Finite Group Databasebeta

← #27466296

Subgroups — #27466296

Order 13568.

872 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20279187C211yesmaximal, characteristic
2#20278411C211yesmaximal, characteristic, complemented
3#20276276C211yesmaximal, characteristic, complemented
4#20276275C211yesmaximal, characteristic, complemented
5#20276121C211yesmaximal, characteristic, complemented
6#20275849C211yesmaximal, characteristic, complemented
7#20275674C211yesmaximal, characteristic, complemented
8#20275400C211yesmaximal, characteristic, complemented
9#20275395C211yesmaximal, characteristic, complemented
10#20275144C211yesmaximal, characteristic
11#20274841C211yesmaximal, characteristic, complemented
12#20274802C211yesmaximal, characteristic, complemented
13#20274489C211yesmaximal, characteristic, complemented
14C2×#16810385C211yesmaximal, characteristic, complemented
15C53×#1874C211yesmaximal, characteristic
16C23×#88322C2211yescharacteristic
17C2×#13133216C2211yescharacteristic
18C2×#13133198—21no—
19C2×#13133198—21no—
20C2×#13133196—21no—
21C22×C4⋊Dic53C2211yescharacteristic
22C2×#13133138C2211yescharacteristic
23C2×#13133138C2211yescharacteristic
24C2×C8×Dic53C2211yescharacteristic
25C2×C8×Dic53C2211yescharacteristic
26#16810559—21no—
27#16810559—21no—
28#16810556—21no—
29#16810543—21no—
30#16810526C2211yescharacteristic
31#16810526C2211yescharacteristic
32#16810522C2211yescharacteristic
33#16810519C2211yescharacteristic
34#16810496—21no—
35#16810492C2211yescharacteristic
36#16810492C2211yescharacteristic
37#16810492C2211yescharacteristic
38#16810483C2211yescharacteristic
39C4×C22⋊Dic53C2211yescharacteristic
40#16810386C2211yescharacteristic
41#16810386C2211yescharacteristic
42#16810386C2211yescharacteristic
43#16810386C2211yescharacteristic
44#16810385C2211yescharacteristic
45#16810385C2211yescharacteristic
46#16810385C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810385C2211yescharacteristic
49#16810385C2211yescharacteristic
50#16810377C2211yescharacteristic
51#16810377C2211yescharacteristic
52C2×#121×C53C2211yescharacteristic
53C22×C4⋊C4×C53C2211yescharacteristic
54C2×#102×C53C2211yescharacteristic
55C2×#102×C53C2211yescharacteristic
56C2×C22⋊C8×C53C2211yescharacteristic
57C53×#340C2211yescharacteristic
58C53×#340C2211yescharacteristic
59C22×Dic106—21no—
60C23×Dic53—21no—
61C22×#88322—22no—
62C22×#88322—22no—
63C22×#88322—41no—
64C22×#88322C2311yescharacteristic
65C22×#88322C2311yescharacteristic
66#13133216—22no—
67#13133216—41no—
68#13133198—44no—
69#13133198—44no—
70#13133196—44no—
71C2×C4⋊Dic53—21no—
72C2×C4⋊Dic53—21no—
73C2×C4⋊Dic53—22no—
74C2×C4⋊Dic53—22no—
75C2×C4⋊Dic53C2311yescharacteristic
76C2×C4⋊Dic53C2311yescharacteristic
77C2×C4⋊Dic53C2311yescharacteristic
78C2×C4⋊Dic53C2311yescharacteristic
79C2×#11719919—21no—
80C2×#11719919—21no—
81C2×#11719919—21no—
82C2×#11719919—21no—
83C2×#11719919—21no—
84C2×#11719919—21no—
85C2×#11719919—21no—
86C2×#11719919—21no—
