Finite Group Databasebeta

← #27276044

Subgroups — #27276044

Order 13568.

836 conjugacy classes of subgroups.

The subgroup lattice has 528 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#20273405C211yesmaximal, characteristic, complemented
4#20273383C211yesmaximal, characteristic, complemented
5#20273214C211yesmaximal, characteristic, complemented
6#20273102C211yesmaximal, characteristic, complemented
7C53×#3508C211yesmaximal, characteristic
8#16810676C2211yescharacteristic
9#16810647C2211yescharacteristic
10C2×C4⋊#88322C2211yescharacteristic
11C2×#13133144—21no—
12C2×#13133138—22no—
13C4×C4⋊Dic53C2211yescharacteristic
14#16810382—21no—
15#16810372—21no—
16#16810369—21no—
17#16810367—22no—
18#16810360—22no—
19Q8⋊#88322—21no—
20D4⋊#88322—21no—
21C53×#559—21no—
22C53×#559C412yescomplemented
23C53×#550—21no—
24C53×#550C412yescomplemented
25C53×#546—22no—
26C53×#546—22no—
27C53×#540—22no—
28C53×#535—21no—
29C53×#533C2211yescharacteristic
30C53×#532—21no—
31C53×#532—21no—
32C2×#127×C53C2211yescharacteristic
33C53×#518—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#13133138—44no—
46C22⋊#88322—21no—
47C4×#88322—21no—
48#13133124—21no—
49C2×#41×C53—22no—
50C2×#41×C53—22no—
51C2×#41×C53D411yescharacteristic
52C22×D4×C53—21no—
53C22×D4×C53—21no—
54C22×D4×C53D412yes—
55C4⋊Q8×C53D411yescharacteristic
56C4⋊Q8×C53D411yescharacteristic
57#127×C53—21no—
58#127×C53—21no—
59#127×C53—22no—
60#127×C53C2×C412yes—
61#127×C53D412yes—
62#125×C53—21no—
63#124×C53—22no—
64#124×C53—22no—
65#124×C53—22no—
66#123×C53—24no—
67#123×C53—24no—
68C22⋊Q8×C53—21no—
69C22⋊Q8×C53D412yes—
70#121×C53—22no—
71#121×C53—22no—
72#121×C53—22no—
73#121×C53—24no—
74#121×C53—22no—
75#121×C53—24no—
76#120×C53—24no—
77C4×Q8×C53—21no—
78C4×Q8×C53C2×C411yescharacteristic
79C4×Q8×C53C2×C412yes—
80C4×Q8×C53D411yescharacteristic
81C4×D4×C53—22no—
82C4×D4×C53—22no—
83C4×D4×C53—22no—
84C4×D4×C53—22no—
85C4×D4×C53—22no—
86C4×D4×C53—22no—
87C4×D4×C53—21no—
88C4×D4×C53—21no—
89C4×D4×C53C2×C411yescharacteristic
90C4×D4×C53D411yescharacteristic
91#117×C53—21no—
92#117×C53—21no—
93#117×C53D411yescharacteristic
94C2×C22⋊C4×C53—24no—
95C2×C42×C53—21no—
96C2×C42×C53C2311yescharacteristic
97C24×C53—22no—
98#41×C53—21no—
99#41×C53—21no—
100#41×C53—21no—
101#41×C53—21no—
102#41×C53—84no—
103#41×C53—82no—
104#41×C53—82no—
105C2×Q8×C53—21no—
106C2×Q8×C53C22⋊C411yescharacteristic
107C2×Q8×C53C22⋊C412yes—
108C2×D4×C53—21no—
109C2×D4×C53—24no—
110C2×D4×C53—24no—
111C2×D4×C53—22no—
112C2×D4×C53—22no—
113C2×D4×C53—22no—
114C2×D4×C53—22no—
115C2×D4×C53—22no—
116C2×D4×C53—42no—
117C2×D4×C53—42no—
118C2×D4×C53—41no—
119C2×D4×C53—41no—
120C2×D4×C53—41no—
121C2×D4×C53—41no—
122C2×D4×C53—82no—
123C2×D4×C53C22⋊C411yescharacteristic
124C2×D4×C53C22⋊C414yes—
125C22×C4×C53—24no—
126C22×C4×C53—24no—
127C22×C4×C53—21no—
128C22×C4×C53—21no—
129C22×C4×C53—21no—
130C22×C4×C53C2×D411yescharacteristic
