Finite Group Databasebeta

← #27417780

Subgroups — #27417780

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×#16810958C211yesmaximal, characteristic, complemented
3C22×#13133150C211yesmaximal, characteristic, complemented
4#20276043C212yesmaximal, complemented
5#20275918C212yesmaximal, complemented
6#20275136C211yesmaximal, characteristic, complemented
7#20275008C211yesmaximal, characteristic
8#20274968C212yesmaximal, complemented
9#20274902C212yesmaximal, complemented
10#20274803C211yesmaximal, characteristic, complemented
11C53×#2487C211yesmaximal, 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#16810958—24no—
26C2×#13133150—24no—
27C2×#13133150C2212yes—
28C2×#13133150C2212yes—
29C2×#13133150C2212yes—
30C2×#13133140C2212yes—
31C2×#13133140C2212yes—
32C2×C22⋊#88322C2211yescharacteristic
33C2×C22⋊#88322C2212yes—
34#16810385C2212yes—
35#16810385C2212yes—
36#16810385C2212yes—
37#16810378C2212yes—
38C2×C4×D4×C53C2211yescharacteristic
39C22×C4⋊C4×C53C2211yescharacteristic
40C53×#400—21no—
41C53×#399—21no—
42C53×#389C2212yes—
43C53×#386—21no—
44C53×#385—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#13133150—24no—
84#13133150—24no—
85#13133150—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
94C4×D4×C53—22no—
95C4×D4×C53—22no—
96C2×C4⋊C4×C53—21no—
97C2×C4⋊C4×C53—22no—
98C2×C4⋊C4×C53—22no—
99C2×C4⋊C4×C53—22no—
100C2×C4⋊C4×C53D411yescharacteristic
101C2×C4⋊C4×C53C2312yes—
102C2×C4⋊C4×C53C2312yes—
103C2×C22⋊C4×C53—21no—
104C2×C22⋊C4×C53—22no—
105C2×C22⋊C4×C53D411yescharacteristic
106C2×C22⋊C4×C53D411yescharacteristic
107C2×C42×C53C2311yescharacteristic
108#95×C53—22no—
109#95×C53—21no—
110#95×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×C53—22no—
123C22×C4×C53—22no—
124C22×C4×C53—42no—
125C22×C4×C53—82no—
126C22×C4×C53C2×D411yescharacteristic
127C22×C4×C53C2×D411yescharacteristic
128C22×C4×C53C2×D411yescharacteristic
129C22×C4×C53C2×D412yes—
130C22×C4×C53C2×D412yes—
131C22×C4×C53C2×D411yescharacteristic
132C22×C4×C53C2411yescharacteristic
133C22×Dic53—21no—
134C22×Dic53—21no—
135C22×Dic53—21no—
136C22×Dic53—21no—
137C22×Dic53—24no—
138C22×Dic53—24no—
139C22×Dic53—82no—
140C22×Dic53—82no—
141C2×Dic106—22no—
142C2×Dic106—22no—
143C2×Dic106—21no—
144C2×Dic106—24no—
145C2×Dic106—82no—
146C2×Dic106C2×D412yes—
147C4⋊C4×C53—22no—
148C4⋊C4×C53—22no—
149C4⋊C4×C53—22no—
150C4⋊C4×C53—22no—
151C4⋊C4×C53—22no—
152C4⋊C4×C53—42no—
153C4⋊C4×C53—82no—
154C22⋊C4×C53—22no—
155C22⋊C4×C53—22no—
156C22⋊C4×C53—42no—
157C22⋊C4×C53—82no—
158C42×C53—21no—
159C42×C53—21no—
160C22⋊Dic53—24no—
161C22⋊Dic53—24no—
162C4⋊Dic53—24no—
163C4⋊Dic53—24no—
164C4⋊Dic53—24no—
165C4⋊Dic53—21no—
166C4⋊Dic53—21no—
167C4⋊Dic53—21no—
168C4⋊Dic53C2×D412yes—
169C4⋊Dic53C2×D412yes—
170C4⋊Dic53C2×D412yes—
171#11719919—24no—
172#11719919—24no—
173#11719919—24no—
174C4×Dic53—24no—
175C2×#88322—28no—
176C2×#88322—22no—
177C2×#88322—22no—
178C2×#88322—22no—
179C2×#88322—22no—
180C2×#88322—22no—