87C2×C22⋊Dic53—21no—
88C2×C22⋊Dic53—21no—
89C2×C22⋊Dic53—21no—
90C2×C22⋊Dic53—21no—
91C2×C22⋊Dic53—21no—
92C2×C22⋊Dic53D411yescharacteristic
93C2×C22⋊Dic53D411yescharacteristic
94C2×C4×Dic53C2311yescharacteristic
95C2×C4×Dic53C2311yescharacteristic
96#13133138—22no—
97#13133138—41no—
98C8×Dic53—22no—
99C8×Dic53—41no—
100#13133124—21no—
101#13133124—21no—
102#13133124—21no—
103#13133124D411yescharacteristic
104#13133124D411yescharacteristic
105C22×D4×C53—21no—
106C22×D4×C53C2311yescharacteristic
107C23×C4×C53—21no—
108C23×C4×C53C2311yescharacteristic
109C22×C8×C53C2311yescharacteristic
110C22×C8×C53C2311yescharacteristic
111#121×C53—22no—
112#121×C53—22no—
113C2×C4⋊C4×C53—21no—
114C2×C4⋊C4×C53—22no—
115C2×C4⋊C4×C53—41no—
116C2×C4⋊C4×C53C2311yescharacteristic
117C2×C4⋊C4×C53C2311yescharacteristic
118C2×C4⋊C4×C53C2311yescharacteristic
119C2×C22⋊C4×C53—21no—
120#102×C53—22no—
121#102×C53—41no—
122C22⋊C8×C53—22no—
123C24×C53—21no—
124C24×C53C2×D411yescharacteristic
125C2×D4×C53—21no—
126C2×D4×C53—21no—
127C2×D4×C53—21no—
128C2×D4×C53—21no—
129C2×D4×C53—21no—
130C2×D4×C53—21no—
131C2×D4×C53—42no—
132C2×D4×C53—42no—
133C2×D4×C53—42no—
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—42no—
142C22×C4×C53—42no—
143C22×C4×C53—81no—
144C22×C4×C53C2×D411yescharacteristic
145C22×C4×C53#4112yes—
146C22×C4×C53#4112yes—
147C22×C4×C53C2411yescharacteristic
148C22×Dic53—21no—
149C22×Dic53—21no—
150C22×Dic53—21no—
151C22×Dic53—21no—
152C22×Dic53—42no—
153C22×Dic53—42no—
154C22×Dic53—42no—
155C22×Dic53C2×D411yescharacteristic
156C22×Dic53C2×D411yescharacteristic
157C22×Dic53C2×D411yescharacteristic
158C22×Dic53C2×D411yescharacteristic
159C2×Dic106—42no—
160C2×Dic106—42no—
161C2×Dic106—81no—
162C2×C8×C53—21no—
163C2×C8×C53—21no—
164C2×C8×C53—21no—
165C2×C8×C53—21no—
166C2×C8×C53—21no—
167C2×C8×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—42no—
174C4⋊C4×C53#4112yes—
175C22⋊C4×C53—42no—
176C22⋊Dic53—42no—
177C22⋊Dic53—42no—
178C22⋊Dic53—42no—
179C22⋊Dic53—42no—
180C22⋊Dic53—42no—
181C22⋊Dic53D814yes—
182C22⋊Dic53SD1614yes—
183C4⋊Dic53—21no—
184C4⋊Dic53—21no—
185C4⋊Dic53—21no—
186C4⋊Dic53—21no—
187C4⋊Dic53—21no—
188C4⋊Dic53—21no—
189C4⋊Dic53—21no—
190C4⋊Dic53—21no—
191C4⋊Dic53—42no—
192C4⋊Dic53—42no—
193#11719919—42no—
194#11719919—42no—
195#11719919—42no—
196#11719919—42no—
197#11719919—42no—
198#11719919—42no—
199#11719919—42no—
200#11719919—81no—
201C4×Dic53—21no—
202C4×Dic53—21no—
203C4×Dic53—21no—
204C4×Dic53—21no—
205C2×#88322—21no—
206C2×#88322—21no—
207C2×#88322—21no—
208C2×#88322—21no—
209C2×#88322—21no—
210C2×#88322—44no—
211C2×#88322#4112yes—
212C23×C53—21no—
213C23×C53—21no—
214C23×C53—21no—
215C23×C53—21no—
216C23×C53—41no—
217C23×C53—41no—
218C23×C53—41no—
219C23×C53—41no—
220C23×C53—41no—
221C23×C53—41no—
222C23×C53—41no—
223C23×C53#12412yes—
224C23×C53C2×D812yes—
225C23×C53C2×SD1612yes—
226C23×C53C22×D411yescharacteristic
227D4×C53—41no—
228D4×C53—41no—
229D4×C53—41no—
