Finite Group Databasebeta

← #27417782

Subgroups — #27417782

Order 13568.

1118 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811330C211yesmaximal, characteristic
2C2×#16810959C211yesmaximal, characteristic, complemented
3C22×#13133151C211yesmaximal, characteristic, complemented
4#20279032C211yesmaximal, characteristic
5#20276044C212yesmaximal, complemented
6#20275919C212yesmaximal, complemented
7#20275138C211yesmaximal, characteristic, complemented
8#20274972C212yesmaximal, complemented
9#20274905C212yesmaximal, complemented
10#20274806C211yesmaximal, characteristic, complemented
11C53×#1869C211yesmaximal, characteristic
12#16811330—24no—
13#16811330—24no—
14C2×#13133217C2211yescharacteristic
15C2×#13133208—21no—
16C2×#13133196C2211yescharacteristic
17C2×#13133196C2211yescharacteristic
18C2×#13133194—21no—
19C2×#13133194—21no—
20C2×C4×Dic106C2211yescharacteristic
21C2×D4×Dic53—21no—
22C22×C4⋊Dic53C2211yescharacteristic
23C22×C4⋊Dic53C2211yescharacteristic
24C22×C4⋊Dic53C2211yescharacteristic
25#16810959—24no—
26C2×#13133151—24no—
27C2×#13133151C2212yes—
28C2×#13133151C2212yes—
29C2×#13133151C2212yes—
30C2×#13133142C2212yes—
31C2×#13133142C2212yes—
32#16810562—21no—
33#16810559—21no—
34#16810521C2212yes—
35#16810503—21no—
36#16810495—21no—
37#16810483C2211yescharacteristic
38#16810483C2212yes—
39#16810386C2212yes—
40#16810386C2212yes—
41#16810386C2212yes—
42C23×C8×C53C2211yescharacteristic
43C2×C4×D4×C53C2211yescharacteristic
44C2×C22⋊C8×C53C2211yescharacteristic
45C2×C22⋊C8×C53C2212yes—
46C53×#336C2212yes—
47C22×Dic106—21no—
48C22×Dic106C2311yescharacteristic
49C23×Dic53—21no—
50C23×Dic53—21no—
51C23×Dic53—21no—
52#13133217—22no—
53#13133217—22no—
54#13133208—28no—
55#13133196—22no—
56#13133196—22no—
57#13133196—22no—
58#13133196D414yes—
59#13133194—28no—
60#13133194—28no—
61C4×Dic106—22no—
62C4×Dic106—22no—
63D4×Dic53—28no—
64C2×C4⋊Dic53—22no—
65C2×C4⋊Dic53—22no—
66C2×C4⋊Dic53—22no—
67C2×C4⋊Dic53—21no—
68C2×C4⋊Dic53—21no—
69C2×C4⋊Dic53—21no—
70C2×C4⋊Dic53—21no—
71C2×C4⋊Dic53—22no—
72C2×C4⋊Dic53—22no—
73C2×C4⋊Dic53D414yes—
74C2×C4⋊Dic53C2312yes—
75C2×C4⋊Dic53C2312yes—
76C2×C4⋊Dic53C2311yescharacteristic
77C2×C4⋊Dic53C2311yescharacteristic
78C2×C4⋊Dic53C2311yescharacteristic
79C2×#11719919—21no—
80C2×#11719919—21no—
81C2×#11719919—21no—
82C2×#11719919—22no—
83C2×C22⋊Dic53—21no—
84C2×C22⋊Dic53—21no—
85C2×C22⋊Dic53—21no—
86C2×C22⋊Dic53—22no—
87C2×C4×Dic53—21no—
88#13133151—24no—
89#13133151—24no—
90#13133151—24no—
91#13133142—24no—
92#13133142—24no—
93#13133124—22no—
94#13133124—21no—
95#13133124—22no—
96C22×D4×C53D411yescharacteristic
97C23×C4×C53C2311yescharacteristic
98C23×C4×C53C2311yescharacteristic
99C22×C8×C53—24no—
100C22×C8×C53—22no—
101C22×C8×C53C2312yes—
102C22×C8×C53C2312yes—
103C4×D4×C53—22no—
104C4×D4×C53—22no—
105C2×C4⋊C4×C53D411yescharacteristic
106C2×C22⋊C4×C53D411yescharacteristic
107C2×C22⋊C4×C53D411yescharacteristic
108C2×C42×C53C2311yescharacteristic
109C22⋊C8×C53—24no—
110C22⋊C8×C53—22no—
111C24×C53C2×D411yescharacteristic
112C24×C53C2×D411yescharacteristic
113C2×D4×C53—22no—
