Finite Group Databasebeta

← #27466039

Subgroups — #27466039

Order 13568.

812 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×#16810701C211yesmaximal, characteristic
2C2×#16810663C211yesmaximal, characteristic, complemented
3#20279012C211yesmaximal, characteristic
4#20276217C211yesmaximal, characteristic, complemented
5#20276211C211yesmaximal, characteristic, complemented
6#20276151C211yesmaximal, characteristic, complemented
7#20276141C211yesmaximal, characteristic, complemented
8#20275445C211yesmaximal, characteristic, complemented
9#20275440C211yesmaximal, characteristic, complemented
10#20274807C211yesmaximal, characteristic, complemented
11#20274801C211yesmaximal, characteristic, complemented
12#20274746C211yesmaximal, characteristic, complemented
13#20274746C211yesmaximal, characteristic, complemented
14#20274746C211yesmaximal, characteristic, complemented
15C53×#2487C211yesmaximal, characteristic
16C22×#11719917C2211yescharacteristic
17C22×#11719917C2211yescharacteristic
18C2×#13133176C2211yescharacteristic
19#16810701—24no—
20#16810663—24no—
21C2×#13133150C2211yescharacteristic
22C2×#13133150C2211yescharacteristic
23C2×#13133150C2211yescharacteristic
24C2×#13133148C2211yescharacteristic
25C2×#13133148C2211yescharacteristic
26C2×#13133148C2211yescharacteristic
27C2×C4⋊#88322C2211yescharacteristic
28C2×C22⋊#88322C2211yescharacteristic
29C2×C22⋊#88322C2211yescharacteristic
30C2×C4×#88322C2211yescharacteristic
31#16810551—21no—
32#16810537—21no—
33#16810521C2211yescharacteristic
34#16810521C2211yescharacteristic
35#16810511—21no—
36#16810504—21no—
37#16810483C2211yescharacteristic
38#16810483C2211yescharacteristic
39#16810483C2211yescharacteristic
40#16810385C2211yescharacteristic
41#16810385C2211yescharacteristic
42#16810385C2211yescharacteristic
43#16810385C2211yescharacteristic
44#16810385C2211yescharacteristic
45#16810385C2211yescharacteristic
46#16810385C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810385C2211yescharacteristic
49#16810385C2211yescharacteristic
50C2×C4×D4×C53C2211yescharacteristic
51C22×C4⋊C4×C53C2211yescharacteristic
52C53×#400—21no—
53C53×#399—21no—
54C53×#389C2211yescharacteristic
55C53×#389C2211yescharacteristic
56C53×#386—21no—
57C53×#385—21no—
58C53×#380C2211yescharacteristic
59C53×#380C2211yescharacteristic
60C53×#380C2211yescharacteristic
61C2×#11719917—22no—
62C2×#11719917—22no—
63C2×#11719917—22no—
64C2×#11719917—22no—
65C2×#11719917—22no—
66C2×#11719917D414yes—
67C22×#88322C2311yescharacteristic
68C22×#88322C2311yescharacteristic
69C22×#88322C2311yescharacteristic
70C22×#88322C2311yescharacteristic
71#13133176—22no—
72#13133176—22no—
73C2×C4⋊Dic53C2311yescharacteristic
74C2×C4⋊Dic53C2311yescharacteristic
75C2×C4⋊Dic53C2311yescharacteristic
76C2×C4⋊Dic53C2311yescharacteristic
77C2×#11719919—21no—
78C2×#11719919—21no—
79C2×C22⋊Dic53—21no—
