Finite Group Databasebeta

← #27389448

Subgroups — #27389448

Order 13568.

1318 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811213C211yesmaximal, characteristic
2#20277312C211yesmaximal, characteristic, complemented
3#20277067C211yesmaximal, characteristic, complemented
4#20275701C211yesmaximal, characteristic, complemented
5#20275551C211yesmaximal, characteristic, complemented
6#20275550C211yesmaximal, characteristic, complemented
7#20275167C214yesmaximal, complemented
8#20274502C214yesmaximal, complemented
9C53×#1996C211yesmaximal, characteristic, complemented
10#16811213—22no—
11#16811213—22no—
12#16811213—22no—
13#16811213—22no—
14C2×#13133212C412yescomplemented
15C2×#13133206C412yescomplemented
16C2×#13133202C412yescomplemented
17C2×#13133192C412yescomplemented
18C2×#13133191C2211yescharacteristic
19C2×#13133187C2212yes—
20C2×#13133177C2212yes—
21C22×#11719921C2211yescharacteristic
22C2×C22⋊C4×D53C2212yes—
23C2×#13133131C2214yes—
24C2×#13133131C2214yes—
25#16810515—22no—
26#16810513—22no—
27#16810483C2211yescharacteristic
28#16810478C2212yes—
29C22⋊C4×Dic53C2212yes—
30C22⋊C4×Dic53C2212yes—
31C4×#11719921—22no—
32C4×C22⋊Dic53C2211yescharacteristic
33#16810470—22no—
34#16810390C2214yes—
35C2×C22⋊Q8×C53C2211yescharacteristic
36C53×#380C2211yescharacteristic
37C4×C22⋊C4×C53C2211yescharacteristic
38C53×#342C2214yes—
39C24×D53C2311yescharacteristic
40C22×D212C2×C412yes—
41C22×D212C2×C412yes—
42C22×D212C2×C411yescharacteristic, complemented
43C22×C22⋊D53C2312yes—
44C23×Dic53C2311yescharacteristic
45C22×C4×D53—22no—
46C22×C4×D53C2×C412yescomplemented
47#13133212—24no—
48#13133212—24no—
49#13133206—24no—
50#13133206—24no—
51#13133202—24no—
52#13133202—24no—
53#13133192—24no—
54#13133192—24no—
55#13133191—21no—
56#13133191—21no—
57#13133191—21no—
58#13133191—21no—
59#13133187—24no—
60#13133187—24no—
61#13133177—24no—
62#13133177—24no—
63C2×C4⋊Dic53—21no—
64C2×C4⋊Dic53C2×C411yescharacteristic, complemented
65C2×#11719919C2×C412yescomplemented
66C2×C22⋊Dic53C2312yes—
67C2×C22⋊Dic53C2312yes—
68C2×#11719921—22no—
69C2×#11719921—22no—
70C2×#11719921—22no—
71C2×#11719921C2×C412yes—
72C2×#11719921C2×C412yes—
73C2×#11719921C2×C412yes—
74C2×#11719921C2×C412yes—
75C2×#11719921C2×C412yescomplemented
76C2×#11719921C2×C412yescomplemented
77C2×#11719921D412yes—
78C2×#11719921D412yes—
79C2×#11719921D412yes—
80C2×#11719921D412yes—
81C22⋊C4×D53—24no—
82C22⋊C4×D53—24no—
83C2×C4×Dic53—21no—
84C2×C4×Dic53—22no—
85C2×C4×Dic53—22no—
86C2×C4×Dic53C2×C412yescomplemented
87#13133131—28no—
88#13133131—28no—
89#13133124—21no—
90#13133124—21no—
91#13133124—22no—
92#13133124—22no—
93C22×Q8×C53C2×C411yescharacteristic, complemented
94C23×C4×C53C2311yescharacteristic
95C22⋊Q8×C53—21no—
96C22⋊Q8×C53—21no—
97C22⋊Q8×C53—21no—
98C22⋊Q8×C53—21no—
99C2×C4⋊C4×C53—21no—
100C2×C4⋊C4×C53C2×C412yescomplemented
101C2×C4⋊C4×C53C2×C411yescharacteristic, complemented
102C2×C22⋊C4×C53C2312yes—
103C2×C22⋊C4×C53C2312yes—
104C2×C22⋊C4×C53C2312yes—
105C2×C42×C53—21no—
106#95×C53—21no—
107#95×C53—21no—
108C24×C53C2411yescharacteristic
109C23×D53—22no—
110C23×D53—22no—
111C23×D53—81no—
112C23×D53—81no—
113C23×D53C22×C412yes—
114C23×D53C22×C412yes—
115C23×D53C2×D412yes—
