Finite Group Databasebeta

← #27420198

Subgroups — #27420198

Order 13568.

1066 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811178C211yesmaximal, characteristic
2C2×#16810918C211yesmaximal, characteristic, complemented
3C4×#13133193C211yesmaximal, characteristic
4#20276055C212yesmaximal, complemented
5#20275864C212yesmaximal, complemented
6#20275668C211yesmaximal, characteristic, complemented
7#20275662C211yesmaximal, characteristic, complemented
8#20275024C212yesmaximal
9#20274913C212yesmaximal
10#20274692C211yesmaximal, characteristic
11C2×C53×#475C211yesmaximal, characteristic, complemented
12#16811178—24no—
13#16811178—24no—
14C2×#13133217C2211yescharacteristic
15C2×#13133208C2212yes—
16C2×#13133197—21no—
17C2×#13133193—21no—
18C2×#13133193C2211yescharacteristic
19C2×Q8×Dic53—21no—
20C2×#13133182—21no—
21C2×#13133175C2212yes—
22C2×C42×D53C2211yescharacteristic
23#16810918—24no—
24C2×#13133150C2212yes—
25C2×C4⋊#88322C2211yescharacteristic
26C2×#13133141C2212yes—
27C2×#13133129C2212yes—
28#16810530C2211yescharacteristic
29#16810519C2211yescharacteristic
30#16810504—21no—
31#16810494—21no—
32#16810480C2211yescharacteristic
33C4×#11719919—21no—
34C4×C4⋊Dic53C2211yescharacteristic
35C4×#11719921—21no—
36#16810386C2212yes—
37#16810386C2212yes—
38#16810385C2212yes—
39#16810385C2212yes—
40#16810377C2212yes—
41C2×C4⋊Q8×C53C2211yescharacteristic
42C2×C4×Q8×C53C2211yescharacteristic
43C53×#475—24no—
44C2×#106×C53C2212yes—
45C2×C4⋊C8×C53C2211yescharacteristic
46C2×#103×C53C2212yes—
47C22×Dic106—21no—
48C22×C4×D53D411yescharacteristic
49C22×C4×D53D411yescharacteristic
50C22×C4×D53C2311yescharacteristic
51C22×#88322C2312yes—
52#13133217—22no—
53#13133217D414yes—
54#13133208—24no—
55#13133208—24no—
56#13133197—28no—
57#13133193—28no—
58#13133193—22no—
59#13133193Q814yescomplemented
60Q8×Dic53—28no—
61#13133182—28no—
62#13133175—24no—
63#13133175—24no—
64C2×C4⋊Dic53—21no—
65C2×C4⋊Dic53—22no—
66C2×C4⋊Dic53C2311yescharacteristic
67C2×C4⋊Dic53C2312yes—
68C2×C4⋊Dic53C2311yescharacteristic
69C2×#11719919—21no—
70C2×#11719919—22no—
71C2×#11719919—21no—
72C2×#11719921—22no—
73C2×#11719921—21no—
74C2×#11719921—21no—
75C2×C4×Dic53—21no—
76C2×C4×Dic53—22no—
77C2×C4×Dic53D411yescharacteristic
78C2×C4×Dic53D411yescharacteristic
79C2×C4×Dic53C2311yescharacteristic
80C42×D53—22no—
81C42×D53—22no—
82#13133150—24no—
83C4⋊#88322—22no—
84#13133141—24no—
85#13133129—24no—
86#13133124—21no—
87#13133124—21no—
88#13133124—21no—
89#13133124—21no—
90#13133124—21no—
91C22×Q8×C53—21no—
92C22×Q8×C53C2311yescharacteristic
