Finite Group Databasebeta

← #27393993

Subgroups — #27393993

Order 13568.

1194 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811155C211yesmaximal, characteristic
2C2×C4⋊C8×D53C211yesmaximal, characteristic, complemented
3#20277120C211yesmaximal, characteristic, complemented
4#20275671C212yesmaximal, complemented
5#20275646C212yesmaximal, complemented
6#20275140C212yesmaximal
7#20275137C212yesmaximal, complemented
8#20274721C211yesmaximal, characteristic, complemented
9#20274717C211yesmaximal, characteristic, complemented
10#20274715C211yesmaximal, characteristic, complemented
11C53×#1802C211yesmaximal, characteristic, complemented
12C22×C8×D53C2212yes—
13#16811155—24no—
14#16811155C418yescomplemented
15C2×#13133210—21no—
16C2×#13133208C2211yescharacteristic
17C2×#13133204C2211yescharacteristic
18C2×#13133199—21no—
19C2×#13133197C2211yescharacteristic
20C2×#13133188C2211yescharacteristic
21C2×C4×Dic106C2211yescharacteristic
22C2×C4×C22⋊D53—21no—
23C2×#13133176—21no—
24C2×#13133175—21no—
25C2×C42×D53C2211yescharacteristic
26C2×C4⋊#88322C2211yescharacteristic
27C2×#13133146C2212yes—
28C4⋊C8×D53—24no—
29C2×#13133142C2212yes—
30C2×#13133140C2212yes—
31C2×#13133138C2212yes—
32#16810493C2211yescharacteristic
33#16810488—21no—
34#16810480—21no—
35#16810480C2212yes—
36C4⋊C4×Dic53—21no—
37C4⋊C4×Dic53C2212yes—
38#16810386C2212yes—
39#16810385C2212yes—
40#16810385C2212yes—
41C2×C4×D4×C53C2211yescharacteristic
42C2×C4⋊C8×C53C2211yescharacteristic
43C2×#102×C53C2212yes—
44C53×#384C2211yescharacteristic
45C53×#340C2212yes—
46C22×Dic106C2311yescharacteristic
47C22×C22⋊D53—21no—
48C22×C4×D53D411yescharacteristic
49C22×C4×D53D411yescharacteristic
50C22×C4×D53C2311yescharacteristic
51C22×#88322C2312yes—
52C2×C8×D53—24no—
53C2×C8×D53—42no—
54C2×C8×D53—42no—
55#13133210—28no—
56#13133208—22no—
57#13133208C2×C414yes—
58#13133204—22no—
59#13133204C2×C414yes—
60#13133199—28no—
61#13133197—22no—
62#13133197C2×C414yes—
63#13133188—22no—
64#13133188C2×C414yes—
65C4×Dic106—22no—
66C4×Dic106C2×C414yes—
67C4×C22⋊D53—28no—
68#13133176—28no—
69#13133175—44no—
70C2×C4⋊Dic53—21no—
71C2×C4⋊Dic53—21no—
72C2×C4⋊Dic53—21no—
73C2×C4⋊Dic53—21no—
74C2×C4⋊Dic53C2312yes—
75C2×C4⋊Dic53C2311yescharacteristic
76C2×#11719919—21no—
77C2×#11719919—21no—
78C2×C22⋊Dic53—21no—
79C2×C22⋊Dic53—21no—
80C2×C22⋊Dic53—21no—
81C2×#11719921—21no—
82C2×#11719921—22no—
83C2×C4×Dic53—21no—
84C2×C4×Dic53—22no—
85C2×C4×Dic53D411yescharacteristic
86C2×C4×Dic53D411yescharacteristic
87C2×C4×Dic53C2311yescharacteristic
88C42×D53C2×C414yes—
89C42×D53D414yes—
90C4⋊#88322—22no—
91#13133146—24no—
92#13133142—42no—
93#13133140—42no—
94#13133138—42no—
95#13133124—22no—
96#13133124—22no—
97C22×D4×C53C2311yescharacteristic
98C23×C4×C53—21no—
99C22×C8×C53C2312yes—
100C4×D4×C53—22no—
