Finite Group Databasebeta

← #27466009

Subgroups — #27466009

Order 13568.

838 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810695C211yesmaximal, 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#20275455C211yesmaximal, characteristic, complemented
9#20275450C211yesmaximal, characteristic, complemented
10#20274807C211yesmaximal, characteristic, complemented
11#20274805C211yesmaximal, characteristic, complemented
12#20274746C211yesmaximal, characteristic, complemented
13#20274742C211yesmaximal, characteristic, complemented
14#20274738C211yesmaximal, characteristic, complemented
15C53×#2486C211yesmaximal, characteristic
16C22×#11719917C2211yescharacteristic
17C23×#88322C2211yescharacteristic
18C2×#13133176C2211yescharacteristic
19#16810695—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×#13133128C2211yescharacteristic
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×#123×C53—21no—
51C2×C4×D4×C53C2211yescharacteristic
52C22×C4⋊C4×C53C2211yescharacteristic
53C53×#398—21no—
54C53×#397—21no—
55C53×#389C2211yescharacteristic
56C53×#389C2211yescharacteristic
57C53×#381—21no—
58C53×#380C2211yescharacteristic
59C53×#380C2211yescharacteristic
60C53×#380C2211yescharacteristic
61C2×#11719917—22no—
62C2×#11719917—22no—
63C2×#11719917D414yes—
64C22×#88322—22no—
65C22×#88322—22no—
66C22×#88322—22no—
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—
93#13133128—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
103#123×C53—44no—
104C4×D4×C53—22no—
105C4×D4×C53—22no—
106C2×C4⋊C4×C53—21no—
107C2×C4⋊C4×C53—21no—
108C2×C4⋊C4×C53—21no—
109C2×C4⋊C4×C53—22no—
110C2×C4⋊C4×C53—22no—
111C2×C4⋊C4×C53D411yescharacteristic
112C2×C4⋊C4×C53C2311yescharacteristic
113C2×C4⋊C4×C53C2311yescharacteristic
114C2×C4⋊C4×C53C2311yescharacteristic
115C2×C4⋊C4×C53C2311yescharacteristic
116C2×C22⋊C4×C53—21no—
117C2×C22⋊C4×C53—21no—
118C2×C22⋊C4×C53—21no—
119C2×C22⋊C4×C53D411yescharacteristic
120C2×C22⋊C4×C53D411yescharacteristic
121C2×C42×C53C2311yescharacteristic
122#95×C53—21no—
123#95×C53—21no—
124#95×C53—21no—
125#95×C53—21no—
126#95×C53—21no—
127C24×C53C2×D411yescharacteristic
128C24×C53C2×D411yescharacteristic
129C2×D4×C53—22no—
130C2×D4×C53—22no—
131C2×D4×C53—41no—
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—21no—
146C22×C4×C53—42no—
147C22×C4×C53—42no—
148C22×C4×C53—42no—
149C22×C4×C53C2×D411yescharacteristic
150C22×C4×C53C2×D411yescharacteristic
151C22×C4×C53C2×D411yescharacteristic
152C22×C4×C53C2×D411yescharacteristic
153C22×C4×C53C2×D412yes—
154C22×C4×C53#4112yes—
155C22×C4×C53C2411yescharacteristic
156C22×Dic53—21no—
157C22×Dic53—21no—
158C22×Dic53—21no—
159C22×Dic53—21no—
160C4⋊C4×C53—22no—
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—21no—
169C4⋊C4×C53—42no—
170C4⋊C4×C53—42no—
171C4⋊C4×C53—42no—
172C22⋊C4×C53—22no—
173C22⋊C4×C53—22no—
174C22⋊C4×C53—42no—
175C22⋊C4×C53—42no—
176C22⋊C4×C53—42no—
