Finite Group Databasebeta

← #27466028

Subgroups — #27466028

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×#16810699C211yesmaximal, characteristic
2#20279026C211yesmaximal, characteristic
3#20279006C211yesmaximal, characteristic
4#20276214C211yesmaximal, characteristic, complemented
5#20276209C211yesmaximal, characteristic, complemented
6#20276149C211yesmaximal, characteristic, complemented
7#20276139C211yesmaximal, characteristic, complemented
8#20275446C211yesmaximal, characteristic, complemented
9#20275441C211yesmaximal, characteristic, complemented
10#20274810C211yesmaximal, characteristic, complemented
11#20274801C211yesmaximal, characteristic, complemented
12#20274749C211yesmaximal, characteristic, complemented
13#20274748C211yesmaximal, characteristic, complemented
14#20274745C211yesmaximal, characteristic, complemented
15C2×C53×#428C211yesmaximal, characteristic, complemented
16C22×#11719912C2211yescharacteristic
17C2×C4×C22⋊D53C2211yescharacteristic
18#16810699—24no—
19C2×#13133151C2211yescharacteristic
20C2×#13133149C2211yescharacteristic
21C2×#13133146C2211yescharacteristic
22C2×C22⋊#88322C2211yescharacteristic
23C2×#13133129C2211yescharacteristic
24C2×C8×Dic53C2211yescharacteristic
25#16810561—21no—
26#16810558—21no—
27#16810551—21no—
28#16810538—21no—
29#16810519C2211yescharacteristic
30#16810519C2211yescharacteristic
31#16810519C2211yescharacteristic
32#16810519C2211yescharacteristic
33#16810514—21no—
34#16810507—21no—
35#16810502—21no—
36#16810496—21no—
37#16810483C2211yescharacteristic
38#16810483C2211yescharacteristic
39#16810480C2211yescharacteristic
40#16810480C2211yescharacteristic
41#16810480C2211yescharacteristic
42#16810480C2211yescharacteristic
43#16810386C2211yescharacteristic
44#16810386C2211yescharacteristic
45#16810385C2211yescharacteristic
46#16810385C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810385C2211yescharacteristic
49#16810385C2211yescharacteristic
50#16810385C2211yescharacteristic
51#16810385C2211yescharacteristic
52#16810385C2211yescharacteristic
53C22×M16×C53C2211yescharacteristic
54C2×#117×C53C2211yescharacteristic
55C22×C4⋊C4×C53C2211yescharacteristic
56C53×#428—24no—
57C2×#107×C53C2211yescharacteristic
58C2×#107×C53C2211yescharacteristic
59C2×#106×C53C2211yescharacteristic
60C2×#106×C53C2211yescharacteristic
61C22×C22⋊D53D411yescharacteristic
62C22×C4×D53C2311yescharacteristic
63C2×#11719912—22no—
64C2×#11719912—22no—
65C2×#11719912—22no—
66C22×#88322C2311yescharacteristic
67C22×#88322C2311yescharacteristic
68C4×C22⋊D53—22no—
69C4×C22⋊D53—22no—
70C2×C4⋊Dic53C2311yescharacteristic
71C2×C4⋊Dic53C2311yescharacteristic
72C2×C4⋊Dic53C2311yescharacteristic
73C2×C4⋊Dic53C2311yescharacteristic
74C2×#11719919—21no—
75C2×#11719919—21no—
76C2×#11719919—21no—
77C2×#11719919—21no—
78C2×#11719919D411yescharacteristic
79C2×C22⋊Dic53—21no—