230D4×C53—41no—
231D4×C53—84no—
232C2×C4×C53—21no—
233C2×C4×C53—21no—
234C2×C4×C53—22no—
235C2×C4×C53—22no—
236C2×C4×C53—22no—
237C2×C4×C53—22no—
238C2×C4×C53—22no—
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—84no—
248C2×C4×C53#12412yes—
249C2×C4×C53C2×#4111yescharacteristic
250C2×C4×C53C2×#4111yescharacteristic
251C2×C4×C53C2×#4111yescharacteristic
252C2×C4×C53C2×#4111yescharacteristic
253C2×C4×C53H(5,𝔽2)11yescharacteristic
254C2×C4×C532−1+411yescharacteristic
255C2×Dic53—21no—
256C2×Dic53—21no—
257C2×Dic53—21no—
258C2×Dic53—21no—
259C2×Dic53—21no—
260C2×Dic53—21no—
261C2×Dic53—21no—
262C2×Dic53—21no—
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—
275C2×Dic53—84no—
276C2×Dic53C2×D812yes—
277C2×Dic53C2×D812yes—
278C2×Dic53C2×SD1612yes—
279C2×Dic53C2×SD1612yes—
280Dic106—84no—
281C8×C53—41no—
282C8×C53—41no—
283#88322—22no—
284#88322—41no—
285#27695—531nomaximal, Sylow
286C22×C53—22no—
287C22×C53—22no—
288C22×C53—22no—
289C22×C53—22no—
290C22×C53—22no—
291C22×C53—22no—
292C22×C53—81no—
293C22×C53—81no—
294C22×C53—81no—
295C22×C53—81no—
296C22×C53—81no—
297C22×C53—81no—
298C22×C53—81no—
299C22×C53#48614yes—
300C22×C53C2×#12411yescharacteristic
301C22×C53#54611yescharacteristic
302C22×C53#54711yescharacteristic
303C22×C53C22×D811yescharacteristic
304C22×C53C22×SD1611yescharacteristic
305C22×C53#57711yescharacteristic
306C22×C53#57811yescharacteristic
307C4×C53—22no—
308C4×C53—42no—
309C4×C53—81no—
310C4×C53—81no—
311C4×C53#52911yescharacteristic
312C4×C53#53911yescharacteristic
313C4×C53#55111yescharacteristic
314C4×C53#55211yescharacteristic
315Dic53—22no—
316Dic53—22no—
317Dic53—22no—
318Dic53—22no—
319Dic53—81no—
320Dic53—81no—
321Dic53—81no—
322Dic53—81no—
323#2501—531no—
324#2440—531no—
325#2099—531no—
326#2099—531no—
327#2071—531no—
328#2023—531no—
329#1990—531no—
330#1934—531no—
331#1934—531no—
332#1893—531no—
333#1874D5311yescharacteristic
334#1834—531no—
335#1823—531no—
336C8×C22⋊C4—531no—
337C2×#340—531no—
338C2×C53—24no—
339C2×C53—161no—
340C2×C53#248811yescharacteristic
341C2×C53C2×#48611yescharacteristic
342C2×C53#313711yescharacteristic
343C2×C53#329411yescharacteristic
344C2×C53#329811yescharacteristic
345C2×C53#333411yescharacteristic
346C2×C53#333511yescharacteristic
347C23×C8—531no—
348C2×C4⋊Q8—531no—
349C2×#123—1061no—
350C2×#123—1061no—
351C2×C22⋊Q8—1061no—
352C2×#121C2×D5311yescharacteristic
353C22×C4⋊C4—531no—
354C22×C4⋊C4C2×D5311yescharacteristic
355C2×#102—531no—
356C2×#102—531no—
357C2×#102C2×D5311yescharacteristic
358C2×#102C2×D5311yescharacteristic
359C2×C22⋊C8C2×D5311yescharacteristic
360C2×C4×C8—531no—
361C2×C4×C8—531no—
362#399—1061no—
363#399—1061no—
364#398—1061no—
365#395—1061no—
366#390—531no—
367#390—531no—
368#389—531no—
369#389—531no—
370#385—1061no—
371#384—531no—
372#384—531no—
373#384—531no—
374#380—531no—
375C4×C22⋊C4—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—
385#340—531no—
386#340C2×D5311yescharacteristic
