Finite Group Databasebeta

← #27362717

Subgroups — #27362717

Order 13568.

964 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285843C211yesmaximal, characteristic, complemented
2#20284715C211yesmaximal, characteristic
3#20284187C211yesmaximal, characteristic, complemented
4#20273985C214yesmaximal, complemented
5#20273825C214yesmaximal, complemented
6#20273400C211yesmaximal, characteristic, complemented
7#20273140C211yesmaximal, characteristic, complemented
8#20273110C211yesmaximal, characteristic, complemented
9C2×C53×#556C211yesmaximal, characteristic, complemented
10C2×#13133248—21no—
11#16811173C2211yescharacteristic
12#16811019C2212yescomplemented
13#16810971C2212yescomplemented
14#16810941—22no—
15#16810936—21no—
16#16810910—22no—
17#16810903C2212yes—
18#16810845—22no—
19#16810833C2212yescomplemented
20#16810823C2212yescomplemented
21#16810818C2212yes—
22#16810757—21no—
23C4×#11719922—22no—
24#16810676C2211yescharacteristic
25#16810676C2211yescharacteristic
26#16810647C2211yescharacteristic
27#16810622—21no—
28#16810369C2214yescomplemented
29Q8⋊#88322C2214yescomplemented
30D4⋊#88322C2214yes—
31C53×#556—22no—
32C53×#556—22no—
33C53×#556—22no—
34C53×#556C2214yes—
35C2×C4⋊Q8×C53C2211yescharacteristic
36C2×#126×C53—22no—
37C2×#124×C53—21no—
38C2×C22⋊Q8×C53—21no—
39C2×C4×Q8×C53—21no—
40C2×C4×Q8×C53C2211yescharacteristic
41C2×C4×D4×C53C2211yescharacteristic
42C2×#117×C53—21no—
43C2×#11719945—21no—
44#13133248—44no—
45C2×#11719925—22no—
46C2×#11719924—22no—
47C2×#11719922—22no—
48C2×#11719917—21no—
49#13133217C2311yescharacteristic
50#13133214C2311yescharacteristic
51#13133205—21no—
52#13133191—21no—
53#13133191—21no—
54C4×D212—21no—
55C4×D212—21no—
56#13133150—22no—
57#13133148—24no—
58C4⋊#88322—21no—
59C4⋊#88322—21no—
60C4⋊#88322C2312yes—
61C4⋊#88322C2312yes—
62#13133140—22no—
63#13133138—22no—
64#13133137—22no—
65#13133137—22no—
66#13133137—24no—
67C22⋊#88322—21no—
68C22⋊#88322—21no—
69C4×#88322—21no—
70C4×#88322C2312yes—
71C22×Q8×C53—21no—
72C22×Q8×C53D411yescharacteristic
73C22×D4×C53D411yescharacteristic
74C23×C4×C53—21no—
75C4⋊Q8×C53—21no—
76C4⋊Q8×C53—21no—
77C4⋊Q8×C53D412yes—
78C4⋊Q8×C53C2312yes—
79#126×C53—28no—
80#126×C53—28no—
81#124×C53—24no—
82#124×C53—24no—
83C22⋊Q8×C53—24no—
84C22⋊Q8×C53—24no—
85C4×Q8×C53—24no—
86C4×Q8×C53—24no—
87C4×Q8×C53—21no—
88C4×Q8×C53—21no—
89C4×Q8×C53D412yes—
90C4×Q8×C53C2312yes—
91C4×D4×C53—21no—
92C4×D4×C53—21no—
93C4×D4×C53D412yes—
94C4×D4×C53C2312yes—
95#117×C53—24no—
96#117×C53—24no—
97C2×C4⋊C4×C53—22no—
98C2×C4⋊C4×C53—21no—
99C2×C4⋊C4×C53—21no—
100C2×C4⋊C4×C53D411yescharacteristic
101C2×C4⋊C4×C53D411yescharacteristic
102C2×C4⋊C4×C53D412yes—
103C2×C22⋊C4×C53—22no—
104C2×C22⋊C4×C53—22no—
105C2×C22⋊C4×C53—21no—
106C2×C42×C53—22no—
107C2×C42×C53—21no—
108C2×C42×C53C2311yescharacteristic
