Finite Group Databasebeta

← C4×#16810801 #27389977

Subgroups — C4×#16810801 #27389977

Order 13568.

1442 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810801C211yesmaximal, characteristic
2C4×#13133247C212yesmaximal, complemented
3C4×#13133204C211yesmaximal, characteristic, complemented
4#20275103C211yesmaximal, characteristic, complemented
5#20275084C212yesmaximal, complemented
6#20274948C212yesmaximal, complemented
7#20274925C211yesmaximal, characteristic, complemented
8#20274669C211yesmaximal, characteristic
9C4×#13133144C212yesmaximal, complemented
10C4×#13133134C211yesmaximal, characteristic, complemented
11C4×H(5,𝔽2)×C53C211yesmaximal, characteristic
12C2×#13133247C2212yes—
13C2×#13133204C2211yescharacteristic
14C2×C4×Dic106C2211yescharacteristic, complemented
15#16810801C418yesdirect product factor
16#16810728—22no—
17#16810723—22no—
18C4×#11719925—22no—
19C4×#11719923—22no—
20#16810666C2212yescomplemented
21C2×#13133144C2212yes—
22C2×#13133144C2212yes—
23#16810631C2212yescomplemented
24#16810624C2212yes—
25C2×#13133134C2211yescharacteristic
26C2×#13133134C2211yescharacteristic
27C4×#11719917C2212yescomplemented
28#16810528C2211yescharacteristic, complemented
29#16810526C2211yescharacteristic
30#16810517—22no—
31#16810488C2211yescharacteristic, complemented
32C4×C22⋊Dic53—22no—
33C42×Dic53C2211yescharacteristic, complemented
34#16810392C2212yescomplemented
35#16810382C2212yescomplemented
36#16810382C2212yescomplemented
37C2×H(5,𝔽2)×C53C2211yescharacteristic
38C53×#520—22no—
39C53×#520C2212yes—
40C53×#518—22no—
41C53×#518—22no—
42C53×#518C2211yescharacteristic
43C4×#41×C53—22no—
44C4×#41×C53C2212yes—
45C2×C4×D4×C53—22no—
46C2×C4×D4×C53—22no—
47C2×C4×D4×C53C2211yescharacteristic
48C22×Dic106C2311yescharacteristic
49#13133247—24no—
50#13133247C2×C418yescomplemented
51C2×#11719925—22no—
52C2×#11719923—22no—
53C2×#11719917C2312yes—
54C2×#11719917C2312yes—
55#13133204—22no—
56#13133204C2×C414yescomplemented
57C4×Dic106—22no—
58C4×Dic106—22no—
59C2×C4⋊Dic53C2311yescharacteristic
60C2×#11719919—21no—
61C2×C22⋊Dic53—22no—
62C2×C22⋊Dic53—22no—
63C2×C4×Dic53—21no—
64C2×C4×Dic53—21no—
65C2×C4×Dic53—21no—
66C2×C4×Dic53C2311yescharacteristic
67C2×C4×Dic53C2311yescharacteristic
68#13133150—22no—
69#13133148—22no—
70#13133144—24no—
71#13133144C2×C418yescomplemented
72#13133141—22no—
73#13133140—22no—
74#13133140—22no—
75#13133138—22no—
76#13133134—22no—
77#13133134C2×C414yescomplemented
78#13133128—22no—
79C4×#88322—22no—
80#13133124—22no—
81#13133124—22no—
82H(5,𝔽2)×C53—24no—
83H(5,𝔽2)×C53—22no—
84H(5,𝔽2)×C53C2×C414yes—
85C2×#41×C53—22no—
86C2×#41×C53C2312yes—
87C22×D4×C53—22no—
88C22×D4×C53—22no—
89C22×D4×C53C2311yescharacteristic
90C23×C4×C53—22no—
