Finite Group Databasebeta

← #27367905

Subgroups — #27367905

Order 13568.

738 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285290C211yesmaximal, characteristic, complemented
2#20284806C211yesmaximal, characteristic, complemented
3C2×D4×#88322C211yesmaximal, characteristic, complemented
4#20274126C212yesmaximal, complemented
5#20274122C212yesmaximal, complemented
6#20274112C212yesmaximal, complemented
7#20274068C212yesmaximal, complemented
8#20273476C211yesmaximal, characteristic, complemented
9#20273411C211yesmaximal, characteristic, complemented
10#20273410C211yesmaximal, characteristic, complemented
11C53×#3109C211yesmaximal, characteristic, complemented
12C23×#88322—21no—
13#16811173C2211yescharacteristic
14#16811026C2211yescharacteristic, complemented
15#16810999C2211yescharacteristic, complemented
16#16810971C2211yescharacteristic, complemented
17#16810969C2211yescharacteristic, complemented
18#16810941—21no—
19#16810913C2212yes—
20#16810857—21no—
21#16810857—21no—
22#16810818C2212yes—
23#16810757—21no—
24#16810733—22no—
25C4×#11719922—21no—
26C4×#11719922—21no—
27D4×#88322—21no—
28D4×#88322—21no—
29D4×#88322C2212yes—
30D4×#88322C2212yes—
31#16810680C2211yescharacteristic
32C2×C4⋊#88322C2211yescharacteristic
33C2×C22⋊#88322—21no—
34C2×C4×#88322C2211yescharacteristic
35#16810375C2212yescomplemented
36#16810373C2212yescomplemented
37#16810372C2212yescomplemented
38#16810371C2212yescomplemented
39#16810367C2212yescomplemented
40#16810366C2212yescomplemented
41C53×#532C2211yescharacteristic
42C2×C4×D4×C53C2211yescharacteristic
43C53×#481—21no—
44C53×#476C2212yes—
45C53×#459C2212yes—
46C53×#447—21no—
47C53×#442—22no—
48C53×#432C2211yescharacteristic
49C2×#11719945—21no—
50C2×#11719922—21no—
51C2×#11719922—21no—
52C2×#11719922—22no—
53C22×#88322—22no—
54C22×#88322—22no—
55C22×#88322—21no—
56C22×#88322—44no—
57C22×#88322D411yescharacteristic
58C22×#88322D411yescharacteristic
59#13133217C2311yescharacteristic
60#13133214C2311yescharacteristic
61#13133205—21no—
62#13133191—21no—
63#13133191—21no—
64C4×D212—21no—
65C4×D212—21no—
66#13133150—21no—
67#13133150—21no—
68#13133148—22no—
69C4⋊#88322—21no—
70C4⋊#88322C2312yes—
71C4⋊#88322C2312yes—
72#13133138—21no—
73#13133138—21no—
74#13133138—22no—
75#13133137—22no—
76#13133137—21no—
77#13133137—21no—
78C22⋊#88322—22no—
79C22⋊#88322—22no—
80C22⋊#88322—21no—
81#13133128—21no—
82C4×#88322D411yescharacteristic
83C4×#88322D411yescharacteristic
84C4×#88322C2311yescharacteristic
85C4×#88322C2311yescharacteristic
86C2×#41×C53—21no—
87C22×D4×C53D411yescharacteristic
88C23×C4×C53—21no—
89C2×D8×C53—22no—
90C4⋊Q8×C53C2311yescharacteristic
91#127×C53C2311yescharacteristic
92#124×C53—21no—
93#121×C53—21no—
94#121×C53—21no—
95C4×D4×C53—21no—
96C4×D4×C53—21no—
97C4×D4×C53—21no—
98C4×D4×C53—21no—
99C4×D4×C53D412yes—
100C4×D4×C53C2312yes—
101C2×C4⋊C4×C53D411yescharacteristic
102C2×C22⋊C4×C53—21no—
