Finite Group Databasebeta

← #27401710

Subgroups — #27401710

Order 13568.

850 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#106×D53C211yesmaximal, characteristic
2#20277588C211yesmaximal, characteristic
3#20277580C211yesmaximal, characteristic
4#20275443C211yesmaximal, characteristic, complemented
5#20275441C211yesmaximal, characteristic, complemented
6#20275300C211yesmaximal, characteristic, complemented
7#20275298C211yesmaximal, characteristic, complemented
8#20275293C211yesmaximal, characteristic, complemented
9#20275291C211yesmaximal, characteristic, complemented
10#20274804C211yesmaximal, characteristic
11#20274800C211yesmaximal, characteristic
12#20274737C211yesmaximal, characteristic
13#106×Dic53C211yesmaximal, characteristic, complemented
14C8×#11719921C211yesmaximal, characteristic
15C4×#106×C53C211yesmaximal, characteristic, complemented
16C22×C8×D53C2211yescharacteristic
17C2×#13133174—21no—
18C2×C4⋊C4×D53C2211yescharacteristic
19C2×C4⋊C4×D53C2211yescharacteristic
20C2×#13133149C2211yescharacteristic
21C2×#13133149C2211yescharacteristic
22C2×#13133148C2211yescharacteristic
23C2×#13133148C2211yescharacteristic
24#106×D53C418yescomplemented
25C2×#13133129C2211yescharacteristic
26C2×C8×Dic53C2211yescharacteristic
27#16810521C2211yescharacteristic
28#16810521C2211yescharacteristic
29#16810519C2211yescharacteristic
30#16810519C2211yescharacteristic
31#16810507—21no—
32#16810507—21no—
33#16810505—21no—
34#16810505—21no—
35#16810485—21no—
36#16810482—21no—
37#16810480C2211yescharacteristic
38#16810480C2211yescharacteristic
39C4⋊C4×Dic53C2211yescharacteristic
40C4⋊C4×Dic53C2211yescharacteristic
41C4×#11719921—21no—
42C4×#11719921C2211yescharacteristic
43#16810386C2211yescharacteristic
44#16810386C2211yescharacteristic
45#16810385C2211yescharacteristic
46#16810385C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810385C2211yescharacteristic
49#16810385C2211yescharacteristic
50#16810385C2211yescharacteristic
51#16810378C2211yescharacteristic
52#16810377C2211yescharacteristic
53C2×#106×C53C2211yescharacteristic
54C2×#106×C53C2211yescharacteristic
55C2×C4×C8×C53C2211yescharacteristic
56C4×C4⋊C4×C53C2211yescharacteristic
57C4×C4⋊C4×C53C2211yescharacteristic
58C53×#340C2211yescharacteristic
59C53×#340C2211yescharacteristic
60C22×C4×D53—21no—
61C22×C4×D53—21no—
62C22×C4×D53C2311yescharacteristic
63C22×#88322C2311yescharacteristic
64C22×#88322C2311yescharacteristic
65C2×C8×D53—22no—
66C2×C8×D53—22no—
67C2×C8×D53C2×C414yes—
68#13133174—44no—
69C2×C4⋊Dic53C2311yescharacteristic
70C2×C4⋊Dic53C2311yescharacteristic
71C2×C4⋊Dic53C2311yescharacteristic
72C2×C4⋊Dic53C2311yescharacteristic
73C2×#11719919—21no—
74C2×#11719919—21no—
75C2×#11719919—21no—
76C2×#11719919—21no—
77C2×#11719919—21no—
78C2×#11719919—21no—
79C4⋊C4×D53—22no—
80C4⋊C4×D53—22no—
