Finite Group Databasebeta

← #27393773

Subgroups — #27393773

Order 13568.

1222 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811177C211yesmaximal, characteristic
2C2×#16810675C211yesmaximal, characteristic
3#384×D53C211yesmaximal, characteristic
4#20275674C212yesmaximal, complemented
5#20275515C212yesmaximal, complemented
6#20275021C212yesmaximal
7#20275019C212yesmaximal
8#20274721C211yesmaximal, characteristic, complemented
9#20274720C211yesmaximal, characteristic, complemented
10#20274717C211yesmaximal, characteristic, complemented
11C53×#1803C211yesmaximal, characteristic, complemented
12#16811177—24no—
13#16811177C418yescomplemented
14C2×#13133216C2211yescharacteristic
15C2×#13133215C2211yescharacteristic
16C2×#13133207—21no—
17C2×#13133197C2211yescharacteristic
18C2×#13133195—21no—
19C2×#13133193C2211yescharacteristic
20C2×C4×Dic106—21no—
21C2×C4×Dic106C2211yescharacteristic
22C2×#13133174—21no—
23C2×C4⋊C4×D53—21no—
24C2×C4⋊C4×D53—21no—
25C2×C4⋊C4×D53C2212yes—
26C2×C42×D53C2211yescharacteristic
27#16810675—24no—
28C2×C4⋊#88322C2211yescharacteristic
29C2×#13133142C2212yes—
30C2×#13133141C2212yes—
31C2×#13133140C2212yes—
32C2×#13133129C2212yes—
33C2×C8×Dic53C2212yes—
34#16810493C2211yescharacteristic
35#16810492—21no—
36#16810492—21no—
37#16810492C2212yes—
38#16810386C2212yes—
39#16810385C2212yes—
40#16810385C2212yes—
41C2×C4×Q8×C53C2211yescharacteristic
42C2×C4⋊C8×C53C2211yescharacteristic
43C2×#103×C53C2212yes—
44C53×#384C2211yescharacteristic
45C53×#340C2212yes—
46C22×Dic106—21no—
47C22×Dic106C2311yescharacteristic
48C22×C4×D53—22no—
49C22×C4×D53D411yescharacteristic
50C22×C4×D53D411yescharacteristic
51C22×C4×D53C2311yescharacteristic
52C22×#88322C2312yes—
53#13133216—22no—
54#13133216C2×C414yescomplemented
55#13133215—22no—
56#13133215C2×C414yescomplemented
57#13133207—28no—
58#13133197—22no—
59#13133197C2×C414yes—
60#13133195—28no—
61#13133193—22no—
62#13133193C2×C414yes—
63C4×Dic106—22no—
64C4×Dic106—28no—
65C4×Dic106C2×C414yescomplemented
66#13133174—28no—
67C2×C4⋊Dic53—21no—
68C2×C4⋊Dic53—21no—
69C2×C4⋊Dic53—21no—
70C2×C4⋊Dic53C2312yes—
71C2×C4⋊Dic53C2311yescharacteristic
72C2×#11719919—21no—
73C2×#11719919—21no—
74C2×#11719919—21no—
75C2×#11719919—21no—
76C2×#11719919—21no—
77C2×#11719919—22no—
78C2×#11719919—22no—
79C2×#11719919—22no—
80C4⋊C4×D53—42no—
81C4⋊C4×D53—42no—
82C4⋊C4×D53—44no—
83C4⋊C4×D53—44no—
84C2×#11719921—21no—
85C2×#11719921—21no—
86C2×C4×Dic53—21no—
87C2×C4×Dic53D411yescharacteristic
88C2×C4×Dic53D411yescharacteristic
89C2×C4×Dic53C2311yescharacteristic
90C42×D53C2×C414yes—
91C42×D53D414yes—
92C4⋊#88322—22no—
93#13133142—42no—
94#13133141—42no—
95#13133140—42no—
96#13133129—24no—
97C8×Dic53—24no—
98C22×Q8×C53C2311yescharacteristic
