Finite Group Databasebeta

← #27350321

Subgroups — #27350321

Order 13568.

944 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×#16811308C211yesmaximal, characteristic
2#20283369C212yesmaximal, complemented
3#20273083C212yesmaximal, complemented
4#20273069C212yesmaximal, complemented
5#20273061C212yesmaximal
6#20273057C211yesmaximal, characteristic, complemented
7#20273048C211yesmaximal, characteristic, complemented
8#20273045C211yesmaximal, characteristic, complemented
9#20273037C211yesmaximal, characteristic, complemented
10#20273028C211yesmaximal, characteristic, complemented
11C2×C53×#324C211yesmaximal, characteristic, complemented
12#16811308—22no—
13#16811308—22no—
14#16811308—22no—
15#16811308—22no—
16C2×#13133212C411yescharacteristic
17C2×#13133212C2211yescharacteristic
18C2×#13133208C411yescharacteristic
19C2×#13133207C411yescharacteristic
20C2×#13133207C2211yescharacteristic
21C2×#13133200C411yescharacteristic
22C2×#13133200C2211yescharacteristic
23C2×#13133197C411yescharacteristic
24C2×#13133196C411yescharacteristic
25C2×#13133182C2211yescharacteristic
26C2×C4×C22⋊D53C411yescharacteristic
27C2×#13133176C411yescharacteristic
28C2×#13133176C2211yescharacteristic
29C2×#13133175C2211yescharacteristic
30#16810681—22no—
31#16810681—22no—
32#16810677C2212yes—
33#16810677C2212yes—
34#16810673C2212yes—
35#16810673C2212yes—
36#16810649—22no—
37#16810587C2212yes—
38#16810578—22no—
39#16810349C2211yescharacteristic
40#16810349C2211yescharacteristic
41#16810349C2212yes—
42#16810348C2212yes—
43#16810348C2212yes—
44#16810346C2212yes—
45#16810346C2212yes—
46C22×C4⋊C4×C53C2211yescharacteristic
47C2×C22⋊C8×C53C2212yes—
48C53×#324C2211yescharacteristic
49C53×#324C2211yescharacteristic
50C53×#324C2212yes—
51C22×Dic106C2×C411yescharacteristic
52C22×C22⋊D53C2×C411yescharacteristic
53C22×C4×D53C2×C411yescharacteristic
54C2×#11719917—22no—
55#13133212—22no—
56#13133212—22no—
57#13133212—22no—
58#13133212—22no—
59#13133208—22no—
60#13133208—22no—
61#13133207—22no—
62#13133207—22no—
63#13133207—22no—
64#13133207—22no—
65#13133200—22no—
66#13133200—22no—
67#13133200—22no—
68#13133200—22no—
69#13133197—22no—
70#13133197—22no—
71#13133196—22no—
72#13133196—22no—
73#13133182—22no—
74#13133182—22no—
75C4×C22⋊D53—21no—
76C4×C22⋊D53—21no—
77C4×C22⋊D53—21no—
78C4×C22⋊D53—21no—
79#13133176—21no—
80#13133176—21no—
81#13133176—21no—
82#13133176—21no—
83#13133176D412yes—
84#13133176D412yes—
85#13133176D412yes—
86#13133176D412yes—
87#13133175—22no—
88#13133175—22no—
89C2×C4⋊Dic53C2×C411yescharacteristic
90C2×C4⋊Dic53C2×C411yescharacteristic
91C2×C4⋊Dic53C2×C411yescharacteristic
92C2×C4⋊Dic53C2×C411yescharacteristic
93C2×C4⋊Dic53C2×C411yescharacteristic
94C2×#11719919C2×C411yescharacteristic
95C2×#11719919C2×C411yescharacteristic
96C2×#11719919C2×C411yescharacteristic
97C2×#11719919C2×C411yescharacteristic
98C2×#11719919C2×C411yescharacteristic
99C2×#11719919C2×C411yescharacteristic
100C2×#11719919C2311yescharacteristic
