Finite Group Databasebeta

← #27276023

Subgroups — #27276023

Order 13568.

772 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20275639C211yesmaximal, characteristic, complemented
2#20274735C211yesmaximal, characteristic, complemented
3#20273396C211yesmaximal, characteristic, complemented
4#20273392C211yesmaximal, characteristic, complemented
5#20273210C211yesmaximal, characteristic, complemented
6#20273102C211yesmaximal, characteristic, complemented
7C53×#3511C211yesmaximal, characteristic
8#16810676C2211yescharacteristic
9#16810647C2211yescharacteristic
10C2×C4⋊#88322C2211yescharacteristic
11C2×#13133144—21no—
12C2×#13133141—22no—
13C4×C4⋊Dic53C2211yescharacteristic
14#16810382—21no—
15#16810372—22no—
16#16810369—21no—
17#16810367—21no—
18#16810357—22no—
19Q8⋊#88322—21no—
20D4⋊#88322—21no—
21C53×#556—21no—
22C53×#555C412yescomplemented
23C53×#554C412yescomplemented
24C53×#553—22no—
25C53×#548—22no—
26C53×#547—21no—
27C53×#542—22no—
28C53×#537—21no—
29C53×#533—21no—
30C53×#532—21no—
31C53×#532C2211yescharacteristic
32C2×C4⋊Q8×C53C2211yescharacteristic
33C53×#520—21no—
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#13133141—44no—
46C22⋊#88322—21no—
47C4×#88322—21no—
48#13133124—21no—
49C2×#41×C53—22no—
50C2×#41×C53—21no—
51C2×#41×C53D411yescharacteristic
52C22×Q8×C53D412yes—
53C4⋊Q8×C53—21no—
54C4⋊Q8×C53—21no—
55C4⋊Q8×C53C2×C412yes—
56C4⋊Q8×C53D411yescharacteristic
57C4⋊Q8×C53D412yes—
58#127×C53D411yescharacteristic
59#126×C53—24no—
60#125×C53—22no—
61#124×C53—22no—
62#124×C53—21no—
63#124×C53—24no—
64#123×C53—24no—
65C22⋊Q8×C53—22no—
66C22⋊Q8×C53—22no—
67C22⋊Q8×C53—22no—
68C22⋊Q8×C53—24no—
69#121×C53—22no—
70#121×C53—24no—
71#121×C53—22no—
72#121×C53—22no—
73#121×C53—21no—
74#121×C53D412yes—
75C4×Q8×C53—22no—
76C4×Q8×C53—22no—
77C4×Q8×C53C2×C411yescharacteristic
78C4×Q8×C53D411yescharacteristic
79C4×D4×C53—22no—
80C4×D4×C53—24no—
81C4×D4×C53—21no—
82C4×D4×C53—22no—
83C4×D4×C53—21no—
84C4×D4×C53—21no—
85C4×D4×C53C2×C411yescharacteristic
86C4×D4×C53C2×C412yes—
87C4×D4×C53D411yescharacteristic
88#117×C53—22no—
89#117×C53—22no—
90#117×C53—21no—
91#117×C53D411yescharacteristic
92C2×C4⋊C4×C53—22no—
93C2×C4⋊C4×C53—21no—
94C2×C4⋊C4×C53—21no—
95C2×C42×C53—21no—
96C2×C42×C53C2311yescharacteristic
97#41×C53—21no—
98#41×C53—21no—
99#41×C53—21no—
100#41×C53—21no—
101#41×C53—41no—
102#41×C53—41no—
103#41×C53—41no—
104#41×C53—41no—
105#41×C53—82no—
106#41×C53—82no—
107C2×Q8×C53—21no—
108C2×Q8×C53—22no—
109C2×Q8×C53—42no—
110C2×Q8×C53—42no—
111C2×Q8×C53C22⋊C411yescharacteristic
112C2×Q8×C53C22⋊C414yes—
113C2×D4×C53—21no—
114C2×D4×C53—22no—
115C2×D4×C53—24no—
116C2×D4×C53—21no—
117C2×D4×C53—21no—
118C2×D4×C53C22⋊C411yescharacteristic
119C2×D4×C53C22⋊C412yes—
120C22×C4×C53—24no—
121C22×C4×C53—22no—
122C22×C4×C53—21no—
123C22×C4×C53—22no—
124C22×C4×C53—21no—
125C22×C4×C53—21no—
126C22×C4×C53C2×D411yescharacteristic