116C23×D53C2×D412yes—
117C2×Q8×C53—21no—
118C2×Q8×C53—21no—
119C2×Q8×C53—81no—
120C22×C4×C53—22no—
121C22×C4×C53—22no—
122C22×C4×C53—81no—
123C22×C4×C53C22×C412yes—
124C22×C4×C53C22×C412yes—
125C22×C4×C53C22×C411yescharacteristic
126C22×C4×C53C22×C411yescharacteristic
127C22×C4×C53C2×D411yescharacteristic
128C22×C4×C53C2×D411yescharacteristic
129C22×C4×C53C2×D411yescharacteristic
130C22×C4×C53C2×D411yescharacteristic
131C2×C22⋊D53—22no—
132C2×C22⋊D53—22no—
133C2×C22⋊D53—24no—
134C2×C22⋊D53—24no—
135C22×Dic53—22no—
136C22×Dic53—22no—
137C22×Dic53—81no—
138C22×Dic53C22×C412yes—
139C22×Dic53C22×C412yes—
140C22×Dic53C2×D412yes—
141C22×Dic53C2×D412yes—
142C2×D212—24no—
143C2×D212—24no—
144C2×D212—21no—
145C2×D212—21no—
146C2×D212—82no—
147C2×D212—82no—
148C2×D212—81no—
149C2×C4×D53—28no—
150C2×C4×D53—24no—
151C2×C4×D53—82no—
152C2×C4×D53—82no—
153C2×C4×D53—82no—
154C4⋊C4×C53—22no—
155C4⋊C4×C53—22no—
156C4⋊C4×C53—21no—
157C4⋊C4×C53—21no—
158C4⋊C4×C53—81no—
159C22⋊C4×C53—22no—
160C22⋊C4×C53—22no—
161C22⋊C4×C53—22no—
162C22⋊C4×C53—24no—
163C22⋊C4×C53#4114yes—
164C42×C53—81no—
165C22⋊Dic53—22no—
166C22⋊Dic53—24no—
167C22⋊Dic53#4114yes—
168#11719921—28no—
169#11719921—24no—
170#11719921—24no—
171#11719921—22no—
172#11719921—22no—
173#11719921—24no—
174#11719921—82no—
175#11719921—82no—
176#11719921C22⋊C414yes—
177#11719921C22⋊C414yes—
178#11719921C22⋊C414yes—
179#11719921C22⋊C414yes—
180C4⋊Dic53—21no—
181C4⋊Dic53—21no—
182C4⋊Dic53—81no—
183#11719919—24no—
184C4×Dic53—24no—
185C4×Dic53—82no—
186C4×Dic53—82no—
187C4×Dic53—81no—
188C23×C53—41no—
189C23×C53—41no—
190C23×C53C23×C411yescharacteristic
191C23×C53C22×D411yescharacteristic
192C23×C53C22×D411yescharacteristic
193C23×C53C2×#4112yes—
194C23×C53C2×#4112yes—
195C22×D53—22no—
196C22×D53—22no—
197C22×D53—24no—
198C22×D53—22no—
199C22×D53—22no—
200C22×D53—22no—
201C22×D53—24no—
202C22×D53—42no—
203C22×D53—42no—
204C22×D53—42no—
205C22×D53—42no—
206C22×D53—42no—
207C22×D53—42no—
208C22×D53—42no—
209C22×D53—42no—
210C22×D53—84no—
211C22×D53—84no—
212C22×D53C2×C22⋊C414yes—
213Q8×C53—81no—
214Q8×C53—81no—
215C2×C4×C53—22no—
216C2×C4×C53—22no—
217C2×C4×C53—24no—
218C2×C4×C53—41no—
219C2×C4×C53—41no—
220C2×C4×C53—41no—
221C2×C4×C53—41no—
222C2×C4×C53—41no—
223C2×C4×C53—41no—
224C2×C4×C53—42no—
225C2×C4×C53—42no—
226C2×C4×C53—42no—
227C2×C4×C53—42no—
228C2×C4×C53—42no—
229C2×C4×C53—42no—
230C2×C4×C53—42no—
231C2×C4×C53C2×C22⋊C411yescharacteristic
232C2×C4×C53C2×C22⋊C411yescharacteristic
233C2×C4×C53C2×C22⋊C411yescharacteristic
234C2×C4×C53C2×C22⋊C411yescharacteristic
235C22⋊D53—44no—
236C22⋊D53—42no—
237C22⋊D53—42no—
238C2×Dic53—22no—
239C2×Dic53—24no—
240C2×Dic53—24no—
241C2×Dic53—42no—
242C2×Dic53—42no—
243C2×Dic53—42no—
244C2×Dic53—42no—
245C2×Dic53—42no—
246C2×Dic53—42no—
247C2×Dic53—42no—
248C2×Dic53—42no—
249C2×Dic53C2×C22⋊C414yes—
250D212—81no—
251D212—81no—
252D212—84no—
253D212—84no—
254C4×D53—84no—
255C4×D53—88no—
256#21422—531nomaximal, Sylow
257C22×C53—21no—
258C22×C53—21no—
259C22×C53—21no—
260C22×C53—21no—
261C22×C53—21no—
262C22×C53—21no—
263C22×C53—42no—
264C22×C53—42no—
265C22×C53C22×C22⋊C411yescharacteristic
266C22×C53C4×#4112yes—
267C22×C53#52512yes—