109C24×C53—21no—
110C2×Q8×C53—21no—
111C2×Q8×C53—22no—
112C2×Q8×C53—22no—
113C2×Q8×C53—22no—
114C2×Q8×C53—22no—
115C2×Q8×C53—44no—
116C2×Q8×C53C2×D412yes—
117C2×D4×C53—21no—
118C2×D4×C53—42no—
119C2×D4×C53C2×D412yes—
120C22×C4×C53—21no—
121C22×C4×C53—21no—
122C22×C4×C53—21no—
123C22×C4×C53—22no—
124C22×C4×C53—22no—
125C22×C4×C53—22no—
126C22×C4×C53—42no—
127C22×C4×C53—42no—
128C22×C4×C53C2×D411yescharacteristic
129C22×C4×C53C2×D411yescharacteristic
130C22×C4×C53C2×D411yescharacteristic
131C2×C22⋊D53—41no—
132#11719945—42no—
133#11719945—42no—
134C2×D212—21no—
135C2×D212—21no—
136C2×D212—21no—
137C2×C4×D53—41no—
138C2×Dic106—21no—
139C4⋊C4×C53—28no—
140C4⋊C4×C53—22no—
141C4⋊C4×C53—22no—
142C4⋊C4×C53—21no—
143C4⋊C4×C53—21no—
144C4⋊C4×C53—22no—
145C4⋊C4×C53—22no—
146C4⋊C4×C53—22no—
147C4⋊C4×C53C2×D412yes—
148C4⋊C4×C53C2×D412yes—
149C4⋊C4×C53C2×D414yes—
150C22⋊C4×C53—28no—
151C22⋊C4×C53—28no—
152C22⋊C4×C53—22no—
153C22⋊C4×C53—22no—
154C42×C53—28no—
155C42×C53—22no—
156C42×C53—22no—
157C42×C53C2×D411yescharacteristic
158C42×C53C2×D411yescharacteristic
159C42×C53C2×D411yescharacteristic
160C42×C53C2411yescharacteristic
161#11719925—44no—
162#11719924—44no—
163#11719922—44no—
164#11719921—41no—
165#11719921—41no—
166C4⋊Dic53—21no—
167C4⋊Dic53—21no—
168#11719917—42no—
169C2×#88322—22no—
170C2×#88322—22no—
171C2×#88322—22no—
172C23×C53—22no—
173C23×C53—42no—
174C23×C53—42no—
175C23×C53—42no—
176C23×C53#12011yescharacteristic
177C22×D53—41no—
178Q8×C53—22no—
179Q8×C53—22no—
180Q8×C53—44no—
181Q8×C53—44no—
182Q8×C53—42no—
183D4×C53—22no—
184D4×C53—22no—
185D4×C53—42no—
186C2×C4×C53—22no—
187C2×C4×C53—22no—
188C2×C4×C53—24no—
189C2×C4×C53—22no—
190C2×C4×C53—22no—
191C2×C4×C53—41no—
192C2×C4×C53—42no—
193C2×C4×C53—44no—
194C2×C4×C53—42no—
195C2×C4×C53—42no—
196C2×C4×C53—42no—
197C2×C4×C53—42no—
198C2×C4×C53—44no—
199C2×C4×C53#12011yescharacteristic
200C2×C4×C53#12011yescharacteristic
201C2×C4×C53#12011yescharacteristic
202C2×C4×C53#13512yes—
203C2×C4×C53AGL(1,ℤ/8)12yes—
204C2×C4×C53C22×D411yescharacteristic
205C2×C4×C53C22×D411yescharacteristic
206C2×C4×C53C22×D411yescharacteristic
207C22⋊D53—82no—
208C2×Dic53—41no—
209D212—42no—
210D212—41no—
211D212—41no—
212D212—42no—
213C4×D53—82no—
214Dic106—41no—
215Dic106—41no—
216#88322—42no—
217#88322—42no—
218#88322—42no—
219#18762—531nomaximal, Sylow
220C22×C53—21no—
221C22×C53—21no—
222C22×C53—42no—
223C22×C53—42no—
224C22×C53—42no—
225C22×C53—84no—
226C22×C53#44912yes—
227C22×C53C2×#12011yescharacteristic
228C2×D53—81no—
229C2×D53—81no—
230C2×D53—81no—
231C4×C53—21no—
232C4×C53—21no—
233C4×C53—21no—
234C4×C53—42no—
235C4×C53—42no—
236C4×C53—44no—
237C4×C53—42no—
238C4×C53#44912yes—
239C4×C53C2×#13511yescharacteristic
240C4×C53C2×AGL(1,ℤ/8)11yescharacteristic
241C4×C53#57512yes—
242Dic53—81no—