93C22×C8×C53C2312yes—
94C4⋊Q8×C53—22no—
95C4⋊Q8×C53D414yes—
96C4×Q8×C53—22no—
97C4×Q8×C53—22no—
98C2×C4⋊C4×C53—21no—
99C2×C4⋊C4×C53—21no—
100C2×C4⋊C4×C53C2311yescharacteristic
101C2×C4⋊C4×C53C2312yes—
102C2×C42×C53—21no—
103C2×C42×C53C2311yescharacteristic
104#106×C53—24no—
105C4⋊C8×C53—22no—
106#103×C53—24no—
107C23×D53C2×D411yescharacteristic
108C2×Q8×C53—24no—
109C2×Q8×C53—21no—
110C2×Q8×C53—21no—
111C2×Q8×C53—21no—
112C2×Q8×C53—21no—
113C2×Q8×C53—21no—
114C2×Q8×C53—42no—
115C2×Q8×C53—42no—
116C2×Q8×C53C2×Q812yes—
117C22×C4×C53—21no—
118C22×C4×C53—22no—
119C22×C4×C53—21no—
120C22×C4×C53C2×D411yescharacteristic
121C22×C4×C53C2×D411yescharacteristic
122C22×C4×C53C2411yescharacteristic
123C22×Dic53—21no—
124C22×Dic53—22no—
125C22×Dic53—21no—
126C22×Dic53C2×D411yescharacteristic
127C22×Dic53C2×D411yescharacteristic
128C22×Dic53C2×D411yescharacteristic
129C2×C4×D53—21no—
130C2×C4×D53—21no—
131C2×C4×D53—21no—
132C2×C4×D53—21no—
133C2×C4×D53—22no—
134C2×C4×D53—22no—
135C2×C4×D53—22no—
136C2×C4×D53—22no—
137C2×C4×D53—41no—
138C2×C4×D53—41no—
139C2×C4×D53C2×Q812yes—
140C2×C4×D53#4112yes—
141C2×Dic106—24no—
142C2×Dic106—42no—
143C2×Dic106—42no—
144C2×C8×C53—22no—
145C2×C8×C53—22no—
146C2×C8×C53—22no—
147C4⋊C4×C53—24no—
148C4⋊C4×C53—22no—
149C4⋊C4×C53—21no—
150C4⋊C4×C53—21no—
151C4⋊C4×C53—42no—
152C4⋊C4×C53C2×D414yes—
153C42×C53—42no—
154C42×C53C2×D412yes—
155C42×C53C2×D412yes—
156#11719921—28no—
157#11719921—24no—
158#11719921—42no—
159C4⋊Dic53—28no—
160C4⋊Dic53—24no—
161C4⋊Dic53—22no—
162C4⋊Dic53—21no—
163C4⋊Dic53—21no—
164C4⋊Dic53—21no—
165C4⋊Dic53C2×D414yes—
166C4⋊Dic53C2×Q812yes—
167#11719919—28no—
168#11719919—24no—
169#11719919—42no—
170C4×Dic53—28no—
171C4×Dic53—22no—
172C4×Dic53—22no—
173C4×Dic53—21no—
174C4×Dic53—21no—
175C4×Dic53—42no—
176C2×#88322—22no—
177C2×#88322—22no—
178C2×#88322—22no—
179C23×C53C22×D411yescharacteristic
180C22×D53—21no—
181C22×D53—21no—
182C22×D53—21no—
183C22×D53—21no—
184C22×D53—21no—
185C22×D53—21no—
186C22×D53C22⋊Q812yes—
187Q8×C53—22no—
188Q8×C53—22no—
189Q8×C53—41no—
190Q8×C53—41no—
191Q8×C53—48no—
192C2×C4×C53—21no—
193C2×C4×C53—21no—
194C2×C4×C53—24no—
195C2×C4×C53—22no—
196C2×C4×C53—41no—
197C2×C4×C53—41no—
198C2×C4×C53—42no—
199C2×C4×C53—42no—
200C2×C4×C53—41no—
201C2×C4×C53—41no—
202C2×C4×C53—41no—
203C2×C4×C53#12012yes—
204C2×C4×C53#12012yes—
205C2×C4×C53#13512yes—
206C2×C4×C53#13712yes—
207C2×C4×C53C22×D411yescharacteristic
208C2×C4×C53C22×D411yescharacteristic
209C2×C4×C53C22×Q811yescharacteristic
210C2×C4×C53C2×#4111yescharacteristic