131C22×C4×C53C2×D411yescharacteristic
132C22×C4×C53C2×D411yescharacteristic
133C22×Dic53—41no—
134C4⋊C4×C53—21no—
135C4⋊C4×C53—22no—
136C4⋊C4×C53—21no—
137C4⋊C4×C53—21no—
138C4⋊C4×C53—21no—
139C4⋊C4×C53—21no—
140C4⋊C4×C53—22no—
141C4⋊C4×C53C22⋊C411yescharacteristic
142C4⋊C4×C53C22⋊C411yescharacteristic
143C4⋊C4×C53C22⋊C412yes—
144C22⋊C4×C53—24no—
145C22⋊C4×C53—24no—
146C22⋊C4×C53—24no—
147C22⋊C4×C53—24no—
148C22⋊C4×C53—21no—
149C22⋊C4×C53—22no—
150C22⋊C4×C53—21no—
151C22⋊C4×C53—21no—
152C42×C53—21no—
153C42×C53—22no—
154C42×C53—21no—
155C42×C53—21no—
156C42×C53C22×C411yescharacteristic
157C42×C53C2×D411yescharacteristic
158C42×C53C2×D411yescharacteristic
159C42×C53C2×D411yescharacteristic
160C4⋊Dic53—22no—
161C4⋊Dic53—41no—
162C4×Dic53—41no—
163C4×Dic53—41no—
164#11719917—41no—
165#11719917—41no—
166C2×#88322—42no—
167C2×#88322—41no—
168C2×#88322—41no—
169C2×#88322—41no—
170C23×C53—21no—
171C23×C53—24no—
172C23×C53—42no—
173C23×C53—42no—
174C23×C53—82no—
175C23×C53—82no—
176C23×C53#12011yescharacteristic
177Q8×C53—21no—
178Q8×C53—21no—
179Q8×C53—42no—
180Q8×C53—41no—
181Q8×C53—41no—
182D4×C53—21no—
183D4×C53—21no—
184D4×C53—44no—
185D4×C53—44no—
186D4×C53—41no—
187D4×C53—41no—
188D4×C53—42no—
189D4×C53—42no—
190D4×C53—42no—
191D4×C53—42no—
192D4×C53—42no—
193D4×C53—42no—
194D4×C53—82no—
195D4×C53—84no—
196D4×C53—84no—
197C2×C4×C53—21no—
198C2×C4×C53—21no—
199C2×C4×C53—21no—
200C2×C4×C53—21no—
201C2×C4×C53—22no—
202C2×C4×C53—22no—
203C2×C4×C53—21no—
204C2×C4×C53—41no—
205C2×C4×C53—42no—
206C2×C4×C53—41no—
207C2×C4×C53—41no—
208C2×C4×C53—41no—
209C2×C4×C53—41no—
210C2×C4×C53—84no—
211C2×C4×C53—84no—
212C2×C4×C53C2×C22⋊C411yescharacteristic
213C2×C4×C53C2×C22⋊C411yescharacteristic
214C2×C4×C53C2×C22⋊C411yescharacteristic
215C2×C4×C53#12011yescharacteristic
216C2×C4×C53#12011yescharacteristic
217C2×C4×C53#12011yescharacteristic
218C2×C4×C53#13511yescharacteristic
219C2×C4×C53#13511yescharacteristic
220C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
221C2×C4×C53#13711yescharacteristic
222C2×Dic53—42no—
223C2×Dic53—81no—
224C2×Dic53—81no—
225#88322—81no—
226#88322—81no—
227#88322—81no—
228#8426—531nomaximal, Sylow
229C22×C53—21no—
230C22×C53—21no—
231C22×C53—41no—
232C22×C53—44no—
233C22×C53—41no—
234C22×C53—44no—
235C22×C53—41no—
236C22×C53—44no—
237C22×C53—162no—
238C22×C53—162no—
239C22×C53#37911yescharacteristic
240C22×C53#44911yescharacteristic
241C22×C53#45211yescharacteristic
242C4×C53—21no—
243C4×C53—21no—
244C4×C53—21no—
245C4×C53—41no—
246C4×C53—42no—
247C4×C53—41no—
248C4×C53—42no—
249C4×C53—41no—
250C4×C53#41611yescharacteristic
251C4×C53#41711yescharacteristic
252C4×C53#44911yescharacteristic
253C4×C53#45211yescharacteristic
254C4×C53#45311yescharacteristic
255C4×C53#45511yescharacteristic
256Dic53—82no—
257#3508D5311yescharacteristic
258#1982—531no—
259#1806—531no—
260#1550—531no—
