Finite Group Databasebeta

← #27391872

Subgroups — #27391872

Order 13568.

876 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810904C211yesmaximal, characteristic
2#20277461C211yesmaximal, characteristic, complemented
3#20277075C211yesmaximal, characteristic
4#20275663C211yesmaximal, characteristic, complemented
5#20275306C211yesmaximal, characteristic, complemented
6#20275285C211yesmaximal, characteristic, complemented
7#20274973C211yesmaximal, characteristic
8#20274959C211yesmaximal, characteristic
9#20274722C211yesmaximal, characteristic
10#20274615C211yesmaximal, characteristic, complemented
11#107×Dic53C211yesmaximal, characteristic, complemented
12#20274574C211yesmaximal, characteristic, complemented
13#20274539C211yesmaximal, characteristic
14#20274529C211yesmaximal, characteristic
15C53×#1949C211yesmaximal, characteristic, complemented
16C2×#13133217C2211yescharacteristic
17C2×C4×D212C2211yescharacteristic
18C2×#13133179C2211yescharacteristic
19C2×#13133179C2211yescharacteristic
20#16810904C418yescomplemented
21C2×#13133151C2211yescharacteristic
22C2×#13133151C2211yescharacteristic
23C2×C4⋊#88322C2211yescharacteristic
24C2×#13133139C2211yescharacteristic
25C2×#13133139C2211yescharacteristic
26C2×#13133137C2211yescharacteristic
27C2×#13133137C2211yescharacteristic
28C2×#13133127C2211yescharacteristic
29C2×C8×Dic53C2211yescharacteristic
30#16810527—21no—
31#16810527—21no—
32#16810519C2211yescharacteristic
33#16810519C2211yescharacteristic
34#16810508—21no—
35#16810508—21no—
36C4⋊C4×Dic53C2211yescharacteristic
37C4⋊C4×Dic53C2211yescharacteristic
38C4⋊C4×Dic53C2211yescharacteristic
39C4⋊C4×Dic53C2211yescharacteristic
40C4×#11719921—21no—
41C4×#11719921—21no—
42#16810470—21no—
43#16810469C2211yescharacteristic
44#95×D53—21no—
45#16810385C2211yescharacteristic
46#16810385C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810385C2211yescharacteristic
49#16810385C2211yescharacteristic
50#16810385C2211yescharacteristic
51#16810377C2211yescharacteristic
52#16810377C2211yescharacteristic
53C2×#107×C53C2211yescharacteristic
54C2×#106×C53C2211yescharacteristic
55C2×C4⋊C8×C53C2211yescharacteristic
56C53×#382C2211yescharacteristic
57C4×C4⋊C4×C53C2211yescharacteristic
58C53×#340C2211yescharacteristic
59C53×#340C2211yescharacteristic
60C22×Dic106—21no—
61C22×D212C2311yescharacteristic
62C22×C4×D53—21no—
63C22×C4×D53—21no—
64C22×C4×D53—21no—
65C22×#88322C2311yescharacteristic
66C22×#88322C2311yescharacteristic
67#13133217—22no—
68#13133217C2×C414yescomplemented
69C4×D212—22no—
70C4×D212C2×C414yes—
71#13133179—22no—
72#13133179—22no—
73#13133179—22no—
74#13133179Q814yescomplemented
75C2×C4⋊Dic53—21no—
76C2×C4⋊Dic53C2311yescharacteristic
77C2×C4⋊Dic53C2311yescharacteristic
78C2×C4⋊Dic53C2311yescharacteristic
79C2×#11719919—21no—
80C2×#11719919—21no—
81C2×#11719921—21no—
82C2×#11719921—21no—
83C2×#11719921—21no—