81C4⋊C4×D53C2×C414yes—
82C4⋊C4×D53C2×C414yes—
83C2×#11719921—21no—
84C2×#11719921—21no—
85C2×#11719921D411yescharacteristic
86C2×#11719921D411yescharacteristic
87C2×C4×Dic53—21no—
88C2×C4×Dic53—21no—
89C2×C4×Dic53C2311yescharacteristic
90#13133149—22no—
91#13133149C2×C414yes—
92#13133148C2×C414yes—
93#13133148C2×C414yes—
94#13133129—22no—
95C8×Dic53—22no—
96#13133124—21no—
97#13133124—21no—
98#13133124—21no—
99#13133124—21no—
100#13133124—21no—
101#13133124—21no—
102#13133124—21no—
103#13133124—21no—
104#13133124D411yescharacteristic
105#13133124D411yescharacteristic
106C22×C8×C53C2311yescharacteristic
107C22×C8×C53C2311yescharacteristic
108C2×C4⋊C4×C53C2311yescharacteristic
109C2×C4⋊C4×C53C2311yescharacteristic
110C2×C4⋊C4×C53C2311yescharacteristic
111C2×C4⋊C4×C53C2311yescharacteristic
112C2×C42×C53—21no—
113C2×C42×C53—21no—
114C2×C42×C53C2311yescharacteristic
115#106×C53—22no—
116#106×C53C2×C414yes—
117C4×C8×C53—22no—
118#95×C53—21no—
119#95×C53—21no—
120C23×D53C2×D411yescharacteristic
121C22×C4×C53—21no—
122C22×C4×C53—21no—
123C22×C4×C53—21no—
124C22×C4×C53—21no—
125C22×C4×C53C2×D411yescharacteristic
126C22×C4×C53C2×D411yescharacteristic
127C22×C4×C53C2411yescharacteristic
128C22×Dic53—21no—
129C22×Dic53—21no—
130C22×Dic53—21no—
131C22×Dic53—21no—
132C22×Dic53C2×D411yescharacteristic
133C22×Dic53C2×D411yescharacteristic
134C22×Dic53C2×D411yescharacteristic
135C2×C4×D53—21no—
136C2×C4×D53—21no—
137C2×C4×D53—21no—
138C2×C4×D53—21no—
139C2×C4×D53—24no—
140C2×C4×D53—24no—
141C2×C4×D53—42no—
142C2×C4×D53—42no—
143C2×C4×D53—81no—
144C2×C4×D53—81no—
145C2×C4×D53C22×C412yes—
146C2×C4×D53#4112yes—
147C2×C8×C53—21no—
148C2×C8×C53—21no—
149C2×C8×C53—21no—
150C2×C8×C53C22×C412yes—
151C2×C8×C53#4112yes—
152C2×C8×C53#4112yes—
153C4⋊C4×C53—21no—
154C4⋊C4×C53—21no—
155C4⋊C4×C53—21no—
156C4⋊C4×C53—21no—
157C4⋊C4×C53—21no—
158C4⋊C4×C53—21no—
159C4⋊C4×C53C22×C412yes—
160C4⋊C4×C53C22×C412yes—
161C42×C53—21no—
162C42×C53—21no—
163C42×C53—42no—
164C42×C53—81no—
165#11719921—22no—
166#11719921—42no—
167#11719921—42no—
168#11719921SD1614yes—
169C4⋊Dic53—21no—
170C4⋊Dic53—21no—
171C4⋊Dic53—21no—
172C4⋊Dic53—21no—
173C4⋊Dic53—21no—
174C4⋊Dic53—21no—
175C4⋊Dic53C22×C412yes—
176C4⋊Dic53C22×C412yes—
177#11719919—24no—
178#11719919—24no—
179#11719919—42no—
180#11719919—42no—
181#11719919—81no—
182#11719919—81no—
183C4×Dic53—21no—
184C4×Dic53—21no—
185C4×Dic53—42no—
186C4×Dic53—81no—
187C2×#88322—21no—
188C2×#88322—21no—
189C2×#88322—21no—
190C2×#88322—21no—
191C2×#88322—21no—
192C2×#88322C22×C412yes—
193C8×D53—28no—
194C23×C53C22×D411yescharacteristic
195C22×D53—21no—
196C22×D53—21no—
197C22×D53—21no—
198C22×D53—21no—
199C22×D53C4×D412yes—
200C22×D53C2×SD1612yes—
201C22×D53#13512yes—
202C2×C4×C53—21no—
203C2×C4×C53—21no—
204C2×C4×C53—22no—
205C2×C4×C53—22no—
206C2×C4×C53—21no—
207C2×C4×C53—21no—