243C2×#556D5311yescharacteristic, complemented
244#3227—531no—
245#3100—531no—
246#3037—531no—
247#1652—534no—
248#1628—534no—
249#1549—531no—
250#1486—531no—
251#1479—531no—
252C2×C53—22no—
253C2×C53—82no—
254C2×C53C2×#44911yescharacteristic
255C2×C53#300511yescharacteristic
256C2×C53#302011yescharacteristic
257D53—161no—
258C2×#137—1061no—
259#556—22no—
260#556—22no—
261#556—22no—
262#556C2×D5314yes—
263#532—531no—
264C2×C4⋊Q8C2×D5311yescharacteristic
265C2×#126—22no—
266C2×#124—21no—
267C2×C22⋊Q8—21no—
268C2×C4×Q8—21no—
269C2×C4×Q8C2×D5311yescharacteristic
270C2×C4×D4C2×D5311yescharacteristic
271C2×#117—21no—
272#498—532no—
273#486—532no—
274#481—1062no—
275#480—1061no—
276#476—1062no—
277#474—532no—
278#464—1062no—
279#462—532no—
280#460—532no—
281#459—532no—
282#447—1061no—
283C4×D8—1062no—
284#431—531no—
285#431—531no—
286#423—531no—
287#417—1061no—
288#332—534no—
289Q8⋊C8—534no—
290D4⋊C8—534no—
291C53#1876211yescharacteristic, complemented, Sylow
292C2×#41—1061no—
293C22×Q8—21no—
294C22×Q8C22⋊D5311yescharacteristic
295C22×D4C22⋊D5311yescharacteristic
296C23×C4—21no—
297#137—2124no—
298C2×Q16—1062no—
299C2×SD16—1062no—
300C2×D8—1062no—
301C2×M16—1061no—
302C4⋊Q8—21no—
303C4⋊Q8—21no—
304C4⋊Q8—531no—
305C4⋊Q8C22⋊D5312yes—
306C4⋊Q8C22×D5312yes—
307#127—531no—
308#126—28no—
309#126—28no—
310#124—24no—
311#124—24no—
312#124—1061no—
313C22⋊Q8—24no—
314C22⋊Q8—24no—
315#121—1061no—
316#121—1061no—
317C4×Q8—24no—
318C4×Q8—24no—
319C4×Q8—21no—
320C4×Q8—21no—
321C4×Q8C22⋊D5312yes—
322C4×Q8C22×D5312yes—
323C4×D4—21no—
324C4×D4—21no—
325C4×D4—1061no—
326C4×D4—1061no—
327C4×D4C22⋊D5312yes—
328C4×D4C22×D5312yes—
329#117—24no—
330#117—24no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4C22⋊D5311yescharacteristic
335C2×C4⋊C4C22⋊D5311yescharacteristic
336C2×C4⋊C4C22⋊D5312yes—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—21no—
340C2×C42—22no—
341C2×C42—21no—
342C2×C42C22×D5311yescharacteristic
343#107—1062no—
344#106—1064no—
345C4⋊C8—532no—
346C4⋊C8—532no—
347C4⋊C8—1061no—
348C4⋊C8—1061no—
349#103—1062no—
350#102—1062no—
351#102—1062no—
352#102—1062no—
353#102—1064no—
354C22⋊C8—1061no—
355C22⋊C8—1061no—
356C4×C8—532no—
357C4×C8—1061no—
358C24—21no—
359#41—2122no—
360#41—2122no—
361C2×Q8—21no—
362C2×Q8—22no—
363C2×Q8—22no—
364C2×Q8—22no—
365C2×Q8—22no—
366C2×Q8—44no—
367C2×Q8—1061no—
368C2×Q8C2×C22⋊D5312yes—
369C2×D4—21no—
370C2×D4—42no—
371C2×D4—1061no—
372C2×D4—1061no—
373C2×D4—1061no—
374C2×D4—2121no—
375C2×D4C2×C22⋊D5312yes—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—2121no—
385C22×C4C2×C22⋊D5311yescharacteristic
386C22×C4C2×C22⋊D5311yescharacteristic
387C22×C4C2×C22⋊D5311yescharacteristic
388Q16—2124no—
389SD16—2124no—
390D8—2124no—
391M16—2122no—
392C2×C8—1062no—
393C2×C8—1062no—
394C2×C8—1062no—
395C4⋊C4—28no—
396C4⋊C4—22no—
397C4⋊C4—22no—