91C23×C4×C53D412yes—
92C4×Q8×C53—22no—
93C4×Q8×C53D412yes—
94C4×Q8×C53D412yes—
95C4×D4×C53—24no—
96C4×D4×C53—24no—
97C4×D4×C53—24no—
98C4×D4×C53—24no—
99C4×D4×C53—22no—
100C4×D4×C53—22no—
101C4×D4×C53—22no—
102C4×D4×C53—22no—
103C4×D4×C53—22no—
104C4×D4×C53—22no—
105C4×D4×C53—24no—
106C4×D4×C53—22no—
107C4×D4×C53D412yes—
108C4×D4×C53D412yes—
109#117×C53—24no—
110#117×C53—22no—
111#117×C53—22no—
112#117×C53C2312yes—
113C2×C4⋊C4×C53—22no—
114C2×C4⋊C4×C53—22no—
115C2×C4⋊C4×C53C2311yescharacteristic
116C2×C22⋊C4×C53—22no—
117C2×C22⋊C4×C53—24no—
118C2×C22⋊C4×C53—22no—
119C2×C22⋊C4×C53D412yes—
120C2×C42×C53—22no—
121C2×C42×C53—22no—
122C2×C42×C53C2311yescharacteristic
123C24×C53—22no—
124C24×C53C2×D412yes—
125#41×C53—28no—
126#41×C53—28no—
127#41×C53—22no—
128#41×C53—22no—
129#41×C53C22×C414yes—
130#41×C53#4114yes—
131C2×Q8×C53C2×D412yes—
132C2×D4×C53—24no—
133C2×D4×C53—28no—
134C2×D4×C53—28no—
135C2×D4×C53—28no—
136C2×D4×C53—22no—
137C2×D4×C53—28no—
138C2×D4×C53—22no—
139C2×D4×C53—22no—
140C2×D4×C53C22×C412yes—
141C2×D4×C53C2×D412yes—
142C2×D4×C53#4112yes—
143C22×C4×C53—22no—
144C22×C4×C53—22no—
145C22×C4×C53—22no—
146C22×C4×C53—22no—
147C22×C4×C53—24no—
148C22×C4×C53—22no—
149C22×C4×C53—22no—
150C22×C4×C53—22no—
151C22×C4×C53—22no—
152C22×C4×C53—44no—
153C22×C4×C53—84no—
154C22×C4×C53C2×D411yescharacteristic
155C22×C4×C53C2×D412yes—
156C22×C4×C53C2×D411yescharacteristic
157C22×C4×C53C2411yescharacteristic
158C22×Dic53—21no—
159C22×Dic53—21no—
160C22×Dic53—21no—
161C22×Dic53—21no—
162C2×Dic106—22no—
163C2×Dic106—22no—
164C2×Dic106—21no—
165C2×Dic106C22×C412yescomplemented
166C4⋊C4×C53—24no—
167C4⋊C4×C53—22no—
168C4⋊C4×C53—22no—
169C4⋊C4×C53—22no—
170C4⋊C4×C53C2×D412yes—
171C4⋊C4×C53C2×D412yes—
172C22⋊C4×C53—24no—
173C22⋊C4×C53—24no—
174C22⋊C4×C53—22no—
175C22⋊C4×C53—22no—
176C22⋊C4×C53—22no—
177C22⋊C4×C53—24no—
178C42×C53—24no—
179C42×C53—22no—
180C42×C53—22no—
181C42×C53—22no—
182C42×C53C2×D412yes—
183C42×C53C2×D412yes—
184C22⋊Dic53—28no—
185C22⋊Dic53—44no—
186#11719925—28no—
187#11719923—28no—
188C4⋊Dic53—22no—
189#11719919—42no—
190C4×Dic53—21no—
191C4×Dic53—22no—
192C4×Dic53—42no—
193C4×Dic53—42no—
194C4×Dic53—41no—
195C4×Dic53—41no—
196C4×Dic53C22×C412yescomplemented
197#11719917—22no—
198#11719917—22no—
199#11719917—22no—
200#11719917C22×C414yescomplemented
201C2×#88322—22no—
202C2×#88322—22no—
203C23×C53—28no—
204C23×C53—22no—
205C23×C53—22no—
206C23×C53—22no—
207C23×C53—42no—
208C23×C53—42no—
209C23×C53—82no—
210C23×C53—82no—
211C23×C53C4×D414yes—
212C23×C53C22×D411yescharacteristic
213Q8×C53—22no—
214Q8×C53C4×D414yes—
215D4×C53—216no—
216D4×C53—24no—
217D4×C53—24no—
218D4×C53—24no—
219D4×C53—22no—
220D4×C53—24no—
221D4×C53C4×D414yes—