208C2×C4×C53—21no—
209C2×C4×C53—41no—
210C2×C4×C53—41no—
211C2×C4×C53—41no—
212C2×C4×C53—41no—
213C2×C4×C53—41no—
214C2×C4×C53—41no—
215C2×C4×C53—41no—
216C2×C4×C53—41no—
217C2×C4×C53—41no—
218C2×C4×C53—41no—
219C2×C4×C53C2×SD1612yes—
220C2×C4×C53C23×C411yescharacteristic
221C2×C4×C53C2×#4111yescharacteristic
222C2×C4×C53C2×#4111yescharacteristic
223C2×C4×C53C2×#4111yescharacteristic
224C2×C4×C532−1+411yescharacteristic
225C2×C4×C532−1+411yescharacteristic
226C2×Dic53—21no—
227C2×Dic53—21no—
228C2×Dic53—21no—
229C2×Dic53—21no—
230C2×Dic53—22no—
231C2×Dic53—22no—
232C2×Dic53—21no—
233C2×Dic53—21no—
234C2×Dic53—21no—
235C2×Dic53—41no—
236C2×Dic53—41no—
237C2×Dic53—41no—
238C2×Dic53—41no—
239C2×Dic53—41no—
240C2×Dic53—41no—
241C2×Dic53—41no—
242C2×Dic53—41no—
243C2×Dic53—41no—
244C2×Dic53—41no—
245C2×Dic53C4×D412yes—
246C2×Dic53C2×SD1612yes—
247C4×D53—22no—
248C4×D53—22no—
249C4×D53—22no—
250C4×D53—22no—
251C4×D53—48no—
252C4×D53—84no—
253C8×C53—41no—
254C8×C53#11714yescomplemented
255#88322—22no—
256#88322—41no—
257#22609—531nomaximal, Sylow
258C22×C53C2×C4×D411yescharacteristic
259C22×C53#54711yescharacteristic
260C22×C53#54711yescharacteristic
261C22×C53C22×SD1611yescharacteristic
262C22×C53C2×#13511yescharacteristic
263C22×C53#57711yescharacteristic
264C22×C53#57711yescharacteristic
265C2×D53—22no—
266C2×D53—22no—
267C2×D53—22no—
268C2×D53—22no—
269C2×D53—22no—
270C2×D53—22no—
271C2×D53C4×SD1614yes—
272C4×C53—22no—
273C4×C53—41no—
274C4×C53—42no—
275C4×C53—42no—
276C4×C53—81no—
277C4×C53—81no—
278C4×C53C2×#11711yescharacteristic
279C4×C53#51911yescharacteristic
280C4×C53#55211yescharacteristic
281C4×C53#55211yescharacteristic
282Dic53—22no—
283Dic53—22no—
284Dic53—41no—
285Dic53—42no—
286Dic53—42no—
287Dic53—81no—
288Dic53—81no—
289C22×#106—531no—
290#2345—531no—
291#2344—531no—
292#1941—531no—
293#1941—531no—
294#1920—531no—
295#1920—531no—
296#1919—531no—
297#1919—531no—
298#1824—531no—
299#1823—531no—
300#1807—531no—
301C4×#106—531no—
302C4×#106D5311yescharacteristic, complemented
303C8×C22⋊C4—531no—
304C2×C53#234411yescharacteristic
305C2×C53C2×C4×SD1611yescharacteristic
306C2×C53#294911yescharacteristic
307C2×C53#333511yescharacteristic
308C2×C53#333511yescharacteristic
309C2×C53#333711yescharacteristic
310C2×C53#333711yescharacteristic
311D53—24no—
312C23×C8—531no—
313C2×#117—1061no—
314C22×C4⋊C4—531no—
315C22×C4⋊C4—531no—
316C2×#106—531no—
317C2×#106—531no—
318C2×#106—531no—
319C2×#106—538no—
320C2×#106—531no—
321C2×#106C2×D5311yescharacteristic
322C2×#106C2×D5311yescharacteristic
323C2×C22⋊C8—531no—
324C2×C4×C8—531no—
325C2×C4×C8C2×D5311yescharacteristic
326#389—531no—
327#389—531no—
328#389—531no—
329#389—531no—
330#387—1061no—
331#387—1061no—
332#387—1061no—
333#387—1061no—
334#381—1061no—
335#380—531no—
336#380—531no—
337#380—1061no—
338C4×C4⋊C4—531no—