211C2×C4×C532−1+411yescharacteristic
212C2×C4×C532−1+411yescharacteristic
213C2×Dic53—21no—
214C2×Dic53—21no—
215C2×Dic53—24no—
216C2×Dic53—21no—
217C2×Dic53—21no—
218C2×Dic53—22no—
219C2×Dic53—21no—
220C2×Dic53—21no—
221C2×Dic53—21no—
222C2×Dic53—21no—
223C2×Dic53—41no—
224C2×Dic53—42no—
225C2×Dic53—42no—
226C2×Dic53—41no—
227C2×Dic53—41no—
228C2×Dic53—41no—
229C2×Dic53—41no—
230C2×Dic53C22⋊Q812yes—
231C4×D53—22no—
232C4×D53—22no—
233C4×D53—22no—
234C4×D53—22no—
235C4×D53—44no—
236C4×D53—44no—
237Dic106—48no—
238C8×C53—42no—
239#88322—42no—
240#24303—531nomaximal, Sylow
241C22×C53C2×#12011yescharacteristic
242C22×C53C2×C22⋊Q811yescharacteristic
243C22×C53#53611yescharacteristic
244C22×C53C2×#13511yescharacteristic
245C22×C53C2×#13711yescharacteristic
246C22×C53#57511yescharacteristic
247C22×C53#57511yescharacteristic
248C2×D53—22no—
249C2×D53—22no—
250C2×D53—22no—
251C2×D53—41no—
252C2×D53—41no—
253C2×D53—41no—
254C2×D53—41no—
255C4×C53—22no—
256C4×C53—42no—
257C4×C53—44no—
258C4×C53—81no—
259C4×C53#45214yes—
260C4×C53#54811yescharacteristic
261C4×C53#54811yescharacteristic
262C4×C53D4×Q811yescharacteristic
263C4×C53D4×Q811yescharacteristic
264Dic53—22no—
265Dic53—42no—
266Dic53—41no—
267Dic53—44no—
268Dic53—41no—
269Dic53—81no—
270C2×#533—531no—
271C2×#477—531no—
272C2×#475D5311yescharacteristic, complemented
273C4×C22⋊Q8—531no—
274#2059—532no—
275#2026—532no—
276#1988—531no—
277#1987—531no—
278#1873—532no—
279#1854—532no—
280#1797—531no—
281C2×C53#240911yescharacteristic
282C2×C53C2×#45211yescharacteristic
283C2×C53#300511yescharacteristic
284C2×C53#307611yescharacteristic
285C2×C53#307811yescharacteristic
286C2×C53#323311yescharacteristic
287C2×C53#323711yescharacteristic
288D53—42no—
289#533—1064no—
290#533—1064no—
291C2×C4⋊Q8—531no—
292C2×C4⋊Q8C2×D5311yescharacteristic
293C2×#125—532no—
294C2×C22⋊Q8—531no—
295C2×C22⋊Q8—1061no—
296C2×C22⋊Q8—1061no—
297C2×C4×Q8—1061no—
298C2×C4×Q8—1061no—
299C2×C4×Q8C2×D5311yescharacteristic
300C2×#117—532no—
301C22×C42—531no—
302#477—1064no—
303#475—24no—
304C2×#107—532no—
305C2×#106C2×D5312yes—
306C2×C4⋊C8—531no—
307C2×C4⋊C8C2×D5311yescharacteristic
308C2×#103—532no—
309C2×#103C2×D5312yes—
310C2×C22⋊C8—532no—
311#391—531no—
312#389—531no—
313#387—1061no—
314#385—1061no—
315#380—531no—
316C4×C4⋊C4—531no—
317C4×C4⋊C4—1061no—
318C4×C22⋊C4—1061no—
319#340—532no—
320#340—532no—
321#340—532no—
322#340—532no—
323#336—532no—
324C53#2430311yescharacteristic, complemented, Sylow
325C22×Q8—21no—
326C22×Q8—1061no—
327C22×Q8C22×D5311yescharacteristic
