Finite Group Databasebeta

← #27432524

Subgroups — #27432524

Order 13568.

824 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810951C211yesmaximal, characteristic
2C2×#16810879C211yesmaximal, characteristic
3#20279193C211yesmaximal, characteristic
4#20276191C211yesmaximal, characteristic, complemented
5#20276123C211yesmaximal, characteristic, complemented
6#20276093C211yesmaximal, characteristic, complemented
7#20276037C211yesmaximal, characteristic, complemented
8#20275478C211yesmaximal, characteristic, complemented
9#20275349C211yesmaximal, characteristic, complemented
10#20275056C211yesmaximal, characteristic, complemented
11#20275027C211yesmaximal, characteristic
12#20274900C211yesmaximal, characteristic
13#20274720C211yesmaximal, characteristic, complemented
14#20274436C211yesmaximal, characteristic
15C53×#2362C211yesmaximal, characteristic
16C22×#11719925C2211yescharacteristic
17C22×#11719917C2211yescharacteristic
18C2×#13133196C2211yescharacteristic
19C22×C4⋊Dic53C2211yescharacteristic
20#16810951—24no—
21#16810879—24no—
22C2×#13133148C2211yescharacteristic
23C2×#13133148C2211yescharacteristic
24C2×#13133148C2211yescharacteristic
25C2×C4⋊#88322C2211yescharacteristic
26C2×C4⋊#88322C2211yescharacteristic
27C2×#13133141C2211yescharacteristic
28C2×#13133141C2211yescharacteristic
29C2×#13133141C2211yescharacteristic
30C2×#13133140C2211yescharacteristic
31C2×#13133140C2211yescharacteristic
32C2×#13133140C2211yescharacteristic
33C2×C22⋊#88322C2211yescharacteristic
34#16810555—21no—
35#16810555—21no—
36#16810529C2211yescharacteristic
37#16810529C2211yescharacteristic
38#16810498—21no—
39#16810498—21no—
40#16810493C2211yescharacteristic
41#16810493C2211yescharacteristic
42#16810385C2211yescharacteristic
43#16810385C2211yescharacteristic
44#16810385C2211yescharacteristic
45#16810385C2211yescharacteristic
46#16810385C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810378C2211yescharacteristic
49#16810378C2211yescharacteristic
50C2×C22⋊Q8×C53C2211yescharacteristic
51C2×C4×Q8×C53C2211yescharacteristic
52C2×#117×C53C2211yescharacteristic
53C53×#391C2211yescharacteristic
54C53×#389C2211yescharacteristic
55C53×#388—21no—
56C53×#388—21no—
57C53×#387—21no—
58C4×C4⋊C4×C53—21no—
59C4×C4⋊C4×C53C2211yescharacteristic
60C4×C22⋊C4×C53C2211yescharacteristic
61C22×Dic106C2311yescharacteristic
62C23×Dic53—21no—
63C2×#11719925—22no—
64C2×#11719925—22no—
65C2×#11719925—41no—
66C2×#11719917—22no—
67C2×#11719917—22no—
68C2×#11719917—41no—
69C22×#88322C2311yescharacteristic
70C22×#88322C2311yescharacteristic
71C22×#88322C2311yescharacteristic
72C22×#88322C2311yescharacteristic
73#13133196—22no—
74#13133196—22no—
75C2×C4⋊Dic53—21no—
76C2×C4⋊Dic53—21no—
77C2×C4⋊Dic53—21no—
78C2×C4⋊Dic53—21no—
79C2×C4⋊Dic53—21no—
80C2×C4⋊Dic53—22no—
81C2×C4⋊Dic53—22no—
82C2×C4⋊Dic53C2311yescharacteristic
83C2×C4⋊Dic53C2311yescharacteristic
84C2×C4⋊Dic53C2311yescharacteristic
85C2×#11719919—21no—
86C2×C22⋊Dic53—21no—
87#13133148—22no—