84C2×C4×Dic53—21no—
85C2×C4×Dic53—21no—
86C2×C4×Dic53—21no—
87C2×C4×Dic53—21no—
88C2×C4×Dic53C2311yescharacteristic
89C2×C4×Dic53C2311yescharacteristic
90#13133151C2×C414yescomplemented
91#13133151C2×C414yescomplemented
92C4⋊#88322—22no—
93#13133139C2×C414yes—
94#13133139C2×C414yes—
95#13133137—22no—
96#13133137—22no—
97#13133127—22no—
98C8×Dic53—22no—
99#13133124—21no—
100#13133124—21no—
101#13133124—21no—
102#13133124—21no—
103#13133124—21no—
104#13133124—21no—
105#13133124D411yescharacteristic
106#13133124D411yescharacteristic
107#13133124D411yescharacteristic
108#13133124D411yescharacteristic
109C22×C8×C53C2311yescharacteristic
110C22×C8×C53C2311yescharacteristic
111C2×C4⋊C4×C53C2311yescharacteristic
112C2×C4⋊C4×C53C2311yescharacteristic
113C2×C4⋊C4×C53C2311yescharacteristic
114C2×C4⋊C4×C53C2311yescharacteristic
115C2×C42×C53—21no—
116C2×C42×C53C2311yescharacteristic
117#107×C53—22no—
118#106×C53—22no—
119C4⋊C8×C53C2×C414yescomplemented
120#95×C53—21no—
121#95×C53—21no—
122#95×C53—21no—
123C23×D53—21no—
124C22×C4×C53—21no—
125C22×C4×C53—21no—
126C22×C4×C53—21no—
127C22×C4×C53—21no—
128C22×C4×C53C2×D411yescharacteristic
129C22×C4×C53C2×D411yescharacteristic
130C22×C4×C53C2411yescharacteristic
131C22×Dic53—21no—
132C22×Dic53—21no—
133C22×Dic53—21no—
134C22×Dic53C2×D411yescharacteristic
135C22×Dic53C2×D411yescharacteristic
136C22×Dic53C2×D411yescharacteristic
137C22×Dic53C2×D411yescharacteristic
138C2×D212—21no—
139C2×D212—21no—
140C2×D212—21no—
141C2×D212C22×C412yes—
142C2×D212C2×Q812yes—
143C2×D212#4112yes—
144C2×C4×D53—24no—
145C2×C4×D53—24no—
146C2×C4×D53—24no—
147C2×C4×D53—81no—
148C2×C4×D53—81no—
149C2×C4×D53—81no—
150C2×C4×D53—81no—
151C2×C4×D53—81no—
152C2×C4×D53—81no—
153C2×Dic106—24no—
154C2×Dic106—81no—
155C2×Dic106—81no—
156C2×C8×C53—21no—
157C2×C8×C53—21no—
158C2×C8×C53—21no—
159C2×C8×C53—21no—
160C2×C8×C53C22×C412yes—
161C2×C8×C53C22×C412yes—
162C4⋊C4×C53—21no—
163C4⋊C4×C53—21no—
164C4⋊C4×C53—21no—
165C4⋊C4×C53—21no—
166C4⋊C4×C53—21no—
167C4⋊C4×C53—21no—
168C4⋊C4×C53C2×Q812yes—
169C4⋊C4×C53#4112yes—
170C42×C53—81no—
171C42×C53C22×C412yes—
172C42×C53#4112yes—
173#11719921—24no—
174#11719921—24no—
175#11719921—24no—
176C4⋊Dic53—24no—
177C4⋊Dic53—21no—
178C4⋊Dic53—21no—
179C4⋊Dic53—21no—
180C4⋊Dic53C22×C412yes—
181C4⋊Dic53C22×C412yes—
182C4⋊Dic53C22×C412yes—
183#11719919—81no—
184#11719919—81no—
185C4×Dic53—21no—
186C4×Dic53—21no—
187C4×Dic53—81no—
188C4×Dic53—81no—
189C4×Dic53—81no—
190C4×Dic53—81no—
191C4×Dic53C2×Q812yes—
192C4×Dic53#4112yes—
193C2×#88322—21no—
194C2×#88322—21no—
195C2×#88322—21no—
196C2×#88322—21no—
197C2×#88322—21no—
198C2×#88322—21no—
199C23×C53C22×D411yescharacteristic
200C22×D53—22no—
201C22×D53—22no—
202C22×D53—22no—
203C22×D53—41no—
204C22×D53—41no—
205C22×D53—41no—
206C22×D53—41no—
207C2×C4×C53—21no—
208C2×C4×C53—21no—
209C2×C4×C53—21no—
210C2×C4×C53—21no—