177C42×C53—21no—
178C42×C53—21no—
179C22⋊Dic53—42no—
180C22⋊Dic53—42no—
181C22⋊Dic53—81no—
182C22⋊Dic53—81no—
183C4⋊Dic53—21no—
184C4⋊Dic53—21no—
185C4⋊Dic53—21no—
186C4⋊Dic53—21no—
187C4⋊Dic53—21no—
188C4⋊Dic53—21no—
189C4⋊Dic53—21no—
190C4⋊Dic53—21no—
191#11719919—42no—
192#11719919—81no—
193C4×Dic53—42no—
194#11719917—28no—
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×#88322—44no—
204C2×#88322C2×D412yes—
205C2×#88322C2×D412yes—
206C2×#88322#4112yes—
207C2×#88322#4112yes—
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×C53AGL(1,ℤ/8)12yes—
222C23×C53C22×D411yescharacteristic
223D4×C53—44no—
224C2×C4×C53—21no—
225C2×C4×C53—21no—
226C2×C4×C53—22no—
227C2×C4×C53—22no—
228C2×C4×C53—22no—
229C2×C4×C53—22no—
230C2×C4×C53—21no—
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—21no—
238C2×C4×C53—21no—
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—41no—
254C2×C4×C53—41no—
255C2×C4×C53—84no—
256C2×C4×C53#12112yes—
257C2×C4×C53C22×D411yescharacteristic
258C2×C4×C53C2×#4111yescharacteristic
259C2×C4×C53C2×#4111yescharacteristic
260C2×C4×C53C2×#4111yescharacteristic
261C2×C4×C532−1+411yescharacteristic
262C2×C4×C532−1+411yescharacteristic
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—
274C2×Dic53—41no—
275#88322—22no—
276#88322—41no—
277#88322—41no—
278#88322#12314yescomplemented
279#27669—531nomaximal, Sylow
280C22×C53—22no—
281C22×C53—22no—
282C22×C53—22no—
283C22×C53—22no—
284C22×C53—22no—
285C22×C53—22no—
286C22×C53—41no—
287C22×C53—41no—
288C22×C53—41no—
289C22×C53—41no—
290C22×C53—41no—
291C22×C53—41no—
292C22×C53—41no—
293C22×C53—41no—
294C22×C53C2×#12111yescharacteristic
295C22×C53#54711yescharacteristic
296C22×C53#54711yescharacteristic
297C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
298C22×C53#57511yescharacteristic
299C22×C53#57711yescharacteristic
300C22×C53#57811yescharacteristic
301C4×C53—22no—
302C4×C53—41no—
303C4×C53—41no—
304C4×C53—41no—
305C4×C53—81no—
306C4×C53—81no—
307C4×C53—81no—
308C4×C53—81no—
309C4×C53C2×#12311yescharacteristic
310C4×C53#53611yescharacteristic
311C4×C53#55211yescharacteristic
312C4×C53#55211yescharacteristic
313Dic53—81no—
314Dic53—81no—
315Dic53—81no—
316Dic53—81no—
317C2×#435—531no—
318C2×#428—531no—
319#2486D5311yescharacteristic
320#2485—531no—
321#2088—531no—
322#2087—531no—
323#2077—531no—
324#2075—531no—
325#1944—531no—
326#1943—531no—
327#1825—531no—
328#1824—531no—
329#1809—531no—
330#1808—531no—
331#1807—531no—
332C2×C53—42no—
333C2×C53—42no—
334C2×C53#248611yescharacteristic
335C2×C53#305511yescharacteristic
336C2×C53#306311yescharacteristic
337C2×C53#334311yescharacteristic
338C2×C53#334511yescharacteristic
339C2×C53#334811yescharacteristic
340C2×C53#334911yescharacteristic
341C22×M16—531no—
342C23×C8—531no—
343C2×#123—21no—
344C2×C4×D4C2×D5311yescharacteristic
345C2×#117—531no—
346C22×C4⋊C4C2×D5311yescharacteristic
347#435—1064no—