101C2×C22⋊Dic53C2×C411yescharacteristic
102C2×C22⋊Dic53C2×C411yescharacteristic
103C2×C22⋊Dic53C2×C411yescharacteristic
104C2×#11719921C2×C411yescharacteristic
105C2×#11719921C2×C411yescharacteristic
106C2×#11719921C2×C411yescharacteristic
107C2×#11719921C2×C411yescharacteristic
108C2×#11719921C2311yescharacteristic
109C2×C4×Dic53C2×C411yescharacteristic
110C2×C4×Dic53C2×C411yescharacteristic
111C2×C4×Dic53C2×C411yescharacteristic
112C2×C4×Dic53C2311yescharacteristic
113C2×C4×Dic53C2311yescharacteristic
114#13133146—22no—
115#13133146—22no—
116#13133146—22no—
117#13133146—22no—
118C22⋊#88322—22no—
119C22⋊#88322C2312yes—
120C22⋊#88322C2312yes—
121#13133127—22no—
122#13133127—22no—
123C8×Dic53—22no—
124C8×Dic53—22no—
125C23×C4×C53C2×C411yescharacteristic
126C23×C4×C53C2×C411yescharacteristic
127C23×C4×C53C2311yescharacteristic
128C22×C8×C53—22no—
129C2×C4⋊C4×C53—22no—
130C2×C4⋊C4×C53—22no—
131C2×C4⋊C4×C53C2×C411yescharacteristic
132C2×C4⋊C4×C53C2×C411yescharacteristic
133C2×C4⋊C4×C53C2311yescharacteristic
134C2×C4⋊C4×C53C2311yescharacteristic
135C22⋊C8×C53—22no—
136C22⋊C8×C53C2312yes—
137C22⋊C8×C53C2312yes—
138C24×C53C22×C411yescharacteristic
139C23×D53C22×C411yescharacteristic
140C22×C4×C53—21no—
141C22×C4×C53—21no—
142C22×C4×C53—21no—
143C22×C4×C53—21no—
144C22×C4×C53—22no—
145C22×C4×C53—41no—
146C22×C4×C53—42no—
147C22×C4×C53—42no—
148C22×C4×C53C22×C411yescharacteristic
149C22×C4×C53C22×C411yescharacteristic
150C22×C4×C53C22×C411yescharacteristic
151C22×C4×C53C22×C411yescharacteristic
152C22×C4×C53C22×C411yescharacteristic
153C22×C4×C53C2×D412yes—
154C22×C4×C53C2×D412yes—
155C22×C4×C53C2411yescharacteristic
156C2×C22⋊D53—21no—
157C2×C22⋊D53—21no—
158C2×C22⋊D53—42no—
159C22×Dic53C22×C411yescharacteristic
160C22×Dic53C22×C411yescharacteristic
161C22×Dic53C22×C411yescharacteristic
162C22×Dic53C22×C411yescharacteristic
163C22×Dic53C22×C411yescharacteristic
164C22×Dic53C22×C411yescharacteristic
165C22×Dic53C22×C411yescharacteristic
166C2×C4×D53—22no—
167C2×C4×D53—21no—
168C2×C4×D53—21no—
169C2×C4×D53—41no—
170C2×Dic106—22no—
171C2×Dic106—22no—
172C2×Dic106—41no—
173C2×C8×C53—22no—
174C2×C8×C53—22no—
175C2×C8×C53—44no—
176C4⋊C4×C53—22no—
177C4⋊C4×C53—22no—
178C4⋊C4×C53—22no—
179C4⋊C4×C53—22no—
180C22⋊Dic53—21no—
181C22⋊Dic53—21no—
182C22⋊Dic53C22⋊C411yescharacteristic
183C22⋊Dic53C22⋊C411yescharacteristic
184C22⋊Dic53C22⋊C411yescharacteristic
185C22⋊Dic53C22⋊C411yescharacteristic
186C22⋊Dic53C22⋊C412yes—
187C22⋊Dic53C22⋊C412yes—
188#11719921—22no—
189#11719921—22no—
190#11719921—22no—
191#11719921—21no—
192#11719921—21no—
193#11719921—22no—
194C4⋊Dic53—22no—
195C4⋊Dic53—21no—
196C4⋊Dic53—21no—
197C4⋊Dic53—21no—
198C4⋊Dic53—21no—
199C4⋊Dic53C22⋊C412yes—
200C4⋊Dic53C22⋊C412yes—
201C4⋊Dic53C22⋊C412yes—
202C4⋊Dic53C22⋊C412yes—
203#11719919—22no—
204#11719919—22no—
205#11719919—21no—
206#11719919—21no—
207#11719919—22no—
208#11719919—22no—
209#11719919—22no—
210#11719919—22no—
211C4×Dic53—21no—