222C2×C4×C53—28no—
223C2×C4×C53—22no—
224C2×C4×C53—24no—
225C2×C4×C53—24no—
226C2×C4×C53—22no—
227C2×C4×C53—22no—
228C2×C4×C53—22no—
229C2×C4×C53—22no—
230C2×C4×C53—41no—
231C2×C4×C53—44no—
232C2×C4×C53—44no—
233C2×C4×C53—42no—
234C2×C4×C53—82no—
235C2×C4×C53—82no—
236C2×C4×C53—82no—
237C2×C4×C53—84no—
238C2×C4×C53—82no—
239C2×C4×C53#12011yescharacteristic
240C2×C4×C53#12012yes—
241C2×C4×C53#12011yescharacteristic
242C2×C4×C53C23×C411yescharacteristic
243C2×C4×C53C22×D412yes—
244C2×C4×C53C2×#4112yes—
245C2×C4×C53C2×#4111yescharacteristic
246C2×Dic53—22no—
247C2×Dic53—22no—
248C2×Dic53—22no—
249C2×Dic53—41no—
250C2×Dic53—41no—
251C2×Dic53—41no—
252C2×Dic53—41no—
253C2×Dic53—41no—
254C2×Dic53—41no—
255C2×Dic53—42no—
256C2×Dic53—41no—
257Dic106—24no—
258Dic106—42no—
259#88322—24no—
260#88322—42no—
261C4×#455—531nomaximal, Sylow
262C22×C53—21no—
263C22×C53—21no—
264C22×C53—24no—
265C22×C53—24no—
266C22×C53—44no—
267C22×C53—42no—
268C22×C53—42no—
269C22×C53—42no—
270C22×C53—42no—
271C22×C53—82no—
272C22×C53—88no—
273C22×C53C2×C4×D411yescharacteristic
274C22×C53C2×#12011yescharacteristic
275C22×C53#52511yescharacteristic
276C4×C53—21no—
277C4×C53—22no—
278C4×C53—24no—
279C4×C53—42no—
280C4×C53—42no—
281C4×C53—42no—
282C4×C53#45512yescomplemented
283C4×C53C2×C4×D412yes—
284C4×C53#52512yes—
285Dic53—42no—
286Dic53—42no—
287Dic53—42no—
288Dic53—81no—
289C4×H(5,𝔽2)D5311yescharacteristic
290C2×#455—531no—
291C4×#137—532no—
292C4×#124—531no—
293#1885—531no—
294#1882—532no—
295#1860—532no—
296#1856—531no—
297#1792—531no—
298C4×#104—532no—
299C4×#100—531no—
300C2×C53—21no—
301C2×C53—21no—
302C2×C53—44no—
303C2×C53—44no—
304C2×C53C4×#12011yescharacteristic
305C2×C53C2×#45511yescharacteristic
306C2×C53#301811yescharacteristic
307C2×H(5,𝔽2)C2×D5311yescharacteristic
308C2×#137—532no—
309C2×#124—531no—
310#520—22no—
311#520C2×D5312yes—
312#518—22no—
313#518—22no—
314#518C2×D5311yescharacteristic
315C4×#41—22no—
316C4×#41C2×D5312yes—
317C2×C4×Q8—531no—
318C2×C4×D4—22no—
319C2×C4×D4—22no—
320C2×C4×D4C2×D5311yescharacteristic
321#455—538no—
322#441—1062no—
323#440—1062no—
324C4×Q16—1062no—
325C4×SD16—1062no—
326#428—532no—
327C2×#104—532no—
328C2×#104—532no—
329#419—532no—
330#417—532no—
331C2×#100—531no—
332C2×#100—531no—
333C4×M16—532no—
334#391—531no—
335#390—531no—
336#388—1062no—
337#382—531no—
338C4×C22⋊C4—1062no—
339C43—531no—
340#343—532no—
341#339—532no—
342#339—532no—
343C53C4×#45511yescharacteristic, complemented, Sylow
344H(5,𝔽2)—24no—
345H(5,𝔽2)—22no—
346H(5,𝔽2)C4×D5314yes—
347C2×#41—22no—
348C2×#41C22×D5312yes—
349C22×Q8—531no—
350C22×D4—22no—
351C22×D4—22no—
352C22×D4C22×D5311yescharacteristic
353C23×C4—22no—
354C23×C4C22⋊D5312yes—
355#137—538no—
356#137—1064no—