99C22×C8×C53C2312yes—
100C4×Q8×C53—22no—
101C4×Q8×C53C2×C414yescomplemented
102C2×C4⋊C4×C53—21no—
103C2×C4⋊C4×C53—21no—
104C2×C4⋊C4×C53—21no—
105C2×C4⋊C4×C53C2312yes—
106C2×C4⋊C4×C53C2311yescharacteristic
107C2×C42×C53—21no—
108C2×C42×C53C2311yescharacteristic
109C4⋊C8×C53—22no—
110#103×C53—42no—
111C23×D53C2×D411yescharacteristic
112C2×Q8×C53—21no—
113C2×Q8×C53—21no—
114C2×Q8×C53—21no—
115C2×Q8×C53—21no—
116C2×Q8×C53—21no—
117C2×Q8×C53C22×C412yes—
118C22×C4×C53—21no—
119C22×C4×C53—21no—
120C22×C4×C53—22no—
121C22×C4×C53C2×D411yescharacteristic
122C22×C4×C53C2×D411yescharacteristic
123C22×C4×C53C2411yescharacteristic
124C22×Dic53—21no—
125C22×Dic53—21no—
126C22×Dic53—22no—
127C22×Dic53C2×D411yescharacteristic
128C22×Dic53C2×D411yescharacteristic
129C22×Dic53C2×D411yescharacteristic
130C2×C4×D53—21no—
131C2×C4×D53—21no—
132C2×C4×D53—21no—
133C2×C4×D53—21no—
134C2×C4×D53—22no—
135C2×C4×D53—22no—
136C2×C4×D53—44no—
137C2×C4×D53—44no—
138C2×C4×D53—44no—
139C2×C4×D53C22⋊C414yes—
140C2×C4×D53C22⋊C414yes—
141C2×C4×D53SD1614yes—
142C2×C4×D53Q1614yes—
143C2×C4×D53C22×C412yes—
144C2×C4×D53C2×D412yes—
145C2×Dic106—21no—
146C2×Dic106—21no—
147C2×Dic106—21no—
148C2×Dic106—21no—
149C2×Dic106—21no—
150C2×Dic106—24no—
151C2×Dic106—42no—
152C2×Dic106—42no—
153C2×Dic106C22×C412yes—
154C2×C8×C53—22no—
155C2×C8×C53—22no—
156C2×C8×C53—22no—
157C4⋊C4×C53—24no—
158C4⋊C4×C53—21no—
159C4⋊C4×C53—22no—
160C4⋊C4×C53—22no—
161C4⋊C4×C53—42no—
162C4⋊C4×C53—42no—
163C4⋊C4×C53C22×C412yes—
164C42×C53—24no—
165C42×C53C22×C412yes—
166C42×C53C2×D412yes—
167#11719921—24no—
168#11719921—24no—
169C4⋊Dic53—21no—
170C4⋊Dic53—24no—
171C4⋊Dic53—22no—
172C4⋊Dic53—22no—
173C4⋊Dic53—42no—
174C4⋊Dic53—42no—
175C4⋊Dic53C22×C412yes—
176#11719919—24no—
177#11719919—24no—
178#11719919—24no—
179#11719919—24no—
180#11719919—24no—
181#11719919—44no—
182#11719919—44no—
183#11719919—44no—
184C4×Dic53—24no—
185C4×Dic53C22⋊C414yes—
186C4×Dic53C22⋊C414yes—
187C4×Dic53C22×C412yes—
188C4×Dic53C2×D412yes—
189C2×#88322—22no—
190C2×#88322—22no—
191C2×#88322—22no—
192C23×C53C22×D411yescharacteristic
193C22×D53—21no—
194C22×D53—21no—
195C22×D53—21no—
196C22×D53—21no—
197C22×D53C2×C22⋊C412yes—
198C22×D53C2×SD1612yes—
199C22×D53C2×Q1612yes—
200Q8×C53—22no—
201Q8×C53—22no—
202Q8×C53—41no—
203Q8×C53—41no—
204C2×C4×C53—21no—
205C2×C4×C53—21no—
206C2×C4×C53—22no—
207C2×C4×C53—22no—
208C2×C4×C53—41no—
209C2×C4×C53—41no—
210C2×C4×C53—41no—
211C2×C4×C53—41no—
212C2×C4×C53—42no—
213C2×C4×C53—42no—
214C2×C4×C53—42no—
215C2×C4×C53C2×C22⋊C412yes—
216C2×C4×C53C2×C22⋊C412yes—
217C2×C4×C53C2×SD1612yes—
218C2×C4×C53C2×Q1612yes—
219C2×C4×C53C23×C411yescharacteristic
220C2×C4×C53C22×D411yescharacteristic
221C2×C4×C53H(5,𝔽2)11yescharacteristic
222C2×C4×C53H(5,𝔽2)11yescharacteristic