212C4×Dic53—21no—
213C4×Dic53—21no—
214C4×Dic53—21no—
215C4×Dic53—21no—
216C4×Dic53—21no—
217C4×Dic53C2×D412yes—
218C4×Dic53C2×D412yes—
219C4×Dic53C2×D412yes—
220C4×Dic53C2×D412yes—
221#11719917—44no—
222C2×#88322—22no—
223C2×#88322—22no—
224C23×C53—21no—
225C23×C53—21no—
226C23×C53—22no—
227C23×C53—41no—
228C23×C53C2×C22⋊C411yescharacteristic
229C23×C53C2×C22⋊C411yescharacteristic
230C23×C53C23×C411yescharacteristic
231C22×D53—21no—
232C22×D53—21no—
233C22×D53—21no—
234C22×D53—41no—
235C22×D53—41no—
236C2×C4×C53—21no—
237C2×C4×C53—21no—
238C2×C4×C53—24no—
239C2×C4×C53—21no—
240C2×C4×C53—21no—
241C2×C4×C53—21no—
242C2×C4×C53—21no—
243C2×C4×C53—21no—
244C2×C4×C53—41no—
245C2×C4×C53—42no—
246C2×C4×C53—41no—
247C2×C4×C53—42no—
248C2×C4×C53—42no—
249C2×C4×C53—42no—
250C2×C4×C53—42no—
251C2×C4×C53—41no—
252C2×C4×C53—41no—
253C2×C4×C53C2×C22⋊C412yes—
254C2×C4×C53#13112yes—
255C2×C4×C53C22×D412yes—
256C22⋊D53—42no—
257C22⋊D53—42no—
258C2×Dic53—21no—
259C2×Dic53—21no—
260C2×Dic53—21no—
261C2×Dic53—21no—
262C2×Dic53—21no—
263C2×Dic53—21no—
264C2×Dic53—21no—
265C2×Dic53—21no—
266C2×Dic53—21no—
267C2×Dic53—41no—
268C2×Dic53—41no—
269C2×Dic53—41no—
270C2×Dic53—41no—
271C2×Dic53C2×C22⋊C412yes—
272C2×Dic53C2×C22⋊C411yescharacteristic
273C2×Dic53C2×C22⋊C411yescharacteristic
274C2×Dic53C2×C22⋊C411yescharacteristic
275C2×Dic53C2×C22⋊C411yescharacteristic
276C2×Dic53C2×C22⋊C412yes—
277C4×D53—42no—
278C4×D53—42no—
279Dic106—44no—
280C8×C53—42no—
281C8×C53—42no—
282#88322—42no—
283#88322—42no—
284#17417—531nomaximal, Sylow
285C22×C53—21no—
286C22×C53—21no—
287C22×C53—21no—
288C22×C53—21no—
289C22×C53—21no—
290C22×C53—21no—
291C22×C53—21no—
292C22×C53—42no—
293C22×C53—42no—
294C22×C53#41012yes—
295C22×C53C22×C22⋊C411yescharacteristic
296C22×C53C2×#13111yescharacteristic
297C22×C53#56811yescharacteristic
298C2×D53—41no—
299C2×D53—41no—
300C2×D53—41no—
301C2×D53—81no—
302C2×D53—81no—
303C4×C53—22no—
304C4×C53—22no—
305C4×C53—22no—
306C4×C53—41no—
307C4×C53—41no—
308C4×C53—41no—
309Dic53—22no—
310Dic53—21no—
311Dic53—21no—
312Dic53—21no—
313Dic53—21no—
314Dic53—22no—
315Dic53—41no—
316Dic53—41no—
317Dic53—41no—
318C2×#552—531no—
319#2924—532no—
320#1473—532no—
321#1470—532no—
322#1468—532no—
323#1467—531no—
324#1464—531no—
325#1463—531no—
326#1461—531no—
327#1458—531no—
328C2×#324D5311yescharacteristic, complemented
329C2×C53—21no—
330C2×C53—21no—
331C2×C53—22no—
332C2×C53—41no—
333C2×C53#288011yescharacteristic
334C2×C53C2×#41011yescharacteristic
335C2×C53#288911yescharacteristic
336D53—81no—
337#552—1062no—
338#552—1062no—
339#552—1062no—
340#552—1062no—
341C2×#126—531no—
342C2×#126—531no—
343C2×#125—531no—
344C2×#125—531no—
345C2×#125—531no—
346C2×#123—531no—
347C2×#123—531no—
348C2×C22⋊Q8—531no—
349C2×C22⋊Q8—531no—
350C2×C4×Q8—531no—
351C2×C4×D4—531no—
352C2×#117—531no—
353C2×#117—531no—
354C2×#117—531no—
355C22×C4⋊C4C2×D5311yescharacteristic
356#433—1062no—
357#433—1062no—
358#432—532no—
359#432—532no—