80C2×C22⋊Dic53—21no—
81C2×C22⋊Dic53—21no—
82C2×C22⋊Dic53—21no—
83C2×C4×Dic53—21no—
84#13133150—22no—
85#13133150—22no—
86#13133150—22no—
87#13133148—22no—
88#13133148—22no—
89#13133148—22no—
90C4⋊#88322—22no—
91C22⋊#88322—22no—
92C22⋊#88322—22no—
93C4×#88322—22no—
94#13133124—21no—
95#13133124—21no—
96#13133124—21no—
97#13133124—21no—
98#13133124—21no—
99C22×D4×C53D411yescharacteristic
100C23×C4×C53—21no—
101C23×C4×C53C2311yescharacteristic
102C23×C4×C53C2311yescharacteristic
103C4×D4×C53—22no—
104C4×D4×C53—22no—
105C2×C4⋊C4×C53—21no—
106C2×C4⋊C4×C53—21no—
107C2×C4⋊C4×C53—21no—
108C2×C4⋊C4×C53—22no—
109C2×C4⋊C4×C53—22no—
110C2×C4⋊C4×C53D411yescharacteristic
111C2×C4⋊C4×C53C2311yescharacteristic
112C2×C4⋊C4×C53C2311yescharacteristic
113C2×C4⋊C4×C53C2311yescharacteristic
114C2×C4⋊C4×C53C2311yescharacteristic
115C2×C22⋊C4×C53—21no—
116C2×C22⋊C4×C53—21no—
117C2×C22⋊C4×C53—21no—
118C2×C22⋊C4×C53D411yescharacteristic
119C2×C22⋊C4×C53D411yescharacteristic
120C2×C42×C53C2311yescharacteristic
121#95×C53—21no—
122#95×C53—21no—
123#95×C53—21no—
124#95×C53—21no—
125#95×C53—21no—
126C24×C53C2×D411yescharacteristic
127C24×C53C2×D411yescharacteristic
128C2×D4×C53—22no—
129C2×D4×C53—22no—
130C2×D4×C53—41no—
131C22×C4×C53—21no—
132C22×C4×C53—21no—
133C22×C4×C53—21no—
134C22×C4×C53—21no—
135C22×C4×C53—21no—
136C22×C4×C53—21no—
137C22×C4×C53—21no—
138C22×C4×C53—21no—
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—42no—
146C22×C4×C53—81no—
147C22×C4×C53—81no—
148C22×C4×C53C2×D411yescharacteristic
149C22×C4×C53C2×D411yescharacteristic
150C22×C4×C53C2×D411yescharacteristic
151C22×C4×C53C2×D412yes—
152C22×C4×C53C2×D411yescharacteristic
153C22×C4×C53#4112yes—
154C22×C4×C53C2411yescharacteristic
155C22×Dic53—21no—
156C22×Dic53—21no—
157C22×Dic53—21no—
158C22×Dic53—21no—
159C4⋊C4×C53—22no—
160C4⋊C4×C53—21no—
161C4⋊C4×C53—21no—
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×C53—42no—
169C4⋊C4×C53—81no—
170C4⋊C4×C53—81no—
171C22⋊C4×C53—22no—
172C22⋊C4×C53—22no—
173C22⋊C4×C53—42no—
174C22⋊C4×C53—81no—
175C22⋊C4×C53—81no—
176C42×C53—21no—
177C42×C53—21no—
178C22⋊Dic53—42no—
179C22⋊Dic53—42no—
180C22⋊Dic53—81no—
181C22⋊Dic53—81no—
182C4⋊Dic53—21no—
183C4⋊Dic53—21no—
184C4⋊Dic53—21no—
185C4⋊Dic53—21no—
186C4⋊Dic53—21no—
187C4⋊Dic53—21no—
188C4⋊Dic53—21no—
189C4⋊Dic53—21no—
190#11719919—42no—
191#11719919—81no—
192C4×Dic53—42no—
193#11719917—28no—
194#11719917—44no—
195C2×#88322—21no—
196C2×#88322—21no—
197C2×#88322—21no—
198C2×#88322—21no—
199C2×#88322—21no—
200C2×#88322—21no—
201C2×#88322—21no—
202C2×#88322—21no—
203C2×#88322C2×D412yes—
204C2×#88322C2×D412yes—
205C2×#88322#4112yes—
206C2×#88322#4112yes—
207C23×C53—21no—
208C23×C53—21no—
209C23×C53—21no—
210C23×C53—21no—
211C23×C53—21no—
212C23×C53—21no—
213C23×C53—21no—
214C23×C53—21no—
215C23×C53—21no—
216C23×C53—21no—
217C23×C53—21no—
218C23×C53—21no—
