Finite Group Databasebeta

← #27276002

Subgroups — #27276002

Order 13568.

1094 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20275643C211yesmaximal, characteristic, complemented
2#20274735C211yesmaximal, characteristic, complemented
3#20273392C211yesmaximal, characteristic, complemented
4#20273383C211yesmaximal, characteristic, complemented
5#20273210C211yesmaximal, characteristic, complemented
6#20273102C211yesmaximal, characteristic, complemented
7C53×#3498C211yesmaximal, characteristic
8#16810676C2211yescharacteristic, complemented
9#16810647C2211yescharacteristic, complemented
10C2×C4⋊#88322C2211yescharacteristic, complemented
11C2×#13133144—21no—
12C2×#13133138—22no—
13C4×C4⋊Dic53C2211yescharacteristic, complemented
14#16810382—21no—
15#16810369—21no—
16#16810367—22no—
17#16810367—21no—
18#16810357—22no—
19Q8⋊#88322—21no—
20D4⋊#88322—21no—
21C2×H(5,𝔽2)×C53—21no—
22C53×#550—21no—
23C53×#550C412yescomplemented
24D42×C53—22no—
25D42×C53—22no—
26D42×C53—21no—
27D42×C53C412yescomplemented
28C53×#535—22no—
29C53×#535—21no—
30C53×#532—21no—
31C53×#532C2211yescharacteristic
32C2×#127×C53—21no—
33C2×#127×C53C2211yescharacteristic
34C4×#41×C53C2211yescharacteristic
35C2×#11719917—21no—
36C22×#88322—21no—
37C2×C4⋊Dic53—21no—
38C2×C4×Dic53—21no—
39C4⋊#88322—21no—
40C4⋊#88322—21no—
41C4⋊#88322—21no—
42C4⋊#88322—21no—
43C4⋊#88322—21no—
44#13133144—42no—
45#13133138—44no—
46C22⋊#88322—21no—
47C4×#88322—21no—
48#13133124—21no—
49H(5,𝔽2)×C53—42no—
50H(5,𝔽2)×C53—42no—
51H(5,𝔽2)×C53—42no—
52H(5,𝔽2)×C53—42no—
53C2×#41×C53—22no—
54C2×#41×C53—22no—
55C2×#41×C53—21no—
56C2×#41×C53D411yescharacteristic
57C22×D4×C53—22no—
58C22×D4×C53—24no—
59C22×D4×C53—22no—
60C22×D4×C53—22no—
61C22×D4×C53—22no—
62C22×D4×C53—21no—
63C22×D4×C53—21no—
64C22×D4×C53D412yes—
65C4⋊Q8×C53D411yescharacteristic
66#127×C53—22no—
67#127×C53—22no—
68#127×C53—21no—
69#127×C53—21no—
70#127×C53—22no—
71#127×C53C2×C412yes—
72#127×C53D411yescharacteristic
73#127×C53D412yes—
74#124×C53—22no—
75#124×C53—21no—
76#121×C53—22no—
77#121×C53—24no—
78#121×C53—22no—
79#121×C53—22no—
80#121×C53—22no—
81#121×C53—24no—
82#121×C53—21no—
83#121×C53D412yes—
84#120×C53—24no—
85#120×C53—24no—
86#120×C53—24no—
87C4×Q8×C53C2×C411yescharacteristic
88C4×Q8×C53D411yescharacteristic
89C4×D4×C53—22no—
90C4×D4×C53—22no—
91C4×D4×C53—21no—
92C4×D4×C53—21no—
93C4×D4×C53—21no—
94C4×D4×C53C2×C411yescharacteristic
95C4×D4×C53C2×C412yes—
96C4×D4×C53D411yescharacteristic
97#117×C53—21no—
98#117×C53D411yescharacteristic
99C2×C42×C53—21no—
100C2×C42×C53C2311yescharacteristic
101C24×C53—24no—
102C24×C53—22no—
103#41×C53—21no—
104#41×C53—21no—
105#41×C53—44no—
106#41×C53—44no—