223C2×C4×C532−1+411yescharacteristic
224C2×C4×C532−1+411yescharacteristic
225C2×Dic53—21no—
226C2×Dic53—21no—
227C2×Dic53—22no—
228C2×Dic53—22no—
229C2×Dic53—41no—
230C2×Dic53—41no—
231C2×Dic53—41no—
232C2×Dic53—41no—
233C2×Dic53—42no—
234C2×Dic53—42no—
235C2×Dic53—42no—
236C2×Dic53C2×C22⋊C412yes—
237C2×Dic53C2×C22⋊C412yes—
238C2×Dic53C2×C22⋊C412yes—
239C2×Dic53C2×SD1612yes—
240C2×Dic53C2×Q1612yes—
241C2×Dic53AGL(1,ℤ/8)12yes—
242C2×Dic53#13712yes—
243C4×D53—22no—
244C4×D53—22no—
245C4×D53—22no—
246C4×D53—22no—
247C4×D53—28no—
248C4×D53—88no—
249C4×D53#103116yescomplemented
250Dic106—22no—
251Dic106—22no—
252Dic106—41no—
253Dic106—41no—
254Dic106—48no—
255C8×C53—42no—
256#88322—42no—
257#21860—531nomaximal, Sylow
258C22×C53C22×C22⋊C411yescharacteristic
259C22×C53#53511yescharacteristic
260C22×C53#53611yescharacteristic
261C22×C53C22×SD1611yescharacteristic
262C22×C53C22×Q1611yescharacteristic
263C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
264C22×C53C2×#13711yescharacteristic
265C2×D53—22no—
266C2×D53—22no—
267C2×D53—22no—
268C2×D53—22no—
269C2×D53—22no—
270C2×D53—22no—
271C2×D53C2×#10314yes—
272C4×C53—22no—
273C4×C53—42no—
274C4×C53—42no—
275C4×C53—82no—
276C4×C53C2×#10314yes—
277C4×C53#52011yescharacteristic
278C4×C53#52011yescharacteristic
279C4×C53#53711yescharacteristic
280C4×C53#53711yescharacteristic
281Dic53—22no—
282Dic53—42no—
283Dic53—42no—
284Dic53—82no—
285Dic53C2×#10314yes—
286Dic53#41714yes—
287C2×#533—531no—
288C2×#431—531no—
289C2×#384—531no—
290#1990—532no—
291#1956—532no—
292#1872—532no—
293#1872—532no—
294#1803—531no—
295#1803—531no—
296#1803—531no—
297#1803D5311yescharacteristic, complemented
298C2×C53#231111yescharacteristic
299C2×C53C22×#10311yescharacteristic
300C2×C53C2×#41711yescharacteristic
301C2×C53#320811yescharacteristic
302C2×C53#320911yescharacteristic
303C2×C53#322911yescharacteristic
304C2×C53#323011yescharacteristic
305D53—24no—
306#533—538no—
307#533—1064no—
308C2×C4⋊Q8—531no—
309C2×C4⋊Q8—531no—
310C2×#125—1061no—
311C2×C22⋊Q8—531no—
312C2×C22⋊Q8—531no—
313C2×C22⋊Q8—1061no—
314C2×C4×Q8—531no—
315C2×C4×Q8—1061no—
316C2×C4×Q8C2×D5311yescharacteristic
317C2×#117—1061no—
318C22×C4⋊C4—532no—
319C22×C4⋊C4—1061no—
320C22×C4⋊C4—1061no—
321C22×C42—531no—
322#431—1064no—
323C2×C4⋊C8—531no—
324C2×C4⋊C8C2×D5311yescharacteristic
325C2×#103—532no—
326C2×#103—532no—
327C2×#103—532no—
328C2×#103C2×D5312yes—
329C2×C22⋊C8—532no—
330C2×C4×C8—532no—
331#384—532no—
332#384—531no—
333#384—1061no—
334#384—1061no—
335#384C2×D5311yescharacteristic
336#340—532no—
337#340—532no—
338#340—532no—
339#340C2×D5312yes—
340C53#2186011yescharacteristic, complemented, Sylow
341C22×Q8—531no—
342C22×Q8—1061no—
343C22×Q8C22×D5311yescharacteristic
344C23×C4—531no—