357C2×Q16—1062no—
358C2×SD16—1062no—
359C2×M16—532no—
360C2×M16—532no—
361#124—534no—
362#124—1062no—
363C4×Q8—22no—
364C4×Q8—1062no—
365C4×Q8—1062no—
366C4×Q8C22⋊D5312yes—
367C4×Q8C22⋊D5312yes—
368C4×D4—24no—
369C4×D4—24no—
370C4×D4—24no—
371C4×D4—24no—
372C4×D4—22no—
373C4×D4—22no—
374C4×D4—24no—
375C4×D4—22no—
376C4×D4—22no—
377C4×D4—22no—
378C4×D4—22no—
379C4×D4—22no—
380C4×D4C22⋊D5312yes—
381C4×D4C22⋊D5312yes—
382#117—24no—
383#117—22no—
384#117—22no—
385#117C22×D5312yes—
386C2×C4⋊C4—22no—
387C2×C4⋊C4—22no—
388C2×C4⋊C4—531no—
389C2×C4⋊C4—1061no—
390C2×C4⋊C4C22×D5311yescharacteristic
391C2×C22⋊C4—22no—
392C2×C22⋊C4—24no—
393C2×C22⋊C4—22no—
394C2×C22⋊C4—1062no—
395C2×C22⋊C4—1062no—
396C2×C22⋊C4C22⋊D5312yes—
397C2×C42—22no—
398C2×C42—22no—
399C2×C42—531no—
400C2×C42—531no—
401C2×C42—1061no—
402C2×C42—1061no—
403C2×C42—1061no—
404C2×C42C22×D5311yescharacteristic
405#107—1062no—
406#106—1062no—
407#104—538no—
408#104—1064no—
409#103—1062no—
410#103—1062no—
411#103—1062no—
412#102—1062no—
413#100—534no—
414#100—1062no—
415#97—1062no—
416C4×C8—1062no—
417#95—1062no—
418#95—1062no—
419C24—22no—
420C24C2×C22⋊D5312yes—
421#41—28no—
422#41—28no—
423#41—22no—
424#41—22no—
425#41C2×C4×D5314yes—
426#41#1171994514yes—
427C2×Q8—532no—
428C2×Q8—1061no—
429C2×Q8—1062no—
430C2×Q8—1062no—
431C2×Q8C2×C22⋊D5312yes—
432C2×D4—24no—
433C2×D4—28no—
434C2×D4—28no—
435C2×D4—28no—
436C2×D4—22no—
437C2×D4—28no—
438C2×D4—22no—
439C2×D4—22no—
440C2×D4C2×C4×D5312yes—
441C2×D4#1171994512yes—
442C2×D4C2×C22⋊D5312yes—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—24no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—44no—
453C22×C4—84no—
454C22×C4—1061no—
455C22×C4—1061no—
456C22×C4—1061no—
457C22×C4—1061no—
458C22×C4C2×C22⋊D5311yescharacteristic
459C22×C4C2×C22⋊D5312yes—
460C22×C4C2×C22⋊D5311yescharacteristic
461C22×C4C23×D5311yescharacteristic
462Q16—1068no—
463SD16—1068no—
464M16—534no—
465M16—1062no—
466M16—1062no—
467M16—1062no—
468C2×C8—1062no—
469C2×C8—1062no—
470C4⋊C4—24no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—1062no—
475C4⋊C4—2122no—
476C4⋊C4C2×C22⋊D5312yes—
477C4⋊C4C2×C22⋊D5312yes—
478C22⋊C4—24no—
479C22⋊C4—24no—
480C22⋊C4—22no—
481C22⋊C4—22no—
482C22⋊C4—22no—
483C22⋊C4—24no—
484C22⋊C4—1068no—
485C22⋊C4—2124no—
486C42—24no—
487C42—22no—
488C42—22no—
489C42—22no—
490C42—532no—
491C42—1061no—
492C42—1062no—
493C42—2122no—
494C42—2122no—
495C42—2121no—
496C42—2121no—
497C42C2×C22⋊D5312yes—
498C42C2×C22⋊D5312yes—
499C23—28no—
500C23—22no—
501C23—22no—
502C23—22no—
503C23—42no—
504C23—42no—
505C23—82no—
506C23—82no—
507C23C4×C22⋊D5314yes—
508C23C22×C22⋊D5311yescharacteristic
509Q8—22no—
510Q8—1064no—
511Q8—2122no—
512Q8C4×C22⋊D5314yes—