328C23×C4—531no—
329C23×C4—531no—
330C23×C4—531no—
331C22×C8—532no—
332C22×C8C22×D5312yes—
333C4⋊Q8—22no—
334C4⋊Q8—534no—
335C4⋊Q8—1062no—
336C4⋊Q8C22⋊D5314yes—
337#125—1064no—
338#125—1064no—
339C22⋊Q8—534no—
340C22⋊Q8—1068no—
341C22⋊Q8—1062no—
342C22⋊Q8—1068no—
343C4×Q8—22no—
344C4×Q8—22no—
345C4×Q8—1068no—
346C4×Q8—1068no—
347#117—1064no—
348#117—1064no—
349C2×C4⋊C4—21no—
350C2×C4⋊C4—21no—
351C2×C4⋊C4—532no—
352C2×C4⋊C4—531no—
353C2×C4⋊C4—531no—
354C2×C4⋊C4—1062no—
355C2×C4⋊C4—1062no—
356C2×C4⋊C4—1061no—
357C2×C4⋊C4—1061no—
358C2×C4⋊C4—1061no—
359C2×C4⋊C4C22×D5311yescharacteristic
360C2×C4⋊C4C22×D5312yes—
361C2×C22⋊C4—1062no—
362C2×C22⋊C4—1061no—
363C2×C22⋊C4—1061no—
364C2×C42—21no—
365C2×C42—531no—
366C2×C42—531no—
367C2×C42—531no—
368C2×C42—1062no—
369C2×C42—1062no—
370C2×C42—1062no—
371C2×C42—1061no—
372C2×C42C22×D5311yescharacteristic
373#107—1064no—
374#106—24no—
375C4⋊C8—22no—
376C4⋊C8—1062no—
377#103—24no—
378#103—1064no—
379C22⋊C8—1064no—
380#95—1061no—
381#95—1061no—
382#95—1061no—
383#95—1061no—
384#95—1061no—
385C24—531no—
386C2×Q8—24no—
387C2×Q8—21no—
388C2×Q8—21no—
389C2×Q8—21no—
390C2×Q8—21no—
391C2×Q8—21no—
392C2×Q8—42no—
393C2×Q8—42no—
394C2×Q8—1064no—
395C2×Q8—2122no—
396C2×Q8—2122no—
397C2×Q8Q8×D5312yes—
398C22×C4—21no—
399C22×C4—22no—
400C22×C4—21no—
401C22×C4—531no—
402C22×C4—532no—
403C22×C4—532no—
404C22×C4—531no—
405C22×C4—531no—
406C22×C4—1061no—
407C22×C4—1061no—
408C22×C4—1061no—
409C22×C4—1061no—
410C22×C4—1061no—
411C22×C4—1062no—
412C22×C4—1062no—
413C22×C4—1062no—
414C22×C4—1062no—
415C22×C4—1062no—
416C22×C4—1061no—
417C22×C4—2121no—
418C22×C4—2121no—
419C22×C4D4×D5311yescharacteristic
420C22×C4D4×D5311yescharacteristic
421C22×C4C23×D5311yescharacteristic
422C2×C8—22no—
423C2×C8—22no—
424C2×C8—22no—
425C2×C8—1062no—
426C2×C8—1062no—
427C2×C8—1062no—
428C4⋊C4—24no—
429C4⋊C4—22no—
430C4⋊C4—21no—
431C4⋊C4—21no—
432C4⋊C4—42no—
433C4⋊C4—534no—
434C4⋊C4—532no—
435C4⋊C4—1064no—
436C4⋊C4—1068no—
437C4⋊C4—1068no—
438C4⋊C4—1064no—
439C4⋊C4—1062no—
440C4⋊C4—1061no—
441C4⋊C4—1061no—
442C4⋊C4—1061no—
443C4⋊C4—2122no—
444C4⋊C4C2×C22⋊D5314yes—
445C22⋊C4—1064no—
446C22⋊C4—1068no—
447C22⋊C4—2122no—
448C42—42no—
449C42—1068no—
450C42—1062no—
451C42—1062no—
452C42—1061no—
453C42—1061no—
454C42—2122no—
455C42D4×D5312yes—
456C42C2×C22⋊D5312yes—
457C23—532no—
458C23—1061no—
459C23—1061no—
460C23—1061no—
461C23—1061no—
462C23—1061no—
463C23—1061no—
464C23C2×D4×D5311yescharacteristic
465Q8—22no—
466Q8—22no—
467Q8—41no—
468Q8—41no—
469Q8—48no—