107#41×C53—44no—
108#41×C53—44no—
109#41×C53—41no—
110#41×C53—41no—
111#41×C53—41no—
112#41×C53—41no—
113#41×C53D812yes—
114#41×C53SD1612yes—
115C2×Q8×C53—21no—
116C2×Q8×C53C22⋊C411yescharacteristic
117C2×D4×C53—21no—
118C2×D4×C53—24no—
119C2×D4×C53—24no—
120C2×D4×C53—22no—
121C2×D4×C53—24no—
122C2×D4×C53—24no—
123C2×D4×C53—22no—
124C2×D4×C53—22no—
125C2×D4×C53—22no—
126C2×D4×C53—21no—
127C2×D4×C53—22no—
128C2×D4×C53—21no—
129C2×D4×C53—22no—
130C2×D4×C53—44no—
131C2×D4×C53—44no—
132C2×D4×C53—42no—
133C2×D4×C53—42no—
134C2×D4×C53—48no—
135C2×D4×C53—48no—
136C2×D4×C53—44no—
137C2×D4×C53—44no—
138C2×D4×C53—41no—
139C2×D4×C53—41no—
140C2×D4×C53—41no—
141C2×D4×C53—41no—
142C2×D4×C53—82no—
143C2×D4×C53—84no—
144C2×D4×C53—82no—
145C2×D4×C53—82no—
146C2×D4×C53C22⋊C411yescharacteristic
147C2×D4×C53C22⋊C412yes—
148C2×D4×C53C22⋊C414yes—
149C22×C4×C53—21no—
150C22×C4×C53—24no—
151C22×C4×C53—21no—
152C22×C4×C53—22no—
153C22×C4×C53—21no—
154C22×C4×C53C2×D411yescharacteristic
155C22×C4×C53C2×D411yescharacteristic
156C22×C4×C53C2×D411yescharacteristic
157C22×Dic53—41no—
158C4⋊C4×C53—21no—
159C4⋊C4×C53—21no—
160C4⋊C4×C53—21no—
161C4⋊C4×C53C22⋊C411yescharacteristic
162C4⋊C4×C53C22⋊C411yescharacteristic
163C4⋊C4×C53C22⋊C412yes—
164C22⋊C4×C53—24no—
165C22⋊C4×C53—24no—
166C22⋊C4×C53—21no—
167C22⋊C4×C53—21no—
168C22⋊C4×C53—22no—
169C22⋊C4×C53—21no—
170C22⋊C4×C53—22no—
171C42×C53—21no—
172C42×C53—21no—
173C42×C53—21no—
174C42×C53C22×C411yescharacteristic
175C42×C53C2×D411yescharacteristic
176C42×C53C2×D411yescharacteristic
177C42×C53C2×D411yescharacteristic
178C4⋊Dic53—22no—
179C4⋊Dic53—41no—
180C4×Dic53—41no—
181C4×Dic53—41no—
182#11719917—41no—
183#11719917—41no—
184C2×#88322—42no—
185C2×#88322—41no—
186C2×#88322—41no—
187C2×#88322—41no—
188C23×C53—21no—
189C23×C53—24no—
190C23×C53—22no—
191C23×C53—48no—
192C23×C53—42no—
193C23×C53—42no—
194C23×C53—82no—
195C23×C53—82no—
196C23×C53—84no—
197C23×C53—84no—
198C23×C53#12011yescharacteristic
199Q8×C53—21no—
200Q8×C53—41no—
201Q8×C53—41no—
202Q8×C53#10212yescomplemented
203D4×C53—22no—
204D4×C53—22no—
205D4×C53—21no—
206D4×C53—44no—
207D4×C53—42no—
208D4×C53—48no—
209D4×C53—44no—
210D4×C53—48no—
211D4×C53—42no—
212D4×C53—42no—
213D4×C53—42no—
214D4×C53—42no—
215D4×C53—41no—
216D4×C53—42no—
217D4×C53—41no—
218D4×C53—42no—
219D4×C53—42no—
220D4×C53—41no—
221D4×C53—42no—
222D4×C53—41no—
223D4×C53—82no—
224D4×C53—84no—
225D4×C53—84no—
226D4×C53#10212yescomplemented
227C2×C4×C53—21no—
228C2×C4×C53—22no—
229C2×C4×C53—22no—
230C2×C4×C53—21no—
231C2×C4×C53—21no—
232C2×C4×C53—22no—
233C2×C4×C53—21no—
234C2×C4×C53—21no—
235C2×C4×C53—41no—
236C2×C4×C53—48no—
237C2×C4×C53—42no—