261#1545—531no—
262#1503—531no—
263#1477—531no—
264C2×C53—21no—
265C2×C53—21no—
266C2×C53—81no—
267C2×C53—84no—
268C2×C53#178811yescharacteristic
269C2×C53#179211yescharacteristic
270C2×C53#179511yescharacteristic
271#559—21no—
272#559Dic5312yescomplemented
273#550—21no—
274#550Dic5312yescomplemented
275#546—22no—
276#546—22no—
277#540—22no—
278#535—21no—
279#533C2×D5311yescharacteristic
280#532—21no—
281#532—21no—
282C2×#127C2×D5311yescharacteristic
283#518—21no—
284C4×#41C2×D5311yescharacteristic
285#431—531no—
286#423—531no—
287C2×C4⋊C8—531no—
288C2×#104—1061no—
289C2×#102—1062no—
290C4×C4⋊C4—531no—
291#339—1061no—
292#333—1061no—
293#332—1061no—
294#331—1062no—
295#328—1062no—
296Q8⋊C8—1061no—
297D4⋊C8—1061no—
298C53#842611yescharacteristic, complemented, Sylow
299C2×#41—22no—
300C2×#41—22no—
301C2×#41C22⋊D5311yescharacteristic
302C22×D4—21no—
303C22×D4—21no—
304C22×D4C22⋊D5312yes—
305C2×M16—1061no—
306C22×C8—1061no—
307C4⋊Q8C22⋊D5311yescharacteristic
308C4⋊Q8C22⋊D5311yescharacteristic
309#127—22no—
310#127—21no—
311#127—21no—
312#127C2×Dic5312yes—
313#127C22⋊D5312yes—
314#125—21no—
315#124—22no—
316#124—22no—
317#124—22no—
318#123—24no—
319#123—24no—
320C22⋊Q8—21no—
321C22⋊Q8C22⋊D5312yes—
322#121—22no—
323#121—22no—
324#121—22no—
325#121—24no—
326#121—24no—
327#121—22no—
328#120—24no—
329C4×Q8—21no—
330C4×Q8C2×Dic5311yescharacteristic
331C4×Q8C2×Dic5312yes—
332C4×Q8C22⋊D5311yescharacteristic
333C4×D4—22no—
334C4×D4—22no—
335C4×D4—22no—
336C4×D4—21no—
337C4×D4—21no—
338C4×D4—22no—
339C4×D4—22no—
340C4×D4—22no—
341C4×D4C2×Dic5311yescharacteristic
342C4×D4C22⋊D5311yescharacteristic
343#117—21no—
344#117—21no—
345#117C22⋊D5311yescharacteristic
346C2×C4⋊C4—1061no—
347C2×C22⋊C4—24no—
348C2×C42—21no—
349C2×C42—1061no—
350C2×C42C22×D5311yescharacteristic
351C4⋊C8—1061no—
352C4⋊C8—1061no—
353C4⋊C8—1061no—
354C4⋊C8—1061no—
355C4⋊C8—1061no—
356#104—2122no—
357#102—2124no—
358C22⋊C8—1061no—
359C4×C8—1061no—
360#95—1061no—
361C24—22no—
362#41—21no—
363#41—21no—
364#41—21no—
365#41—21no—
366#41—84no—
367#41—82no—
368#41—82no—
369C2×Q8—21no—
370C2×Q8C22⋊Dic5311yescharacteristic
371C2×Q8C22⋊Dic5312yes—
372C2×D4—21no—
373C2×D4—24no—
374C2×D4—24no—
375C2×D4—22no—
376C2×D4—22no—
377C2×D4—22no—
378C2×D4—22no—
379C2×D4—22no—
380C2×D4—42no—
381C2×D4—42no—
382C2×D4—41no—
383C2×D4—41no—
384C2×D4—41no—
385C2×D4—41no—
386C2×D4—82no—
387C2×D4C22⋊Dic5311yescharacteristic
388C2×D4C22⋊Dic5314yes—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—2121no—
395C22×C4C2×C22⋊D5311yescharacteristic
396C22×C4C2×C22⋊D5311yescharacteristic
397C22×C4C2×C22⋊D5311yescharacteristic
398M16—2121no—
399M16—2121no—
400C2×C8—2122no—
401C2×C8—2121no—
402C2×C8—2121no—
403C2×C8—2121no—
404C4⋊C4—21no—
405C4⋊C4—22no—
406C4⋊C4—21no—
407C4⋊C4—21no—
408C4⋊C4—21no—
409C4⋊C4—21no—
410C4⋊C4—22no—
411C4⋊C4—1062no—
412C4⋊C4—2121no—
413C4⋊C4C22⋊Dic5311yescharacteristic
