Finite Group Databasebeta

← #27276016

Subgroups — #27276016

Order 13568.

756 conjugacy classes of subgroups.

The subgroup lattice has 506 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20275664C211yesmaximal, characteristic, complemented
2#20274735C211yesmaximal, characteristic, complemented
3#20273400C211yesmaximal, 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—
12C4×C4⋊Dic53C2211yescharacteristic
13#16810385—22no—
14#16810382—21no—
15#16810369—21no—
16#16810369—22no—
17#16810367—21no—
18#16810357—22no—
19Q8⋊#88322—21no—
20D4⋊#88322—21no—
21C53×#556C412yescomplemented
22C53×#555—21no—
23C53×#554—21no—
24C53×#553—22no—
25C53×#548—22no—
26C53×#547C412yescomplemented
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—
45C22⋊#88322—21no—
46C4×#88322—21no—
47#13133124—21no—
48C2×#41×C53—22no—
49C2×#41×C53—21no—
50C2×#41×C53D411yescharacteristic
51C22×Q8×C53—21no—
52C4⋊Q8×C53—21no—
53C4⋊Q8×C53—21no—
54C4⋊Q8×C53C2×C412yes—
55C4⋊Q8×C53D412yes—
56C4⋊Q8×C53D411yescharacteristic
57#127×C53D411yescharacteristic
58#126×C53—24no—
59#125×C53—22no—
60#124×C53—24no—
61#124×C53—21no—
62#124×C53—22no—
63#123×C53—24no—
64C22⋊Q8×C53—22no—
65C22⋊Q8×C53—24no—
66C22⋊Q8×C53—22no—
67C22⋊Q8×C53—22no—
68#121×C53—22no—
69#121×C53—22no—
70#121×C53—24no—
71#121×C53—22no—
72#121×C53—21no—
73#121×C53D412yes—
74C4×Q8×C53—22no—
75C4×Q8×C53—22no—
76C4×Q8×C53C2×C411yescharacteristic
77C4×Q8×C53D411yescharacteristic
78C4×D4×C53—22no—
79C4×D4×C53—24no—
80C4×D4×C53—22no—
81C4×D4×C53—21no—
82C4×D4×C53—21no—
83C4×D4×C53—21no—
84C4×D4×C53C2×C411yescharacteristic
85C4×D4×C53C2×C412yes—
86C4×D4×C53D411yescharacteristic
87#117×C53—22no—
88#117×C53—22no—
89#117×C53—21no—
90#117×C53D411yescharacteristic
91C2×C4⋊C4×C53—22no—
92C2×C4⋊C4×C53—21no—
93C2×C4⋊C4×C53D412yes—
94C2×C42×C53—21no—
95C2×C42×C53C2311yescharacteristic
96#41×C53—21no—
97#41×C53—21no—
98#41×C53—21no—
99#41×C53—21no—
100#41×C53—41no—
101#41×C53—41no—
102#41×C53—41no—
103#41×C53—41no—
104#41×C53—84no—
105C2×Q8×C53—22no—
106C2×Q8×C53—21no—
107C2×Q8×C53—22no—
108C2×Q8×C53—82no—
109C2×Q8×C53C22⋊C411yescharacteristic
110C2×D4×C53—21no—
111C2×D4×C53—22no—
112C2×D4×C53—24no—
113C2×D4×C53—21no—
114C2×D4×C53—21no—
115C2×D4×C53C22⋊C411yescharacteristic
116C2×D4×C53C22⋊C412yes—
117C22×C4×C53—24no—
118C22×C4×C53—22no—
119C22×C4×C53—21no—
120C22×C4×C53—22no—
121C22×C4×C53—21no—
122C22×C4×C53—21no—
123C22×C4×C53C2×D411yescharacteristic
124C22×C4×C53C2×D411yescharacteristic
125C22×C4×C53C2×D411yescharacteristic
126C22×Dic53—41no—
127C4⋊C4×C53—24no—
128C4⋊C4×C53—24no—
129C4⋊C4×C53—22no—
130C4⋊C4×C53—21no—
131C4⋊C4×C53—22no—
132C4⋊C4×C53—22no—
133C4⋊C4×C53—21no—
134C4⋊C4×C53—21no—
135C4⋊C4×C53C22⋊C411yescharacteristic