339C4×C4⋊C4—531no—
340C4×C4⋊C4C2×D5311yescharacteristic
341C4×C4⋊C4C2×D5311yescharacteristic
342C4×C22⋊C4—531no—
343C4×C22⋊C4—1061no—
344#340—531no—
345#340—531no—
346#340—531no—
347#340—531no—
348#340—531no—
349#340—531no—
350#340—531no—
351#340—531no—
352#340C2×D5311yescharacteristic
353#340C2×D5311yescharacteristic
354#336—531no—
355#336—531no—
356C53#2260911yescharacteristic, complemented, Sylow
357C23×C4—531no—
358C23×C4—1061no—
359C23×C4—1061no—
360C22×C8—531no—
361C22×C8—534no—
362C22×C8—531no—
363C22×C8—1062no—
364C22×C8—1062no—
365C22×C8C22×D5311yescharacteristic
366C22×C8C22×D5311yescharacteristic
367#117—2124no—
368C2×C4⋊C4—531no—
369C2×C4⋊C4—531no—
370C2×C4⋊C4—534no—
371C2×C4⋊C4—534no—
372C2×C4⋊C4—531no—
373C2×C4⋊C4—531no—
374C2×C4⋊C4—1062no—
375C2×C4⋊C4—1062no—
376C2×C4⋊C4—1061no—
377C2×C4⋊C4—1061no—
378C2×C4⋊C4—1061no—
379C2×C4⋊C4—1061no—
380C2×C4⋊C4—1061no—
381C2×C4⋊C4—1061no—
382C2×C4⋊C4C22×D5311yescharacteristic
383C2×C4⋊C4C22×D5311yescharacteristic
384C2×C4⋊C4C22×D5311yescharacteristic
385C2×C4⋊C4C22×D5311yescharacteristic
386C2×C22⋊C4—531no—
387C2×C22⋊C4—531no—
388C2×C22⋊C4—1061no—
389C2×C22⋊C4—1061no—
390C2×C42—21no—
391C2×C42—21no—
392C2×C42—531no—
393C2×C42—1061no—
394C2×C42—1061no—
395C2×C42C22×D5311yescharacteristic
396#106—22no—
397#106—534no—
398#106—534no—
399#106—534no—
400#106—1062no—
401#106C4×D5314yes—
402C22⋊C8—1062no—
403C4×C8—22no—
404C4×C8—1062no—
405#95—21no—
406#95—21no—
407#95—531no—
408#95—531no—
409#95—1061no—
410#95—1061no—
411#95—1061no—
412#95—1061no—
413#95—1061no—
414#95—1061no—
415#95—1061no—
416#95—1061no—
417C24—531no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—531no—
423C22×C4—531no—
424C22×C4—532no—
425C22×C4—532no—
426C22×C4—531no—
427C22×C4—1064no—
428C22×C4—1064no—
429C22×C4—1061no—
430C22×C4—1061no—
431C22×C4—1061no—
432C22×C4—1061no—
433C22×C4—1061no—
434C22×C4—1061no—
435C22×C4—1061no—
436C22×C4—1061no—
437C22×C4—4241no—
438C22×C4—4241no—
439C22×C4—2122no—
440C22×C4—2122no—
441C22×C4D4×D5311yescharacteristic
442C22×C4D4×D5311yescharacteristic
443C22×C4C23×D5311yescharacteristic
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—21no—
447C2×C8—532no—
448C2×C8—1061no—
449C2×C8—1061no—
450C2×C8—1068no—
451C2×C8—1061no—
452C2×C8—1061no—
453C2×C8—1061no—
454C2×C8C2×C4×D5312yes—
455C2×C8#1171994512yes—
456C2×C8#1171994512yes—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—532no—
464C4⋊C4—532no—
465C4⋊C4—1064no—
466C4⋊C4—1064no—
467C4⋊C4—1061no—
468C4⋊C4—1061no—
469C4⋊C4—1061no—
470C4⋊C4—1061no—
471C4⋊C4—1061no—
472C4⋊C4—1061no—
473C4⋊C4—4241no—
474C4⋊C4—4241no—
475C4⋊C4—2122no—
476C4⋊C4—2122no—
477C4⋊C4C2×C4×D5312yes—
478C4⋊C4C2×C4×D5312yes—
479C22⋊C4—534no—
480C22⋊C4—1062no—
481C22⋊C4—2122no—
482C22⋊C4—2122no—
483C42—21no—
484C42—21no—
485C42—42no—
486C42—81no—
487C42—1061no—
488C42—1061no—
