Finite Group Databasebeta

← #27466294

Subgroups — #27466294

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#20279182C211yesmaximal, characteristic
2#20278418C211yesmaximal, characteristic, complemented
3#20276276C211yesmaximal, characteristic, complemented
4#20276275C211yesmaximal, characteristic, complemented
5#20276122C211yesmaximal, characteristic, complemented
6#20275850C211yesmaximal, characteristic, complemented
7#20275674C211yesmaximal, characteristic, complemented
8#20275399C211yesmaximal, characteristic, complemented
9#20275396C211yesmaximal, characteristic, complemented
10#20275141C211yesmaximal, characteristic
11#20275030C211yesmaximal, characteristic
12#20274841C211yesmaximal, characteristic, complemented
13#20274489C211yesmaximal, characteristic, complemented
14C2×#16810385C211yesmaximal, characteristic, complemented
15C53×#1823C211yesmaximal, characteristic, complemented
16C23×#88322C2211yescharacteristic
17C2×#13133216C2211yescharacteristic
18C2×#13133200—21no—
19C2×#13133200—21no—
20C2×#13133194—21no—
21C2×#13133191C2211yescharacteristic
22C22×C4⋊Dic53C2211yescharacteristic
23C2×#13133139C2211yescharacteristic
24C2×#13133139C2211yescharacteristic
25C2×#13133137C2211yescharacteristic
26C2×#13133137C2211yescharacteristic
27C2×C8×Dic53C2211yescharacteristic
28C2×C8×Dic53C2211yescharacteristic
29#16810558—21no—
30#16810558—21no—
31#16810557—21no—
32#16810544—21no—
33#16810524C2211yescharacteristic
34#16810524C2211yescharacteristic
35#16810522C2211yescharacteristic
36#16810519C2211yescharacteristic
37#16810495—21no—
38#16810492C2211yescharacteristic
39#16810492C2211yescharacteristic
40#16810492C2211yescharacteristic
41C4×C22⋊Dic53C2211yescharacteristic
42#16810386C2211yescharacteristic
43#16810386C2211yescharacteristic
44#16810385C2211yescharacteristic
45#16810385C2211yescharacteristic
46#16810385C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810385C2211yescharacteristic
49#16810385C2211yescharacteristic
50#16810377C2211yescharacteristic
51#16810377C2211yescharacteristic
52C22×C4⋊C4×C53C2211yescharacteristic
53C2×C22⋊C8×C53C2211yescharacteristic
54C53×#380C2211yescharacteristic
55C53×#340C2211yescharacteristic
56C53×#340C2211yescharacteristic
57C53×#340C2211yescharacteristic
58C53×#340C2211yescharacteristic
59C22×Dic106—21no—
60C22×D212C2311yescharacteristic
61C22×C22⋊D53—21no—
62C23×Dic53—21no—
63C22×#88322—22no—
64C22×#88322—22no—
65C22×#88322—41no—
66C22×#88322C2311yescharacteristic
67C22×#88322C2311yescharacteristic
68#13133216—22no—
69#13133216—41no—
70#13133200—44no—
71#13133200—44no—
72#13133194—44no—
73#13133191—22no—
74#13133191—22no—
75C2×C4⋊Dic53—21no—
76C2×C4⋊Dic53—21no—
77C2×C4⋊Dic53—22no—
78C2×C4⋊Dic53—41no—
79C2×C4⋊Dic53C2311yescharacteristic
80C2×C4⋊Dic53C2311yescharacteristic
81C2×C4⋊Dic53C2311yescharacteristic
82C2×#11719919—21no—
83C2×#11719919—21no—
84C2×#11719919—21no—
85C2×#11719919—21no—
86C2×#11719919—21no—
87C2×#11719919—21no—
88C2×#11719919—21no—
89C2×#11719919—21no—
90C2×C22⋊Dic53D411yescharacteristic
91C2×C22⋊Dic53D411yescharacteristic