238C2×C4×C53—42no—
239C2×C4×C53—42no—
240C2×C4×C53—41no—
241C2×C4×C53—41no—
242C2×C4×C53C2×C22⋊C411yescharacteristic
243C2×C4×C53C2×C22⋊C411yescharacteristic
244C2×C4×C53C2×C22⋊C411yescharacteristic
245C2×C4×C53#12011yescharacteristic
246C2×C4×C53#12011yescharacteristic
247C2×C4×C53#12011yescharacteristic
248C2×C4×C53C2×D811yescharacteristic
249C2×C4×C53C2×SD1611yescharacteristic
250C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
251C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
252C2×Dic53—42no—
253C2×Dic53—81no—
254C2×Dic53—81no—
255#88322—81no—
256#88322—81no—
257#88322—81no—
258#8420—531nomaximal, Sylow
259C22×C53—21no—
260C22×C53—21no—
261C22×C53—22no—
262C22×C53—41no—
263C22×C53—41no—
264C22×C53—44no—
265C22×C53—42no—
266C22×C53—44no—
267C22×C53—42no—
268C22×C53—44no—
269C22×C53—42no—
270C22×C53—162no—
271C22×C53—162no—
272C22×C53—168no—
273C22×C53#37911yescharacteristic
274C22×C53#44711yescharacteristic
275C22×C53#45011yescharacteristic
276C4×C53—21no—
277C4×C53—21no—
278C4×C53—21no—
279C4×C53—22no—
280C4×C53—41no—
281C4×C53—41no—
282C4×C53—42no—
283C4×C53C2×#10211yescharacteristic
284C4×C53#41811yescharacteristic
285C4×C53#44711yescharacteristic
286C4×C53#45011yescharacteristic
287C4×C53#45311yescharacteristic
288C4×C53#45511yescharacteristic
289Dic53—82no—
290#3498D5311yescharacteristic
291#1982—531no—
292#1806—531no—
293#1547—531no—
294#1545—531no—
295#1502—531no—
296#1477—531no—
297C2×C53—21no—
298C2×C53—21no—
299C2×C53—42no—
300C2×C53—84no—
301C2×C53—82no—
302C2×C53#178611yescharacteristic
303C2×C53#179211yescharacteristic
304C2×C53#179511yescharacteristic
305C2×H(5,𝔽2)—21no—
306#550—21no—
307#550Dic5312yescomplemented
308D42—22no—
309D42—22no—
310D42—21no—
311D42Dic5312yescomplemented
312#535—22no—
313#535—21no—
314#532—21no—
315#532C2×D5311yescharacteristic
316C2×#127—21no—
317C2×#127C2×D5311yescharacteristic
318C4×#41C2×D5311yescharacteristic
319#431—531no—
320#423—531no—
321C2×C4⋊C8—531no—
322C2×#104—1061no—
323C2×#102—1062no—
324C4×C4⋊C4—531no—
325#339—1061no—
326#332—1061no—
327#331—1062no—
328#331—1061no—
329#327—1062no—
330Q8⋊C8—1061no—
331D4⋊C8—1061no—
332C53#842011yescharacteristic, complemented, Sylow
333H(5,𝔽2)—42no—
334H(5,𝔽2)—42no—
335H(5,𝔽2)—42no—
336H(5,𝔽2)—42no—
337C2×#41—22no—
338C2×#41—22no—
339C2×#41—21no—
340C2×#41C22⋊D5311yescharacteristic
341C22×D4—22no—
342C22×D4—24no—
343C22×D4—22no—
344C22×D4—22no—
345C22×D4—22no—
346C22×D4—21no—
347C22×D4—21no—
348C22×D4C22⋊D5312yes—
349C2×M16—1061no—
350C22×C8—1061no—
351C4⋊Q8C22⋊D5311yescharacteristic
352#127—22no—
353#127—22no—
354#127—22no—
355#127—21no—
356#127—21no—
357#127C2×Dic5312yes—
358#127C22⋊D5311yescharacteristic