489C42—4241no—
490C42—2122no—
491C23—532no—
492C23—532no—
493C23—532no—
494C23—1061no—
495C23—1061no—
496C23—1061no—
497C23—1061no—
498C23C2×D4×D5311yescharacteristic
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—532no—
517C2×C4—532no—
518C2×C4—1061no—
519C2×C4—1061no—
520C2×C4—1061no—
521C2×C4—1061no—
522C2×C4—1062no—
523C2×C4—1062no—
524C2×C4—1062no—
525C2×C4—1062no—
526C2×C4—1061no—
527C2×C4—1062no—
528C2×C4—1062no—
529C2×C4—1061no—
530C2×C4—1061no—
531C2×C4—4244no—
532C2×C4—2128no—
533C2×C4—2121no—
534C2×C4—2121no—
535C2×C4—2121no—
536C2×C4—2121no—
537C2×C4—2121no—
538C2×C4—2121no—
539C2×C4—2121no—
540C2×C4—2121no—
541C2×C4—2121no—
542C2×C4—2121no—
543C2×C4SD16×D5312yes—
544C2×C4C22×C4×D5311yescharacteristic
545C2×C4#41×D5311yescharacteristic
546C2×C4C2×#1171994511yescharacteristic
547C2×C4C2×#1171994511yescharacteristic
548C2×C4#1313326411yescharacteristic
549C2×C4#1313326511yescharacteristic
550C8—41no—
551C8—1062no—
552C8—2121no—
553C8#1313317414yescomplemented
554C22—534no—
555C22—1062no—
556C22—1062no—
557C22—1062no—
558C22—1062no—
559C22—1062no—
560C22—1062no—
561C22C4×D4×D5311yescharacteristic
562C22#1681127411yescharacteristic
563C22#1681127811yescharacteristic
564C22C2×SD16×D5311yescharacteristic
565C22#135×D5311yescharacteristic
566C22#1681142911yescharacteristic
567C22#1681143011yescharacteristic
568C4—22no—
569C4—42no—
570C4—41no—
571C4—42no—
572C4—81no—
573C4—81no—
574C4—1062no—
575C4—1062no—
576C4—4241no—
577C4—4241no—
578C4—2122no—
579C4—2121no—
580C4—2122no—
581C4C2×#1313317411yescharacteristic
582C4#1681109911yescharacteristic
583C4#1681131211yescharacteristic
584C4#1681131411yescharacteristic
585C2—1064no—
586C2#2027758011yescharacteristic
587C2C4×SD16×D5311yescharacteristic
588C2#2028356911yescharacteristic
589C2#2028706811yescharacteristic
590C2#2028707411yescharacteristic
591C2#2028709511yescharacteristic
592C2#2028709911yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC4×#106×C53C2
socleC23×C53C22×D4
prime radicalC23×D53C2×D4
cosocleC22×C4×C53C24
hypercenterC4×#106D53
hypocenterC53#22609
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ C4×#106

Complements

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

NH ≅ G/Ntype
4. #20275443C2semidirect
5. #20275441C2semidirect
6. #20275300C2semidirect
7. #20275298C2semidirect
8. #20275293C2semidirect
9. #20275291C2semidirect
13. #106×Dic53C2semidirect
15. C4×#106×C53C2semidirect
24. #106×D53C4semidirect
254. C8×C53#117semidirect
302. C4×#106D53semidirect
356. C53#22609semidirect
553. C8#13133174semidirect

Maximal subgroups

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

subgroupindex
C2×#106×D532
#202775882
#202775802
#202754432
#202754412
#202753002
#202752982
#202752932
#202752912
#202748042
#202748002
#202747372
#106×Dic532
C8×#117199212
C4×#106×C532
#2260953