103C2×C42×C53C2311yescharacteristic
104#106×C53—22no—
105C4⋊C8×C53C2312yes—
106#102×C53—22no—
107#102×C53—22no—
108C22⋊C8×C53—21no—
109#97×C53—21no—
110C24×C53—21no—
111#41×C53—82no—
112C2×Q8×C53—21no—
113C2×D4×C53—24no—
114C2×D4×C53—21no—
115C2×D4×C53—21no—
116C2×D4×C53—21no—
117C2×D4×C53—21no—
118C2×D4×C53—41no—
119C2×D4×C53C2×D412yes—
120C22×C4×C53—21no—
121C22×C4×C53—22no—
122C22×C4×C53—22no—
123C22×C4×C53—42no—
124C22×C4×C53—42no—
125C22×C4×C53—41no—
126C22×C4×C53C2×D411yescharacteristic
127C22×C4×C53C2×D411yescharacteristic
128C22×C4×C53#4111yescharacteristic
129C2×C22⋊D53—41no—
130#11719945—82no—
131C2×D212—21no—
132C2×D212—21no—
133C2×D212—21no—
134C2×C4×D53—41no—
135C2×Dic106—21no—
136D8×C53—44no—
137C2×C8×C53—22no—
138C4⋊C4×C53—21no—
139C4⋊C4×C53—21no—
140C4⋊C4×C53—21no—
141C4⋊C4×C53C2×D412yes—
142C22⋊C4×C53—22no—
143C22⋊C4×C53—22no—
144C22⋊C4×C53—41no—
145C22⋊C4×C53—41no—
146C42×C53C2×D411yescharacteristic
147C42×C53C2×D411yescharacteristic
148C42×C53#4111yescharacteristic
149C42×C53C2411yescharacteristic
150#11719922—42no—
151#11719922—44no—
152#11719922—42no—
153#11719921—41no—
154#11719921—41no—
155C4⋊Dic53—21no—
156C4⋊Dic53—21no—
157C2×#88322—22no—
158C2×#88322—22no—
159C2×#88322—22no—
160C2×#88322—41no—
161C2×#88322—44no—
162C2×#88322—42no—
163C2×#88322—44no—
164C2×#88322C2×D411yescharacteristic
165C2×#88322C2×D411yescharacteristic
166C2×#88322C2×D411yescharacteristic
167C2×#88322C2×D411yescharacteristic
168C23×C53—24no—
169C23×C53—22no—
170C23×C53—42no—
171C23×C53—42no—
172C23×C53—41no—
173C23×C53#12111yescharacteristic
174C22×D53—41no—
175Q8×C53—81no—
176D4×C53—24no—
177D4×C53—22no—
178D4×C53—42no—
179D4×C53—41no—
180D4×C53—41no—
181D4×C53—81no—
182D4×C53—82no—
183D4×C53#13514yes—
184C2×C4×C53—22no—
185C2×C4×C53—22no—
186C2×C4×C53—21no—
187C2×C4×C53—21no—
188C2×C4×C53—22no—
189C2×C4×C53—44no—
190C2×C4×C53—42no—
191C2×C4×C53—42no—
192C2×C4×C53—42no—
193C2×C4×C53—41no—
194C2×C4×C53—82no—
195C2×C4×C53#12111yescharacteristic
196C2×C4×C53#12111yescharacteristic
197C2×C4×C53#12111yescharacteristic
198C2×C4×C53C22×D411yescharacteristic
199C2×C4×C53C22×D411yescharacteristic
200C2×C4×C53C2×#4111yescharacteristic
201C22⋊D53—82no—
202C2×Dic53—41no—
203D212—42no—
204D212—41no—
205D212—41no—
206D212—81no—
207C4×D53—82no—
208Dic106—81no—
209C8×C53—42no—
210#88322—21no—
211#88322—21no—
212#88322—21no—
213#88322—42no—
214#88322—42no—
215#88322AGL(1,ℤ/8)12yes—
216#19258—531nomaximal, Sylow
217C22×C53—21no—
218C22×C53—21no—
219C22×C53—24no—
220C22×C53—48no—
221C22×C53—42no—
222C22×C53—42no—
223C22×C53—81no—
224C22×C53—81no—
225C22×C53—81no—
226C22×C53#47212yes—
227C22×C53C2×#12111yescharacteristic
228C2×D53—81no—
229C2×D53—81no—
230C2×D53—81no—
231C4×C53—21no—
232C4×C53—21no—
233C4×C53—21no—
234C4×C53—21no—
235C4×C53—42no—
236C4×C53—81no—
237C4×C53C2×#13512yes—
238C4×C53C2×AGL(1,ℤ/8)11yescharacteristic
239C4×C53#57511yescharacteristic