359#127C22⋊D5312yes—
360#124—22no—
361#124—21no—
362#121—22no—
363#121—24no—
364#121—24no—
365#121—21no—
366#121—22no—
367#121—22no—
368#121—22no—
369#121C22⋊D5312yes—
370#120—24no—
371#120—24no—
372#120—24no—
373C4×Q8C2×Dic5311yescharacteristic
374C4×Q8C22⋊D5311yescharacteristic
375C4×D4—22no—
376C4×D4—21no—
377C4×D4—21no—
378C4×D4—22no—
379C4×D4—21no—
380C4×D4C2×Dic5311yescharacteristic
381C4×D4C2×Dic5312yes—
382C4×D4C22⋊D5311yescharacteristic
383#117—21no—
384#117C22⋊D5311yescharacteristic
385C2×C4⋊C4—1061no—
386C2×C42—21no—
387C2×C42—1061no—
388C2×C42C22×D5311yescharacteristic
389C4⋊C8—1061no—
390C4⋊C8—1061no—
391C4⋊C8—1061no—
392C4⋊C8—1061no—
393C4⋊C8—1061no—
394#104—2122no—
395#102—2124no—
396C22⋊C8—1061no—
397C4×C8—1061no—
398#95—1061no—
399C24—24no—
400C24—22no—
401#41—21no—
402#41—21no—
403#41—44no—
404#41—44no—
405#41—44no—
406#41—44no—
407#41—41no—
408#41—41no—
409#41—41no—
410#41—41no—
411#41#1171992212yes—
412#41#1171992312yes—
413C2×Q8—21no—
414C2×Q8C22⋊Dic5311yescharacteristic
415C2×D4—21no—
416C2×D4—24no—
417C2×D4—24no—
418C2×D4—22no—
419C2×D4—24no—
420C2×D4—24no—
421C2×D4—22no—
422C2×D4—22no—
423C2×D4—22no—
424C2×D4—21no—
425C2×D4—22no—
426C2×D4—21no—
427C2×D4—22no—
428C2×D4—44no—
429C2×D4—44no—
430C2×D4—42no—
431C2×D4—42no—
432C2×D4—48no—
433C2×D4—48no—
434C2×D4—44no—
435C2×D4—44no—
436C2×D4—41no—
437C2×D4—41no—
438C2×D4—41no—
439C2×D4—41no—
440C2×D4—82no—
441C2×D4—84no—
442C2×D4—82no—
443C2×D4—82no—
444C2×D4C22⋊Dic5311yescharacteristic
445C2×D4C22⋊Dic5312yes—
446C2×D4C22⋊Dic5314yes—
447C22×C4—21no—
448C22×C4—24no—
449C22×C4—21no—
450C22×C4—22no—
451C22×C4—21no—
452C22×C4—2121no—
453C22×C4C2×C22⋊D5311yescharacteristic
454C22×C4C2×C22⋊D5311yescharacteristic
455C22×C4C2×C22⋊D5311yescharacteristic
456M16—2121no—
457M16—2121no—
458C2×C8—2122no—
459C2×C8—2121no—
460C2×C8—2121no—
461C2×C8—2121no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—1062no—
466C4⋊C4—2121no—
467C4⋊C4C22⋊Dic5311yescharacteristic
468C4⋊C4C22⋊Dic5311yescharacteristic
469C4⋊C4C22⋊Dic5312yes—
470C22⋊C4—24no—
471C22⋊C4—24no—
472C22⋊C4—21no—
473C22⋊C4—21no—
474C22⋊C4—22no—
475C22⋊C4—21no—
476C22⋊C4—22no—
477C42—21no—
478C42—21no—
479C42—21no—
480C42—2121no—
481C42—2121no—
482C42C22×Dic5311yescharacteristic
483C42C2×C22⋊D5311yescharacteristic
484C42C2×C22⋊D5311yescharacteristic
485C42C2×C22⋊D5311yescharacteristic
486C23—21no—
487C23—24no—
488C23—22no—
489C23—48no—
490C23—42no—
491C23—42no—
492C23—82no—
493C23—82no—
494C23—84no—
495C23—84no—
496C23#1313318611yescharacteristic
497Q8—21no—
498Q8—41no—
499Q8—41no—
500Q8#1313313812yescomplemented
501D4—22no—
502D4—22no—
503D4—21no—
504D4—44no—
505D4—42no—