345C23×C4—531no—
346C23×C4—531no—
347C23×C4—1062no—
348C22×C8—532no—
349C22×C8C22×D5312yes—
350C4⋊Q8—534no—
351C4⋊Q8—534no—
352C4⋊Q8—1062no—
353C4⋊Q8—1062no—
354#125—1068no—
355C22⋊Q8—534no—
356C22⋊Q8—534no—
357C22⋊Q8—1062no—
358C22⋊Q8—1068no—
359C22⋊Q8—1062no—
360C4×Q8—22no—
361C4×Q8—534no—
362C4×Q8—1068no—
363C4×Q8—1062no—
364C4×Q8C4×D5314yescomplemented
365#117—1068no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—531no—
370C2×C4⋊C4—532no—
371C2×C4⋊C4—1061no—
372C2×C4⋊C4—1061no—
373C2×C4⋊C4—1061no—
374C2×C4⋊C4—1061no—
375C2×C4⋊C4—1061no—
376C2×C4⋊C4—1061no—
377C2×C4⋊C4—1062no—
378C2×C4⋊C4—1062no—
379C2×C4⋊C4—1062no—
380C2×C4⋊C4—1061no—
381C2×C4⋊C4—1061no—
382C2×C4⋊C4—2122no—
383C2×C4⋊C4—2122no—
384C2×C4⋊C4—2124no—
385C2×C4⋊C4—2124no—
386C2×C4⋊C4C22×D5312yes—
387C2×C4⋊C4C22×D5311yescharacteristic
388C2×C22⋊C4—1061no—
389C2×C22⋊C4—1061no—
390C2×C42—21no—
391C2×C42—531no—
392C2×C42—531no—
393C2×C42—531no—
394C2×C42—534no—
395C2×C42—534no—
396C2×C42—1061no—
397C2×C42C22×D5311yescharacteristic
398C4⋊C8—22no—
399C4⋊C8—1062no—
400#103—42no—
401#103—2122no—
402#103—2122no—
403#103—2122no—
404C22⋊C8—1064no—
405C4×C8—1064no—
406C24—531no—
407C2×Q8—21no—
408C2×Q8—21no—
409C2×Q8—21no—
410C2×Q8—21no—
411C2×Q8—21no—
412C2×Q8—532no—
413C2×Q8—1064no—
414C2×Q8—1061no—
415C2×Q8—1061no—
416C2×Q8—1061no—
417C2×Q8—1061no—
418C2×Q8—1061no—
419C2×Q8—2122no—
420C2×Q8—2122no—
421C2×Q8C2×C4×D5312yes—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—22no—
425C22×C4—531no—
426C22×C4—531no—
427C22×C4—534no—
428C22×C4—534no—
429C22×C4—534no—
430C22×C4—531no—
431C22×C4—534no—
432C22×C4—532no—
433C22×C4—532no—
434C22×C4—1062no—
435C22×C4—1062no—
436C22×C4—1061no—
437C22×C4—1061no—
438C22×C4—1061no—
439C22×C4—1061no—
440C22×C4—1061no—
441C22×C4—1061no—
442C22×C4—1062no—
443C22×C4—2124no—
444C22×C4—2124no—
445C22×C4—2124no—
446C22×C4D4×D5311yescharacteristic
447C22×C4D4×D5311yescharacteristic
448C22×C4C23×D5311yescharacteristic
449C2×C8—22no—
450C2×C8—22no—
451C2×C8—22no—
452C2×C8—1062no—
453C2×C8—1062no—
454C2×C8—1062no—
455C4⋊C4—24no—
456C4⋊C4—21no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—42no—
460C4⋊C4—42no—
461C4⋊C4—532no—
462C4⋊C4—1064no—
463C4⋊C4—1064no—
464C4⋊C4—1064no—
465C4⋊C4—1064no—
466C4⋊C4—1064no—
467C4⋊C4—1064no—
468C4⋊C4—1061no—
469C4⋊C4—1062no—
470C4⋊C4—1062no—
471C4⋊C4—2122no—
472C4⋊C4—2124no—
473C4⋊C4—2124no—
474C4⋊C4—2124no—
475C4⋊C4—2122no—
476C4⋊C4C2×C4×D5312yes—
477C22⋊C4—1064no—
478C22⋊C4—1064no—
479C42—24no—
480C42—534no—
481C42—534no—
482C42—532no—
483C42—532no—
484C42—1064no—
485C42C2×C4×D5312yes—
486C42D4×D5312yes—
487C23—532no—
488C23—532no—
489C23—532no—
490C23—1061no—
491C23—1061no—
492C23—1061no—
493C23—1061no—
494C23C2×D4×D5311yescharacteristic