181C2×#88322#4114yes—
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—42no—
208C2×C4×C53—42no—
209C2×C4×C53—42no—
210C2×C4×C53—42no—
211C2×C4×C53—42no—
212C2×C4×C53—42no—
213C2×C4×C53—84no—
214C2×C4×C53#12012yes—
215C2×C4×C53C2×Q1612yes—
216C2×C4×C53C22×D411yescharacteristic
217C2×C4×C53C22×D411yescharacteristic
218C2×C4×C53C2×#4112yes—
219C2×C4×C532−1+412yes—
220C2×Dic53—22no—
221C2×Dic53—22no—
222C2×Dic53—22no—
223C2×Dic53—22no—
224C2×Dic53—42no—
225C2×Dic53—42no—
226C2×Dic53—42no—
227C2×Dic53—42no—
228C2×Dic53—84no—
229C2×Dic53—84no—
230Dic106—24no—
231Dic106—41no—
232Dic106—41no—
233Dic106—84no—
234#88322—24no—
235#88322—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—82no—
257C4×C53—82no—
258C4×C53#52511yescharacteristic
259C4×C53#53611yescharacteristic
260C4×C53#54712yes—
261Dic53—42no—
262Dic53—42no—
263Dic53—42no—
264Dic53—42no—
265C2×#554—531no—
266C2×#484—531no—
267C22×#107—531no—
268#2487D5311yescharacteristic
269#2057—532no—
270#2036—532no—
271#1892—531no—
272#1869—531no—
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×C8—531no—
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⋊C4C2×D5311yescharacteristic
298#484—1064no—
299C2×#107—532no—
300C2×#107—532no—
301C2×#107—532no—
302C2×#107—1064no—
303C2×#103—532no—
304C2×#103—532no—
305C2×C22⋊C8—531no—
306C2×C22⋊C8—532no—
307#400—21no—
308#399—21no—
309#389C2×D5312yes—
310#386—21no—
311#385—21no—
312#380C2×D5311yescharacteristic
313#380C2×D5312yes—
314#340—532no—
315#340—532no—
316#340—532no—
317#336—532no—
318C53#2409811yescharacteristic, complemented, Sylow
319C22×Q8—531no—
320C22×Q8—1061no—
321C22×D4C22⋊D5311yescharacteristic
322C23×C4—21no—
323C23×C4—1061no—
324C23×C4—1061no—
325C23×C4C22×D5311yescharacteristic
326C23×C4C22×D5311yescharacteristic
327C22×C8—532no—
328C22×C8—532no—
329C22×C8—1064no—
330C22×C8—1062no—
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—22no—
346C2×C4⋊C4—22no—
347C2×C4⋊C4—21no—
348C2×C4⋊C4—22no—
349C2×C4⋊C4—534no—
350C2×C4⋊C4—531no—
351C2×C4⋊C4—531no—
352C2×C4⋊C4—531no—
353C2×C4⋊C4—1062no—
354C2×C4⋊C4—1062no—
355C2×C4⋊C4—1062no—
356C2×C4⋊C4—1061no—
357C2×C4⋊C4—1061no—
358C2×C4⋊C4—1061no—
359C2×C4⋊C4—1061no—
360C2×C4⋊C4—1061no—
361C2×C4⋊C4—1061no—
362C2×C4⋊C4C22⋊D5311yescharacteristic
363C2×C4⋊C4C22×D5312yes—
364C2×C4⋊C4C22×D5312yes—
365C2×C22⋊C4—21no—
366C2×C22⋊C4—22no—
367C2×C22⋊C4—1061no—
368C2×C22⋊C4—1061no—
369C2×C22⋊C4C22⋊D5311yescharacteristic
370C2×C22⋊C4C22⋊D5311yescharacteristic
371C2×C42—1061no—
372C2×C42C22×D5311yescharacteristic
373#107—1064no—
374#107—1064no—
375#107—1064no—
376#103—1064no—
377#103—1064no—
378C22⋊C8—1064no—
379C22⋊C8—1062no—
380#95—22no—
381#95—21no—
382#95—22no—
383C24C2×C22⋊D5311yescharacteristic
384C24C2×C22⋊D5311yescharacteristic
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×D4#1171992514yes—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—42no—
403C22×C4—82no—
404C22×C4—1064no—
405C22×C4—1064no—
406C22×C4—1061no—
407C22×C4—1061no—