348#428—1064no—
349C2×#107—531no—
350C2×#107—531no—
351C2×#107—531no—
352C2×#106—531no—
353C2×#106—531no—
354C2×#106—531no—
355C2×C4⋊C8—531no—
356C2×C22⋊C8—531no—
357C2×C22⋊C8—531no—
358C2×#97—531no—
359#398—21no—
360#397—21no—
361#397—1061no—
362#394—1061no—
363#389—531no—
364#389—531no—
365#389C2×D5311yescharacteristic
366#389C2×D5311yescharacteristic
367#388—1061no—
368#387—1061no—
369#381—21no—
370#380—531no—
371#380—531no—
372#380—531no—
373#380C2×D5311yescharacteristic
374#380C2×D5311yescharacteristic
375#380C2×D5311yescharacteristic
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—
385#340—531no—
386C53#2766911yescharacteristic, complemented, Sylow
387C22×D4C22⋊D5311yescharacteristic
388C23×C4—21no—
389C23×C4C22×D5311yescharacteristic
390C23×C4C22×D5311yescharacteristic
391C2×M16—534no—
392C2×M16—1062no—
393C2×M16—1062no—
394C22×C8—531no—
395C22×C8—531no—
396C22×C8—531no—
397C22×C8—531no—
398C22×C8—1062no—
399C22×C8—1062no—
400C22×C8—1062no—
401#123—44no—
402C4×D4—22no—
403C4×D4—22no—
404#117—1062no—
405#117—1062no—
406C2×C4⋊C4—22no—
407C2×C4⋊C4—22no—
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—531no—
415C2×C4⋊C4—1061no—
416C2×C4⋊C4—1061no—
417C2×C4⋊C4C22⋊D5311yescharacteristic
418C2×C4⋊C4C22×D5311yescharacteristic
419C2×C4⋊C4C22×D5311yescharacteristic
420C2×C4⋊C4C22×D5311yescharacteristic
421C2×C4⋊C4C22×D5311yescharacteristic
422C2×C22⋊C4—21no—
423C2×C22⋊C4—21no—
424C2×C22⋊C4—21no—
425C2×C22⋊C4—1061no—
426C2×C22⋊C4—1061no—
427C2×C22⋊C4—1061no—
428C2×C22⋊C4—1061no—
429C2×C22⋊C4C22⋊D5311yescharacteristic
430C2×C22⋊C4C22⋊D5311yescharacteristic
431C2×C42—1061no—
432C2×C42C22×D5311yescharacteristic
433#107—1062no—
434#107—1062no—
435#107—1062no—
436#106—1062no—
437#106—1062no—
438#106—1062no—
439C4⋊C8—1062no—
440C22⋊C8—1062no—
441C22⋊C8—1062no—
442#97—1062no—
443#95—21no—
444#95—21no—
445#95—21no—
446#95—21no—
447#95—21no—
448#95—1061no—
449#95—1061no—
450#95—1061no—
451#95—1061no—
452#95—1061no—
453C24C2×C22⋊D5311yescharacteristic
454C24C2×C22⋊D5311yescharacteristic
455C2×D4—22no—
456C2×D4—22no—
457C2×D4—41no—
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—21no—
471C22×C4—21no—
472C22×C4—42no—
473C22×C4—42no—
474C22×C4—42no—
475C22×C4—1061no—
476C22×C4—1061no—
477C22×C4—1061no—
478C22×C4—1061no—
479C22×C4D4×D5312yes—
480C22×C4#1171994712yes—
481C22×C4C2×C22⋊D5311yescharacteristic
482C22×C4C2×C22⋊D5311yescharacteristic
483C22×C4C2×C22⋊D5311yescharacteristic
484C22×C4C2×C22⋊D5311yescharacteristic
485C22×C4C23×D5311yescharacteristic
486M16—1068no—
487C2×C8—532no—
488C2×C8—532no—
489C2×C8—532no—
490C2×C8—532no—
491C2×C8—1061no—
492C2×C8—1061no—
493C2×C8—1061no—
494C2×C8—1061no—
495C2×C8—1061no—
496C2×C8—1061no—
497C2×C8—1061no—
498C2×C8—1061no—
499C2×C8—2124no—
500C4⋊C4—22no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—42no—
510C4⋊C4—42no—
511C4⋊C4—42no—
512C4⋊C4—1061no—
513C4⋊C4—1061no—
514C4⋊C4—1061no—