92C2×#11719921—21no—
93C2×#11719921—21no—
94C2×#11719921—21no—
95C2×#11719921—21no—
96C2×#11719921—21no—
97C2×C4×Dic53C2311yescharacteristic
98C2×C4×Dic53C2311yescharacteristic
99#13133139—22no—
100#13133139—41no—
101#13133137—22no—
102#13133137—41no—
103C8×Dic53—22no—
104C8×Dic53—41no—
105#13133124—21no—
106#13133124—21no—
107#13133124D411yescharacteristic
108#13133124D411yescharacteristic
109C23×C4×C53—21no—
110C23×C4×C53C2311yescharacteristic
111C22×C8×C53C2311yescharacteristic
112C22×C8×C53C2311yescharacteristic
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—
121C22⋊C8×C53—22no—
122#95×C53—21no—
123C24×C53C2×D411yescharacteristic
124C23×D53—21no—
125C22×C4×C53—21no—
126C22×C4×C53—21no—
127C22×C4×C53—21no—
128C22×C4×C53—21no—
129C22×C4×C53—21no—
130C22×C4×C53—21no—
131C22×C4×C53—21no—
132C22×C4×C53—21no—
133C22×C4×C53—42no—
134C22×C4×C53—42no—
135C22×C4×C53—42no—
136C22×C4×C53C2×D411yescharacteristic
137C22×C4×C53#4112yes—
138C22×C4×C53#4112yes—
139C22×C4×C53C2411yescharacteristic
140C2×C22⋊D53—42no—
141C2×C22⋊D53—42no—
142C2×C22⋊D53—42no—
143C22×Dic53—21no—
144C22×Dic53—21no—
145C22×Dic53—21no—
146C22×Dic53—42no—
147C22×Dic53—42no—
148C22×Dic53—81no—
149C22×Dic53C2×D411yescharacteristic
150C22×Dic53C2×D411yescharacteristic
151C22×Dic53C2×D411yescharacteristic
152C22×Dic53C2×D411yescharacteristic
153C2×D212—21no—
154C2×D212—21no—
155C2×D212—21no—
156C2×D212—21no—
157C2×D212—21no—
158C2×D212—21no—
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—21no—
174C4⋊C4×C53—21no—
175C4⋊C4×C53—21no—
176C4⋊C4×C53—42no—
177C22⋊C4×C53—42no—
178C22⋊Dic53D814yes—
179C22⋊Dic53SD1614yes—
180#11719921—42no—
181#11719921—42no—
182#11719921—42no—
183#11719921—42no—
184#11719921—42no—
185C4⋊Dic53—21no—
186C4⋊Dic53—21no—
187C4⋊Dic53—21no—
188C4⋊Dic53—21no—
189C4⋊Dic53—21no—
190C4⋊Dic53—42no—
191C4⋊Dic53—42no—
192C4⋊Dic53#4112yes—
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#12412yes—
217C23×C53C2×D812yes—
218C23×C53C2×SD1612yes—
219C23×C53C22×D411yescharacteristic
220C22×D53—41no—
221C22×D53—41no—
222C22×D53—41no—
223C22×D53—41no—
224C22×D53—41no—
225C22×D53—41no—
226C22×D53—41no—
227C2×C4×C53—21no—
228C2×C4×C53—21no—
229C2×C4×C53—22no—
230C2×C4×C53—22no—
231C2×C4×C53—22no—
232C2×C4×C53—22no—
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—41no—
241C2×C4×C53—41no—
242C2×C4×C53—41no—
243C2×C4×C53—41no—
244C2×C4×C53—41no—
245C2×C4×C53—84no—
246C2×C4×C53#12412yes—
247C2×C4×C53C2×#4111yescharacteristic
248C2×C4×C53C2×#4111yescharacteristic
249C2×C4×C53C2×#4111yescharacteristic
250C2×C4×C53C2×#4111yescharacteristic
251C2×C4×C53H(5,𝔽2)11yescharacteristic
252C2×C4×C532−1+411yescharacteristic
253C22⋊D53—84no—
254C2×Dic53—21no—
255C2×Dic53—21no—
256C2×Dic53—21no—
257C2×Dic53—21no—
258C2×Dic53—21no—
259C2×Dic53—21no—
260C2×Dic53—21no—
261C2×Dic53—21no—
262C2×Dic53—22no—
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—84no—
272C2×Dic53C2×D812yes—
273C2×Dic53C2×D812yes—
274C2×Dic53C2×SD1612yes—
275C2×Dic53C2×SD1612yes—
276D212—41no—
277D212—41no—
278D212—41no—