114C2×D4×C53—22no—
115C2×D4×C53Q1614yes—
116C22×C4×C53—22no—
117C22×C4×C53—22no—
118C22×C4×C53—21no—
119C22×C4×C53—21no—
120C22×C4×C53—22no—
121C22×C4×C53—22no—
122C22×C4×C53C2×D411yescharacteristic
123C22×C4×C53C2×D411yescharacteristic
124C22×C4×C53C2×D411yescharacteristic
125C22×C4×C53C2×D412yes—
126C22×C4×C53C2×D412yes—
127C22×C4×C53C2×D411yescharacteristic
128C22×C4×C53C2411yescharacteristic
129C22×Dic53—21no—
130C22×Dic53—21no—
131C22×Dic53—21no—
132C22×Dic53—21no—
133C22×Dic53—24no—
134C22×Dic53—24no—
135C22×Dic53—22no—
136C22×Dic53—22no—
137C22×Dic53—42no—
138C22×Dic53—82no—
139C22×Dic53—82no—
140C22×Dic53—82no—
141C2×Dic106—22no—
142C2×Dic106—22no—
143C2×Dic106—21no—
144C2×Dic106—24no—
145C2×Dic106—82no—
146C2×Dic106C2×D412yes—
147C2×C8×C53—28no—
148C2×C8×C53—22no—
149C2×C8×C53—22no—
150C2×C8×C53—22no—
151C2×C8×C53—22no—
152C2×C8×C53—22no—
153C2×C8×C53#4114yes—
154C4⋊C4×C53—22no—
155C22⋊C4×C53—22no—
156C22⋊C4×C53—22no—
157C42×C53—21no—
158C42×C53—21no—
159C22⋊Dic53—24no—
160C22⋊Dic53—24no—
161C22⋊Dic53—42no—
162C22⋊Dic53—82no—
163C4⋊Dic53—24no—
164C4⋊Dic53—24no—
165C4⋊Dic53—24no—
166C4⋊Dic53—21no—
167C4⋊Dic53—21no—
168C4⋊Dic53—21no—
169C4⋊Dic53—22no—
170C4⋊Dic53—22no—
171C4⋊Dic53—22no—
172C4⋊Dic53—22no—
173C4⋊Dic53—42no—
174C4⋊Dic53C2×D412yes—
175C4⋊Dic53C2×D412yes—
176C4⋊Dic53C2×D412yes—
177#11719919—24no—
178#11719919—24no—
179#11719919—24no—
180#11719919—82no—
181C4×Dic53—24no—
182C23×C53—22no—
183C23×C53—22no—
184C23×C53—21no—
185C23×C53—22no—
186C23×C53—21no—
187C23×C53—22no—
188C23×C53#12012yes—
189C23×C53C2×Q1612yes—
190C23×C53C2×Q1612yes—
191C23×C53#13712yes—
192C23×C53C22×D411yescharacteristic
193D4×C53—28no—
194C2×C4×C53—21no—
195C2×C4×C53—22no—
196C2×C4×C53—21no—
197C2×C4×C53—22no—
198C2×C4×C53—22no—
199C2×C4×C53—24no—
200C2×C4×C53—21no—
201C2×C4×C53—21no—
202C2×C4×C53—22no—
203C2×C4×C53—22no—
204C2×C4×C53—41no—
205C2×C4×C53—41no—
206C2×C4×C53—42no—
207C2×C4×C53#12012yes—
208C2×C4×C53C2×Q1612yes—
209C2×C4×C53C22×D411yescharacteristic
210C2×C4×C53C22×D411yescharacteristic
211C2×C4×C53C2×#4112yes—
212C2×C4×C532−1+412yes—
213C2×Dic53—22no—
214C2×Dic53—22no—
215C2×Dic53—22no—
216C2×Dic53—22no—
217C2×Dic53—42no—
218C2×Dic53—42no—
219C2×Dic53—42no—
220C2×Dic53—42no—
221C2×Dic53—42no—
222C2×Dic53—42no—
223C2×Dic53—42no—
224C2×Dic53—42no—
225C2×Dic53—42no—
226C2×Dic53—42no—
227C2×Dic53—84no—
228C2×Dic53—84no—
229C2×Dic53—84no—
230Dic106—24no—
231Dic106—41no—
232Dic106—41no—
233Dic106—84no—
234C8×C53—24no—
235C8×C53—42no—
236#24098—531nomaximal, Sylow
237C22×C53—24no—
238C22×C53—24no—
239C22×C53—22no—
240C22×C53—22no—
241C22×C53—24no—
242C22×C53—22no—
243C22×C53—42no—
244C22×C53—41no—
245C22×C53—41no—
246C22×C53#45114yes—
247C22×C53C2×#12011yescharacteristic
248C22×C53#54712yes—
249C22×C53C22×Q1611yescharacteristic
250C22×C53C2×#13711yescharacteristic
251C22×C53#57612yes—
252C4×C53—22no—
253C4×C53—22no—
254C4×C53—41no—
255C4×C53—41no—
256C4×C53#52511yescharacteristic
257C4×C53#53611yescharacteristic
258C4×C53#54712yes—
259Dic53—42no—
260Dic53—42no—
261Dic53—42no—
262Dic53—42no—