101C4×D4×C53C2×C414yes—
102C2×C4⋊C4×C53—21no—
103C2×C4⋊C4×C53—21no—
104C2×C4⋊C4×C53C2312yes—
105C2×C4⋊C4×C53C2311yescharacteristic
106C2×C22⋊C4×C53—21no—
107C2×C42×C53C2311yescharacteristic
108C4⋊C8×C53—22no—
109#102×C53—42no—
110C24×C53—21no—
111C23×D53C2×D411yescharacteristic
112C2×D4×C53—21no—
113C2×D4×C53—21no—
114C2×D4×C53—21no—
115C2×D4×C53—21no—
116C2×D4×C53—21no—
117C2×D4×C53C22×C412yes—
118C22×C4×C53—21no—
119C22×C4×C53—24no—
120C22×C4×C53—22no—
121C22×C4×C53—42no—
122C22×C4×C53—42no—
123C22×C4×C53C2×D411yescharacteristic
124C22×C4×C53C2×D411yescharacteristic
125C22×C4×C53C2411yescharacteristic
126C2×C22⋊D53—24no—
127C2×C22⋊D53—42no—
128C2×C22⋊D53—42no—
129C22×Dic53—21no—
130C22×Dic53—21no—
131C22×Dic53—22no—
132C22×Dic53C2×D411yescharacteristic
133C22×Dic53C2×D411yescharacteristic
134C22×Dic53C2×D411yescharacteristic
135C2×C4×D53—22no—
136C2×C4×D53—22no—
137C2×C4×D53—22no—
138C2×C4×D53—22no—
139C2×C4×D53—21no—
140C2×C4×D53—21no—
141C2×C4×D53—21no—
142C2×C4×D53—21no—
143C2×C4×D53C22⋊C414yes—
144C2×C4×D53C22⋊C414yes—
145C2×C4×D53C22×C412yes—
146C2×C4×D53C2×D412yes—
147C2×Dic106—21no—
148C2×Dic106—21no—
149C2×Dic106—21no—
150C2×Dic106—21no—
151C2×Dic106—21no—
152C2×Dic106C22×C412yes—
153C2×C8×C53—22no—
154C2×C8×C53—22no—
155C2×C8×C53—22no—
156C4⋊C4×C53—21no—
157C4⋊C4×C53—22no—
158C4⋊C4×C53—22no—
159C4⋊C4×C53—42no—
160C4⋊C4×C53—42no—
161C4⋊C4×C53C22×C412yes—
162C22⋊C4×C53—24no—
163C42×C53C22×C412yes—
164C42×C53C2×D412yes—
165C22⋊Dic53—24no—
166C22⋊Dic53—24no—
167C22⋊Dic53—24no—
168#11719921—24no—
169#11719921—44no—
170C4⋊Dic53—21no—
171C4⋊Dic53—24no—
172C4⋊Dic53—24no—
173C4⋊Dic53—22no—
174C4⋊Dic53—22no—
175C4⋊Dic53—42no—
176C4⋊Dic53—42no—
177C4⋊Dic53C22×C412yes—
178#11719919—24no—
179#11719919—24no—
180C4×Dic53—24no—
181C4×Dic53—44no—
182C4×Dic53C22⋊C414yes—
183C4×Dic53C22⋊C414yes—
184C4×Dic53C22×C412yes—
185C4×Dic53C2×D412yes—
186C2×#88322—22no—
187C2×#88322—22no—
188C2×#88322—22no—
189C8×D53—48no—
190C23×C53—22no—
191C23×C53—41no—
192C23×C53—41no—
193C23×C53—41no—
194C23×C53—41no—
195C23×C53—41no—
196C23×C53—41no—
197C23×C53C22×D411yescharacteristic
198C22×D53—21no—
199C22×D53—21no—
200C22×D53—21no—
201C22×D53—21no—
202C22×D53C2×C22⋊C412yes—
203C22×D53AGL(1,ℤ/8)12yes—
204C22×D53#13712yes—
205D4×C53—22no—
206D4×C53—22no—
207D4×C53—41no—
208D4×C53—41no—
209C2×C4×C53—21no—
210C2×C4×C53—21no—
211C2×C4×C53—22no—
212C2×C4×C53—41no—
213C2×C4×C53—41no—
214C2×C4×C53—48no—
215C2×C4×C53—42no—
216C2×C4×C53—42no—
217C2×C4×C53—42no—
218C2×C4×C53C2×C22⋊C412yes—
219C2×C4×C53C2×C22⋊C412yes—
220C2×C4×C53#13512yes—
221C2×C4×C53#13512yes—
222C2×C4×C53C23×C411yescharacteristic
223C2×C4×C53C22×D411yescharacteristic
224C2×C4×C53H(5,𝔽2)11yescharacteristic
225C2×C4×C53H(5,𝔽2)11yescharacteristic
226C2×C4×C532−1+411yescharacteristic