506D4—48no—
507D4—44no—
508D4—48no—
509D4—42no—
510D4—42no—
511D4—41no—
512D4—42no—
513D4—41no—
514D4—42no—
515D4—42no—
516D4—41no—
517D4—42no—
518D4—41no—
519D4—42no—
520D4—42no—
521D4—82no—
522D4—84no—
523D4—84no—
524D4#1313313812yescomplemented
525C2×C4—21no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—22no—
533C2×C4—41no—
534C2×C4—48no—
535C2×C4—42no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—4241no—
541C2×C4—4241no—
542C2×C4—2122no—
543C2×C4C2×C22⋊Dic5311yescharacteristic
544C2×C4C2×C22⋊Dic5311yescharacteristic
545C2×C4C2×C22⋊Dic5311yescharacteristic
546C2×C4#1313318611yescharacteristic
547C2×C4#1313318611yescharacteristic
548C2×C4#1313318611yescharacteristic
549C2×C4C2×#1171992211yescharacteristic
550C2×C4C2×#1171992311yescharacteristic
551C2×C4#1313324311yescharacteristic
552C2×C4#1313324311yescharacteristic
553C8—4241no—
554C8—4241no—
555C8—4241no—
556C22—21no—
557C22—21no—
558C22—22no—
559C22—41no—
560C22—41no—
561C22—44no—
562C22—42no—
563C22—44no—
564C22—42no—
565C22—44no—
566C22—42no—
567C22—162no—
568C22—162no—
569C22—168no—
570C22#1681047911yescharacteristic
571C22#1681075711yescharacteristic
572C22#1681077411yescharacteristic
573C4—22no—
574C4—21no—
575C4—21no—
576C4—21no—
577C4—42no—
578C4—41no—
579C4—41no—
580C4—4242no—
581C4C2×#1313313811yescharacteristic
582C4#1681062811yescharacteristic
583C4#1681075711yescharacteristic
584C4#1681077411yescharacteristic
585C4#1681079111yescharacteristic
586C4#1681080111yescharacteristic
587C2—21no—
588C2—21no—
589C2—42no—
590C2—84no—
591C2—82no—
592C2#2027464611yescharacteristic
593C2#2027466911yescharacteristic
594C2#2027468511yescharacteristic
subgroupG/H
centerC22#16810479
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
Fitting, prime radicalC53×#3498C2
socleC22×C53#379
cosocleC2×C42×C53C23
hypercenter#3498D53
hypocenterC53#8420
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3498

Complements

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

NH ≅ G/Ntype
1. #20275643C2semidirect
2. #20274735C2semidirect
3. #20273392C2semidirect
4. #20273383C2semidirect
5. #20273210C2semidirect
6. #20273102C2semidirect
8. #16810676C22semidirect
9. #16810647C22semidirect
10. C2×C4⋊#88322C22semidirect
13. C4×C4⋊Dic53C22semidirect
23. C53×#550C4semidirect
27. D42×C53C4semidirect
202. Q8×C53#102semidirect
226. D4×C53#102semidirect
307. #550Dic53semidirect
311. D42Dic53semidirect
332. C53#8420semidirect
500. Q8#13133138semidirect
524. D4#13133138semidirect

Maximal subgroups

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

subgroupindex
#202756432
#202747352
#202733922
#202733832
#202732102
#202731022
C53×#34982
#842053