414C4⋊C4C22⋊Dic5311yescharacteristic
415C4⋊C4C22⋊Dic5312yes—
416C22⋊C4—24no—
417C22⋊C4—24no—
418C22⋊C4—24no—
419C22⋊C4—24no—
420C22⋊C4—21no—
421C22⋊C4—22no—
422C22⋊C4—21no—
423C22⋊C4—21no—
424C42—21no—
425C42—22no—
426C42—21no—
427C42—21no—
428C42—2121no—
429C42—2121no—
430C42C22×Dic5311yescharacteristic
431C42C2×C22⋊D5311yescharacteristic
432C42C2×C22⋊D5311yescharacteristic
433C42C2×C22⋊D5311yescharacteristic
434C23—24no—
435C23—21no—
436C23—42no—
437C23—42no—
438C23—82no—
439C23—82no—
440C23#1313318611yescharacteristic
441Q8—21no—
442Q8—21no—
443Q8—42no—
444Q8—41no—
445Q8—41no—
446D4—21no—
447D4—21no—
448D4—44no—
449D4—44no—
450D4—41no—
451D4—41no—
452D4—42no—
453D4—42no—
454D4—42no—
455D4—42no—
456D4—42no—
457D4—42no—
458D4—82no—
459D4—84no—
460D4—84no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—41no—
469C2×C4—42no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—84no—
475C2×C4—84no—
476C2×C4—4241no—
477C2×C4—4241no—
478C2×C4—2122no—
479C2×C4C2×C22⋊Dic5311yescharacteristic
480C2×C4C2×C22⋊Dic5311yescharacteristic
481C2×C4C2×C22⋊Dic5311yescharacteristic
482C2×C4#1313318611yescharacteristic
483C2×C4#1313318611yescharacteristic
484C2×C4#1313318611yescharacteristic
485C2×C4#1313323711yescharacteristic
486C2×C4#1313323711yescharacteristic
487C2×C4#1313324211yescharacteristic
488C2×C4#1313324711yescharacteristic
489C8—4241no—
490C8—4241no—
491C8—4241no—
492C22—21no—
493C22—21no—
494C22—41no—
495C22—44no—
496C22—41no—
497C22—44no—
498C22—41no—
499C22—44no—
500C22—162no—
501C22—162no—
502C22#1681047911yescharacteristic
503C22#1681076811yescharacteristic
504C22#1681078511yescharacteristic
505C4—21no—
506C4—21no—
507C4—21no—
508C4—41no—
509C4—41no—
510C4—42no—
511C4—41no—
512C4—42no—
513C4—4242no—
514C4#1681061911yescharacteristic
515C4#1681062411yescharacteristic
516C4#1681076811yescharacteristic
517C4#1681078511yescharacteristic
518C4#1681079111yescharacteristic
519C4#1681080111yescharacteristic
520C2—21no—
521C2—21no—
522C2—84no—
523C2—81no—
524C2#2027465511yescharacteristic
525C2#2027466911yescharacteristic
526C2#2027468511yescharacteristic
subgroupG/H
centerC22#16810479
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
Fitting, prime radicalC53×#3508C2
socleC22×C53#379
cosocleC2×C42×C53C23
hypercenter#3508D53
hypocenterC53#8426
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3508

Complements

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

NH ≅ G/Ntype
1. #20275643C2semidirect
2. #20274735C2semidirect
3. #20273405C2semidirect
4. #20273383C2semidirect
5. #20273214C2semidirect
6. #20273102C2semidirect
22. C53×#559C4semidirect
24. C53×#550C4semidirect
272. #559Dic53semidirect
274. #550Dic53semidirect
298. C53#8426semidirect

Maximal subgroups

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

subgroupindex
#202756432
#202747352
#202734052
#202733832
#202732142
#202731022
C53×#35082
#842653