136C4⋊C4×C53C22⋊C414yes—
137C4⋊C4×C53C22⋊C411yescharacteristic
138C4⋊C4×C53C22⋊C412yes—
139C22⋊C4×C53—24no—
140C22⋊C4×C53—24no—
141C22⋊C4×C53—24no—
142C22⋊C4×C53—21no—
143C22⋊C4×C53—21no—
144C22⋊C4×C53—22no—
145C22⋊C4×C53—21no—
146C22⋊C4×C53—22no—
147C42×C53—24no—
148C42×C53—21no—
149C42×C53—21no—
150C42×C53—21no—
151C42×C53C22×C411yescharacteristic
152C42×C53C2×D411yescharacteristic
153C42×C53C2×D411yescharacteristic
154C42×C53C2×D411yescharacteristic
155C4⋊Dic53—22no—
156C4⋊Dic53—41no—
157C4×Dic53—41no—
158C4×Dic53—41no—
159#11719917—41no—
160#11719917—41no—
161C2×#88322—42no—
162C2×#88322—41no—
163C2×#88322—41no—
164C2×#88322—41no—
165C23×C53—21no—
166C23×C53—22no—
167C23×C53#12011yescharacteristic
168Q8×C53—21no—
169Q8×C53—21no—
170Q8×C53—44no—
171Q8×C53—41no—
172Q8×C53—41no—
173Q8×C53—82no—
174D4×C53—21no—
175D4×C53—21no—
176D4×C53—42no—
177D4×C53—41no—
178D4×C53—42no—
179D4×C53—41no—
180D4×C53—42no—
181D4×C53—41no—
182D4×C53—41no—
183D4×C53—41no—
184D4×C53—41no—
185D4×C53—84no—
186C2×C4×C53—21no—
187C2×C4×C53—21no—
188C2×C4×C53—21no—
189C2×C4×C53—24no—
190C2×C4×C53—22no—
191C2×C4×C53—21no—
192C2×C4×C53—21no—
193C2×C4×C53—41no—
194C2×C4×C53—42no—
195C2×C4×C53—42no—
196C2×C4×C53—42no—
197C2×C4×C53—41no—
198C2×C4×C53—41no—
199C2×C4×C53—41no—
200C2×C4×C53—41no—
201C2×C4×C53—82no—
202C2×C4×C53—84no—
203C2×C4×C53C2×C22⋊C411yescharacteristic
204C2×C4×C53C2×C22⋊C411yescharacteristic
205C2×C4×C53C2×C22⋊C411yescharacteristic
206C2×C4×C53#12011yescharacteristic
207C2×C4×C53#12011yescharacteristic
208C2×C4×C53#12011yescharacteristic
209C2×C4×C53#13511yescharacteristic
210C2×C4×C53#13511yescharacteristic
211C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
212C2×C4×C53#13711yescharacteristic
213C2×Dic53—42no—
214C2×Dic53—81no—
215C2×Dic53—81no—
216#88322—81no—
217#88322—81no—
218#88322—81no—
219#8422—531nomaximal, Sylow
220C22×C53—21no—
221C22×C53—21no—
222C22×C53—41no—
223C22×C53—42no—
224C22×C53—41no—
225C22×C53—42no—
226C22×C53—41no—
227C22×C53—42no—
228C22×C53#37911yescharacteristic
229C22×C53#44911yescharacteristic
230C22×C53#45211yescharacteristic
231C4×C53—21no—
232C4×C53—21no—
233C4×C53—21no—
234C4×C53—41no—
235C4×C53—41no—
236C4×C53—44no—
237C4×C53—41no—
238C4×C53—42no—
239C4×C53#41611yescharacteristic
240C4×C53#41711yescharacteristic
241C4×C53#44911yescharacteristic
242C4×C53#45211yescharacteristic
243C4×C53#45411yescharacteristic
244C4×C53#45611yescharacteristic
245Dic53—82no—
246#3511D5311yescharacteristic
247#1987—531no—
248#1806—531no—
249#1549—531no—
250#1547—531no—
251#1502—531no—
252#1477—531no—
253C2×C53—21no—
254C2×C53—21no—
255C2×C53—81no—
256C2×C53—82no—
257C2×C53#178811yescharacteristic
258C2×C53#179311yescharacteristic
