Finite Group Databasebeta

← #27393790

Subgroups — #27393790

Order 13568.

1138 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811155C211yesmaximal, characteristic
2C2×#16810675C211yesmaximal, characteristic, complemented
3#20277120C211yesmaximal, characteristic, complemented
4#20275678C212yesmaximal, complemented
5#20275525C212yesmaximal, complemented
6#20275034C212yesmaximal
7#20275026C212yesmaximal, complemented
8#20274721C211yesmaximal, characteristic, complemented
9#20274717C211yesmaximal, characteristic, complemented
10#20274715C211yesmaximal, characteristic, complemented
11C53×#1802C211yesmaximal, characteristic, complemented
12#16811155—24no—
13#16811155C418yescomplemented
14C2×#13133210—21no—
15C2×#13133208C2211yescharacteristic
16C2×#13133204C2211yescharacteristic
17C2×#13133199—21no—
18C2×#13133197C2211yescharacteristic
19C2×#13133188C2211yescharacteristic
20C2×C4×Dic106C2211yescharacteristic
21C2×C4×C22⋊D53—21no—
22C2×#13133176—21no—
23C2×#13133175C2212yes—
24C2×C42×D53C2211yescharacteristic
25#16810675—24no—
26C2×C4⋊#88322C2211yescharacteristic
27C2×#13133142C2212yes—
28C2×#13133140C2212yes—
29C2×#13133138C2212yes—
30C2×#13133129C2212yes—
31C2×C8×Dic53C2212yes—
32#16810493C2211yescharacteristic
33#16810488C2212yes—
34#16810480—21no—
35#16810480—21no—
36C4⋊C4×Dic53—21no—
37C4⋊C4×Dic53—21no—
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—
52#13133210—28no—
53#13133208—22no—
54#13133208C2×C414yes—
55#13133204—22no—
56#13133204C2×C414yes—
57#13133199—28no—
58#13133197—22no—
59#13133197C2×C414yes—
60#13133188—22no—
61#13133188C2×C414yes—
62C4×Dic106—22no—
63C4×Dic106C2×C414yes—
64C4×C22⋊D53—28no—
65#13133176—28no—
66#13133175—42no—
67#13133175—42no—
68C2×C4⋊Dic53—21no—
69C2×C4⋊Dic53—21no—
70C2×C4⋊Dic53—21no—
71C2×C4⋊Dic53—21no—
72C2×C4⋊Dic53C2312yes—
73C2×C4⋊Dic53C2311yescharacteristic
74C2×#11719919—21no—
75C2×#11719919—21no—
76C2×C22⋊Dic53—21no—
77C2×C22⋊Dic53—21no—
78C2×C22⋊Dic53—21no—
79C2×#11719921—21no—
80C2×#11719921—22no—
81C2×C4×Dic53—21no—
82C2×C4×Dic53—22no—
83C2×C4×Dic53D411yescharacteristic
84C2×C4×Dic53D411yescharacteristic
85C2×C4×Dic53C2311yescharacteristic
86C42×D53C2×C414yes—
87C42×D53D414yes—
88C4⋊#88322—22no—
89#13133142—42no—
90#13133140—42no—
91#13133138—42no—
92#13133129—24no—
93C8×Dic53—24no—
94#13133124—22no—
95#13133124—22no—
96C22×D4×C53C2311yescharacteristic
97C23×C4×C53—21no—
98C22×C8×C53C2312yes—
99C4×D4×C53—22no—
100C4×D4×C53C2×C414yes—
101C2×C4⋊C4×C53—21no—
102C2×C4⋊C4×C53—21no—
103C2×C4⋊C4×C53C2312yes—
104C2×C4⋊C4×C53C2311yescharacteristic
105C2×C22⋊C4×C53—21no—
106C2×C42×C53C2311yescharacteristic
107C4⋊C8×C53—22no—
108#102×C53—42no—