80C2×C22⋊Dic53—21no—
81C2×C22⋊Dic53D411yescharacteristic
82C2×#11719921—21no—
83C2×#11719921—21no—
84C2×#11719921—21no—
85C2×#11719921—21no—
86C2×#11719921D411yescharacteristic
87C2×C4×Dic53C2311yescharacteristic
88#13133151—22no—
89#13133149—22no—
90#13133146—22no—
91C22⋊#88322—22no—
92#13133129—22no—
93C8×Dic53—22no—
94#13133124—21no—
95#13133124—21no—
96#13133124—21no—
97#13133124—21no—
98#13133124—21no—
99#13133124—21no—
100#13133124—21no—
101#13133124—21no—
102#13133124—21no—
103#13133124—21no—
104C23×C4×C53—21no—
105C23×C4×C53C2311yescharacteristic
106C2×M16×C53—22no—
107C2×M16×C53—22no—
108C2×M16×C53D414yes—
109C22×C8×C53C2311yescharacteristic
110C22×C8×C53C2311yescharacteristic
111#117×C53—22no—
112#117×C53—22no—
113C2×C4⋊C4×C53—21no—
114C2×C4⋊C4×C53—22no—
115C2×C4⋊C4×C53—22no—
116C2×C4⋊C4×C53C2311yescharacteristic
117C2×C4⋊C4×C53C2311yescharacteristic
118C2×C4⋊C4×C53C2311yescharacteristic
119C2×C4⋊C4×C53C2311yescharacteristic
120C2×C22⋊C4×C53—21no—
121C2×C42×C53—21no—
122#107×C53—22no—
123#107×C53—22no—
124#106×C53—22no—
125#106×C53—22no—
126C24×C53C2×D411yescharacteristic
127C23×D53C2×D411yescharacteristic
128C22×C4×C53—21no—
129C22×C4×C53—21no—
130C22×C4×C53—21no—
131C22×C4×C53—21no—
132C22×C4×C53—21no—
133C22×C4×C53—21no—
134C22×C4×C53—21no—
135C22×C4×C53—21no—
136C22×C4×C53—42no—
137C22×C4×C53—81no—
138C22×C4×C53—81no—
139C22×C4×C53C2×D411yescharacteristic
140C22×C4×C53C2×D412yes—
141C22×C4×C53#4112yes—
142C22×C4×C53C2411yescharacteristic
143C2×C22⋊D53—22no—
144C2×C22⋊D53—22no—
145C2×C22⋊D53—41no—
146C22×Dic53—21no—
147C22×Dic53—21no—
148C22×Dic53—21no—
149C22×Dic53—21no—
150C22×Dic53C2×D411yescharacteristic
151C22×Dic53C2×D411yescharacteristic
152C22×Dic53C2×D411yescharacteristic
153C2×C4×D53—21no—
154C2×C4×D53—21no—
155C2×C4×D53—21no—
156C2×C4×D53—21no—
157C2×C4×D53—21no—
158C2×C4×D53—21no—
159M16×C53—28no—
160C2×C8×C53—21no—
161C2×C8×C53—21no—
162C2×C8×C53C2×D412yes—
163C2×C8×C53C2×D412yes—
164C2×C8×C53#4112yes—
165C2×C8×C53#4112yes—
166C4⋊C4×C53—21no—
167C4⋊C4×C53—21no—
168C4⋊C4×C53—21no—
169C4⋊C4×C53—21no—
170C4⋊C4×C53—21no—
171C4⋊C4×C53—21no—
172C4⋊C4×C53—21no—
173C4⋊C4×C53—21no—
174C4⋊C4×C53—42no—
175C22⋊C4×C53—42no—
176C42×C53—42no—
177C22⋊Dic53—22no—
178C22⋊Dic53—42no—
179C22⋊Dic53—81no—
180#11719921—22no—
181#11719921—42no—
182#11719921—81no—
183#11719921—81no—
184#11719921—81no—
185C4⋊Dic53—21no—
186C4⋊Dic53—21no—
187C4⋊Dic53—21no—
188C4⋊Dic53—21no—
189C4⋊Dic53—21no—
190C4⋊Dic53—21no—
191C4⋊Dic53—21no—
192C4⋊Dic53—21no—
193#11719919—22no—
194#11719919—42no—
195#11719919—81no—
196#11719919—81no—
197#11719919—81no—
198C4×Dic53—21no—
199C4×Dic53—21no—
200C2×#88322—21no—
201C2×#88322—21no—
202C2×#88322—21no—
203C2×#88322—21no—
204C2×#88322—21no—
205C2×#88322—21no—
206#11719912—44no—
207C23×C53—21no—
208C23×C53—21no—
209C23×C53—21no—
210C23×C53—21no—
211C23×C53—21no—
212C23×C53—21no—