268C22×C53#53211yescharacteristic
269C22×C53#53211yescharacteristic
270C2×D53—24no—
271C2×D53—24no—
272C2×D53—24no—
273C2×D53—44no—
274C2×D53—44no—
275C2×D53—42no—
276C2×D53—42no—
277C2×D53—42no—
278C2×D53—82no—
279C2×D53—82no—
280C2×D53—84no—
281C2×D53—84no—
282C4×C53—21no—
283C4×C53—21no—
284C4×C53—21no—
285C4×C53—21no—
286C4×C53—44no—
287C4×C53—42no—
288C4×C53—42no—
289C4×C53—82no—
290Dic53—24no—
291Dic53—44no—
292Dic53—42no—
293Dic53—82no—
294C2×#538—531no—
295#2312—531no—
296#2273—531no—
297#1996—531no—
298#1996D5311yescharacteristic, complemented
299#1964—531no—
300#1964—531no—
301#1898—534no—
302C4×#99—534no—
303C2×C53—22no—
304C2×C53—21no—
305C2×C53—21no—
306C2×C53—22no—
307C2×C53#229711yescharacteristic
308C2×C53#288311yescharacteristic
309C2×C53#288311yescharacteristic
310D53—44no—
311D53—82no—
312#538—1062no—
313#538—1062no—
314#538—1062no—
315#538—1062no—
316C2×#126—532no—
317C2×#124—532no—
318C2×#123—532no—
319C2×C22⋊Q8C2×D5311yescharacteristic
320C2×#121—531no—
321C2×#121—532no—
322C2×#120—532no—
323C2×C4×D4—532no—
324C22×C22⋊C4—531no—
325C22×C22⋊C4—532no—
326C2×#99—534no—
327C2×#99—534no—
328#388—1062no—
329#388—1062no—
330#380—531no—
331#380C2×D5311yescharacteristic
332#379—532no—
333C4×C22⋊C4—531no—
334C4×C22⋊C4—532no—
335C4×C22⋊C4—532no—
336C4×C22⋊C4—1062no—
337C4×C22⋊C4—1062no—
338C4×C22⋊C4C2×D5311yescharacteristic
339#342—534no—
340#342C2×D5314yes—
341C53#2142211yescharacteristic, complemented, Sylow
342C25—531no—
343C22×Q8C4×D5311yescharacteristic, complemented
344C22×D4—532no—
345C22×D4—532no—
346C22×D4—532no—
347C22×D4—531no—
348C23×C4—531no—
349C23×C4—532no—
350C23×C4—1062no—
351C23×C4C22×D5311yescharacteristic
352#126—1064no—
353#126—1064no—
354#124—1064no—
355#124—1064no—
356#123—1064no—
357#123—1064no—
358C22⋊Q8—21no—
359C22⋊Q8—21no—
360C22⋊Q8—21no—
361C22⋊Q8—21no—
362#121—1064no—
363#121—1064no—
364#121—1061no—
365#121—1061no—
366#121—1061no—
367#121—1061no—
368#120—1064no—
369#120—1064no—
370C4×D4—1064no—
371C4×D4—1064no—
372C2×C4⋊C4—21no—
373C2×C4⋊C4—532no—
374C2×C4⋊C4—531no—
375C2×C4⋊C4—1061no—
376C2×C4⋊C4C4×D5312yescomplemented
377C2×C4⋊C4C4×D5311yescharacteristic, complemented
378C2×C22⋊C4—532no—
379C2×C22⋊C4—532no—
380C2×C22⋊C4—532no—
381C2×C22⋊C4—532no—
382C2×C22⋊C4—532no—
383C2×C22⋊C4—532no—
384C2×C22⋊C4—532no—
385C2×C22⋊C4—532no—
386C2×C22⋊C4—532no—
387C2×C22⋊C4—532no—
388C2×C22⋊C4—532no—
389C2×C22⋊C4—532no—
390C2×C22⋊C4—1064no—
391C2×C22⋊C4—1062no—
392C2×C22⋊C4—1064no—
393C2×C22⋊C4—1062no—
394C2×C22⋊C4—1062no—
395C2×C22⋊C4C22×D5312yes—
396C2×C22⋊C4C22×D5312yes—
397C2×C22⋊C4C22×D5312yes—
398C2×C42—21no—
399C2×C42—532no—
400C2×C42—1062no—
401C2×C42—1062no—
402C2×C42—1061no—
403#99—1068no—
404#99—1068no—
405#95—21no—
406#95—21no—
407#95—1062no—
408#95—1062no—
409#95—1061no—
410#95—1061no—
411C24—532no—
412C24—532no—
413C24—532no—
414C24—532no—
415C24—1062no—
416C24—1062no—
417C24—4241no—
418C24—4241no—
419C24C23×D5311yescharacteristic
420C2×Q8—21no—
421C2×Q8—21no—
422C2×Q8—81no—
423C2×D4—1064no—
424C2×D4—1064no—
425C2×D4—1062no—
426C2×D4—1062no—
427C2×D4—1064no—
428C2×D4—1064no—
429C2×D4—1061no—
430C2×D4—1061no—
431C2×D4—4241no—
432C2×D4—4242no—
