋊C4—1064no—
445C4⋊C4—1064no—
446C4⋊C4—1061no—
447C4⋊C4—1061no—
448C4⋊C4—1061no—
449C22⋊C4—22no—
450C22⋊C4—22no—
451C22⋊C4—42no—
452C22⋊C4—44no—
453C22⋊C4—1064no—
454C22⋊C4—1064no—
455C42—21no—
456C42—21no—
457C42—1064no—
458C23—22no—
459C23—22no—
460C23—22no—
461C23—21no—
462C23—21no—
463C23—22no—
464C23#1313318512yes—
465C23C2×#1171992312yes—
466C23C2×#1171992312yes—
467C23#1313324712yes—
468C23C22×C22⋊D5311yescharacteristic
469Q8—1064no—
470Q8—4244no—
471Q8—2121no—
472Q8—2121no—
473D4—28no—
474C2×C4—22no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—24no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—22no—
484C2×C4—42no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—84no—
494C2×C4—1062no—
495C2×C4—1062no—
496C2×C4—1062no—
497C2×C4—1062no—
498C2×C4—4244no—
499C2×C4—4244no—
500C2×C4—2122no—
501C2×C4—2122no—
502C2×C4—2122no—
503C2×C4—2122no—
504C2×C4#1313318512yes—
505C2×C4C2×#1171992312yes—
506C2×C4C2×D4×D5311yescharacteristic
507C2×C4C2×D4×D5311yescharacteristic
508C2×C4#41×D5312yes—
509C2×C4#1313326412yes—
510C8—1064no—
511C8—2122no—
512C22—22no—
513C22—22no—
514C22—24no—
515C22—24no—
516C22—24no—
517C22—22no—
518C22—41no—
519C22—41no—
520C22—42no—
521C22#1681076114yes—
522C22C2×#1313318511yescharacteristic
523C22#1681127212yes—
524C22C22×#1171992311yescharacteristic
525C22C2×#1313324711yescharacteristic
526C22#1681143112yes—
527C4—22no—
528C4—22no—
529C4—41no—
530C4—41no—
531C4—82no—
532C4—82no—
533C4—2122no—
534C4—2122no—
535C4—2122no—
536C4—2122no—
537C4#1681113111yescharacteristic
538C4#1681119511yescharacteristic
539C4#1681127112yes—
540C2—24no—
541C2—42no—
542C2#2027804711yescharacteristic
543C2C2×#1681076111yescharacteristic
544C2#2028399011yescharacteristic
545C2#2028707212yes—
546C2#2028717012yes—
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2486C2
socleC23×C53C22×D4
prime radicalC22×D4×C53D4
hypercenter#2486D53
hypocenterC53#24097
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #2486

Complements

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

NH ≅ G/Ntype
2. C2×#16810953C2semidirect
3. C22×#13133148C2semidirect
4. #20276048C2semidirect
5. #20275911C2semidirect
6. #20275136C2semidirect
8. #20274982C2semidirect
9. #20274907C2semidirect
10. #20274803C2semidirect
320. C53#24097semidirect

Maximal subgroups

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

subgroupindex
C2×#168113302
C2×#168109532
C22×#131331482
#202760482
#202759112
#202751362
#202750082
#202749822
#202749072
#202748032
C53×#24862
#2409753