263Dic53—82no—
264Dic53—82no—
265C2×#554—531no—
266C2×#484—531no—
267C22×#107—531no—
268#2487—531no—
269#2057—532no—
270#2036—532no—
271#1892—531no—
272#1869D5311yescharacteristic
273#1863—532no—
274#1852—532no—
275#1824—531no—
276C2×C53—24no—
277C2×C53—42no—
278C2×C53#240611yescharacteristic
279C2×C53C2×#45111yescharacteristic
280C2×C53#300811yescharacteristic
281C2×C53#333812yes—
282C2×C53#334612yes—
283C23×C8C2×D5311yescharacteristic
284#554—1064no—
285#554—1064no—
286C2×C4⋊Q8—531no—
287C2×#125—1061no—
288C2×C22⋊Q8—531no—
289C2×C22⋊Q8—531no—
290C2×C22⋊Q8—1061no—
291C2×C22⋊Q8—1061no—
292C2×C4×Q8—531no—
293C2×C4×D4—1061no—
294C2×C4×D4C2×D5311yescharacteristic
295C22×C4⋊C4—531no—
296C22×C4⋊C4—531no—
297C22×C4⋊C4—531no—
298#484—1064no—
299C2×#107—532no—
300C2×#107—532no—
301C2×#107—532no—
302C2×#107—1064no—
303C2×#103—532no—
304C2×#103—532no—
305C2×C22⋊C8C2×D5311yescharacteristic
306C2×C22⋊C8C2×D5312yes—
307#400—1061no—
308#399—1061no—
309#389—532no—
310#386—1061no—
311#385—1061no—
312#380—531no—
313#380—532no—
314#340—532no—
315#340—532no—
316#340—532no—
317#336C2×D5312yes—
318C53#2409811yescharacteristic, complemented, Sylow
319C22×Q8—531no—
320C22×Q8—1061no—
321C22×D4D21211yescharacteristic
322C23×C4—1061no—
323C23×C4—1061no—
324C23×C4—1061no—
325C23×C4C22×D5311yescharacteristic
326C23×C4C22×D5311yescharacteristic
327C22×C8—22no—
328C22×C8—24no—
329C22×C8C22×D5312yes—
330C22×C8C22×D5312yes—
331C4⋊Q8—1062no—
332C4⋊Q8—1062no—
333#125—1068no—
334C22⋊Q8—534no—
335C22⋊Q8—1068no—
336C22⋊Q8—1068no—
337C22⋊Q8—1062no—
338C22⋊Q8—1062no—
339C22⋊Q8—1062no—
340C4×Q8—1062no—
341C4×Q8—1062no—
342C4×D4—22no—
343C4×D4—22no—
344C4×D4—1068no—
345C2×C4⋊C4—534no—
346C2×C4⋊C4—531no—
347C2×C4⋊C4—531no—
348C2×C4⋊C4—531no—
349C2×C4⋊C4—532no—
350C2×C4⋊C4—532no—
351C2×C4⋊C4—1062no—
352C2×C4⋊C4—1062no—
353C2×C4⋊C4—1061no—
354C2×C4⋊C4—1061no—
355C2×C4⋊C4—1061no—
356C2×C4⋊C4—1061no—
357C2×C4⋊C4—1061no—
358C2×C4⋊C4—1061no—
359C2×C4⋊C4—1062no—
360C2×C4⋊C4—1062no—
361C2×C4⋊C4—1062no—
362C2×C4⋊C4—1062no—
363C2×C4⋊C4—1061no—
364C2×C4⋊C4D21211yescharacteristic
365C2×C22⋊C4—1061no—
366C2×C22⋊C4—1061no—
367C2×C22⋊C4—1062no—
368C2×C22⋊C4—1061no—
369C2×C22⋊C4D21211yescharacteristic
370C2×C22⋊C4D21211yescharacteristic
371C2×C42—1061no—
372C2×C42C22×D5311yescharacteristic
373#107—1064no—
374#107—1064no—
375#107—1064no—
376#103—1064no—
377#103—1064no—
378C22⋊C8—22no—
379C22⋊C8—24no—
380#95—1062no—
381#95—1061no—
382#95—1062no—
383C24C2×D21211yescharacteristic
384C24C2×D21211yescharacteristic
385C2×Q8—532no—
386C2×Q8—1064no—
387C2×Q8—1061no—
388C2×Q8—1062no—
389C2×Q8—1062no—
390C2×Q8—4242no—
391C2×D4—22no—
392C2×D4—22no—
393C2×D4Dic21214yes—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—1064no—
401C22×C4—1064no—
402C22×C4—1061no—
403C22×C4—1061no—
404C22×C4—1061no—
405C22×C4—1061no—
406C22×C4—1062no—
407C22×C4—1062no—
408C22×C4—4242no—
409C22×C4—4242no—
410C22×C4—4242no—
411C22×C4—2122no—
412C22×C4C2×D21211yescharacteristic
413C22×C4C2×D21211yescharacteristic
414C22×C4C2×D21211yescharacteristic
415C22×C4C2×D21211yescharacteristic