259C2×C53#179511yescharacteristic
260#556Dic5312yescomplemented
261#555—21no—
262#554—21no—
263#553—22no—
264#548—22no—
265#547Dic5312yescomplemented
266#542—22no—
267#537—21no—
268#533—21no—
269#532—21no—
270#532C2×D5311yescharacteristic
271C2×C4⋊Q8C2×D5311yescharacteristic
272#520—21no—
273C4×#41C2×D5311yescharacteristic
274#431—531no—
275#423—531no—
276C2×C4⋊C8—531no—
277C2×#104—1061no—
278C4×C4⋊C4—531no—
279#340—1062no—
280#339—1061no—
281#332—1061no—
282#332—1062no—
283#331—1061no—
284#327—1062no—
285Q8⋊C8—1061no—
286D4⋊C8—1061no—
287C53#842211yescharacteristic, complemented, Sylow
288C2×#41—22no—
289C2×#41—21no—
290C2×#41C22⋊D5311yescharacteristic
291C22×Q8—21no—
292C2×M16—1061no—
293C22×C8—1061no—
294C4⋊Q8—21no—
295C4⋊Q8—21no—
296C4⋊Q8C2×Dic5312yes—
297C4⋊Q8C22⋊D5312yes—
298C4⋊Q8C22⋊D5311yescharacteristic
299#127C22⋊D5311yescharacteristic
300#126—24no—
301#125—22no—
302#124—24no—
303#124—22no—
304#124—21no—
305#123—24no—
306C22⋊Q8—22no—
307C22⋊Q8—24no—
308C22⋊Q8—22no—
309C22⋊Q8—22no—
310#121—22no—
311#121—22no—
312#121—24no—
313#121—22no—
314#121—21no—
315#121C22⋊D5312yes—
316C4×Q8—22no—
317C4×Q8—22no—
318C4×Q8C2×Dic5311yescharacteristic
319C4×Q8C22⋊D5311yescharacteristic
320C4×D4—22no—
321C4×D4—24no—
322C4×D4—22no—
323C4×D4—21no—
324C4×D4—21no—
325C4×D4—21no—
326C4×D4C2×Dic5311yescharacteristic
327C4×D4C2×Dic5312yes—
328C4×D4C22⋊D5311yescharacteristic
329#117—22no—
330#117—22no—
331#117—21no—
332#117C22⋊D5311yescharacteristic
333C2×C4⋊C4—21no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—1061no—
336C2×C4⋊C4C22⋊D5312yes—
337C2×C42—21no—
338C2×C42—1061no—
339C2×C42C22×D5311yescharacteristic
340C4⋊C8—1061no—
341C4⋊C8—1061no—
342C4⋊C8—1061no—
343C4⋊C8—1061no—
344C4⋊C8—1061no—
345#104—2122no—
346C22⋊C8—1061no—
347C4×C8—1061no—
348#95—1061no—
349#41—21no—
350#41—21no—
351#41—21no—
352#41—21no—
353#41—41no—
354#41—41no—
355#41—41no—
356#41—41no—
357#41—84no—
358C2×Q8—22no—
359C2×Q8—21no—
360C2×Q8—22no—
361C2×Q8—82no—
362C2×Q8C22⋊Dic5311yescharacteristic
363C2×D4—21no—
364C2×D4—22no—
365C2×D4—24no—
366C2×D4—21no—
367C2×D4—21no—
368C2×D4C22⋊Dic5311yescharacteristic
369C2×D4C22⋊Dic5312yes—
370C22×C4—24no—
371C22×C4—22no—
372C22×C4—21no—
373C22×C4—22no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—2121no—
377C22×C4C2×C22⋊D5311yescharacteristic
378C22×C4C2×C22⋊D5311yescharacteristic
379C22×C4C2×C22⋊D5311yescharacteristic
380M16—2121no—
381M16—2121no—
382C2×C8—2122no—
383C2×C8—2121no—
384C2×C8—2121no—
385C2×C8—2121no—
386C4⋊C4—24no—
387C4⋊C4—24no—
388C4⋊C4—22no—
389C4⋊C4—21no—
390C4⋊C4—22no—
391C4⋊C4—22no—
392C4⋊C4—21no—
393C4⋊C4—21no—
394C4⋊C4—1062no—
395C4⋊C4—2121no—
396C4⋊C4C22⋊Dic5311yescharacteristic
397C4⋊C4C22⋊Dic5314yes—
398C4⋊C4C22⋊Dic5311yescharacteristic