109C24×C53—21no—
110C23×D53C2×D411yescharacteristic
111C2×D4×C53—21no—
112C2×D4×C53—21no—
113C2×D4×C53—21no—
114C2×D4×C53—21no—
115C2×D4×C53—21no—
116C2×D4×C53C22×C412yes—
117C22×C4×C53—21no—
118C22×C4×C53—24no—
119C22×C4×C53—22no—
120C22×C4×C53—42no—
121C22×C4×C53—42no—
122C22×C4×C53C2×D411yescharacteristic
123C22×C4×C53C2×D411yescharacteristic
124C22×C4×C53C2411yescharacteristic
125C2×C22⋊D53—24no—
126C2×C22⋊D53—42no—
127C2×C22⋊D53—42no—
128C22×Dic53—21no—
129C22×Dic53—21no—
130C22×Dic53—22no—
131C22×Dic53C2×D411yescharacteristic
132C22×Dic53C2×D411yescharacteristic
133C22×Dic53C2×D411yescharacteristic
134C2×C4×D53—22no—
135C2×C4×D53—22no—
136C2×C4×D53—21no—
137C2×C4×D53—21no—
138C2×C4×D53—21no—
139C2×C4×D53—21no—
140C2×C4×D53—41no—
141C2×C4×D53—41no—
142C2×C4×D53C22⋊C414yes—
143C2×C4×D53C22⋊C414yes—
144C2×C4×D53C22×C412yes—
145C2×C4×D53C2×D412yes—
146C2×Dic106—21no—
147C2×Dic106—21no—
148C2×Dic106—21no—
149C2×Dic106—21no—
150C2×Dic106—21no—
151C2×Dic106C22×C412yes—
152C2×C8×C53—22no—
153C2×C8×C53—22no—
154C2×C8×C53—22no—
155C4⋊C4×C53—21no—
156C4⋊C4×C53—22no—
157C4⋊C4×C53—22no—
158C4⋊C4×C53—42no—
159C4⋊C4×C53—42no—
160C4⋊C4×C53C22×C412yes—
161C22⋊C4×C53—24no—
162C42×C53C22×C412yes—
163C42×C53C2×D412yes—
164C22⋊Dic53—24no—
165C22⋊Dic53—24no—
166C22⋊Dic53—24no—
167#11719921—24no—
168#11719921—44no—
169C4⋊Dic53—21no—
170C4⋊Dic53—24no—
171C4⋊Dic53—24no—
172C4⋊Dic53—22no—
173C4⋊Dic53—22no—
174C4⋊Dic53—42no—
175C4⋊Dic53—42no—
176C4⋊Dic53C22×C412yes—
177#11719919—24no—
178#11719919—24no—
179C4×Dic53—24no—
180C4×Dic53—44no—
181C4×Dic53C22⋊C414yes—
182C4×Dic53C22⋊C414yes—
183C4×Dic53C22×C412yes—
184C4×Dic53C2×D412yes—
185C2×#88322—22no—
186C2×#88322—22no—
187C2×#88322—22no—
188C23×C53—22no—
189C23×C53—41no—
190C23×C53—41no—
191C23×C53—41no—
192C23×C53—41no—
193C23×C53—41no—
194C23×C53—41no—
195C23×C53C22×D411yescharacteristic
196C22×D53—21no—
197C22×D53—21no—
198C22×D53—21no—
199C22×D53—21no—
200C22×D53—21no—
201C22×D53—21no—
202C22×D53C2×C22⋊C412yes—
203D4×C53—22no—
204D4×C53—22no—
205D4×C53—41no—
206D4×C53—41no—
207C2×C4×C53—21no—
208C2×C4×C53—21no—
209C2×C4×C53—22no—
210C2×C4×C53—41no—
211C2×C4×C53—41no—
212C2×C4×C53—48no—
213C2×C4×C53—42no—
214C2×C4×C53—42no—
215C2×C4×C53—42no—
216C2×C4×C53C2×C22⋊C412yes—
217C2×C4×C53C2×C22⋊C412yes—
218C2×C4×C53#13512yes—
219C2×C4×C53#13512yes—
220C2×C4×C53C23×C411yescharacteristic
221C2×C4×C53C22×D411yescharacteristic
222C2×C4×C53H(5,𝔽2)11yescharacteristic
223C2×C4×C53H(5,𝔽2)11yescharacteristic
224C2×C4×C532−1+411yescharacteristic
225C2×C4×C532−1+411yescharacteristic
226C22⋊D53—48no—
227C2×Dic53—21no—
228C2×Dic53—21no—
229C2×Dic53—21no—
230C2×Dic53—21no—