127C22×C4×C53C2×D411yescharacteristic
128C22×C4×C53C2×D411yescharacteristic
129C22×Dic53—41no—
130C4⋊C4×C53—24no—
131C4⋊C4×C53—24no—
132C4⋊C4×C53—22no—
133C4⋊C4×C53—22no—
134C4⋊C4×C53—21no—
135C4⋊C4×C53—22no—
136C4⋊C4×C53—22no—
137C4⋊C4×C53—21no—
138C4⋊C4×C53—21no—
139C4⋊C4×C53C22⋊C411yescharacteristic
140C4⋊C4×C53C22⋊C411yescharacteristic
141C4⋊C4×C53C22⋊C412yes—
142C22⋊C4×C53—24no—
143C22⋊C4×C53—24no—
144C22⋊C4×C53—24no—
145C22⋊C4×C53—21no—
146C22⋊C4×C53—21no—
147C22⋊C4×C53—22no—
148C22⋊C4×C53—21no—
149C22⋊C4×C53—22no—
150C42×C53—24no—
151C42×C53—21no—
152C42×C53—21no—
153C42×C53—21no—
154C42×C53C22×C411yescharacteristic
155C42×C53C2×D411yescharacteristic
156C42×C53C2×D411yescharacteristic
157C42×C53C2×D411yescharacteristic
158C4⋊Dic53—22no—
159C4⋊Dic53—41no—
160C4×Dic53—41no—
161C4×Dic53—41no—
162#11719917—41no—
163#11719917—41no—
164C2×#88322—42no—
165C2×#88322—41no—
166C2×#88322—41no—
167C2×#88322—41no—
168C23×C53—21no—
169C23×C53—22no—
170C23×C53#12011yescharacteristic
171Q8×C53—21no—
172Q8×C53—21no—
173Q8×C53—44no—
174Q8×C53—41no—
175Q8×C53—41no—
176Q8×C53—42no—
177Q8×C53—42no—
178D4×C53—21no—
179D4×C53—21no—
180D4×C53—42no—
181D4×C53—41no—
182D4×C53—42no—
183D4×C53—41no—
184D4×C53—42no—
185D4×C53—41no—
186D4×C53—41no—
187D4×C53—41no—
188D4×C53—41no—
189D4×C53—84no—
190C2×C4×C53—21no—
191C2×C4×C53—21no—
192C2×C4×C53—21no—
193C2×C4×C53—24no—
194C2×C4×C53—22no—
195C2×C4×C53—21no—
196C2×C4×C53—21no—
197C2×C4×C53—41no—
198C2×C4×C53—42no—
199C2×C4×C53—42no—
200C2×C4×C53—42no—
201C2×C4×C53—41no—
202C2×C4×C53—41no—
203C2×C4×C53—41no—
204C2×C4×C53—41no—
205C2×C4×C53—82no—
206C2×C4×C53—84no—
207C2×C4×C53C2×C22⋊C411yescharacteristic
208C2×C4×C53C2×C22⋊C411yescharacteristic
209C2×C4×C53C2×C22⋊C411yescharacteristic
210C2×C4×C53#12011yescharacteristic
211C2×C4×C53#12011yescharacteristic
212C2×C4×C53#12011yescharacteristic
213C2×C4×C53#13511yescharacteristic
214C2×C4×C53#13511yescharacteristic
215C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
216C2×C4×C53#13711yescharacteristic
217C2×Dic53—42no—
218C2×Dic53—81no—
219C2×Dic53—81no—
220#88322—81no—
221#88322—81no—
222#88322—81no—
223#8423—531nomaximal, Sylow
224C22×C53—21no—
225C22×C53—21no—
226C22×C53—41no—
227C22×C53—42no—
228C22×C53—41no—
229C22×C53—42no—
230C22×C53—41no—
231C22×C53—42no—
232C22×C53#37911yescharacteristic
233C22×C53#44911yescharacteristic
234C22×C53#45211yescharacteristic
235C4×C53—21no—
236C4×C53—21no—
237C4×C53—21no—
238C4×C53—41no—
239C4×C53—44no—
240C4×C53—41no—
241C4×C53—41no—
242C4×C53—42no—
243C4×C53#41611yescharacteristic
244C4×C53#41711yescharacteristic
245C4×C53#44911yescharacteristic
246C4×C53#45211yescharacteristic
247C4×C53#45311yescharacteristic
248C4×C53#45511yescharacteristic
249Dic53—82no—
250#3511D5311yescharacteristic
251#1981—531no—
252#1806—531no—
253#1548—531no—
254#1547—531no—
255#1502—531no—