213C23×C53#12112yes—
214C23×C53C22×D411yescharacteristic
215C22×D53—21no—
216C22×D53—21no—
217C22×D53—21no—
218C22×D53—21no—
219C22×D53—21no—
220C22×D53—21no—
221C22×D53—21no—
222C2×C4×C53—21no—
223C2×C4×C53—21no—
224C2×C4×C53—22no—
225C2×C4×C53—22no—
226C2×C4×C53—22no—
227C2×C4×C53—22no—
228C2×C4×C53—41no—
229C2×C4×C53—41no—
230C2×C4×C53—41no—
231C2×C4×C53—41no—
232C2×C4×C53—41no—
233C2×C4×C53—41no—
234C2×C4×C53—41no—
235C2×C4×C53—41no—
236C2×C4×C53—41no—
237C2×C4×C53—41no—
238C2×C4×C53—41no—
239C2×C4×C53—41no—
240C2×C4×C53—84no—
241C2×C4×C53#12112yes—
242C2×C4×C53C22×D411yescharacteristic
243C2×C4×C53C2×#4111yescharacteristic
244C2×C4×C53C2×#4111yescharacteristic
245C2×C4×C53C2×#4111yescharacteristic
246C2×C4×C532−1+411yescharacteristic
247C2×C4×C532−1+411yescharacteristic
248C22⋊D53—44no—
249C2×Dic53—21no—
250C2×Dic53—21no—
251C2×Dic53—21no—
252C2×Dic53—21no—
253C2×Dic53—21no—
254C2×Dic53—21no—
255C2×Dic53—21no—
256C2×Dic53—21no—
257C2×Dic53—41no—
258C2×Dic53—41no—
259C2×Dic53—41no—
260C2×Dic53—41no—
261C2×Dic53—41no—
262C2×Dic53—41no—
263C2×Dic53—41no—
264C2×Dic53—41no—
265C2×Dic53—41no—
266C2×Dic53—41no—
267C2×Dic53—41no—
268C2×Dic53—41no—
269C2×Dic53#13712yes—
270C4×D53—41no—
271C4×D53—41no—
272C4×D53—41no—
273C4×D53—41no—
274C8×C53—22no—
275C8×C53#12314yes—
276#88322—41no—
277#88322—41no—
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×C53C2×#12111yescharacteristic
287C22×C53#54711yescharacteristic
288C22×C53#54711yescharacteristic
289C22×C53C2×#13711yescharacteristic
290C22×C53#57511yescharacteristic
291C22×C53#57611yescharacteristic
292C22×C53#57711yescharacteristic
293C2×D53—41no—
294C2×D53—41no—
295C2×D53—41no—
296C2×D53—41no—
297C2×D53—41no—
298C2×D53—41no—
299C2×D53—41no—
300C4×C53—22no—
301C4×C53—81no—
302C4×C53—81no—
303C4×C53—81no—
304C4×C53—81no—
305C4×C53C2×#12311yescharacteristic
306C4×C53#53611yescharacteristic
307C4×C53#55211yescharacteristic
308C4×C53#55211yescharacteristic
309Dic53—22no—
310Dic53—41no—
311Dic53—41no—
312Dic53—81no—
313Dic53—81no—
314Dic53—81no—
315Dic53—81no—
316C2×#436—531no—
317C2×#428D5311yescharacteristic, complemented
318#2487—531no—
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#248611yescharacteristic
333C2×C53#305611yescharacteristic
334C2×C53#306211yescharacteristic
335C2×C53#335111yescharacteristic
336C2×C53#335511yescharacteristic
337C2×C53#335611yescharacteristic
338C2×C53#335711yescharacteristic
339D53—81no—
340C22×M16—531no—
341C22×M16C2×D5311yescharacteristic
342C2×C4×D4—531no—
343C2×#117C2×D5311yescharacteristic
344C22×C4⋊C4C2×D5311yescharacteristic
345#436—1064no—
346#428—24no—
347C2×#107—531no—
348C2×#107C2×D5311yescharacteristic
349C2×#107C2×D5311yescharacteristic
350C2×#106—531no—
351C2×#106C2×D5311yescharacteristic
352C2×#106C2×D5311yescharacteristic
353C2×C4⋊C8—531no—
354C2×C22⋊C8—531no—
355C2×C22⋊C8—531no—
356C2×C4×C8—531no—