387#340C2×D5311yescharacteristic
388#336—531no—
389#336—531no—
390C53#2769511yescharacteristic, complemented, Sylow
391C22×Q8—1061no—
392C22×D4—21no—
393C22×D4C22×D5311yescharacteristic
394C23×C4—21no—
395C23×C4—1061no—
396C23×C4C22×D5311yescharacteristic
397C22×C8—531no—
398C22×C8—531no—
399C22×C8—1062no—
400C22×C8—1062no—
401C22×C8—2121no—
402C22×C8C22×D5311yescharacteristic
403C22×C8C22×D5311yescharacteristic
404C4⋊Q8—1062no—
405C4⋊Q8—2121no—
406#123—2124no—
407#123—2124no—
408C22⋊Q8—2124no—
409#121—22no—
410#121—22no—
411C2×C4⋊C4—22no—
412C2×C4⋊C4—21no—
413C2×C4⋊C4—41no—
414C2×C4⋊C4—531no—
415C2×C4⋊C4—531no—
416C2×C4⋊C4—531no—
417C2×C4⋊C4—531no—
418C2×C4⋊C4—1062no—
419C2×C4⋊C4—1062no—
420C2×C4⋊C4—1061no—
421C2×C4⋊C4—1061no—
422C2×C4⋊C4—1061no—
423C2×C4⋊C4—1061no—
424C2×C4⋊C4—1061no—
425C2×C4⋊C4—1061no—
426C2×C4⋊C4—1061no—
427C2×C4⋊C4—1061no—
428C2×C4⋊C4—1061no—
429C2×C4⋊C4—1061no—
430C2×C4⋊C4C22×D5311yescharacteristic
431C2×C4⋊C4C22×D5311yescharacteristic
432C2×C4⋊C4C22×D5311yescharacteristic
433C2×C22⋊C4—21no—
434C2×C22⋊C4—531no—
435C2×C22⋊C4—531no—
436C2×C22⋊C4—1061no—
437C2×C22⋊C4—1061no—
438C2×C22⋊C4—1061no—
439C2×C22⋊C4—1061no—
440C2×C22⋊C4—1061no—
441C2×C42—531no—
442C2×C42—531no—
443#102—22no—
444#102—41no—
445#102—1062no—
446#102—2121no—
447C22⋊C8—22no—
448C4×C8—1062no—
449C4×C8—2121no—
450#95—531no—
451#95—531no—
452#95—1061no—
453#95—1061no—
454#95—1061no—
455C24—21no—
456C24D4×D5311yescharacteristic
457C2×Q8—4241no—
458C2×Q8—2122no—
459C2×Q8—2122no—
460C2×D4—21no—
461C2×D4—21no—
462C2×D4—21no—
463C2×D4—21no—
464C2×D4—21no—
465C2×D4—21no—
466C2×D4—42no—
467C2×D4—42no—
468C2×D4—42no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—42no—
477C22×C4—42no—
478C22×C4—81no—
479C22×C4—531no—
480C22×C4—531no—
481C22×C4—531no—
482C22×C4—531no—
483C22×C4—1061no—
484C22×C4—1061no—
485C22×C4—1061no—
486C22×C4—1061no—
487C22×C4—2122no—
488C22×C4—2122no—
489C22×C4—2122no—
490C22×C4#1171994512yes—
491C22×C4D4×D5311yescharacteristic
492C22×C4#1171994712yes—
493C22×C4C23×D5311yescharacteristic
494C2×C8—21no—
495C2×C8—21no—
496C2×C8—21no—
497C2×C8—21no—
498C2×C8—21no—
499C2×C8—21no—
500C2×C8—532no—
501C2×C8—1061no—
502C2×C8—1061no—
503C2×C8—1061no—
504C2×C8—1061no—
505C2×C8—1061no—
506C2×C8—2124no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—21no—
512C4⋊C4—42no—
513C4⋊C4—1061no—
514C4⋊C4—1061no—
515C4⋊C4—1061no—
516C4⋊C4—1061no—
517C4⋊C4—1061no—
518C4⋊C4—1061no—
519C4⋊C4—1061no—
520C4⋊C4—1061no—
521C4⋊C4—4241no—
522C4⋊C4—2122no—
523C4⋊C4—2122no—
524C4⋊C4—2122no—
525C4⋊C4—2122no—
526C4⋊C4—2122no—
527C4⋊C4—2122no—
528C4⋊C4—2122no—
529C4⋊C4—2122no—
530C4⋊C4—2122no—
531C4⋊C4#1171994712yes—
532C22⋊C4—42no—
533C22⋊C4—534no—
534C22⋊C4—534no—
535C22⋊C4—2122no—
536C22⋊C4—2122no—
537C22⋊C4—2122no—
538C22⋊C4—2122no—
539C22⋊C4—2122no—
540C42—1061no—
541C42—1061no—
542C42—1061no—
543C42—1061no—
544C23—21no—
545C23—21no—
546C23—21no—
547C23—21no—
548C23—41no—
549C23—41no—
550C23—41no—