360#431—532no—
361#431—532no—
362#424—1062no—
363#407—532no—
364C2×C22⋊C8C2×D5312yes—
365#405—1062no—
366#324—531no—
367#324—531no—
368#324—532no—
369#324—532no—
370#324—532no—
371#324C2×D5311yescharacteristic
372#324C2×D5311yescharacteristic
373#324C2×D5312yes—
374#323—532no—
375#323—532no—
376C53#1741711yescharacteristic, complemented, Sylow
377C22×Q8—531no—
378C22×D4—531no—
379C23×C4—531no—
380C23×C4C4×D5311yescharacteristic
381C23×C4C4×D5311yescharacteristic
382C23×C4C22×D5311yescharacteristic
383C2×M16—1062no—
384C22×C8—22no—
385#126—1062no—
386#126—1062no—
387#126—1062no—
388#126—1062no—
389#125—1062no—
390#125—1062no—
391#125—1062no—
392#125—1062no—
393#125—1062no—
394#125—1062no—
395#123—1062no—
396#123—1062no—
397#123—1062no—
398#123—1062no—
399C22⋊Q8—1062no—
400C22⋊Q8—1062no—
401C22⋊Q8—1062no—
402C22⋊Q8—1062no—
403C4×Q8—1062no—
404C4×Q8—1062no—
405C4×D4—1061no—
406C4×D4—1061no—
407C4×D4—1061no—
408C4×D4—1061no—
409#117—532no—
410#117—532no—
411#117—532no—
412#117—532no—
413#117—1062no—
414#117—1062no—
415#117—1061no—
416#117—1061no—
417#117—1061no—
418#117—1061no—
419C2×C4⋊C4—22no—
420C2×C4⋊C4—22no—
421C2×C4⋊C4—531no—
422C2×C4⋊C4—531no—
423C2×C4⋊C4—531no—
424C2×C4⋊C4—531no—
425C2×C4⋊C4—531no—
426C2×C4⋊C4—531no—
427C2×C4⋊C4—531no—
428C2×C4⋊C4—531no—
429C2×C4⋊C4—531no—
430C2×C4⋊C4—531no—
431C2×C4⋊C4—531no—
432C2×C4⋊C4—531no—
433C2×C4⋊C4C4×D5311yescharacteristic
434C2×C4⋊C4C4×D5311yescharacteristic
435C2×C4⋊C4C22×D5311yescharacteristic
436C2×C4⋊C4C22×D5311yescharacteristic
437C2×C22⋊C4—531no—
438C2×C22⋊C4—531no—
439C2×C22⋊C4—531no—
440C2×C22⋊C4—531no—
441C2×C22⋊C4—531no—
442C2×C22⋊C4—531no—
443C2×C22⋊C4—531no—
444C2×C22⋊C4—531no—
445C2×C42—531no—
446C2×C42—531no—
447C2×C42—531no—
448C2×C42—531no—
449C2×C42—531no—
450C4⋊C8—1062no—
451C4⋊C8—1062no—
452C4⋊C8—1062no—
453C4⋊C8—1062no—
454C22⋊C8—22no—
455C22⋊C8—532no—
456C22⋊C8—532no—
457C22⋊C8—1062no—
458C22⋊C8C22×D5312yes—
459C22⋊C8C22×D5312yes—
460#97—1062no—
461#97—1062no—
462C4×C8—1062no—
463C4×C8—1062no—
464C24—531no—
465C24C2×C4×D5311yescharacteristic
466C2×Q8—1062no—
467C2×Q8—1062no—
468C2×Q8—2121no—
469C2×D4—1061no—
470C2×D4—1061no—
471C2×D4—2122no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—22no—
477C22×C4—41no—
478C22×C4—42no—
479C22×C4—42no—
480C22×C4—531no—
481C22×C4—531no—
482C22×C4—531no—
483C22×C4—531no—
484C22×C4—531no—
485C22×C4—531no—
486C22×C4—531no—
487C22×C4—1061no—
488C22×C4—1061no—
489C22×C4—1062no—
490C22×C4—2121no—
491C22×C4C2×C4×D5311yescharacteristic
492C22×C4C2×C4×D5311yescharacteristic
493C22×C4C2×C4×D5311yescharacteristic
494C22×C4C2×C4×D5311yescharacteristic
495C22×C4C2×C4×D5311yescharacteristic
496C22×C4D4×D5312yes—
497C22×C4D4×D5312yes—
498C22×C4C23×D5311yescharacteristic
499M16—2124no—
500C2×C8—22no—
501C2×C8—22no—
502C2×C8—44no—
503C2×C8—1062no—
504C2×C8—1062no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—532no—
510C4⋊C4—532no—
511C4⋊C4—532no—
512C4⋊C4—532no—
513C4⋊C4—1062no—