357#400—1061no—
358#399—1061no—
359#397—1061no—
360#394—1061no—
361#389—531no—
362#389—531no—
363#389—531no—
364#389—531no—
365#388—1061no—
366#387—1061no—
367#386—1061no—
368#385—1061no—
369#380—531no—
370#380—531no—
371#380—531no—
372#380—531no—
373#380—531no—
374#380—531no—
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×D4—531no—
387C23×C4—21no—
388C23×C4—531no—
389C23×C4C22×D5311yescharacteristic
390C2×M16—22no—
391C2×M16—22no—
392C2×M16—1062no—
393C2×M16—1062no—
394C2×M16—1062no—
395C2×M16C22⋊D5314yes—
396C22×C8—531no—
397C22×C8—531no—
398C22×C8C22×D5311yescharacteristic
399C22×C8C22×D5311yescharacteristic
400C4×D4—1062no—
401C4×D4—1062no—
402#117—22no—
403#117—22no—
404C2×C4⋊C4—22no—
405C2×C4⋊C4—22no—
406C2×C4⋊C4—21no—
407C2×C4⋊C4—531no—
408C2×C4⋊C4—531no—
409C2×C4⋊C4—531no—
410C2×C4⋊C4—531no—
411C2×C4⋊C4—531no—
412C2×C4⋊C4—1061no—
413C2×C4⋊C4—1061no—
414C2×C4⋊C4—1061no—
415C2×C4⋊C4—1061no—
416C2×C4⋊C4C22×D5311yescharacteristic
417C2×C4⋊C4C22×D5311yescharacteristic
418C2×C4⋊C4C22×D5311yescharacteristic
419C2×C4⋊C4C22×D5311yescharacteristic
420C2×C22⋊C4—21no—
421C2×C22⋊C4—531no—
422C2×C22⋊C4—531no—
423C2×C22⋊C4—1061no—
424C2×C22⋊C4—1061no—
425C2×C22⋊C4—1061no—
426C2×C22⋊C4—1061no—
427C2×C22⋊C4—1061no—
428C2×C22⋊C4—1061no—
429C2×C42—21no—
430C2×C42—531no—
431#107—22no—
432#107—22no—
433#107—1062no—
434#106—22no—
435#106—22no—
436#106—1062no—
437C4⋊C8—1062no—
438C22⋊C8—1062no—
439C22⋊C8—1062no—
440C4×C8—1062no—
441#95—1061no—
442#95—1061no—
443#95—1061no—
444#95—1061no—
445#95—1061no—
446#95—1061no—
447#95—1061no—
448#95—1061no—
449#95—1061no—
450#95—1061no—
451C24—531no—
452C24D4×D5311yescharacteristic
453C2×D4—1062no—
454C2×D4—1062no—
455C2×D4—2121no—
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—42no—
465C22×C4—81no—
466C22×C4—81no—
467C22×C4—531no—
468C22×C4—531no—
469C22×C4—531no—
470C22×C4—1061no—
471C22×C4—1061no—
472C22×C4—1061no—
473C22×C4—1061no—
474C22×C4—1061no—
475C22×C4—1061no—
476C22×C4—1061no—
477C22×C4—1061no—
478C22×C4—1061no—
479C22×C4—1061no—
480C22×C4D4×D5311yescharacteristic
481C22×C4#1171994712yes—
482C22×C4C2×C22⋊D5312yes—
483C22×C4C23×D5311yescharacteristic
484M16—28no—
485M16—2124no—
486C2×C8—21no—
487C2×C8—21no—
488C2×C8—1061no—
489C2×C8—1061no—
490C2×C8—1061no—
491C2×C8—1061no—
492C2×C8—1061no—
493C2×C8—1061no—
494C2×C8#1171994512yes—
495C2×C8#1171994512yes—
496C2×C8C2×C22⋊D5312yes—
497C2×C8C2×C22⋊D5312yes—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—42no—
507C4⋊C4—1062no—
508C4⋊C4—1061no—
509C4⋊C4—1061no—
510C4⋊C4—1061no—
511C4⋊C4—1061no—
512C4⋊C4—1061no—
513C4⋊C4—1061no—
514C4⋊C4—1061no—
515C4⋊C4—1061no—
516C4⋊C4—4241no—
517C4⋊C4—4241no—
518C4⋊C4—4241no—
519C4⋊C4—2122no—
520C22⋊C4—42no—
521C22⋊C4—1062no—
522C22⋊C4—1062no—
523C22⋊C4—4241no—
524C22⋊C4—4241no—
525C22⋊C4—4241no—
526C22⋊C4—4241no—
527C22⋊C4—2122no—
528C22⋊C4—2122no—