240Dic53—81no—
241#3169—531no—
242#3109—531no—
243#3109D5311yescharacteristic, complemented
244C2×C8×D4—531no—
245#1677—532no—
246#1677—532no—
247#1675—532no—
248#1667—532no—
249#1568—531no—
250#1551—531no—
251#1551—531no—
252C2×C53—22no—
253C2×C53—44no—
254C2×C53—161no—
255C2×C53#303611yescharacteristic
256C2×C53#305011yescharacteristic
257C2×C53C2×#47211yescharacteristic
258D53—161no—
259C23×C8—1061no—
260#532—531no—
261#532C2×D5311yescharacteristic
262C2×C4×D4C2×D5311yescharacteristic
263#500—531no—
264#493—531no—
265#486—531no—
266#486—531no—
267#481—21no—
268#481—1061no—
269#476—532no—
270#476C2×D5312yes—
271#466—1061no—
272#466—1061no—
273#459—532no—
274#459C2×D5312yes—
275#447—21no—
276#447—1061no—
277#442—22no—
278#442—1062no—
279C4×D8—1061no—
280C4×D8—1061no—
281C8×D4—532no—
282C8×D4—532no—
283C8×D4—1061no—
284C8×D4—1061no—
285#432—531no—
286#432C2×D5311yescharacteristic
287C2×C4⋊C8—531no—
288C2×C22⋊C8—1061no—
289C2×C4×C8—531no—
290#335—532no—
291#334—532no—
292#333—532no—
293#333—532no—
294#331—532no—
295#331—532no—
296C53#1925811yescharacteristic, complemented, Sylow
297C2×#41—21no—
298C2×#41—1061no—
299C22×D4C22⋊D5311yescharacteristic
300C23×C4—21no—
301C2×D8—22no—
302C2×D8—1061no—
303C2×D8—1061no—
304C2×D8—1062no—
305C22×C8—531no—
306C22×C8—531no—
307C22×C8—1062no—
308C22×C8—1062no—
309C22×C8—1061no—
310C22×C8—2124no—
311C4⋊Q8—531no—
312C4⋊Q8C22×D5311yescharacteristic
313#127—531no—
314#127C22×D5311yescharacteristic
315#124—21no—
316#124—1061no—
317#121—21no—
318#121—21no—
319#121—1061no—
320#121—1061no—
321C4×D4—21no—
322C4×D4—21no—
323C4×D4—21no—
324C4×D4—21no—
325C4×D4—1061no—
326C4×D4—1061no—
327C4×D4C22⋊D5312yes—
328C4×D4C22×D5312yes—
329C2×C4⋊C4C22⋊D5311yescharacteristic
330C2×C22⋊C4—21no—
331C2×C42C22×D5311yescharacteristic
332#107—1061no—
333#107—1061no—
334#106—22no—
335#106—1062no—
336C4⋊C8—532no—
337C4⋊C8—532no—
338C4⋊C8—1061no—
339C4⋊C8C22×D5312yes—
340#102—22no—
341#102—22no—
342#102—1061no—
343#102—1061no—
344#102—1061no—
345#102—1061no—
346#102—1062no—
347#102—1062no—
348C22⋊C8—21no—
349C22⋊C8—1062no—
350C22⋊C8—1062no—
351C22⋊C8—1061no—
352#97—21no—
353#97—1061no—
354C4×C8—531no—
355C4×C8—531no—
356C4×C8—531no—
357C4×C8—531no—
358C24—21no—
359#41—82no—
360#41—4242no—
361C2×Q8—21no—
362C2×Q8—1061no—
363C2×D4—24no—
364C2×D4—21no—
365C2×D4—21no—
366C2×D4—21no—
367C2×D4—21no—
368C2×D4—41no—
369C2×D4—1061no—
370C2×D4—1061no—
371C2×D4—1061no—
372C2×D4—2121no—
373C2×D4C2×C22⋊D5312yes—
374C22×C4—21no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—41no—
380C22×C4—2121no—
381C22×C4D4×D5311yescharacteristic
382C22×C4#1171994711yescharacteristic
383C22×C4C2×C22⋊D5311yescharacteristic
384D8—44no—
385D8—2122no—
386D8—2122no—
387D8—2124no—
388C2×C8—22no—
389C2×C8—531no—
390C2×C8—531no—
391C2×C8—531no—
392C2×C8—531no—
393C2×C8—1062no—
394C2×C8—1062no—
395C2×C8—1062no—
396C2×C8—2124no—