211C2×C4×C53—22no—
212C2×C4×C53—22no—
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×C53—41no—
220C2×C4×C53—41no—
221C2×C4×C53—41no—
222C2×C4×C53—41no—
223C2×C4×C53C4×D412yes—
224C2×C4×C53C4×D412yes—
225C2×C4×C53C23×C411yescharacteristic
226C2×C4×C53C22×Q811yescharacteristic
227C2×C4×C53C2×#4111yescharacteristic
228C2×C4×C53C2×#4111yescharacteristic
229C2×C4×C53C2×#4111yescharacteristic
230C2×C4×C53C2×#4111yescharacteristic
231C2×Dic53—21no—
232C2×Dic53—21no—
233C2×Dic53—21no—
234C2×Dic53—21no—
235C2×Dic53—22no—
236C2×Dic53—22no—
237C2×Dic53—22no—
238C2×Dic53—21no—
239C2×Dic53—21no—
240C2×Dic53—21no—
241C2×Dic53—21no—
242C2×Dic53—21no—
243C2×Dic53—21no—
244C2×Dic53—41no—
245C2×Dic53—41no—
246C2×Dic53—41no—
247C2×Dic53—41no—
248C2×Dic53—41no—
249C2×Dic53—41no—
250C2×Dic53C22⋊Q812yes—
251C2×Dic53C22⋊Q812yes—
252D212—22no—
253D212—22no—
254D212—22no—
255D212C4×Q814yes—
256C4×D53—84no—
257C4×D53—84no—
258C4×D53—84no—
259Dic106—84no—
260C8×C53—22no—
261C8×C53—22no—
262#88322—41no—
263#88322—41no—
264#21624—531nomaximal, Sylow
265C22×C53C2×C4×D411yescharacteristic
266C22×C53C2×C22⋊Q811yescharacteristic
267C22×C53#52511yescharacteristic
268C22×C53#57611yescharacteristic
269C22×C53#57611yescharacteristic
270C22×C53#57711yescharacteristic
271C22×C53#57811yescharacteristic
272C2×D53—24no—
273C2×D53—42no—
274C2×D53—42no—
275C2×D53—42no—
276C2×D53—42no—
277C2×D53—42no—
278C2×D53—42no—
279C4×C53—22no—
280C4×C53—22no—
281C4×C53—42no—
282C4×C53—42no—
283C4×C53—81no—
284C4×C53—81no—
285C4×C53C2×C4×Q811yescharacteristic
286C4×C53C4×#4111yescharacteristic
287C4×C53#52911yescharacteristic
288C4×C53#53311yescharacteristic
289Dic53—22no—
290Dic53—22no—
291Dic53—41no—
292Dic53—41no—
293Dic53—42no—
294Dic53—42no—
295Dic53—42no—
296C2×#474—531no—
297#2331—531no—
298#2275—531no—
299#1987—531no—
300#1949D5311yescharacteristic, complemented
301#1921—531no—
302#1918—531no—
303#1864—531no—
304#1862—531no—
305#1804—531no—
306#1776—531no—
307C4×#107—531no—
308#1768—531no—
309#1762—531no—
310C4×#102—531no—
311C2×C53C4×C22⋊Q811yescharacteristic
312C2×C53#294711yescharacteristic
313C2×C53#294811yescharacteristic
314C2×C53#308111yescharacteristic
315C2×C53#308311yescharacteristic
316C2×C53#311811yescharacteristic
317C2×C53#312411yescharacteristic
318D53—44no—
319C2×C4⋊Q8—531no—
320C2×C4×D4—531no—
321C2×C4×D4—531no—
322C2×C4×D4—531no—
323#474—538no—
324C2×#107—531no—
325C2×#107—531no—
326C2×#107C2×D5311yescharacteristic
327C2×#106C2×D5311yescharacteristic
328C2×C4⋊C8—531no—
329C2×C4⋊C8C2×D5311yescharacteristic
330C2×#102—531no—
331C2×#102—531no—
332C2×#102—531no—
333C2×#102—531no—
334C2×#97—531no—
335C2×C4×C8—531no—
336#391—1061no—
337#391—1061no—
338#389—531no—
339#389—531no—
340#387—1061no—
341#387—1061no—
342#382C2×D5311yescharacteristic
343C4×C4⋊C4—531no—
344C4×C4⋊C4—531no—
345C4×C4⋊C4—531no—
346C4×C4⋊C4—531no—
347C4×C4⋊C4C2×D5311yescharacteristic