88#13133148—22no—
89#13133148—22no—
90C4⋊#88322—22no—
91C4⋊#88322—41no—
92#13133141—22no—
93#13133141—22no—
94#13133141—22no—
95#13133140—22no—
96#13133140—22no—
97#13133140—41no—
98C22⋊#88322—22no—
99#13133124—21no—
100#13133124—21no—
101#13133124—21no—
102#13133124—21no—
103C22×Q8×C53C2311yescharacteristic
104C23×C4×C53C2311yescharacteristic
105C22⋊Q8×C53—22no—
106C22⋊Q8×C53D414yes—
107C4×Q8×C53—22no—
108C4×Q8×C53—41no—
109#117×C53—22no—
110#117×C53—41no—
111C2×C4⋊C4×C53—21no—
112C2×C4⋊C4×C53—21no—
113C2×C4⋊C4×C53—21no—
114C2×C4⋊C4×C53C2311yescharacteristic
115C2×C4⋊C4×C53C2311yescharacteristic
116C2×C4⋊C4×C53C2311yescharacteristic
117C2×C22⋊C4×C53—21no—
118C2×C22⋊C4×C53—21no—
119C2×C22⋊C4×C53D411yescharacteristic
120C2×C22⋊C4×C53D411yescharacteristic
121C2×C42×C53—21no—
122C2×C42×C53—21no—
123C2×C42×C53—21no—
124C2×C42×C53C2311yescharacteristic
125C2×C42×C53C2311yescharacteristic
126#95×C53—21no—
127#95×C53—21no—
128#95×C53—21no—
129#95×C53—21no—
130#95×C53D411yescharacteristic
131#95×C53D411yescharacteristic
132C24×C53C2×D411yescharacteristic
133C2×Q8×C53—21no—
134C2×Q8×C53—21no—
135C2×Q8×C53—21no—
136C2×Q8×C53—21no—
137C2×Q8×C53—21no—
138C2×Q8×C53C2×D412yes—
139C22×C4×C53—21no—
140C22×C4×C53—21no—
141C22×C4×C53—21no—
142C22×C4×C53—21no—
143C22×C4×C53—21no—
144C22×C4×C53—21no—
145C22×C4×C53—21no—
146C22×C4×C53—21no—
147C22×C4×C53C2×D411yescharacteristic
148C22×C4×C53C2×D411yescharacteristic
149C22×C4×C53C2×D411yescharacteristic
150C22×C4×C53C2×D412yes—
151C22×C4×C53C2×D411yescharacteristic
152C22×C4×C53C2×D411yescharacteristic
153C22×C4×C53#4112yes—
154C22×C4×C53C2411yescharacteristic
155C22×Dic53—21no—
156C22×Dic53—21no—
157C22×Dic53—21no—
158C22×Dic53—21no—
159C22×Dic53—42no—
160C22×Dic53—81no—
161C22×Dic53—81no—
162C2×Dic106—21no—
163C2×Dic106—21no—
164C2×Dic106—21no—
165C2×Dic106—21no—
166C2×Dic106—21no—
167C2×Dic106—21no—
168C4⋊C4×C53—24no—
169C4⋊C4×C53—21no—
170C4⋊C4×C53—21no—
171C4⋊C4×C53—21no—
172C4⋊C4×C53—21no—
173C4⋊C4×C53—42no—
174C4⋊C4×C53—42no—
175C4⋊C4×C53C2×D412yes—
176C4⋊C4×C53#4112yes—
177C22⋊C4×C53—24no—
178C22⋊C4×C53—22no—
179C22⋊C4×C53—22no—
180C22⋊C4×C53—42no—
181C42×C53—21no—
182C42×C53—21no—
183C42×C53—21no—
184C42×C53—21no—
185C42×C53—42no—
186C42×C53—42no—
187C42×C53—42no—
188C22⋊Dic53—42no—
189#11719925—44no—
190C4⋊Dic53—21no—
191C4⋊Dic53—21no—
192C4⋊Dic53—21no—
193C4⋊Dic53—21no—
194C4⋊Dic53—21no—
195C4⋊Dic53—21no—
196C4⋊Dic53—42no—
197C4⋊Dic53—81no—
198C4⋊Dic53—81no—
199C4⋊Dic53—81no—
200C4⋊Dic53—81no—
201#11719919—42no—
202#11719917—44no—
203C2×#88322—21no—
204C2×#88322—21no—
205C2×#88322—21no—
206C2×#88322—21no—
207C2×#88322—21no—
208C2×#88322—21no—
209C2×#88322—21no—
210C2×#88322—21no—
211C2×#88322—21no—
212C2×#88322—21no—
213C2×#88322—21no—
214C2×#88322#4112yes—
215C23×C53—21no—
216C23×C53—21no—
217C23×C53—21no—
218C23×C53—21no—
219C23×C53—21no—
220C23×C53—21no—
221C23×C53#12112yes—
222C23×C53C22×D411yescharacteristic
223Q8×C53—22no—
224Q8×C53—22no—
225Q8×C53—41no—