231C2×Dic53—21no—
232C2×Dic53—21no—
233C2×Dic53—22no—
234C2×Dic53—22no—
235C2×Dic53—41no—
236C2×Dic53—41no—
237C2×Dic53—41no—
238C2×Dic53—41no—
239C2×Dic53—42no—
240C2×Dic53—42no—
241C2×Dic53—42no—
242C2×Dic53C2×C22⋊C412yes—
243C2×Dic53C2×C22⋊C412yes—
244C2×Dic53C2×C22⋊C412yes—
245C4×D53—22no—
246C4×D53—22no—
247C4×D53—22no—
248C4×D53—22no—
249C4×D53—28no—
250C4×D53—44no—
251Dic106—22no—
252Dic106—22no—
253Dic106—41no—
254Dic106—41no—
255C8×C53—42no—
256#88322—42no—
257#21862—531nomaximal, Sylow
258C22×C53—42no—
259C22×C53—42no—
260C22×C53—42no—
261C22×C53—81no—
262C22×C53—81no—
263C22×C53—81no—
264C22×C53—81no—
265C22×C53C22×C22⋊C411yescharacteristic
266C22×C53#53511yescharacteristic
267C22×C53#53611yescharacteristic
268C22×C53C2×#13511yescharacteristic
269C22×C53C2×#13511yescharacteristic
270C22×C53#57511yescharacteristic
271C22×C53#57511yescharacteristic
272C2×D53—22no—
273C2×D53—22no—
274C2×D53—22no—
275C2×D53—41no—
276C2×D53—41no—
277C2×D53—41no—
278C2×D53—41no—
279C4×C53—22no—
280C4×C53—42no—
281C4×C53—82no—
282C4×C53#41614yes—
283C4×C53#52011yescharacteristic
284C4×C53#52011yescharacteristic
285C4×C53#53711yescharacteristic
286C4×C53#53711yescharacteristic
287Dic53—22no—
288Dic53—22no—
289Dic53—22no—
290Dic53—42no—
291Dic53—42no—
292Dic53—82no—
293C2×#529—531no—
294C2×#431—531no—
295#2285—531no—
296#1991—532no—
297#1958—532no—
298#1875—532no—
299#1873—532no—
300#1803—531no—
301#1803—531no—
302#1802—531no—
303#1802D5311yescharacteristic, complemented
304C2×C53—82no—
305C2×C53#231111yescharacteristic
306C2×C53C2×#41611yescharacteristic
307C2×C53#289611yescharacteristic
308C2×C53#321211yescharacteristic
309C2×C53#321211yescharacteristic
310C2×C53#323311yescharacteristic
311C2×C53#323311yescharacteristic
312D53—42no—
313#529—538no—
314#529—1064no—
315C2×#126—1061no—
316C2×#125—531no—
317C2×#124—531no—
318C2×#123—1061no—
319C2×C22⋊Q8—531no—
320C2×#121—531no—
321C2×C4×Q8—531no—
322C2×C4×D4—1061no—
323C2×C4×D4C2×D5311yescharacteristic
324C2×#117—532no—
325C2×#117—1061no—
326C22×C42—531no—
327#431—1064no—
328C2×C4⋊C8—531no—
329C2×C4⋊C8C2×D5311yescharacteristic
330C2×#103—532no—
331C2×#103—532no—
332C2×#102—532no—
333C2×#102C2×D5312yes—
334C2×C22⋊C8—532no—
335C2×C4×C8—532no—
336#384—531no—
337#384C2×D5311yescharacteristic
338#382—532no—
339#380—1061no—
340#380—1061no—
341C4×C4⋊C4—1061no—
342C4×C4⋊C4—1061no—
343#340—532no—
344#340—532no—
345#340—532no—
346#340C2×D5312yes—
347C53#2186211yescharacteristic, complemented, Sylow
348C22×Q8—531no—
349C22×D4—1061no—
350C22×D4C22×D5311yescharacteristic
351C23×C4—21no—
352C23×C4—531no—
353C23×C4—531no—
354C23×C4—531no—
355C22×C8—532no—