348C4×C22⋊C4—1061no—
349C4×C22⋊C4—1061no—
350C4×C22⋊C4—1061no—
351#376—531no—
352C2×#95—1061no—
353#340—531no—
354#340—531no—
355#340—531no—
356#340—531no—
357#340—531no—
358#340—531no—
359#340C2×D5311yescharacteristic
360#340C2×D5311yescharacteristic
361#336—531no—
362#336—531no—
363C53#2162411yescharacteristic, complemented, Sylow
364C22×Q8—1061no—
365C22×D4—531no—
366C23×C4—1061no—
367C23×C4—1061no—
368C23×C4—1061no—
369C22×C8—531no—
370C22×C8—531no—
371C22×C8C22×D5311yescharacteristic
372C22×C8C22×D5311yescharacteristic
373C4⋊Q8—534no—
374C4⋊Q8—1062no—
375C4×D4—534no—
376C4×D4—534no—
377C4×D4—1062no—
378C4×D4—1062no—
379C4×D4—1062no—
380C4×D4—1062no—
381C2×C4⋊C4—531no—
382C2×C4⋊C4—531no—
383C2×C4⋊C4—531no—
384C2×C4⋊C4—1061no—
385C2×C4⋊C4—1061no—
386C2×C4⋊C4—1061no—
387C2×C4⋊C4C22×D5311yescharacteristic
388C2×C4⋊C4C22×D5311yescharacteristic
389C2×C4⋊C4C22×D5311yescharacteristic
390C2×C4⋊C4C22×D5311yescharacteristic
391C2×C22⋊C4—1061no—
392C2×C22⋊C4—1061no—
393C2×C22⋊C4—1061no—
394C2×C42—21no—
395C2×C42—531no—
396C2×C42—531no—
397C2×C42—1061no—
398C2×C42—1061no—
399C2×C42—1061no—
400C2×C42—1061no—
401C2×C42C22×D5311yescharacteristic
402#107—22no—
403#107—534no—
404#107—534no—
405#106—22no—
406C4⋊C8—1062no—
407C4⋊C8C4×D5314yescomplemented
408#102—534no—
409#102—534no—
410#102—1062no—
411#102—1062no—
412#97—1062no—
413C4×C8—1062no—
414#95—21no—
415#95—21no—
416#95—21no—
417#95—531no—
418#95—531no—
419#95—531no—
420#95—531no—
421#95—1061no—
422#95—1061no—
423#95—1061no—
424#95—1061no—
425#95—1061no—
426#95—1061no—
427C24—1061no—
428C2×Q8—1064no—
429C2×Q8—4241no—
430C2×Q8—4241no—
431C2×D4—532no—
432C2×D4—532no—
433C2×D4—532no—
434C2×D4—1061no—
435C2×D4—1061no—
436C2×D4—1061no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—531no—
442C22×C4—531no—
443C22×C4—531no—
444C22×C4—531no—
445C22×C4—1064no—
446C22×C4—1064no—
447C22×C4—1061no—
448C22×C4—1061no—
449C22×C4—1061no—
450C22×C4—1064no—
451C22×C4—4241no—
452C22×C4—4241no—
453C22×C4—4241no—
454C22×C4—4241no—
455C22×C4—4241no—
456C22×C4—4241no—
457C22×C4D4×D5311yescharacteristic
458C22×C4D4×D5311yescharacteristic
459C22×C4C23×D5311yescharacteristic
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—1061no—
465C2×C8—1061no—
466C2×C8—1061no—
467C2×C8—1061no—
468C2×C8—1061no—
469C2×C8—1061no—
470C2×C8C2×C4×D5312yes—
471C2×C8C2×C4×D5312yes—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—532no—
479C4⋊C4—532no—
480C4⋊C4—532no—
481C4⋊C4—1064no—
482C4⋊C4—1061no—
483C4⋊C4—1061no—
484C4⋊C4—1061no—
485C4⋊C4—4241no—
486C4⋊C4—4241no—
487C4⋊C4#1171994712yes—
488C4⋊C4Q8×D5312yes—
489C22⋊C4—1064no—
490C22⋊C4—1064no—
491C22⋊C4—1064no—
492C42—81no—
493C42—532no—
494C42—532no—
495C42—1061no—
496C42—1061no—
497C42—4241no—
498C42—4241no—
499C42—4241no—
500C42—4241no—
501C42C2×C4×D5312yes—
502C42#1171994912yes—
503C23—1062no—
504C23—1062no—
505C23—1062no—