226Q8×C53—41no—
227C2×C4×C53—21no—
228C2×C4×C53—21no—
229C2×C4×C53—21no—
230C2×C4×C53—21no—
231C2×C4×C53—21no—
232C2×C4×C53—21no—
233C2×C4×C53—22no—
234C2×C4×C53—22no—
235C2×C4×C53—22no—
236C2×C4×C53—22no—
237C2×C4×C53—22no—
238C2×C4×C53—22no—
239C2×C4×C53—22no—
240C2×C4×C53—21no—
241C2×C4×C53—21no—
242C2×C4×C53—21no—
243C2×C4×C53—21no—
244C2×C4×C53—21no—
245C2×C4×C53—21no—
246C2×C4×C53—41no—
247C2×C4×C53—41no—
248C2×C4×C53—41no—
249C2×C4×C53—41no—
250C2×C4×C53—41no—
251C2×C4×C53—41no—
252C2×C4×C53—41no—
253C2×C4×C53—41no—
254C2×C4×C53—41no—
255C2×C4×C53#12112yes—
256C2×C4×C53#13712yes—
257C2×C4×C53#13712yes—
258C2×C4×C53C22×D411yescharacteristic
259C2×C4×C53C2×#4111yescharacteristic
260C2×C4×C53C2×#4111yescharacteristic
261C2×C4×C53C2×#4111yescharacteristic
262C2×C4×C53H(5,𝔽2)11yescharacteristic
263C2×C4×C532−1+411yescharacteristic
264C2×Dic53—41no—
265C2×Dic53—41no—
266C2×Dic53—41no—
267C2×Dic53—41no—
268C2×Dic53—41no—
269C2×Dic53—41no—
270C2×Dic53—41no—
271C2×Dic53—41no—
272C2×Dic53—41no—
273C2×Dic53—41no—
274C2×Dic53—41no—
275C2×Dic53—41no—
276C2×Dic53—84no—
277Dic106—41no—
278Dic106—41no—
279Dic106—41no—
280Dic106—41no—
281#88322—22no—
282#88322—41no—
283#88322—41no—
284#88322—41no—
285#25219—531nomaximal, Sylow
286C22×C53—22no—
287C22×C53—22no—
288C22×C53—22no—
289C22×C53—41no—
290C22×C53—41no—
291C22×C53—41no—
292C22×C53—41no—
293C22×C53C2×#12111yescharacteristic
294C22×C53#54611yescharacteristic
295C22×C53#54711yescharacteristic
296C22×C53C2×#13711yescharacteristic
297C22×C53C2×#13711yescharacteristic
298C22×C53#57711yescharacteristic
299C22×C53#57711yescharacteristic
300C4×C53—22no—
301C4×C53—22no—
302C4×C53—22no—
303C4×C53—41no—
304C4×C53—42no—
305C4×C53—41no—
306C4×C53—42no—
307C4×C53—42no—
308C4×C53—81no—
309C4×C53#52911yescharacteristic
310C4×C53#53911yescharacteristic
311C4×C53#54611yescharacteristic
312C4×C53#54711yescharacteristic
313Dic53—81no—
314Dic53—81no—
315Dic53—81no—
316Dic53—81no—
317C2×#483—531no—
318C2×#470—531no—
319#2502—531no—
320#2362D5311yescharacteristic
321#2084—531no—
322#2072—531no—
323#2066—531no—
324#2056—531no—
325#1948—531no—
326#1927—531no—
327#1878—531no—
328#1873—531no—
329#1851—531no—
330#1803—531no—
331#1736—531no—
332C2×C53—42no—
333C2×C53#242511yescharacteristic
334C2×C53#303911yescharacteristic
335C2×C53#304311yescharacteristic
336C2×C53#331611yescharacteristic
337C2×C53#333111yescharacteristic
338C2×C53#334411yescharacteristic
339C2×C53#335611yescharacteristic
340C22×Q16—531no—
341C22×M16—531no—
342C2×C22⋊Q8—531no—
343C2×C22⋊Q8C2×D5311yescharacteristic
344C2×C4×Q8C2×D5311yescharacteristic
345C2×#117C2×D5311yescharacteristic
346C22×C4⋊C4—531no—
347#483—1064no—
348#470—1064no—
349C2×#106—531no—
350C2×#106—531no—
351C2×#106—531no—
352C2×C4⋊C8—531no—
353C2×C4⋊C8—531no—
354C2×#103—531no—
355C2×#103—531no—
356C2×#103—531no—
357C2×#103—531no—
358C2×#103—531no—
359C2×#103—531no—
360C2×C22⋊C8—531no—
361#398—1061no—
362#398—1061no—
363#391—531no—
364#391—531no—