408C22×C4—1061no—
409C22×C4—1061no—
410C22×C4—4242no—
411C22×C4—4242no—
412C22×C4D4×D5312yes—
413C22×C4D4×D5312yes—
414C22×C4C2×C22⋊D5311yescharacteristic
415C22×C4C2×C22⋊D5311yescharacteristic
416C22×C4C2×C22⋊D5311yescharacteristic
417C22×C4C2×C22⋊D5311yescharacteristic
418C22×C4C23×D5311yescharacteristic
419C2×C8—534no—
420C2×C8—1068no—
421C2×C8—1062no—
422C2×C8—1062no—
423C2×C8—1062no—
424C2×C8—1062no—
425C2×C8—1062no—
426C4⋊C4—22no—
427C4⋊C4—22no—
428C4⋊C4—22no—
429C4⋊C4—22no—
430C4⋊C4—22no—
431C4⋊C4—42no—
432C4⋊C4—82no—
433C4⋊C4—532no—
434C4⋊C4—532no—
435C4⋊C4—532no—
436C4⋊C4—1064no—
437C4⋊C4—1064no—
438C4⋊C4—1064no—
439C4⋊C4—1064no—
440C4⋊C4—1064no—
441C4⋊C4—1064no—
442C4⋊C4—1061no—
443C4⋊C4—1061no—
444C4⋊C4—1061no—
445C22⋊C4—22no—
446C22⋊C4—22no—
447C22⋊C4—42no—
448C22⋊C4—82no—
449C22⋊C4—1064no—
450C22⋊C4—1064no—
451C42—21no—
452C42—21no—
453C42—1064no—
454C23—22no—
455C23—22no—
456C23—22no—
457C23—21no—
458C23—21no—
459C23—22no—
460C23#1313318512yes—
461C23C2×#1171992512yes—
462C23C2×#1171992512yes—
463C23#1313324712yes—
464C23C22×C22⋊D5311yescharacteristic
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—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—84no—
490C2×C4—1062no—
491C2×C4—1062no—
492C2×C4—1062no—
493C2×C4—1062no—
494C2×C4—4244no—
495C2×C4—4244no—
496C2×C4—2122no—
497C2×C4—2122no—
498C2×C4—2122no—
499C2×C4—2122no—
500C2×C4#1313318512yes—
501C2×C4C2×#1171992512yes—
502C2×C4C2×D4×D5311yescharacteristic
503C2×C4C2×D4×D5311yescharacteristic
504C2×C4#41×D5312yes—
505C2×C4#1313326412yes—
506C8—1064no—
507C8—2122no—
508C22—22no—
509C22—22no—
510C22—24no—
511C22—24no—
512C22—22no—
513C22—24no—
514C22—41no—
515C22—41no—
516C22—42no—
517C22#1681077814yes—
518C22C2×#1313318511yescharacteristic
519C22#1681127212yes—
520C22C22×#1171992511yescharacteristic
521C22C2×#1313324711yescharacteristic
522C22#1681142612yes—
523C4—22no—
524C4—22no—
525C4—41no—
526C4—41no—
527C4—82no—
528C4—82no—
529C4—2122no—
530C4—2122no—
531C4—2122no—
532C4—2122no—
533C4#1681113111yescharacteristic
534C4#1681119511yescharacteristic
535C4#1681127112yes—
536C2—24no—
537C2—42no—
538C2#2027804711yescharacteristic
539C2C2×#1681077811yescharacteristic
540C2#2028398111yescharacteristic
541C2#2028710312yes—
542C2#2028720012yes—
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2487C2
socleC23×C53C22×D4
prime radicalC22×D4×C53D4
hypercenter#2487D53
hypocenterC53#24098
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #2487

Complements

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

NH ≅ G/Ntype
2. C2×#16810958C2semidirect
3. C22×#13133150C2semidirect
4. #20276043C2semidirect
5. #20275918C2semidirect
6. #20275136C2semidirect
8. #20274968C2semidirect
9. #20274902C2semidirect
10. #20274803C2semidirect
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×#168109582
C22×#131331502
#202760432
#202759182
#202751362
#202750082
#202749682
#202749022
#202748032
C53×#24872
#2409853