551C23—41no—
552C23—41no—
553C23—41no—
554C23—41no—
555C23#1313320312yes—
556C23D8×D5312yes—
557C23SD16×D5312yes—
558C23C2×D4×D5311yescharacteristic
559Q8—4244no—
560D4—41no—
561D4—41no—
562D4—41no—
563D4—41no—
564D4—84no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
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—84no—
581C2×C4—532no—
582C2×C4—532no—
583C2×C4—532no—
584C2×C4—532no—
585C2×C4—1061no—
586C2×C4—1061no—
587C2×C4—1061no—
588C2×C4—1061no—
589C2×C4—1061no—
590C2×C4—1061no—
591C2×C4—1061no—
592C2×C4—1061no—
593C2×C4—4244no—
594C2×C4—2121no—
595C2×C4—2121no—
596C2×C4—2121no—
597C2×C4—2121no—
598C2×C4—2121no—
599C2×C4—2121no—
600C2×C4—2121no—
601C2×C4—2121no—
602C2×C4—2121no—
603C2×C4—2121no—
604C2×C4—2121no—
605C2×C4—2121no—
606C2×C4#1313320312yes—
607C2×C4#41×D5311yescharacteristic
608C2×C4C2×#1171994711yescharacteristic
609C2×C4C2×#1171994711yescharacteristic
610C2×C4C2×#1171994511yescharacteristic
611C2×C4#1313326211yescharacteristic
612C2×C4#1313326411yescharacteristic
613C8—41no—
614C8—41no—
615C8—1062no—
616C8—2121no—
617C22—22no—
618C22—22no—
619C22—22no—
620C22—22no—
621C22—22no—
622C22—22no—
623C22—81no—
624C22—81no—
625C22—81no—
626C22—81no—
627C22—81no—
628C22—81no—
629C22—81no—
630C22#1681096814yes—
631C22C2×#1313320311yescharacteristic
632C22#1681126011yescharacteristic
633C22#1681127111yescharacteristic
634C22C2×D8×D5311yescharacteristic
635C22C2×SD16×D5311yescharacteristic
636C22#1681143311yescharacteristic
637C22#1681143911yescharacteristic
638C4—22no—
639C4—42no—
640C4—81no—
641C4—81no—
642C4—1062no—
643C4—1062no—
644C4—1062no—
645C4—1062no—
646C4—4241no—
647C4—4241no—
648C4—4241no—
649C4—4241no—
650C4#1681115411yescharacteristic
651C4#1681121811yescharacteristic
652C4#1681130211yescharacteristic
653C4#1681131311yescharacteristic
654C2—24no—
655C2—161no—
656C2#2027903911yescharacteristic
657C2C2×#1681096811yescharacteristic
658C2#2028508311yescharacteristic
659C2#2028652111yescharacteristic
660C2#2028657311yescharacteristic
661C2#2028705711yescharacteristic
662C2#2028707011yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#1874C2
socleC23×C53C22×D4
prime radicalC2×#121×C53C22
hypercenter#1874D53
hypocenterC53#27695
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1874

Complements

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

NH ≅ G/Ntype
2. #20278411C2semidirect
3. #20276276C2semidirect
4. #20276275C2semidirect
5. #20276121C2semidirect
6. #20275849C2semidirect
7. #20275674C2semidirect
8. #20275400C2semidirect
9. #20275395C2semidirect
11. #20274841C2semidirect
12. #20274802C2semidirect
13. #20274489C2semidirect
14. C2×#16810385C2semidirect
390. C53#27695semidirect

Maximal subgroups

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

subgroupindex
#202791872
#202784112
#202762762
#202762752
#202761212
#202758492
#202756742
#202754002
#202753952
#202751442
#202748412
#202748022
#202744892
C2×#168103852
C53×#18742
#2769553