256#1477—531no—
257C2×C53—21no—
258C2×C53—21no—
259C2×C53—81no—
260C2×C53—82no—
261C2×C53#178811yescharacteristic
262C2×C53#179211yescharacteristic
263C2×C53#179611yescharacteristic
264#556—21no—
265#555Dic5312yescomplemented
266#554Dic5312yescomplemented
267#553—22no—
268#548—22no—
269#547—21no—
270#542—22no—
271#537—21no—
272#533—21no—
273#532—21no—
274#532C2×D5311yescharacteristic
275C2×C4⋊Q8C2×D5311yescharacteristic
276#520—21no—
277C4×#41C2×D5311yescharacteristic
278#431—531no—
279#423—531no—
280C2×C4⋊C8—531no—
281C2×#104—1061no—
282C2×#103—1062no—
283C4×C4⋊C4—531no—
284#339—1061no—
285#333—1062no—
286#332—1061no—
287#331—1061no—
288#327—1062no—
289Q8⋊C8—1061no—
290D4⋊C8—1061no—
291C53#842311yescharacteristic, complemented, Sylow
292C2×#41—22no—
293C2×#41—21no—
294C2×#41C22⋊D5311yescharacteristic
295C22×Q8C22⋊D5312yes—
296C2×M16—1061no—
297C22×C8—1061no—
298C4⋊Q8—21no—
299C4⋊Q8—21no—
300C4⋊Q8C2×Dic5312yes—
301C4⋊Q8C22⋊D5312yes—
302C4⋊Q8C22⋊D5311yescharacteristic
303#127C22⋊D5311yescharacteristic
304#126—24no—
305#125—22no—
306#124—22no—
307#124—24no—
308#124—21no—
309#123—24no—
310C22⋊Q8—22no—
311C22⋊Q8—22no—
312C22⋊Q8—22no—
313C22⋊Q8—24no—
314#121—22no—
315#121—24no—
316#121—22no—
317#121—22no—
318#121—21no—
319#121C22⋊D5312yes—
320C4×Q8—22no—
321C4×Q8—22no—
322C4×Q8C2×Dic5311yescharacteristic
323C4×Q8C22⋊D5311yescharacteristic
324C4×D4—22no—
325C4×D4—24no—
326C4×D4—22no—
327C4×D4—21no—
328C4×D4—21no—
329C4×D4—21no—
330C4×D4C2×Dic5311yescharacteristic
331C4×D4C2×Dic5312yes—
332C4×D4C22⋊D5311yescharacteristic
333#117—22no—
334#117—22no—
335#117—21no—
336#117C22⋊D5311yescharacteristic
337C2×C4⋊C4—21no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—1061no—
341C2×C42—21no—
342C2×C42—1061no—
343C2×C42C22×D5311yescharacteristic
344C4⋊C8—1061no—
345C4⋊C8—1061no—
346C4⋊C8—1061no—
347C4⋊C8—1061no—
348C4⋊C8—1061no—
349#104—2122no—
350#103—2124no—
351C22⋊C8—1061no—
352C4×C8—1061no—
353#95—1061no—
354#41—21no—
355#41—21no—
356#41—21no—
357#41—21no—
358#41—41no—
359#41—41no—
360#41—41no—
361#41—41no—
362#41—82no—
363#41—82no—
364C2×Q8—21no—
365C2×Q8—22no—
366C2×Q8—42no—
367C2×Q8—42no—
368C2×Q8C22⋊Dic5311yescharacteristic
369C2×Q8C22⋊Dic5314yes—
370C2×D4—21no—
371C2×D4—22no—
372C2×D4—24no—
373C2×D4—21no—
374C2×D4—21no—
375C2×D4C22⋊Dic5311yescharacteristic
376C2×D4C22⋊Dic5312yes—
377C22×C4—24no—
378C22×C4—22no—
379C22×C4—21no—
380C22×C4—22no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—2121no—
384C22×C4C2×C22⋊D5311yescharacteristic
385C22×C4C2×C22⋊D5311yescharacteristic
386C22×C4C2×C22⋊D5311yescharacteristic
387M16—2121no—
388M16—2121no—
389C2×C8—2122no—
390C2×C8—2121no—
391C2×C8—2121no—
392C2×C8—2121no—
393C4⋊C4—24no—
394C4⋊C4—24no—
395C4⋊C4—22no—
396C4⋊C4—22no—
397C4⋊C4—21no—
398C4⋊C4—22no—
399C4⋊C4—22no—
400C4⋊C4—21no—
401C4⋊C4—21no—
402C4⋊C4—1062no—
403C4⋊C4—2121no—