470Q8—2128no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—24no—
474C2×C4—22no—
475C2×C4—41no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—532no—
483C2×C4—1061no—
484C2×C4—1061no—
485C2×C4—1062no—
486C2×C4—1062no—
487C2×C4—1062no—
488C2×C4—1062no—
489C2×C4—1061no—
490C2×C4—1061no—
491C2×C4—1064no—
492C2×C4—1062no—
493C2×C4—1061no—
494C2×C4—1061no—
495C2×C4—1061no—
496C2×C4—1061no—
497C2×C4—2124no—
498C2×C4—2121no—
499C2×C4—2121no—
500C2×C4—2121no—
501C2×C4—2121no—
502C2×C4—2121no—
503C2×C4—2122no—
504C2×C4—2122no—
505C2×C4—2124no—
506C2×C4#1313318512yes—
507C2×C4#1313318512yes—
508C2×C4#1313323812yes—
509C2×C4#1313324612yes—
510C2×C4C2×D4×D5311yescharacteristic
511C2×C4C22×C22⋊D5311yescharacteristic
512C2×C4C2×Q8×D5311yescharacteristic
513C2×C4#41×D5311yescharacteristic
514C2×C4#1313326411yescharacteristic
515C2×C4#1313326511yescharacteristic
516C8—42no—
517C8—2122no—
518C22—1062no—
519C22—1062no—
520C22—1062no—
521C22—2121no—
522C22—2121no—
523C22—2121no—
524C22—2121no—
525C22C2×#1313318511yescharacteristic
526C22C22⋊Q8×D5311yescharacteristic
527C22#1681119611yescharacteristic
528C22C2×#1313323811yescharacteristic
529C22C2×#1313324611yescharacteristic
530C22#1681142211yescharacteristic
531C22#1681142211yescharacteristic
532C4—22no—
533C4—44no—
534C4—42no—
535C4—81no—
536C4—1062no—
537C4—4241no—
538C4—2121no—
539C4—2121no—
540C4—2124no—
541C4—2122no—
542C4#1681078214yes—
543C4#1681128011yescharacteristic
544C4#1681128311yescharacteristic
545C4#1681128611yescharacteristic
546C4Q8×C22⋊D5311yescharacteristic
547C2—2122no—
548C2#2027807611yescharacteristic
549C2C2×#1681078211yescharacteristic
550C2#2028395611yescharacteristic
551C2#2028449811yescharacteristic
552C2#2028451511yescharacteristic
553C2#2028589811yescharacteristic
554C2#2028592811yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC2×C53×#475C2
socleC23×C53C22×D4
prime radicalC23×D53C2×D4
hypercenterC2×#475D53
hypocenterC53#24303
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#475

Complements

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

NH ≅ G/Ntype
2. C2×#16810918C2semidirect
4. #20276055C2semidirect
5. #20275864C2semidirect
6. #20275668C2semidirect
7. #20275662C2semidirect
11. C2×C53×#475C2semidirect
59. #13133193Q8semidirect
272. C2×#475D53semidirect
324. C53#24303semidirect

Maximal subgroups

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

subgroupindex
C2×#168111782
C2×#168109182
C4×#131331932
#202760552
#202758642
#202756682
#202756622
#202750242
#202749132
#202746922
C2×C53×#4752
#2430353