365#391C2×D5311yescharacteristic
366#389C2×D5311yescharacteristic
367#388—21no—
368#388—21no—
369#387—21no—
370#385—1061no—
371#385—1061no—
372#384—531no—
373#384—531no—
374C4×C4⋊C4—21no—
375C4×C4⋊C4C2×D5311yescharacteristic
376C4×C22⋊C4C2×D5311yescharacteristic
377#340—531no—
378#340—531no—
379#340—531no—
380#340—531no—
381#340—531no—
382#340—531no—
383#336—531no—
384#336—531no—
385C53#2521911yescharacteristic, complemented, Sylow
386C22×Q8—531no—
387C22×Q8C22×D5311yescharacteristic
388C23×C4—1061no—
389C23×C4C22×D5311yescharacteristic
390C2×Q16—1062no—
391C2×Q16—1062no—
392C2×Q16—2121no—
393C2×M16—1062no—
394C2×M16—1062no—
395C2×M16—2121no—
396C22×C8—531no—
397C22×C8—531no—
398C22×C8—531no—
399C22×C8—531no—
400C22⋊Q8—22no—
401C22⋊Q8—1062no—
402C22⋊Q8—1062no—
403C22⋊Q8D21214yes—
404C4×Q8—22no—
405C4×Q8—41no—
406#117—22no—
407#117—41no—
408C2×C4⋊C4—21no—
409C2×C4⋊C4—21no—
410C2×C4⋊C4—21no—
411C2×C4⋊C4—531no—
412C2×C4⋊C4—531no—
413C2×C4⋊C4—531no—
414C2×C4⋊C4—1062no—
415C2×C4⋊C4—1062no—
416C2×C4⋊C4—1061no—
417C2×C4⋊C4—1061no—
418C2×C4⋊C4—1061no—
419C2×C4⋊C4—1061no—
420C2×C4⋊C4—1061no—
421C2×C4⋊C4—1061no—
422C2×C4⋊C4C22×D5311yescharacteristic
423C2×C4⋊C4C22×D5311yescharacteristic
424C2×C4⋊C4C22×D5311yescharacteristic
425C2×C22⋊C4—21no—
426C2×C22⋊C4—21no—
427C2×C22⋊C4—1061no—
428C2×C22⋊C4C22⋊D5311yescharacteristic
429C2×C22⋊C4C22⋊D5311yescharacteristic
430C2×C42—21no—
431C2×C42—21no—
432C2×C42—21no—
433C2×C42C22×D5311yescharacteristic
434C2×C42C22×D5311yescharacteristic
435#106—1062no—
436#106—1062no—
437#106—1062no—
438C4⋊C8—1062no—
439C4⋊C8—2121no—
440#103—1062no—
441#103—1062no—
442#103—1062no—
443#103—1062no—
444#103—1062no—
445#103—2121no—
446C22⋊C8—1062no—
447#95—21no—
448#95—21no—
449#95—21no—
450#95—21no—
451#95—1061no—
452#95—1061no—
453#95—1061no—
454#95—1061no—
455#95C22⋊D5311yescharacteristic
456#95C22⋊D5311yescharacteristic
457C24C2×C22⋊D5311yescharacteristic
458C2×Q8—21no—
459C2×Q8—21no—
460C2×Q8—21no—
461C2×Q8—21no—
462C2×Q8—21no—
463C2×Q8—1061no—
464C2×Q8—1061no—
465C2×Q8—1061no—
466C2×Q8—1061no—
467C2×Q8—1061no—
468C2×Q8—1061no—
469C2×Q8C2×D21212yes—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—1061no—
479C22×C4—1061no—
480C22×C4—1061no—
481C22×C4—1061no—
482C22×C4—4241no—
483C22×C4—4241no—
484C22×C4—2122no—
485C22×C4C2×D21212yes—
486C22×C4#1171994512yes—
487C22×C4C2×C22⋊D5311yescharacteristic
488C22×C4C2×C22⋊D5311yescharacteristic
489C22×C4C2×C22⋊D5311yescharacteristic
490C22×C4C2×C22⋊D5311yescharacteristic
491C22×C4C2×C22⋊D5311yescharacteristic
492C22×C4C23×D5311yescharacteristic
493Q16—2124no—
494M16—2124no—
495C2×C8—532no—
496C2×C8—1061no—
497C2×C8—1061no—
498C2×C8—1061no—
499C2×C8—1061no—
500C2×C8—1061no—
501C2×C8—1061no—
502C2×C8—1061no—
503C2×C8—1061no—
504C2×C8—1061no—
505C2×C8—1061no—
506C2×C8—1061no—
507C4⋊C4—24no—
508C4⋊C4—21no—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—21no—
512C4⋊C4—42no—
513C4⋊C4—42no—
514C4⋊C4—1061no—
515C4⋊C4—1061no—
516C4⋊C4—1061no—
517C4⋊C4—1061no—