506C23—2121no—
507C23—2121no—
508C23—2121no—
509C23—2121no—
510C23C2×D4×D5311yescharacteristic
511Q8—4244no—
512D4—534no—
513D4—1062no—
514D4—1062no—
515D4—1062no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—532no—
533C2×C4—532no—
534C2×C4—1061no—
535C2×C4—1061no—
536C2×C4—1061no—
537C2×C4—1061no—
538C2×C4—1062no—
539C2×C4—1062no—
540C2×C4—1062no—
541C2×C4—1061no—
542C2×C4—1061no—
543C2×C4—1061no—
544C2×C4—1061no—
545C2×C4—1061no—
546C2×C4—1061no—
547C2×C4—4244no—
548C2×C4—4244no—
549C2×C4—4244no—
550C2×C4—2121no—
551C2×C4—2121no—
552C2×C4—2121no—
553C2×C4—2121no—
554C2×C4—2121no—
555C2×C4—2121no—
556C2×C4#1313317912yes—
557C2×C4#1313317912yes—
558C2×C4C22×C4×D5311yescharacteristic
559C2×C4C2×Q8×D5311yescharacteristic
560C2×C4#41×D5311yescharacteristic
561C2×C4#41×D5311yescharacteristic
562C2×C4C2×#1171994711yescharacteristic
563C2×C4C2×#1171994911yescharacteristic
564C8—22no—
565C8—22no—
566C8—2121no—
567C8—2121no—
568C22—1064no—
569C22—2122no—
570C22—2122no—
571C22—2122no—
572C22—2122no—
573C22—2122no—
574C22—2122no—
575C22C2×#1313317911yescharacteristic
576C22C22⋊Q8×D5311yescharacteristic
577C22#1681113311yescharacteristic
578C22#1681142511yescharacteristic
579C22#1681142511yescharacteristic
580C22#1681143311yescharacteristic
581C22#1681143911yescharacteristic
582C4—22no—
583C4—22no—
584C4—42no—
585C4—42no—
586C4—81no—
587C4—81no—
588C4—1062no—
589C4—1062no—
590C4—2121no—
591C4—2122no—
592C4—2122no—
593C4—2122no—
594C4—2121no—
595C4C4×Q8×D5311yescharacteristic
596C4C4×#1171994711yescharacteristic
597C4#1681115811yescharacteristic
598C4#1681117811yescharacteristic
599C2—2124no—
600C2#2027724511yescharacteristic
601C2#2028355611yescharacteristic
602C2#2028356611yescharacteristic
603C2#2028454311yescharacteristic
604C2#2028456911yescharacteristic
605C2#2028489611yescharacteristic
606C2#2028497611yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1949C2
socleC23×C53C22×D4
prime radicalC22×D212C23
cosocleC22×C4×C53C24
hypercenter#1949D53
hypocenterC53#21624
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1949

Complements

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

NH ≅ G/Ntype
2. #20277461C2semidirect
4. #20275663C2semidirect
5. #20275306C2semidirect
6. #20275285C2semidirect
10. #20274615C2semidirect
11. #107×Dic53C2semidirect
12. #20274574C2semidirect
15. C53×#1949C2semidirect
20. #16810904C4semidirect
68. #13133217C2×C4semidirect
74. #13133179Q8semidirect
90. #13133151C2×C4semidirect
91. #13133151C2×C4semidirect
119. C4⋊C8×C53C2×C4semidirect
300. #1949D53semidirect
363. C53#21624semidirect
407. C4⋊C8C4×D53semidirect

Maximal subgroups

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

subgroupindex
C2×#168109042
#202774612
#202770752
#202756632
#202753062
#202752852
#202749732
#202749592
#202747222
#202746152
#107×Dic532
#202745742
#202745392
#202745292
C53×#19492
#2162453