513D4—216no—
514D4—24no—
515D4—24no—
516D4—24no—
517D4—24no—
518D4—22no—
519D4C4×C22⋊D5314yes—
520C2×C4—28no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—41no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—42no—
532C2×C4—82no—
533C2×C4—82no—
534C2×C4—82no—
535C2×C4—84no—
536C2×C4—82no—
537C2×C4—1062no—
538C2×C4—1062no—
539C2×C4—1062no—
540C2×C4—2122no—
541C2×C4—2121no—
542C2×C4—2121no—
543C2×C4—2121no—
544C2×C4—2121no—
545C2×C4—2121no—
546C2×C4—2121no—
547C2×C4—2121no—
548C2×C4#1313318612yes—
549C2×C4#1313318611yescharacteristic
550C2×C4#1313318611yescharacteristic
551C2×C4C22×C4×D5311yescharacteristic
552C2×C4C22×C22⋊D5312yes—
553C2×C4C2×#1171994512yes—
554C2×C4C2×#1171994511yescharacteristic
555C8—1064no—
556C8—2122no—
557C22—21no—
558C22—21no—
559C22—24no—
560C22—24no—
561C22—44no—
562C22—42no—
563C22—42no—
564C22—42no—
565C22—42no—
566C22—82no—
567C22—88no—
568C22C2×C4×C22⋊D5311yescharacteristic
569C22C2×#1313318611yescharacteristic
570C22#1681113411yescharacteristic
571C4—21no—
572C4—22no—
573C4—24no—
574C4—42no—
575C4—42no—
576C4—42no—
577C4—4241no—
578C4—2122no—
579C4—2122no—
580C4—2122no—
581C4#1681080112yesdirect product factor
582C4C2×C4×C22⋊D5312yes—
583C4#1681113412yes—
584C2—21no—
585C2—21no—
586C2—44no—
587C2—44no—
588C2C4×#1313318611yescharacteristic
589C2C2×#1681080111yescharacteristic
590C2#2028406411yescharacteristic
subgroupG/H
centerC2×C4#13133186
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC4×H(5,𝔽2)×C53C2
socleC22×C53C2×#120
prime radicalC2×H(5,𝔽2)×C53C22
cosocleC22×C4×C53C24
hypercenterC4×H(5,𝔽2)D53
hypocenterC53C4×#455
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C4×H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
2. C4×#13133247C2semidirect
3. C4×#13133204C2semidirect
4. #20275103C2semidirect
5. #20275084C2semidirect
6. #20274948C2semidirect
7. #20274925C2semidirect
9. C4×#13133144C2semidirect
10. C4×#13133134C2semidirect
14. C2×C4×Dic106C22semidirect
15. #16810801C4direct (N × H)
20. #16810666C22semidirect
23. #16810631C22semidirect
27. C4×#11719917C22semidirect
28. #16810528C22semidirect
31. #16810488C22semidirect
33. C42×Dic53C22semidirect
34. #16810392C22semidirect
35. #16810382C22semidirect
36. #16810382C22semidirect
50. #13133247C2×C4semidirect
56. #13133204C2×C4semidirect
71. #13133144C2×C4semidirect
77. #13133134C2×C4semidirect
165. C2×Dic106C22×C4semidirect
196. C4×Dic53C22×C4semidirect
200. #11719917C22×C4semidirect
282. C4×C53#455semidirect
343. C53C4×#455semidirect
581. C4#16810801direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168108012
C4×#131332472
C4×#131332042
#202751032
#202750842
#202749482
#202749252
#202746692
C4×#131331442
C4×#131331342
C4×H(5,𝔽2)×C532
C4×#45553