433C2×D4—4242no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—81no—
437C22×C4—532no—
438C22×C4—532no—
439C22×C4—532no—
440C22×C4—532no—
441C22×C4—1064no—
442C22×C4—1068no—
443C22×C4—1062no—
444C22×C4—1062no—
445C22×C4—4242no—
446C22×C4—4242no—
447C22×C4—4242no—
448C22×C4—4241no—
449C22×C4C2×C4×D5312yes—
450C22×C4C2×C4×D5312yes—
451C22×C4C2×C4×D5311yescharacteristic
452C22×C4C2×C4×D5311yescharacteristic
453C22×C4D4×D5311yescharacteristic
454C22×C4D4×D5311yescharacteristic
455C22×C4D4×D5311yescharacteristic
456C22×C4D4×D5311yescharacteristic
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—81no—
462C4⋊C4—1064no—
463C4⋊C4—1061no—
464C4⋊C4—1061no—
465C4⋊C4—4241no—
466C22⋊C4—22no—
467C22⋊C4—22no—
468C22⋊C4—22no—
469C22⋊C4—24no—
470C22⋊C4—534no—
471C22⋊C4—534no—
472C22⋊C4—534no—
473C22⋊C4—534no—
474C22⋊C4—534no—
475C22⋊C4—1064no—
476C22⋊C4—1064no—
477C22⋊C4—1064no—
478C22⋊C4—1068no—
479C22⋊C4—1062no—
480C22⋊C4—1062no—
481C22⋊C4—1062no—
482C22⋊C4—1064no—
483C22⋊C4—4242no—
484C22⋊C4—4242no—
485C22⋊C4#1171994914yes—
486C42—81no—
487C42—1064no—
488C42—4241no—
489C42—4242no—
490C42—4242no—
491C23—41no—
492C23—41no—
493C23—534no—
494C23—1064no—
495C23—1062no—
496C23—1062no—
497C23—1062no—
498C23—1062no—
499C23—1064no—
500C23—1062no—
501C23—4244no—
502C23—4244no—
503C23—2122no—
504C23—2122no—
505C23—2122no—
506C23—2122no—
507C23—2122no—
508C23—2122no—
509C23—2122no—
510C23—2122no—
511C23C22×C4×D5311yescharacteristic
512C23C2×D4×D5311yescharacteristic
513C23C2×D4×D5311yescharacteristic
514C23#41×D5312yes—
515C23C2×#1171994912yes—
516Q8—81no—
517Q8—81no—
518D4—4241no—
519D4—4241no—
520D4—4244no—
521D4—4244no—
522D4—2124no—
523D4—2122no—
524D4—2122no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—24no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—534no—
542C2×C4—1062no—
543C2×C4—1064no—
544C2×C4—1064no—
545C2×C4—4244no—
546C2×C4—4248no—
547C2×C4—2122no—
548C2×C4—2122no—
549C2×C4—2122no—
550C2×C4—2122no—
551C2×C4—2122no—
552C2×C4—2122no—
553C2×C4—2122no—
554C2×C4—2122no—
555C2×C4C22⋊C4×D5311yescharacteristic
556C2×C4C22⋊C4×D5311yescharacteristic
557C2×C4C22⋊C4×D5311yescharacteristic
558C2×C4C22⋊C4×D5311yescharacteristic
559C22—21no—
560C22—21no—
561C22—21no—
562C22—21no—
563C22—21no—
564C22—21no—
565C22—42no—
566C22—42no—
567C22—1064no—
568C22—1064no—
569C22—1064no—
570C22—4244no—
571C22—4242no—
572C22—4242no—
573C22—4244no—
574C22—2124no—
575C22—2124no—
576C22—2122no—
577C22—2122no—
578C22—2122no—
579C22C2×C22⋊C4×D5311yescharacteristic
580C22C4×#1171994912yes—
581C22#1681113612yes—
582C22#1681117211yescharacteristic
583C22#1681117411yescharacteristic
584C4—21no—
585C4—21no—
586C4—21no—
587C4—21no—
588C4—44no—
589C4—42no—
590C4—42no—
591C4—82no—
592C4—1064no—
593C4—4242no—
594C4—2124no—
595C4—2122no—
596C2—22no—
597C2—21no—
598C2—21no—
599C2—22no—
600C2—4242no—
601C2—2124no—
602C2#2027720411yescharacteristic
603C2#2028317311yescharacteristic
604C2#2028317411yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D53
commutatorC23×C53C23×C4
FrattiniC24C23×D53
FittingC53×#1996C2
socleC22×C53C22×C22⋊C4
prime radicalC2×#16811213C2
cosocleC24×C53C24
hypercenter#1996D53
hypocenterC53#21422
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1996