404C4⋊C4C22⋊Dic5311yescharacteristic
405C4⋊C4C22⋊Dic5311yescharacteristic
406C4⋊C4C22⋊Dic5312yes—
407C22⋊C4—24no—
408C22⋊C4—24no—
409C22⋊C4—24no—
410C22⋊C4—21no—
411C22⋊C4—21no—
412C22⋊C4—22no—
413C22⋊C4—21no—
414C22⋊C4—22no—
415C42—24no—
416C42—21no—
417C42—21no—
418C42—21no—
419C42—2121no—
420C42—2121no—
421C42C22×Dic5311yescharacteristic
422C42C2×C22⋊D5311yescharacteristic
423C42C2×C22⋊D5311yescharacteristic
424C42C2×C22⋊D5311yescharacteristic
425C23—21no—
426C23—22no—
427C23#1313318611yescharacteristic
428Q8—21no—
429Q8—21no—
430Q8—44no—
431Q8—41no—
432Q8—41no—
433Q8—42no—
434Q8—42no—
435D4—21no—
436D4—21no—
437D4—42no—
438D4—41no—
439D4—42no—
440D4—41no—
441D4—42no—
442D4—41no—
443D4—41no—
444D4—41no—
445D4—41no—
446D4—84no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—22no—
452C2×C4—21no—
453C2×C4—24no—
454C2×C4—41no—
455C2×C4—42no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—82no—
463C2×C4—84no—
464C2×C4—4241no—
465C2×C4—4241no—
466C2×C4—2122no—
467C2×C4C2×C22⋊Dic5311yescharacteristic
468C2×C4C2×C22⋊Dic5311yescharacteristic
469C2×C4C2×C22⋊Dic5311yescharacteristic
470C2×C4#1313318611yescharacteristic
471C2×C4#1313318611yescharacteristic
472C2×C4#1313318611yescharacteristic
473C2×C4#1313323711yescharacteristic
474C2×C4#1313323711yescharacteristic
475C2×C4#1313324211yescharacteristic
476C2×C4#1313324711yescharacteristic
477C8—4241no—
478C8—4241no—
479C8—4241no—
480C22—21no—
481C22—21no—
482C22—41no—
483C22—42no—
484C22—41no—
485C22—42no—
486C22—41no—
487C22—42no—
488C22#1681047911yescharacteristic
489C22#1681076811yescharacteristic
490C22#1681078511yescharacteristic
491C4—21no—
492C4—21no—
493C4—21no—
494C4—41no—
495C4—41no—
496C4—42no—
497C4—41no—
498C4—44no—
499C4—4242no—
500C4#1681061911yescharacteristic
501C4#1681062411yescharacteristic
502C4#1681076811yescharacteristic
503C4#1681078511yescharacteristic
504C4#1681079111yescharacteristic
505C4#1681080111yescharacteristic
506C2—21no—
507C2—21no—
508C2—81no—
509C2—82no—
510C2#2027465511yescharacteristic
511C2#2027466911yescharacteristic
512C2#2027469011yescharacteristic
subgroupG/H
centerC22#16810479
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
Fitting, prime radicalC53×#3511C2
socleC22×C53#379
cosocleC2×C42×C53C23
hypercenter#3511D53
hypocenterC53#8423
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3511

Complements

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

NH ≅ G/Ntype
1. #20275639C2semidirect
2. #20274735C2semidirect
3. #20273396C2semidirect
4. #20273392C2semidirect
5. #20273210C2semidirect
6. #20273102C2semidirect
22. C53×#555C4semidirect
23. C53×#554C4semidirect
265. #555Dic53semidirect
266. #554Dic53semidirect
291. C53#8423semidirect

Maximal subgroups

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

subgroupindex
#202756392
#202747352
#202733962
#202733922
#202732102
#202731022
C53×#35112
#842353