227C2×C4×C532−1+411yescharacteristic
228C22⋊D53—48no—
229C2×Dic53—21no—
230C2×Dic53—21no—
231C2×Dic53—21no—
232C2×Dic53—21no—
233C2×Dic53—21no—
234C2×Dic53—21no—
235C2×Dic53—22no—
236C2×Dic53—22no—
237C2×Dic53—41no—
238C2×Dic53—41no—
239C2×Dic53—41no—
240C2×Dic53—41no—
241C2×Dic53—42no—
242C2×Dic53—42no—
243C2×Dic53—42no—
244C2×Dic53C2×C22⋊C412yes—
245C2×Dic53C2×C22⋊C412yes—
246C2×Dic53C2×C22⋊C412yes—
247C4×D53—28no—
248C4×D53—22no—
249C4×D53—22no—
250C4×D53—22no—
251C4×D53—22no—
252C4×D53—28no—
253Dic106—22no—
254Dic106—22no—
255Dic106—41no—
256Dic106—41no—
257C8×C53—42no—
258#88322—42no—
259#21887—531nomaximal, Sylow
260C22×C53—42no—
261C22×C53—42no—
262C22×C53—42no—
263C22×C53—81no—
264C22×C53—81no—
265C22×C53—81no—
266C22×C53—81no—
267C22×C53C22×C22⋊C411yescharacteristic
268C22×C53#53711yescharacteristic
269C22×C53#53711yescharacteristic
270C22×C53C2×#13511yescharacteristic
271C22×C53C2×#13511yescharacteristic
272C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
273C22×C53C2×#13711yescharacteristic
274C2×D53—22no—
275C2×D53—22no—
276C2×D53—22no—
277C2×D53—22no—
278C2×D53—22no—
279C2×D53—22no—
280C2×D53#41714yes—
281C4×C53—22no—
282C4×C53—42no—
283C4×C53—82no—
284C4×C53#41614yes—
285C4×C53#52011yescharacteristic
286C4×C53#52011yescharacteristic
287C4×C53#53511yescharacteristic
288C4×C53#53611yescharacteristic
289Dic53—22no—
290Dic53—22no—
291Dic53—22no—
292Dic53—42no—
293Dic53—42no—
294Dic53—82no—
295C2×#529—531no—
296C22×C4⋊C8—531no—
297#2285—531no—
298#1989—532no—
299#1983—532no—
300#1893—532no—
301#1892—532no—
302#1803—531no—
303#1803—531no—
304#1802—531no—
305#1802D5311yescharacteristic, complemented
306C2×C53—82no—
307C2×C53#231111yescharacteristic
308C2×C53C2×#41611yescharacteristic
309C2×C53C2×#41711yescharacteristic
310C2×C53#324711yescharacteristic
311C2×C53#324711yescharacteristic
312C2×C53#324811yescharacteristic
313C2×C53#324811yescharacteristic
314D53—24no—
315C23×C8—532no—
316#529—538no—
317#529—1064no—
318C2×#126—1061no—
319C2×#125—531no—
320C2×#124—531no—
321C2×#123—1061no—
322C2×C22⋊Q8—531no—
323C2×#121—531no—
324C2×C4×Q8—531no—
325C2×C4×D4—1061no—
326C2×C4×D4C2×D5311yescharacteristic
327C2×#117—1061no—
328C2×#117—1061no—
329C22×C42—531no—
330C2×C4⋊C8—532no—
331C2×C4⋊C8—531no—
332C2×C4⋊C8—1064no—
333C2×C4⋊C8C2×D5311yescharacteristic
334C2×#103—532no—
335C2×#103—532no—
336C2×#102—532no—
337C2×#102C2×D5312yes—
338#384—531no—
339#384C2×D5311yescharacteristic
340#382—1061no—
341#380—532no—
342#380—1061no—
343C4×C4⋊C4—532no—
344C4×C4⋊C4—1061no—
345#340—532no—
346#340—532no—
347#340—532no—
348#340C2×D5312yes—
349C53#2188711yescharacteristic, complemented, Sylow
350C22×Q8—531no—
351C22×D4—1061no—
352C22×D4C22×D5311yescharacteristic
353C23×C4—21no—
354C23×C4—531no—
355C23×C4—531no—