Complements

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

NH ≅ G/Ntype
2. #20277312C2semidirect
3. #20277067C2semidirect
4. #20275701C2semidirect
5. #20275551C2semidirect
6. #20275550C2semidirect
7. #20275167C2semidirect
8. #20274502C2semidirect
9. C53×#1996C2semidirect
14. C2×#13133212C4semidirect
15. C2×#13133206C4semidirect
16. C2×#13133202C4semidirect
17. C2×#13133192C4semidirect
42. C22×D212C2×C4semidirect
46. C22×C4×D53C2×C4semidirect
64. C2×C4⋊Dic53C2×C4semidirect
65. C2×#11719919C2×C4semidirect
75. C2×#11719921C2×C4semidirect
76. C2×#11719921C2×C4semidirect
86. C2×C4×Dic53C2×C4semidirect
93. C22×Q8×C53C2×C4semidirect
100. C2×C4⋊C4×C53C2×C4semidirect
101. C2×C4⋊C4×C53C2×C4semidirect
298. #1996D53semidirect
341. C53#21422semidirect
343. C22×Q8C4×D53semidirect
376. C2×C4⋊C4C4×D53semidirect
377. C2×C4⋊C4C4×D53semidirect

Maximal subgroups

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

subgroupindex
C2×#168112132
#202773122
#202770672
#202757012
#202755512
#202755502
#202751672
#202745022
C53×#19962
#2142253