356C22×C8C22×D5312yes—
357#126—1068no—
358#125—534no—
359#125—1062no—
360#124—534no—
361#124—1062no—
362#123—1068no—
363C22⋊Q8—534no—
364C22⋊Q8—1062no—
365#121—534no—
366#121—1062no—
367C4×Q8—534no—
368C4×Q8—1062no—
369C4×D4—22no—
370C4×D4—1068no—
371C4×D4C4×D5314yes—
372#117—1068no—
373#117—2122no—
374#117—2122no—
375C2×C4⋊C4—21no—
376C2×C4⋊C4—21no—
377C2×C4⋊C4—531no—
378C2×C4⋊C4—532no—
379C2×C4⋊C4—1061no—
380C2×C4⋊C4—1061no—
381C2×C4⋊C4—1061no—
382C2×C4⋊C4—1061no—
383C2×C4⋊C4—1061no—
384C2×C4⋊C4—1061no—
385C2×C4⋊C4C22×D5312yes—
386C2×C4⋊C4C22×D5311yescharacteristic
387C2×C22⋊C4—21no—
388C2×C22⋊C4—1061no—
389C2×C22⋊C4—1061no—
390C2×C22⋊C4—1061no—
391C2×C22⋊C4—1061no—
392C2×C22⋊C4—1062no—
393C2×C42—531no—
394C2×C42—531no—
395C2×C42—531no—
396C2×C42—534no—
397C2×C42—534no—
398C2×C42—1061no—
399C2×C42—1062no—
400C2×C42C22×D5311yescharacteristic
401C4⋊C8—22no—
402C4⋊C8—1062no—
403#103—2122no—
404#103—2122no—
405#102—42no—
406#102—2122no—
407C22⋊C8—1064no—
408C4×C8—1064no—
409#95—1062no—
410#95—1062no—
411C24—21no—
412C24—531no—
413C2×Q8—532no—
414C2×Q8—1061no—
415C2×Q8—1061no—
416C2×Q8—1061no—
417C2×Q8—1061no—
418C2×Q8—1061no—
419C2×D4—21no—
420C2×D4—21no—
421C2×D4—21no—
422C2×D4—21no—
423C2×D4—21no—
424C2×D4—1064no—
425C2×D4—2122no—
426C2×D4—2122no—
427C2×D4C2×C4×D5312yes—
428C22×C4—21no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—531no—
434C22×C4—531no—
435C22×C4—531no—
436C22×C4—534no—
437C22×C4—532no—
438C22×C4—532no—
439C22×C4—534no—
440C22×C4—1062no—
441C22×C4—1062no—
442C22×C4—1061no—
443C22×C4—1061no—
444C22×C4—1061no—
445C22×C4—1061no—
446C22×C4—1061no—
447C22×C4—1061no—
448C22×C4—1062no—
449C22×C4—2121no—
450C22×C4—2121no—
451C22×C4D4×D5311yescharacteristic
452C22×C4D4×D5311yescharacteristic
453C22×C4C23×D5311yescharacteristic
454C2×C8—22no—
455C2×C8—22no—
456C2×C8—22no—
457C2×C8—1062no—
458C2×C8—1062no—
459C2×C8—1062no—
460C4⋊C4—21no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—42no—
464C4⋊C4—42no—
465C4⋊C4—532no—
466C4⋊C4—1064no—
467C4⋊C4—1064no—
468C4⋊C4—1064no—
469C4⋊C4—1064no—
470C4⋊C4—1061no—
471C4⋊C4—1062no—
472C4⋊C4—1062no—
473C4⋊C4—2122no—
474C4⋊C4—2122no—
475C4⋊C4C2×C4×D5312yes—
476C22⋊C4—24no—
477C22⋊C4—1064no—
478C22⋊C4—1064no—
479C22⋊C4—1064no—
480C22⋊C4—1064no—
481C22⋊C4—2124no—
482C42—534no—
483C42—534no—
484C42—532no—
485C42—532no—
486C42—1064no—
487C42—2124no—
488C42C2×C4×D5312yes—
489C42D4×D5312yes—
490C23—22no—
491C23—41no—
492C23—41no—
493C23—41no—
494C23—41no—
495C23—41no—
496C23—41no—
497C23—532no—
498C23—1061no—
499C23—1061no—
500C23—1061no—
501C23—1061no—
502C23—1061no—
503C23—1061no—
504C23C2×D4×D5311yescharacteristic
505Q8—1062no—
506Q8—1062no—