529C42—42no—
530C42—1061no—
531C42—1061no—
532C23—21no—
533C23—21no—
534C23—21no—
535C23—21no—
536C23—21no—
537C23—21no—
538C23—1061no—
539C23—1061no—
540C23—1061no—
541C23—1061no—
542C23—1061no—
543C23—1061no—
544C23—1061no—
545C23#1313318812yes—
546C23C2×D4×D5311yescharacteristic
547D4—2124no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—84no—
567C2×C4—532no—
568C2×C4—1061no—
569C2×C4—1061no—
570C2×C4—1061no—
571C2×C4—1061no—
572C2×C4—1061no—
573C2×C4—1061no—
574C2×C4—1061no—
575C2×C4—1061no—
576C2×C4—2121no—
577C2×C4—2121no—
578C2×C4—2121no—
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—2121no—
592C2×C4#1313318812yes—
593C2×C4C22×C22⋊D5311yescharacteristic
594C2×C4C2×#1171994711yescharacteristic
595C2×C4C2×#1171994511yescharacteristic
596C2×C4C2×#1171994511yescharacteristic
597C2×C4#1313326511yescharacteristic
598C2×C4#1313326511yescharacteristic
599C8—22no—
600C8—2121no—
601C8—2121no—
602C8#1313320014yes—
603C22—22no—
604C22—22no—
605C22—22no—
606C22—41no—
607C22—41no—
608C22—41no—
609C22—41no—
610C22—2121no—
611C22—2121no—
612C22—2121no—
613C22—2121no—
614C22—2121no—
615C22—2121no—
616C22—2121no—
617C22C2×#1313318811yescharacteristic
618C22#1681127411yescharacteristic
619C22#1681127411yescharacteristic
620C22#137×D5311yescharacteristic
621C22#1681142211yescharacteristic
622C22#1681142711yescharacteristic
623C22#1681143311yescharacteristic
624C4—22no—
625C4—81no—
626C4—81no—
627C4—81no—
628C4—81no—
629C4—1062no—
630C4—4241no—
631C4—4241no—
632C4—4241no—
633C4—4241no—
634C4—2121no—
635C4—2121no—
636C4C2×#1313320011yescharacteristic
637C4#1681119711yescharacteristic
638C4#1681130811yescharacteristic
639C4#1681130811yescharacteristic
640C2—42no—
641C2—4241no—
642C2#2027901711yescharacteristic
643C2#2028433911yescharacteristic
644C2#2028438911yescharacteristic
645C2#2028727611yescharacteristic
646C2#2028731711yescharacteristic
647C2#2028732611yescharacteristic
648C2#2028733711yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC2×C53×#428C2
socleC23×C53C22×D4
prime radicalC22×C22⋊D53D4
hypercenterC2×#428D53
hypocenterC53#27671
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#428

Complements

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

NH ≅ G/Ntype
4. #20276214C2semidirect
5. #20276209C2semidirect
6. #20276149C2semidirect
7. #20276139C2semidirect
8. #20275446C2semidirect
9. #20275441C2semidirect
10. #20274810C2semidirect
11. #20274801C2semidirect
12. #20274749C2semidirect
13. #20274748C2semidirect
14. #20274745C2semidirect
15. C2×C53×#428C2semidirect
317. C2×#428D53semidirect
385. C53#27671semidirect

Maximal subgroups

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

subgroupindex
C2×#168106992
#202790262
#202790062
#202762142
#202762092
#202761492
#202761392
#202754462
#202754412
#202748102
#202748012
#202747492
#202747482
#202747452
C2×C53×#4282
#2767153