514C4⋊C4—1061no—
515C4⋊C4—1061no—
516C4⋊C4—1061no—
517C4⋊C4—1061no—
518C4⋊C4—1061no—
519C4⋊C4—1061no—
520C4⋊C4—1062no—
521C4⋊C4—1062no—
522C4⋊C4—1062no—
523C4⋊C4—1062no—
524C4⋊C4—1062no—
525C4⋊C4—1062no—
526C22⋊C4—531no—
527C22⋊C4—531no—
528C22⋊C4—531no—
529C22⋊C4—531no—
530C22⋊C4—532no—
531C22⋊C4—532no—
532C22⋊C4—1062no—
533C22⋊C4—1061no—
534C22⋊C4—1061no—
535C22⋊C4—1062no—
536C22⋊C4—1061no—
537C22⋊C4—1061no—
538C22⋊C4—1062no—
539C22⋊C4—1062no—
540C42—532no—
541C42—532no—
542C42—532no—
543C42—532no—
544C42—1061no—
545C42—1061no—
546C42—1061no—
547C42—1061no—
548C42—1061no—
549C42—1061no—
550C23—22no—
551C23—21no—
552C23—21no—
553C23—41no—
554C23—1061no—
555C23—1061no—
556C23—1061no—
557C23—2121no—
558C23—2121no—
559C23C22⋊C4×D5311yescharacteristic
560C23C22⋊C4×D5311yescharacteristic
561C23C22×C4×D5311yescharacteristic
562Q8—2124no—
563D4—2122no—
564D4—2122no—
565C2×C4—21no—
566C2×C4—24no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—41no—
574C2×C4—42no—
575C2×C4—41no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—531no—
583C2×C4—531no—
584C2×C4—531no—
585C2×C4—531no—
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—1061no—
597C2×C4—2122no—
598C2×C4—2122no—
599C2×C4—2121no—
600C2×C4—2121no—
601C2×C4—2121no—
602C2×C4—2121no—
603C2×C4C22⋊C4×D5312yes—
604C2×C4#1313322312yes—
605C2×C4C2×D4×D5312yes—
606C8—42no—
607C8—42no—
608C8—2122no—
609C8—2122no—
610C22—21no—
611C22—21no—
612C22—21no—
613C22—21no—
614C22—21no—
615C22—21no—
616C22—21no—
617C22—42no—
618C22—42no—
619C22—4241no—
620C22—4241no—
621C22—2121no—
622C22—2121no—
623C22—2121no—
624C22#1681059512yes—
625C22C2×C22⋊C4×D5311yescharacteristic
626C22C2×#1313322311yescharacteristic
627C22#1681138911yescharacteristic
628C4—22no—
629C4—22no—
630C4—22no—
631C4—41no—
632C4—41no—
633C4—41no—
634C4—1061no—
635C4—1061no—
636C4—1062no—
637C4—1061no—
638C4—1061no—
639C4—1062no—
640C4—2121no—
641C4—2121no—
642C4—2121no—
643C2—21no—
644C2—21no—
645C2—22no—
646C2—41no—
647C2—4241no—
648C2#2028316111yescharacteristic
649C2C2×#1681059511yescharacteristic
650C2#2028319511yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D53
commutatorC23×C53C23×C4
FrattiniC22×C4C23×D53
FittingC2×C53×#324C2
socleC22×C53C22×C22⋊C4
prime radicalC22×C22⋊D53C2×C4
cosocleC22×C4×C53C24
hypercenterC2×#324D53
hypocenterC53#17417
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#324

Complements

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

NH ≅ G/Ntype
2. #20283369C2semidirect
3. #20273083C2semidirect
4. #20273069C2semidirect
6. #20273057C2semidirect
7. #20273048C2semidirect
8. #20273045C2semidirect
9. #20273037C2semidirect
10. #20273028C2semidirect
11. C2×C53×#324C2semidirect
328. C2×#324D53semidirect
376. C53#17417semidirect

Maximal subgroups

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

subgroupindex
C2×#168113082
#202833692
#202730832
#202730692
#202730612
#202730572
#202730482
#202730452
#202730372
#202730282
C2×C53×#3242
#1741753