416C22×C4D4×D5312yes—
417C22×C4D4×D5312yes—
418C22×C4C23×D5311yescharacteristic
419C2×C8—28no—
420C2×C8—22no—
421C2×C8—22no—
422C2×C8—22no—
423C2×C8—22no—
424C2×C8—22no—
425C2×C8#1171994714yes—
426C4⋊C4—22no—
427C4⋊C4—532no—
428C4⋊C4—532no—
429C4⋊C4—532no—
430C4⋊C4—1064no—
431C4⋊C4—1064no—
432C4⋊C4—1064no—
433C4⋊C4—1064no—
434C4⋊C4—1064no—
435C4⋊C4—1064no—
436C4⋊C4—1061no—
437C4⋊C4—1061no—
438C4⋊C4—1061no—
439C4⋊C4—1062no—
440C4⋊C4—1062no—
441C4⋊C4—1062no—
442C4⋊C4—1062no—
443C4⋊C4—4242no—
444C4⋊C4—2122no—
445C22⋊C4—22no—
446C22⋊C4—22no—
447C22⋊C4—1064no—
448C22⋊C4—1064no—
449C22⋊C4—4242no—
450C22⋊C4—2122no—
451C42—21no—
452C42—21no—
453C42—1064no—
454C23—22no—
455C23—22no—
456C23—22no—
457C23—21no—
458C23—21no—
459C23—22no—
460C23#1313318712yes—
461C23C2×Dic21212yes—
462C23C2×Dic21212yes—
463C23#1313325012yes—
464C23C22×D21211yescharacteristic
465Q8—1064no—
466Q8—4244no—
467Q8—2121no—
468Q8—2121no—
469D4—28no—
470C2×C4—22no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—24no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—22no—
480C2×C4—42no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—1062no—
484C2×C4—1062no—
485C2×C4—1062no—
486C2×C4—1062no—
487C2×C4—4244no—
488C2×C4—4244no—
489C2×C4—4244no—
490C2×C4—2122no—
491C2×C4—2122no—
492C2×C4—2122no—
493C2×C4—2122no—
494C2×C4—2122no—
495C2×C4—2122no—
496C2×C4—2122no—
497C2×C4—2122no—
498C2×C4—2122no—
499C2×C4—2122no—
500C2×C4#1313318712yes—
501C2×C4C2×Dic21212yes—
502C2×C4C2×D4×D5311yescharacteristic
503C2×C4C2×D4×D5311yescharacteristic
504C2×C4C2×#1171994712yes—
505C2×C4#1313326412yes—
506C8—24no—
507C8—42no—
508C22—22no—
509C22—22no—
510C22—24no—
511C22—24no—
512C22—22no—
513C22—24no—
514C22—41no—
515C22—41no—
516C22—42no—
517C22#1681078114yes—
518C22C2×#1313318711yescharacteristic
519C22#1681127712yes—
520C22C22×Dic21211yescharacteristic
521C22C2×#1313325011yescharacteristic
522C22#1681142812yes—
523C4—22no—
524C4—22no—
525C4—41no—
526C4—41no—
527C4—4242no—
528C4—4242no—
529C4—2122no—
530C4—2122no—
531C4—2122no—
532C4—2122no—
533C4#1681112911yescharacteristic
534C4#1681119511yescharacteristic
535C4#1681127312yes—
536C2—24no—
537C2—42no—
538C2#2027804911yescharacteristic
539C2C2×#1681078111yescharacteristic
540C2#2028398511yescharacteristic
541C2#2028710812yes—
542C2#2028720212yes—
subgroupG/H
centerC23C22×D212
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#1869C2
socleC23×C53C22×D4
prime radicalC22×D4×C53D4
hypercenter#1869D53
hypocenterC53#24098
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #1869

Complements

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

NH ≅ G/Ntype
2. C2×#16810959C2semidirect
3. C22×#13133151C2semidirect
5. #20276044C2semidirect
6. #20275919C2semidirect
7. #20275138C2semidirect
8. #20274972C2semidirect
9. #20274905C2semidirect
10. #20274806C2semidirect
318. C53#24098semidirect

Maximal subgroups

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

subgroupindex
C2×#168113302
C2×#168109592
C22×#131331512
#202790322
#202760442
#202759192
#202751382
#202749722
#202749052
#202748062
C53×#18692
#2409853