356C23×C4—531no—
357C22×C8—532no—
358C22×C8—1064no—
359C22×C8—2122no—
360C22×C8—2122no—
361C22×C8C22×D5312yes—
362#126—1068no—
363#125—534no—
364#125—1062no—
365#124—534no—
366#124—1062no—
367#123—1068no—
368C22⋊Q8—534no—
369C22⋊Q8—1062no—
370#121—534no—
371#121—1062no—
372C4×Q8—534no—
373C4×Q8—1062no—
374C4×D4—22no—
375C4×D4—1068no—
376C4×D4C4×D5314yes—
377#117—1068no—
378#117—2124no—
379C2×C4⋊C4—21no—
380C2×C4⋊C4—21no—
381C2×C4⋊C4—531no—
382C2×C4⋊C4—532no—
383C2×C4⋊C4—1061no—
384C2×C4⋊C4—1061no—
385C2×C4⋊C4—1061no—
386C2×C4⋊C4—1061no—
387C2×C4⋊C4—1061no—
388C2×C4⋊C4—1061no—
389C2×C4⋊C4C22×D5312yes—
390C2×C4⋊C4C22×D5311yescharacteristic
391C2×C22⋊C4—21no—
392C2×C22⋊C4—1061no—
393C2×C22⋊C4—1061no—
394C2×C22⋊C4—1061no—
395C2×C22⋊C4—1061no—
396C2×C22⋊C4—1062no—
397C2×C42—531no—
398C2×C42—531no—
399C2×C42—531no—
400C2×C42—534no—
401C2×C42—534no—
402C2×C42—1061no—
403C2×C42—1062no—
404C2×C42C22×D5311yescharacteristic
405C4⋊C8—22no—
406C4⋊C8—1064no—
407C4⋊C8—1062no—
408#103—2122no—
409#103—2122no—
410#102—42no—
411#102—2122no—
412#95—1062no—
413#95—1062no—
414C24—21no—
415C24—531no—
416C2×Q8—532no—
417C2×Q8—1061no—
418C2×Q8—1061no—
419C2×Q8—1061no—
420C2×Q8—1061no—
421C2×Q8—1061no—
422C2×D4—21no—
423C2×D4—21no—
424C2×D4—21no—
425C2×D4—21no—
426C2×D4—21no—
427C2×D4—1064no—
428C2×D4—2122no—
429C2×D4—2122no—
430C2×D4C2×C4×D5312yes—
431C22×C4—21no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—531no—
437C22×C4—531no—
438C22×C4—531no—
439C22×C4—534no—
440C22×C4—534no—
441C22×C4—532no—
442C22×C4—532no—
443C22×C4—1062no—
444C22×C4—1062no—
445C22×C4—1062no—
446C22×C4—1062no—
447C22×C4—1061no—
448C22×C4—1061no—
449C22×C4—1061no—
450C22×C4—1061no—
451C22×C4—1061no—
452C22×C4—1061no—
453C22×C4—1062no—
454C22×C4D4×D5311yescharacteristic
455C22×C4D4×D5311yescharacteristic
456C22×C4C23×D5311yescharacteristic
457C2×C8—22no—
458C2×C8—22no—
459C2×C8—22no—
460C2×C8—1062no—
461C2×C8—1062no—
462C2×C8—1062no—
463C2×C8—2128no—
464C4⋊C4—21no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—42no—
468C4⋊C4—42no—
469C4⋊C4—532no—
470C4⋊C4—1064no—
471C4⋊C4—1064no—
472C4⋊C4—1064no—
473C4⋊C4—1064no—
474C4⋊C4—1061no—
475C4⋊C4—1062no—
476C4⋊C4—1062no—
477C4⋊C4—2122no—
478C4⋊C4—2122no—
479C4⋊C4C2×C4×D5312yes—
480C22⋊C4—24no—
481C22⋊C4—1064no—
482C22⋊C4—1064no—
483C22⋊C4—1064no—
484C22⋊C4—1064no—
485C22⋊C4—2124no—
486C42—534no—
487C42—534no—
488C42—532no—
489C42—532no—
490C42—1064no—
491C42—2124no—
492C42C2×C4×D5312yes—
493C42D4×D5312yes—
494C23—22no—
495C23—41no—
496C23—41no—
497C23—41no—
498C23—41no—
499C23—41no—
500C23—41no—
501C23—532no—
502C23—532no—
503C23—532no—
504C23—1061no—
505C23—1061no—
506C23—1061no—
507C23—1061no—
508C23C2×D4×D5311yescharacteristic
509Q8—1062no—
510Q8—1062no—
511Q8—2121no—
512Q8—2121no—
513D4—22no—
514D4—22no—