219C23×C53—21no—
220C23×C53#12112yes—
221C23×C53C22×D411yescharacteristic
222D4×C53—44no—
223C2×C4×C53—21no—
224C2×C4×C53—21no—
225C2×C4×C53—21no—
226C2×C4×C53—21no—
227C2×C4×C53—22no—
228C2×C4×C53—22no—
229C2×C4×C53—22no—
230C2×C4×C53—22no—
231C2×C4×C53—21no—
232C2×C4×C53—21no—
233C2×C4×C53—21no—
234C2×C4×C53—21no—
235C2×C4×C53—21no—
236C2×C4×C53—21no—
237C2×C4×C53—41no—
238C2×C4×C53—41no—
239C2×C4×C53—41no—
240C2×C4×C53—41no—
241C2×C4×C53—41no—
242C2×C4×C53—41no—
243C2×C4×C53—41no—
244C2×C4×C53—41no—
245C2×C4×C53—41no—
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—84no—
254C2×C4×C53#12112yes—
255C2×C4×C53#13712yes—
256C2×C4×C53C22×D411yescharacteristic
257C2×C4×C53C2×#4111yescharacteristic
258C2×C4×C53C2×#4111yescharacteristic
259C2×C4×C53C2×#4111yescharacteristic
260C2×C4×C532−1+411yescharacteristic
261C2×C4×C532−1+411yescharacteristic
262C2×Dic53—41no—
263C2×Dic53—41no—
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—
274#88322—22no—
275#88322—41no—
276#88322—41no—
277#88322#12314yes—
278#27671—531nomaximal, Sylow
279C22×C53—22no—
280C22×C53—22no—
281C22×C53—22no—
282C22×C53—41no—
283C22×C53—41no—
284C22×C53—41no—
285C22×C53—41no—
286C22×C53—41no—
287C22×C53—41no—
288C22×C53—41no—
289C22×C53—41no—
290C22×C53—41no—
291C22×C53—41no—
292C22×C53—41no—
293C22×C53C2×#12111yescharacteristic
294C22×C53#54711yescharacteristic
295C22×C53#54711yescharacteristic
296C22×C53C2×#13711yescharacteristic
297C22×C53#57511yescharacteristic
298C22×C53#57611yescharacteristic
299C22×C53#57711yescharacteristic
300C4×C53—22no—
301C4×C53—22no—
302C4×C53—41no—
303C4×C53—41no—
304C4×C53—81no—
305C4×C53—81no—
306C4×C53—81no—
307C4×C53—81no—
308C4×C53C2×#12311yescharacteristic
309C4×C53#53611yescharacteristic
310C4×C53#55211yescharacteristic
311C4×C53#55211yescharacteristic
312Dic53—81no—
313Dic53—81no—
314Dic53—81no—
315Dic53—81no—
316C2×#436—531no—
317C2×#428—531no—
318#2487D5311yescharacteristic
319#2485—531no—
320#2088—531no—
321#2087—531no—
322#2077—531no—
323#2075—531no—
324#1942—531no—
325#1941—531no—
326#1825—531no—
327#1823—531no—
328#1809—531no—
329#1809—531no—
330#1809—531no—
331C2×C53—42no—
332C2×C53—81no—
333C2×C53#248611yescharacteristic
334C2×C53#305611yescharacteristic
335C2×C53#306211yescharacteristic
336C2×C53#335111yescharacteristic
337C2×C53#335511yescharacteristic
338C2×C53#335611yescharacteristic
339C2×C53#335711yescharacteristic
340C22×M16—531no—
341C22×M16—531no—
342C2×C4×D4C2×D5311yescharacteristic
343C2×#117—531no—
344C22×C4⋊C4C2×D5311yescharacteristic
345#436—1064no—
346#428—1064no—
347C2×#107—531no—
348C2×#107—531no—
349C2×#107—531no—
350C2×#106—531no—
351C2×#106—531no—
352C2×#106—531no—
353C2×C4⋊C8—531no—
354C2×C22⋊C8—531no—
355C2×C22⋊C8—531no—
356C2×C4×C8—531no—
357#400—21no—
358#399—21no—
359#397—1061no—
360#394—1061no—
361#389—531no—
362#389—531no—