507Q8—2121no—
508Q8—2121no—
509D4—22no—
510D4—22no—
511D4—41no—
512D4—41no—
513D4—2128no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—22no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—48no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—532no—
524C2×C4—532no—
525C2×C4—532no—
526C2×C4—1061no—
527C2×C4—1061no—
528C2×C4—1061no—
529C2×C4—1061no—
530C2×C4—1061no—
531C2×C4—1061no—
532C2×C4—1062no—
533C2×C4—1062no—
534C2×C4—1068no—
535C2×C4—1062no—
536C2×C4—1062no—
537C2×C4—1062no—
538C2×C4—1062no—
539C2×C4—2124no—
540C2×C4—2122no—
541C2×C4—2122no—
542C2×C4—2122no—
543C2×C4—2121no—
544C2×C4—2121no—
545C2×C4—2121no—
546C2×C4—2121no—
547C2×C4C22⋊C4×D5312yes—
548C2×C4C22⋊C4×D5312yes—
549C2×C4#1313323612yes—
550C2×C4#1313323812yes—
551C2×C4C22×C4×D5311yescharacteristic
552C2×C4C2×D4×D5311yescharacteristic
553C2×C4#1313326211yescharacteristic
554C2×C4#1313326311yescharacteristic
555C2×C4#1313326411yescharacteristic
556C2×C4#1313326511yescharacteristic
557C8—42no—
558C8—2122no—
559C22—42no—
560C22—42no—
561C22—42no—
562C22—81no—
563C22—81no—
564C22—81no—
565C22—81no—
566C22—1062no—
567C22—1062no—
568C22—1062no—
569C22—2121no—
570C22—2121no—
571C22—2121no—
572C22—2121no—
573C22C2×C22⋊C4×D5311yescharacteristic
574C22#1681118911yescharacteristic
575C22#1681119611yescharacteristic
576C22C2×#1313323611yescharacteristic
577C22C2×#1313323811yescharacteristic
578C22#1681141911yescharacteristic
579C22#1681142211yescharacteristic
580C4—22no—
581C4—42no—
582C4—82no—
583C4—1062no—
584C4—1062no—
585C4—1062no—
586C4—4242no—
587C4—2122no—
588C4—2122no—
589C4#1681061614yes—
590C4#1681110311yescharacteristic
591C4#1681110411yescharacteristic
592C4#1681120211yescharacteristic
593C4#1681120311yescharacteristic
594C2—82no—
595C2—2122no—
596C2#2027730411yescharacteristic
597C2C2×#1681061611yescharacteristic
598C2#2028323111yescharacteristic
599C2#2028568911yescharacteristic
600C2#2028569011yescharacteristic
601C2#2028589211yescharacteristic
602C2#2028589311yescharacteristic
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#21862
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×#16810675C2semidirect
3. #20277120C2semidirect
4. #20275678C2semidirect
5. #20275525C2semidirect
7. #20275026C2semidirect
8. #20274721C2semidirect
9. #20274717C2semidirect
10. #20274715C2semidirect
11. C53×#1802C2semidirect
13. #16811155C4semidirect
303. #1802D53semidirect
347. C53#21862semidirect

Maximal subgroups

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

subgroupindex
C2×#168111552
C2×#168106752
#202771202
#202756782
#202755252
#202750342
#202750262
#202747212
#202747172
#202747152
C53×#18022
#2186253