279D212—41no—
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#48614yes—
293C22×C53C2×#12411yescharacteristic
294C22×C53#54611yescharacteristic
295C22×C53#54711yescharacteristic
296C22×C53C22×D811yescharacteristic
297C22×C53C22×SD1611yescharacteristic
298C22×C53#57711yescharacteristic
299C22×C53#57811yescharacteristic
300C2×D53—81no—
301C2×D53—81no—
302C2×D53—81no—
303C2×D53—81no—
304C2×D53—81no—
305C2×D53—81no—
306C2×D53—81no—
307C4×C53—22no—
308C4×C53—81no—
309C4×C53—81no—
310C4×C53—81no—
311C4×C53—81no—
312C4×C53#52911yescharacteristic
313C4×C53#53911yescharacteristic
314C4×C53#55111yescharacteristic
315C4×C53#55211yescharacteristic
316Dic53—22no—
317Dic53—22no—
318Dic53—22no—
319Dic53—22no—
320Dic53—42no—
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#1874—531no—
334#1834—531no—
335#1823D5311yescharacteristic, complemented
336C8×C22⋊C4—531no—
337C2×#340—531no—
338C2×C53—24no—
339C2×C53#248811yescharacteristic
340C2×C53C2×#48611yescharacteristic
341C2×C53#313711yescharacteristic
342C2×C53#329411yescharacteristic
343C2×C53#329811yescharacteristic
344C2×C53#333411yescharacteristic
345C2×C53#333511yescharacteristic
346D53—161no—
347C23×C8—531no—
348C2×C4⋊Q8—531no—
349C2×#123—1061no—
350C2×#123—1061no—
351C2×C22⋊Q8—1061no—
352C2×#121—531no—
353C22×C4⋊C4—531no—
354C22×C4⋊C4C2×D5311yescharacteristic
355C2×#102—531no—
356C2×#102—531no—
357C2×#102—531no—
358C2×#102—531no—
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#380C2×D5311yescharacteristic
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#340C2×D5311yescharacteristic
385#340C2×D5311yescharacteristic
386#340C2×D5311yescharacteristic
387#340C2×D5311yescharacteristic
388#336—531no—
389#336—531no—
390C53#2769511yescharacteristic, complemented, Sylow
391C22×Q8—1061no—
392C22×D4—531no—
393C22×D4—1061no—
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—1062no—
410#121—1062no—
411C2×C4⋊C4—22no—
412C2×C4⋊C4—22no—
413C2×C4⋊C4—21no—
414C2×C4⋊C4—531no—
415C2×C4⋊C4—531no—
416C2×C4⋊C4—531no—
417C2×C4⋊C4—1062no—
418C2×C4⋊C4—1061no—
419C2×C4⋊C4—1061no—
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—2121no—
429C2×C4⋊C4C22×D5311yescharacteristic
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—1062no—
444#102—1062no—
445#102—2121no—
446#102—2121no—
447C22⋊C8—22no—
448C4×C8—1062no—
449C4×C8—2121no—
450#95—21no—
451#95—531no—
452#95—531no—
453#95—1061no—
454#95—1061no—
455C24—1061no—
456C24D4×D5311yescharacteristic
457C2×Q8—4241no—
458C2×Q8—2122no—
459C2×Q8—2122no—
460C2×D4—1061no—
461C2×D4—1061no—
462C2×D4—1061no—
463C2×D4—1061no—
464C2×D4—1061no—
465C2×D4—1061no—
466C2×D4—2122no—
467C2×D4—2122no—
468C2×D4—2122no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—42no—
478C22×C4—42no—
479C22×C4—42no—
480C22×C4—531no—
481C22×C4—531no—
482C22×C4—531no—
483C22×C4—531no—
484C22×C4—1061no—
485C22×C4—1061no—
486C22×C4—1061no—
487C22×C4—4241no—
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—21no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—42no—
516C4⋊C4—532no—
517C4⋊C4—1061no—
518C4⋊C4—1061no—