515C4⋊C4—1061no—
516C4⋊C4—1061no—
517C4⋊C4—1061no—
518C4⋊C4—1061no—
519C4⋊C4—1061no—
520C4⋊C4—4241no—
521C4⋊C4—2122no—
522C22⋊C4—22no—
523C22⋊C4—22no—
524C22⋊C4—42no—
525C22⋊C4—42no—
526C22⋊C4—42no—
527C22⋊C4—4241no—
528C22⋊C4—4241no—
529C22⋊C4—2122no—
530C22⋊C4—2122no—
531C42—21no—
532C42—21no—
533C42—2122no—
534C23—21no—
535C23—21no—
536C23—21no—
537C23—21no—
538C23—21no—
539C23—21no—
540C23—21no—
541C23—21no—
542C23—21no—
543C23—21no—
544C23—21no—
545C23—21no—
546C23#1313318912yes—
547C23#1313324312yes—
548C23C22×C22⋊D5311yescharacteristic
549D4—44no—
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—21no—
563C2×C4—21no—
564C2×C4—21no—
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—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—84no—
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—2121no—
593C2×C4—2121no—
594C2×C4#1313318912yes—
595C2×C4C2×D4×D5311yescharacteristic
596C2×C4#41×D5311yescharacteristic
597C2×C4#41×D5311yescharacteristic
598C2×C4C2×#1171994711yescharacteristic
599C2×C4#1313326411yescharacteristic
600C2×C4#1313326411yescharacteristic
601C8—534no—
602C8—1062no—
603C8—2121no—
604C8—2121no—
605C22—22no—
606C22—22no—
607C22—22no—
608C22—22no—
609C22—22no—
610C22—22no—
611C22—41no—
612C22—41no—
613C22—41no—
614C22—41no—
615C22—41no—
616C22—41no—
617C22—41no—
618C22—41no—
619C22C2×#1313318911yescharacteristic
620C22#1681127211yescharacteristic
621C22#1681127211yescharacteristic
622C22C2×#1313324311yescharacteristic
623C22#1681142111yescharacteristic
624C22#1681143111yescharacteristic
625C22#1681143811yescharacteristic
626C4—22no—
627C4—41no—
628C4—41no—
629C4—41no—
630C4—81no—
631C4—81no—
632C4—81no—
633C4—81no—
634C4—4241no—
635C4—4241no—
636C4—4241no—
637C4—4241no—
638C4#123×D5311yescharacteristic
639C4#1681119511yescharacteristic
640C4#1681131011yescharacteristic
641C4#1681131011yescharacteristic
642C2—42no—
643C2—42no—
644C2#2027902011yescharacteristic
645C2#2028433011yescharacteristic
646C2#2028440211yescharacteristic
647C2#2028714911yescharacteristic
648C2#2028718511yescharacteristic
649C2#2028722411yescharacteristic
650C2#2028724511yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2486C2
socleC23×C53C22×D4
prime radicalC22×D4×C53D4
hypercenter#2486D53
hypocenterC53#27669
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #2486

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. #20275455C2semidirect
9. #20275450C2semidirect
10. #20274807C2semidirect
11. #20274805C2semidirect
12. #20274746C2semidirect
13. #20274742C2semidirect
14. #20274738C2semidirect
278. #88322#123semidirect
386. C53#27669semidirect

Maximal subgroups

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

subgroupindex
C2×#168106952
C2×#168106632
#202790122
#202762172
#202762112
#202761512
#202761412
#202754552
#202754502
#202748072
#202748052
#202747462
#202747422
#202747382
C53×#24862
#2766953