363#389C2×D5311yescharacteristic
364#389C2×D5311yescharacteristic
365#388—1061no—
366#387—1061no—
367#386—21no—
368#385—21no—
369#380—531no—
370#380—531no—
371#380—531no—
372#380C2×D5311yescharacteristic
373#380C2×D5311yescharacteristic
374#380C2×D5311yescharacteristic
375#340—531no—
376#340—531no—
377#340—531no—
378#340—531no—
379#340—531no—
380#340—531no—
381#340—531no—
382#340—531no—
383#340—531no—
384#340—531no—
385C53#2767111yescharacteristic, complemented, Sylow
386C22×D4C22⋊D5311yescharacteristic
387C23×C4—21no—
388C23×C4C22×D5311yescharacteristic
389C23×C4C22×D5311yescharacteristic
390C2×M16—534no—
391C2×M16—1062no—
392C2×M16—1062no—
393C2×M16—1062no—
394C2×M16—1062no—
395C2×M16—1062no—
396C22×C8—531no—
397C22×C8—531no—
398C22×C8—531no—
399C22×C8—531no—
400C4×D4—22no—
401C4×D4—22no—
402#117—1062no—
403#117—1062no—
404C2×C4⋊C4—22no—
405C2×C4⋊C4—22no—
406C2×C4⋊C4—21no—
407C2×C4⋊C4—21no—
408C2×C4⋊C4—21no—
409C2×C4⋊C4—531no—
410C2×C4⋊C4—531no—
411C2×C4⋊C4—531no—
412C2×C4⋊C4—531no—
413C2×C4⋊C4—1061no—
414C2×C4⋊C4—1061no—
415C2×C4⋊C4C22⋊D5311yescharacteristic
416C2×C4⋊C4C22×D5311yescharacteristic
417C2×C4⋊C4C22×D5311yescharacteristic
418C2×C4⋊C4C22×D5311yescharacteristic
419C2×C4⋊C4C22×D5311yescharacteristic
420C2×C22⋊C4—21no—
421C2×C22⋊C4—21no—
422C2×C22⋊C4—21no—
423C2×C22⋊C4—1061no—
424C2×C22⋊C4—1061no—
425C2×C22⋊C4—1061no—
426C2×C22⋊C4—1061no—
427C2×C22⋊C4C22⋊D5311yescharacteristic
428C2×C22⋊C4C22⋊D5311yescharacteristic
429C2×C42—1061no—
430C2×C42C22×D5311yescharacteristic
431#107—1062no—
432#107—1062no—
433#107—1062no—
434#106—1062no—
435#106—1062no—
436#106—1062no—
437C4⋊C8—1062no—
438C22⋊C8—1062no—
439C22⋊C8—1062no—
440C4×C8—1062no—
441#95—21no—
442#95—21no—
443#95—21no—
444#95—21no—
445#95—21no—
446#95—1061no—
447#95—1061no—
448#95—1061no—
449#95—1061no—
450#95—1061no—
451C24C2×C22⋊D5311yescharacteristic
452C24C2×C22⋊D5311yescharacteristic
453C2×D4—22no—
454C2×D4—22no—
455C2×D4—41no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—42no—
471C22×C4—81no—
472C22×C4—81no—
473C22×C4—1061no—
474C22×C4—1061no—
475C22×C4—1061no—
476C22×C4—1061no—
477C22×C4D4×D5312yes—
478C22×C4#1171994712yes—
479C22×C4C2×C22⋊D5311yescharacteristic
480C22×C4C2×C22⋊D5311yescharacteristic
481C22×C4C2×C22⋊D5311yescharacteristic
482C22×C4C2×C22⋊D5311yescharacteristic
483C22×C4C23×D5311yescharacteristic
484M16—1068no—
485M16—2124no—
486C2×C8—532no—
487C2×C8—532no—
488C2×C8—532no—
489C2×C8—532no—
490C2×C8—1061no—
491C2×C8—1061no—
492C2×C8—1061no—
493C2×C8—1061no—
494C2×C8—1061no—
495C2×C8—1061no—
496C2×C8—1061no—
497C2×C8—1061no—
498C4⋊C4—22no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—42no—
508C4⋊C4—81no—
509C4⋊C4—81no—
510C4⋊C4—1061no—
511C4⋊C4—1061no—
512C4⋊C4—1061no—
513C4⋊C4—1061no—
514C4⋊C4—1061no—
515C4⋊C4—1061no—
516C4⋊C4—1061no—
517C4⋊C4—1061no—
518C4⋊C4—4241no—
519C4⋊C4—2122no—
520C22⋊C4—22no—