399C4⋊C4C22⋊Dic5312yes—
400C22⋊C4—24no—
401C22⋊C4—24no—
402C22⋊C4—24no—
403C22⋊C4—21no—
404C22⋊C4—21no—
405C22⋊C4—22no—
406C22⋊C4—21no—
407C22⋊C4—22no—
408C42—24no—
409C42—21no—
410C42—21no—
411C42—21no—
412C42—2121no—
413C42—2121no—
414C42C22×Dic5311yescharacteristic
415C42C2×C22⋊D5311yescharacteristic
416C42C2×C22⋊D5311yescharacteristic
417C42C2×C22⋊D5311yescharacteristic
418C23—21no—
419C23—22no—
420C23#1313318611yescharacteristic
421Q8—21no—
422Q8—21no—
423Q8—44no—
424Q8—41no—
425Q8—41no—
426Q8—82no—
427D4—21no—
428D4—21no—
429D4—42no—
430D4—41no—
431D4—42no—
432D4—41no—
433D4—42no—
434D4—41no—
435D4—41no—
436D4—41no—
437D4—41no—
438D4—84no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—22no—
444C2×C4—21no—
445C2×C4—24no—
446C2×C4—41no—
447C2×C4—42no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—82no—
455C2×C4—84no—
456C2×C4—4241no—
457C2×C4—4241no—
458C2×C4—2122no—
459C2×C4C2×C22⋊Dic5311yescharacteristic
460C2×C4C2×C22⋊Dic5311yescharacteristic
461C2×C4C2×C22⋊Dic5311yescharacteristic
462C2×C4#1313318611yescharacteristic
463C2×C4#1313318611yescharacteristic
464C2×C4#1313318611yescharacteristic
465C2×C4#1313323711yescharacteristic
466C2×C4#1313323711yescharacteristic
467C2×C4#1313324211yescharacteristic
468C2×C4#1313324711yescharacteristic
469C8—4241no—
470C8—4241no—
471C8—4241no—
472C22—21no—
473C22—21no—
474C22—41no—
475C22—42no—
476C22—41no—
477C22—42no—
478C22—41no—
479C22—42no—
480C22#1681047911yescharacteristic
481C22#1681076811yescharacteristic
482C22#1681078511yescharacteristic
483C4—21no—
484C4—21no—
485C4—21no—
486C4—41no—
487C4—42no—
488C4—41no—
489C4—41no—
490C4—44no—
491C4—4242no—
492C4#1681061911yescharacteristic
493C4#1681062411yescharacteristic
494C4#1681076811yescharacteristic
495C4#1681078511yescharacteristic
496C4#1681079611yescharacteristic
497C4#1681080611yescharacteristic
498C2—21no—
499C2—21no—
500C2—81no—
501C2—82no—
502C2#2027465511yescharacteristic
503C2#2027467411yescharacteristic
504C2#2027468511yescharacteristic
subgroupG/H
centerC22#16810479
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
Fitting, prime radicalC53×#3511C2
socleC22×C53#379
cosocleC2×C42×C53C23
hypercenter#3511D53
hypocenterC53#8422
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. #20275664C2semidirect
2. #20274735C2semidirect
3. #20273400C2semidirect
4. #20273392C2semidirect
5. #20273210C2semidirect
6. #20273102C2semidirect
21. C53×#556C4semidirect
26. C53×#547C4semidirect
260. #556Dic53semidirect
265. #547Dic53semidirect
287. C53#8422semidirect

Maximal subgroups

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

subgroupindex
#202756642
#202747352
#202734002
#202733922
#202732102
#202731022
C53×#35112
#842253