397C2×C8—2121no—
398C2×C8—2124no—
399C2×C8—2122no—
400C4⋊C4—21no—
401C4⋊C4—21no—
402C4⋊C4—21no—
403C4⋊C4—1061no—
404C4⋊C4—1061no—
405C4⋊C4C2×C22⋊D5312yes—
406C22⋊C4—22no—
407C22⋊C4—22no—
408C22⋊C4—41no—
409C22⋊C4—41no—
410C22⋊C4—2121no—
411C22⋊C4—2121no—
412C42D4×D5311yescharacteristic
413C42#1171994711yescharacteristic
414C42C2×C22⋊D5311yescharacteristic
415C42C23×D5311yescharacteristic
416C23—24no—
417C23—22no—
418C23—42no—
419C23—42no—
420C23—41no—
421C23—2121no—
422C23#1313318911yescharacteristic
423Q8—81no—
424Q8—4241no—
425D4—24no—
426D4—22no—
427D4—42no—
428D4—41no—
429D4—41no—
430D4—81no—
431D4—82no—
432D4—4241no—
433D4—4242no—
434D4—2122no—
435D4—2121no—
436D4—2121no—
437D4#1313323714yes—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—22no—
443C2×C4—44no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—41no—
448C2×C4—82no—
449C2×C4—4242no—
450C2×C4—2121no—
451C2×C4#1313318911yescharacteristic
452C2×C4#1313318911yescharacteristic
453C2×C4#1313318911yescharacteristic
454C2×C4C2×D4×D5311yescharacteristic
455C2×C4C22×C22⋊D5311yescharacteristic
456C2×C4C2×#1171994711yescharacteristic
457C8—42no—
458C8—532no—
459C8—1061no—
460C8—1061no—
461C8—1061no—
462C8—2122no—
463C8—2122no—
464C22—21no—
465C22—21no—
466C22—24no—
467C22—48no—
468C22—42no—
469C22—42no—
470C22—81no—
471C22—81no—
472C22—81no—
473C22—4241no—
474C22—4241no—
475C22—4241no—
476C22#1681089112yes—
477C22C2×#1313318911yescharacteristic
478C4—21no—
479C4—21no—
480C4—21no—
481C4—21no—
482C4—42no—
483C4—81no—
484C4—4241no—
485C4C2×#1313323712yes—
486C4AGL(1,ℤ/8)×D5311yescharacteristic
487C4#1681141911yescharacteristic
488C2—22no—
489C2—44no—
490C2—161no—
491C2—8481no—
492C2#2028417611yescharacteristic
493C2#2028429311yescharacteristic
494C2C2×#1681089111yescharacteristic
subgroupG/H
centerC22C2×#13133189
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#3109C2
socleC22×C53C2×#121
hypercenter#3109D53
hypocenterC53#19258
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3109

Complements

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

NH ≅ G/Ntype
1. #20285290C2semidirect
2. #20284806C2semidirect
3. C2×D4×#88322C2semidirect
4. #20274126C2semidirect
5. #20274122C2semidirect
6. #20274112C2semidirect
7. #20274068C2semidirect
8. #20273476C2semidirect
9. #20273411C2semidirect
10. #20273410C2semidirect
11. C53×#3109C2semidirect
14. #16811026C22semidirect
15. #16810999C22semidirect
16. #16810971C22semidirect
17. #16810969C22semidirect
35. #16810375C22semidirect
36. #16810373C22semidirect
37. #16810372C22semidirect
38. #16810371C22semidirect
39. #16810367C22semidirect
40. #16810366C22semidirect
243. #3109D53semidirect
296. C53#19258semidirect

Maximal subgroups

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

subgroupindex
#202852902
#202848062
C2×D4×#883222
#202741262
#202741222
#202741122
#202740682
#202734762
#202734112
#202734102
C53×#31092
#1925853