521C22⋊C4—22no—
522C22⋊C4—42no—
523C22⋊C4—81no—
524C22⋊C4—81no—
525C22⋊C4—4241no—
526C22⋊C4—4241no—
527C22⋊C4—2122no—
528C22⋊C4—2122no—
529C42—21no—
530C42—21no—
531C42—2122no—
532C23—21no—
533C23—21no—
534C23—21no—
535C23—21no—
536C23—21no—
537C23—21no—
538C23—21no—
539C23—21no—
540C23—21no—
541C23—21no—
542C23—21no—
543C23—21no—
544C23—21no—
545C23#1313318912yes—
546C23C22×C22⋊D5311yescharacteristic
547D4—44no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—84no—
579C2×C4—2121no—
580C2×C4—2121no—
581C2×C4—2121no—
582C2×C4—2121no—
583C2×C4—2121no—
584C2×C4—2121no—
585C2×C4—2121no—
586C2×C4—2121no—
587C2×C4—2121no—
588C2×C4—2121no—
589C2×C4—2121no—
590C2×C4—2121no—
591C2×C4#1313318912yes—
592C2×C4#1313324812yes—
593C2×C4C2×D4×D5311yescharacteristic
594C2×C4#41×D5311yescharacteristic
595C2×C4#41×D5311yescharacteristic
596C2×C4C2×#1171994711yescharacteristic
597C2×C4#1313326411yescharacteristic
598C2×C4#1313326411yescharacteristic
599C8—534no—
600C8—1062no—
601C8—2121no—
602C8—2121no—
603C22—22no—
604C22—22no—
605C22—22no—
606C22—41no—
607C22—41no—
608C22—41no—
609C22—41no—
610C22—41no—
611C22—41no—
612C22—41no—
613C22—41no—
614C22—41no—
615C22—41no—
616C22—41no—
617C22C2×#1313318911yescharacteristic
618C22#1681127211yescharacteristic
619C22#1681127211yescharacteristic
620C22C2×#1313324811yescharacteristic
621C22#1681142111yescharacteristic
622C22#1681142611yescharacteristic
623C22#1681143411yescharacteristic
624C4—22no—
625C4—22no—
626C4—41no—
627C4—41no—
628C4—81no—
629C4—81no—
630C4—81no—
631C4—81no—
632C4—4241no—
633C4—4241no—
634C4—4241no—
635C4—4241no—
636C4#123×D5311yescharacteristic
637C4#1681119511yescharacteristic
638C4#1681131011yescharacteristic
639C4#1681131011yescharacteristic
640C2—42no—
641C2—81no—
642C2#2027902011yescharacteristic
643C2#2028434111yescharacteristic
644C2#2028439111yescharacteristic
645C2#2028726911yescharacteristic
646C2#2028731511yescharacteristic
647C2#2028733011yescharacteristic
648C2#2028734111yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2487C2
socleC23×C53C22×D4
prime radicalC22×D4×C53D4
hypercenter#2487D53
hypocenterC53#27671
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #2487

Complements

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

NH ≅ G/Ntype
2. C2×#16810663C2semidirect
4. #20276217C2semidirect
5. #20276211C2semidirect
6. #20276151C2semidirect
7. #20276141C2semidirect
8. #20275445C2semidirect
9. #20275440C2semidirect
10. #20274807C2semidirect
11. #20274801C2semidirect
12. #20274746C2semidirect
13. #20274746C2semidirect
14. #20274746C2semidirect
385. C53#27671semidirect

Maximal subgroups

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

subgroupindex
C2×#168107012
C2×#168106632
#202790122
#202762172
#202762112
#202761512
#202761412
#202754452
#202754402
#202748072
#202748012
#202747462
C53×#24872
#2767153