519C4⋊C4—1061no—
520C4⋊C4—1061no—
521C4⋊C4—1061no—
522C4⋊C4—4241no—
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—2122no—
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—2121no—
549C23—2121no—
550C23—2121no—
551C23—2121no—
552C23—2121no—
553C23—2121no—
554C23—2121no—
555C23#1313320312yes—
556C23D8×D5312yes—
557C23SD16×D5312yes—
558C23C2×D4×D5311yescharacteristic
559Q8—4244no—
560D4—4244no—
561D4—2121no—
562D4—2121no—
563D4—2121no—
564D4—2121no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
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—41no—
582C2×C4—41no—
583C2×C4—84no—
584C2×C4—532no—
585C2×C4—532no—
586C2×C4—532no—
587C2×C4—532no—
588C2×C4—1061no—
589C2×C4—1061no—
590C2×C4—1061no—
591C2×C4—1061no—
592C2×C4—1061no—
593C2×C4—1061no—
594C2×C4—1061no—
595C2×C4—1061no—
596C2×C4—1062no—
597C2×C4—4244no—
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×C4#41×D5311yescharacteristic
609C2×C4C2×#1171994711yescharacteristic
610C2×C4C2×#1171994511yescharacteristic
611C2×C4#1313326311yescharacteristic
612C2×C4#1313326411yescharacteristic
613C8—41no—
614C8—41no—
615C8—1062no—
616C8—2121no—
617C22—22no—
618C22—22no—
619C22—22no—
620C22—22no—
621C22—22no—
622C22—22no—
623C22—4241no—
624C22—4241no—
625C22—4241no—
626C22—4241no—
627C22—4241no—
628C22—4241no—
629C22—4241no—
630C22#1681096814yes—
631C22C2×#1313320311yescharacteristic
632C22#1681126511yescharacteristic
633C22#1681127111yescharacteristic
634C22C2×D8×D5311yescharacteristic
635C22C2×SD16×D5311yescharacteristic
636C22#1681143211yescharacteristic
637C22#1681143911yescharacteristic
638C4—22no—
639C4—81no—
640C4—81no—
641C4—81no—
642C4—81no—
643C4—1062no—
644C4—1062no—
645C4—1062no—
646C4—1062no—
647C4—4241no—
648C4—4241no—
649C4—2122no—
650C4#1681115011yescharacteristic
651C4#1681121711yescharacteristic
652C4#1681129711yescharacteristic
653C4#1681130611yescharacteristic
654C2—24no—
655C2—8481no—
656C2#2027903711yescharacteristic
657C2C2×#1681096811yescharacteristic
658C2#2028507611yescharacteristic
659C2#2028652611yescharacteristic
660C2#2028657811yescharacteristic
661C2#2028705711yescharacteristic
662C2#2028706511yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#1823C2
socleC23×C53C22×D4
prime radicalC2×#13133191C22
hypercenter#1823D53
hypocenterC53#27695
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1823

Complements

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

NH ≅ G/Ntype
2. #20278418C2semidirect
3. #20276276C2semidirect
4. #20276275C2semidirect
5. #20276122C2semidirect
6. #20275850C2semidirect
7. #20275674C2semidirect
8. #20275399C2semidirect
9. #20275396C2semidirect
12. #20274841C2semidirect
13. #20274489C2semidirect
14. C2×#16810385C2semidirect
15. C53×#1823C2semidirect
335. #1823D53semidirect
390. C53#27695semidirect

Maximal subgroups

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

subgroupindex
#202791822
#202784182
#202762762
#202762752
#202761222
#202758502
#202756742
#202753992
#202753962
#202751412
#202750302
#202748412
#202744892
C2×#168103852
C53×#18232
#2769553