515D4—41no—
516D4—41no—
517D4—2128no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—22no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—48no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—532no—
528C2×C4—532no—
529C2×C4—532no—
530C2×C4—1068no—
531C2×C4—1061no—
532C2×C4—1061no—
533C2×C4—1061no—
534C2×C4—1061no—
535C2×C4—1061no—
536C2×C4—1061no—
537C2×C4—1062no—
538C2×C4—1062no—
539C2×C4—1068no—
540C2×C4—1062no—
541C2×C4—1062no—
542C2×C4—1062no—
543C2×C4—1062no—
544C2×C4—2122no—
545C2×C4—2122no—
546C2×C4—2122no—
547C2×C4—2121no—
548C2×C4—2121no—
549C2×C4—2121no—
550C2×C4—2121no—
551C2×C4C22⋊C4×D5312yes—
552C2×C4C22⋊C4×D5312yes—
553C2×C4#1313323612yes—
554C2×C4#1313323812yes—
555C2×C4C22×C4×D5311yescharacteristic
556C2×C4C2×D4×D5311yescharacteristic
557C2×C4#1313326211yescharacteristic
558C2×C4#1313326311yescharacteristic
559C2×C4#1313326411yescharacteristic
560C2×C4#1313326511yescharacteristic
561C8—42no—
562C8—2122no—
563C22—42no—
564C22—42no—
565C22—42no—
566C22—81no—
567C22—81no—
568C22—81no—
569C22—81no—
570C22—534no—
571C22—1062no—
572C22—1062no—
573C22—1062no—
574C22—1062no—
575C22—1062no—
576C22—1062no—
577C22C2×C22⋊C4×D5311yescharacteristic
578C22#1681120211yescharacteristic
579C22#1681120311yescharacteristic
580C22C2×#1313323611yescharacteristic
581C22C2×#1313323811yescharacteristic
582C22AGL(1,ℤ/8)×D5311yescharacteristic
583C22#137×D5311yescharacteristic
584C4—22no—
585C4—42no—
586C4—82no—
587C4—1062no—
588C4—1062no—
589C4—1062no—
590C4—4242no—
591C4—2122no—
592C4—2122no—
593C4#1681061614yes—
594C4#1681110311yescharacteristic
595C4#1681110411yescharacteristic
596C4#1681118911yescharacteristic
597C4#1681119611yescharacteristic
598C2—82no—
599C2—1064no—
600C2#2027730411yescharacteristic
601C2C2×#1681061611yescharacteristic
602C2#417×D5311yescharacteristic
603C2#2028603711yescharacteristic
604C2#2028604311yescharacteristic
605C2#2028604811yescharacteristic
606C2#2028605411yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1802C2
socleC23×C53C22×D4
prime radicalC2×#13133188C22
cosocleC22×C4×C53C24
hypercenter#1802D53
hypocenterC53#21887
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1802

Complements

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

NH ≅ G/Ntype
2. C2×C4⋊C8×D53C2semidirect
3. #20277120C2semidirect
4. #20275671C2semidirect
5. #20275646C2semidirect
7. #20275137C2semidirect
8. #20274721C2semidirect
9. #20274717C2semidirect
10. #20274715C2semidirect
11. C53×#1802C2semidirect
14. #16811155C4semidirect
305. #1802D53semidirect
349. C53#21887semidirect

Maximal subgroups

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

subgroupindex
C2×#168111552
C2×C4⋊C8×D532
#202771202
#202756712
#202756462
#202751402
#202751372
#202747212
#202747172
#202747152
C53×#18022
#2188753