398C4⋊C4—21no—
399C4⋊C4—21no—
400C4⋊C4—22no—
401C4⋊C4—22no—
402C4⋊C4—22no—
403C4⋊C4—1061no—
404C4⋊C4—1061no—
405C4⋊C4C2×C22⋊D5312yes—
406C4⋊C4C2×C22⋊D5314yes—
407C4⋊C4C2×C22⋊D5312yes—
408C22⋊C4—28no—
409C22⋊C4—28no—
410C22⋊C4—22no—
411C22⋊C4—22no—
412C22⋊C4—2121no—
413C22⋊C4—2121no—
414C42—28no—
415C42—22no—
416C42—22no—
417C42C2×C22⋊D5311yescharacteristic
418C42C2×C22⋊D5311yescharacteristic
419C42C2×C22⋊D5311yescharacteristic
420C42C23×D5311yescharacteristic
421C23—22no—
422C23—42no—
423C23—42no—
424C23—42no—
425C23—2121no—
426C23#1313318611yescharacteristic
427Q8—22no—
428Q8—22no—
429Q8—44no—
430Q8—42no—
431Q8—44no—
432Q8—2121no—
433Q8—2121no—
434D4—22no—
435D4—22no—
436D4—42no—
437D4—4242no—
438D4—2122no—
439D4—2121no—
440D4—2121no—
441D4—2122no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—24no—
447C2×C4—41no—
448C2×C4—42no—
449C2×C4—44no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—44no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—4242no—
456C2×C4—2121no—
457C2×C4#1313318611yescharacteristic
458C2×C4#1313318611yescharacteristic
459C2×C4#1313318611yescharacteristic
460C2×C4#1313323712yes—
461C2×C4#1313324212yes—
462C2×C4C22×C22⋊D5311yescharacteristic
463C2×C4C22×C22⋊D5311yescharacteristic
464C2×C4C22×C22⋊D5311yescharacteristic
465C8—2122no—
466C8—2122no—
467C8—2122no—
468C22—21no—
469C22—21no—
470C22—42no—
471C22—42no—
472C22—42no—
473C22—84no—
474C22—4241no—
475C22—4241no—
476C22—4241no—
477C22#1681076812yes—
478C22C2×#1313318611yescharacteristic
479C4—21no—
480C4—21no—
481C4—21no—
482C4—42no—
483C4—42no—
484C4—44no—
485C4—42no—
486C4—4241no—
487C4#1681076812yes—
488C4C2×#1313323711yescharacteristic
489C4C2×#1313324211yescharacteristic
490C4#1681142112yes—
491C2—22no—
492C2—82no—
493C2—8481no—
494C2C2×#1681076811yescharacteristic
495C2#2028395711yescharacteristic
496C2#2028407911yescharacteristic
subgroupG/H
centerC22C2×#13133186
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC2×C53×#556C2
socleC22×C53C2×#120
prime radical#20284715C2
hypercenterC2×#556D53
hypocenterC53#18762
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#556

Complements

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

NH ≅ G/Ntype
1. #20285843C2semidirect
3. #20284187C2semidirect
4. #20273985C2semidirect
5. #20273825C2semidirect
6. #20273400C2semidirect
7. #20273140C2semidirect
8. #20273110C2semidirect
9. C2×C53×#556C2semidirect
12. #16811019C22semidirect
13. #16810971C22semidirect
19. #16810833C22semidirect
20. #16810823C22semidirect
28. #16810369C22semidirect
29. Q8⋊#88322C22semidirect
243. C2×#556D53semidirect
291. C53#18762semidirect

Maximal subgroups

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

subgroupindex
#202858432
#202847152
#202841872
#202739852
#202738252
#202734002
#202731402
#202731102
C2×C53×#5562
#1876253