518C4⋊C4—1061no—
519C4⋊C4—1061no—
520C4⋊C4—4241no—
521C4⋊C4—4241no—
522C4⋊C4—4241no—
523C4⋊C4—4241no—
524C4⋊C4—2122no—
525C4⋊C4—2122no—
526C4⋊C4C2×D21212yes—
527C4⋊C4#1171994712yes—
528C22⋊C4—24no—
529C22⋊C4—22no—
530C22⋊C4—22no—
531C22⋊C4—42no—
532C22⋊C4—2122no—
533C42—21no—
534C42—21no—
535C42—21no—
536C42—21no—
537C42—42no—
538C42—42no—
539C42—42no—
540C23—21no—
541C23—21no—
542C23—21no—
543C23—21no—
544C23—21no—
545C23—21no—
546C23#1313319112yes—
547C23C22×C22⋊D5311yescharacteristic
548Q8—22no—
549Q8—22no—
550Q8—41no—
551Q8—41no—
552Q8—2121no—
553Q8—2121no—
554Q8—2121no—
555Q8—2121no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—21no—
567C2×C4—22no—
568C2×C4—21no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—4244no—
585C2×C4—2121no—
586C2×C4—2121no—
587C2×C4—2121no—
588C2×C4—2121no—
589C2×C4—2121no—
590C2×C4—2121no—
591C2×C4—2121no—
592C2×C4—2121no—
593C2×C4—2121no—
594C2×C4—2121no—
595C2×C4—2121no—
596C2×C4—2121no—
597C2×C4#1313319112yes—
598C2×C4#1313324712yes—
599C2×C4#1313324712yes—
600C2×C4C22×D21211yescharacteristic
601C2×C4#41×D5311yescharacteristic
602C2×C4C2×#1171994711yescharacteristic
603C2×C4C2×#1171994511yescharacteristic
604C2×C4#1313326211yescharacteristic
605C2×C4#1313326411yescharacteristic
606C8—1062no—
607C8—2121no—
608C8—2121no—
609C8—2121no—
610C22—22no—
611C22—22no—
612C22—22no—
613C22—41no—
614C22—41no—
615C22—41no—
616C22—41no—
617C22C2×#1313319111yescharacteristic
618C22#1681126711yescharacteristic
619C22#1681127211yescharacteristic
620C22C2×#1313324711yescharacteristic
621C22C2×#1313324711yescharacteristic
622C22#1681143111yescharacteristic
623C22#1681143111yescharacteristic
624C4—22no—
625C4—22no—
626C4—22no—
627C4—42no—
628C4—41no—
629C4—42no—
630C4—41no—
631C4—42no—
632C4—81no—
633C4—4241no—
634C4—4241no—
635C4—4241no—
636C4—4241no—
637C4#1681115311yescharacteristic
638C4#1681121511yescharacteristic
639C4#1681126811yescharacteristic
640C4#1681127711yescharacteristic
641C2—42no—
642C2#2027826011yescharacteristic
643C2#2028420211yescharacteristic
644C2#2028423611yescharacteristic
645C2#2028682611yescharacteristic
646C2#2028703111yescharacteristic
647C2#2028717011yescharacteristic
648C2#2028732411yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2362C2
socleC23×C53C22×D4
prime radicalC24×C53C2×D4
hypercenter#2362D53
hypocenterC53#25219
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2362

Complements

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

NH ≅ G/Ntype
4. #20276191C2semidirect
5. #20276123C2semidirect
6. #20276093C2semidirect
7. #20276037C2semidirect
8. #20275478C2semidirect
9. #20275349C2semidirect
10. #20275056C2semidirect
13. #20274720C2semidirect
385. C53#25219semidirect

Maximal subgroups

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

subgroupindex
C2×#168109512
C2×#168108792
#202791932
#202761912
#202761232
#202760932
#202760372
#202754782
#202753492
#202750562
#202750272
#202749002
#202747202
#202744362
C53×#23622
#2521953