495Q8—22no—
496Q8—22no—
497Q8—41no—
498Q8—41no—
499Q8—1062no—
500Q8—1062no—
501Q8—2128no—
502Q8—2121no—
503Q8—2121no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—532no—
516C2×C4—532no—
517C2×C4—532no—
518C2×C4—532no—
519C2×C4—532no—
520C2×C4—532no—
521C2×C4—5316no—
522C2×C4—532no—
523C2×C4—1068no—
524C2×C4—1061no—
525C2×C4—1061no—
526C2×C4—1062no—
527C2×C4—1062no—
528C2×C4—1062no—
529C2×C4—1062no—
530C2×C4—1062no—
531C2×C4—1062no—
532C2×C4—4248no—
533C2×C4—2122no—
534C2×C4—2122no—
535C2×C4—2122no—
536C2×C4—2121no—
537C2×C4—2121no—
538C2×C4—2121no—
539C2×C4—2121no—
540C2×C4C22⋊C4×D5312yes—
541C2×C4C22⋊C4×D5312yes—
542C2×C4SD16×D5312yes—
543C2×C4Q16×D5312yes—
544C2×C4C22×C4×D5311yescharacteristic
545C2×C4C2×D4×D5311yescharacteristic
546C2×C4#1313326211yescharacteristic
547C2×C4#1313326311yescharacteristic
548C2×C4#1313326411yescharacteristic
549C2×C4#1313326511yescharacteristic
550C8—42no—
551C8—2122no—
552C22—534no—
553C22—1062no—
554C22—1062no—
555C22—1062no—
556C22—1062no—
557C22—1062no—
558C22—1062no—
559C22C2×C22⋊C4×D5311yescharacteristic
560C22#1681118911yescharacteristic
561C22#1681119611yescharacteristic
562C22C2×SD16×D5311yescharacteristic
563C22C2×Q16×D5311yescharacteristic
564C22AGL(1,ℤ/8)×D5311yescharacteristic
565C22#137×D5311yescharacteristic
566C4—22no—
567C4—42no—
568C4—42no—
569C4—82no—
570C4—534no—
571C4—534no—
572C4—1062no—
573C4—4242no—
574C4—2122no—
575C4—2122no—
576C4#103×D5314yes—
577C4#1681110311yescharacteristic
578C4#1681110411yescharacteristic
579C4#1681120211yescharacteristic
580C4#1681120311yescharacteristic
581C2—1064no—
582C2#2027730411yescharacteristic
583C2C2×#103×D5311yescharacteristic
584C2#417×D5311yescharacteristic
585C2#2028565511yescharacteristic
586C2#2028566211yescharacteristic
587C2#2028585811yescharacteristic
588C2#2028586511yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1803C2
socleC23×C53C22×D4
prime radicalC23×D53C2×D4
cosocleC22×C4×C53C24
hypercenter#1803D53
hypocenterC53#21860
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1803

Complements

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

NH ≅ G/Ntype
4. #20275674C2semidirect
5. #20275515C2semidirect
8. #20274721C2semidirect
9. #20274720C2semidirect
10. #20274717C2semidirect
11. C53×#1803C2semidirect
13. #16811177C4semidirect
54. #13133216C2×C4semidirect
56. #13133215C2×C4semidirect
65. C4×Dic106C2×C4semidirect
101. C4×Q8×C53C2×C4semidirect
249. C4×D53#103semidirect
297. #1803D53semidirect
340. C53#21860semidirect
364. C4×Q8C4×D53semidirect

Maximal subgroups

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

subgroupindex
C2×#168111772
C2×#168106752
#384×D532
#202756742
#202755152
#202750212
#202750192
#202747212
#202747202
#202747172
C53×#18032
#2186053