Finite Group Databasebeta

← #27370180

Subgroups — #27370180

Order 13568.

694 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#20285633C211yesmaximal, characteristic
2#20284837C211yesmaximal, characteristic, complemented
3#20283804C211yesmaximal, characteristic, complemented
4#20274253C212yesmaximal, complemented
5#20274197C212yesmaximal, complemented
6#20274129C212yesmaximal, complemented
7#20274073C212yesmaximal, complemented
8#20273498C211yesmaximal, characteristic, complemented
9#20273429C211yesmaximal, characteristic, complemented
10#20273408C211yesmaximal, characteristic, complemented
11C2×C53×#460C211yesmaximal, characteristic, complemented
12#16811173C2211yescharacteristic
13#16810999C2212yes—
14#16810969C2212yes—
15#16810936—21no—
16#16810918C2211yescharacteristic, complemented
17#16810918C2211yescharacteristic, complemented
18#16810823C2211yescharacteristic, complemented
19#16810823C2211yescharacteristic, complemented
20#16810815—22no—
21#16810762—21no—
22#16810757—21no—
23#16810757—21no—
24#16810754—22no—
25#16810754C2212yescomplemented
26Q8×#88322C2212yescomplemented
27#16810721—21no—
28#16810721—21no—
29#16810680—21no—
30#16810680C2211yescharacteristic
31#16810680C2211yescharacteristic
32#16810676—21no—
33#16810676C2211yescharacteristic
34#16810375C2212yescomplemented
35#16810373C2212yescomplemented
36#16810371C2212yescomplemented
37#16810367C2212yes—
38#16810367C2212yes—
39#16810366C2212yescomplemented
40C22×SD16×C53—21no—
41C2×#127×C53C2211yescharacteristic
42C2×C4×Q8×C53C2211yescharacteristic
43C53×#460—21no—
44C53×#460—21no—
45C53×#460C2212yes—
46C53×#460C2212yes—
47C2×C4⋊C8×C53C2211yescharacteristic
48C2×#102×C53—21no—
49C2×#11719945—21no—
50C2×#11719924—21no—
51C2×#11719924—21no—
52C2×#11719922—22no—
53C2×#11719922—22no—
54#13133217C2311yescharacteristic
55#13133214C2311yescharacteristic
56#13133205—21no—
57#13133191—21no—
58#13133191—21no—
59C4×D212—21no—
60C4×D212—21no—
61#13133150—21no—
62#13133150—21no—
63C4⋊#88322—22no—
64C4⋊#88322—22no—
65C4⋊#88322D412yes—
66C4⋊#88322C2312yes—
67C4⋊#88322C2311yescharacteristic
68C4⋊#88322C2311yescharacteristic
69#13133141—21no—
70#13133141—21no—
71#13133138—22no—
72#13133138—22no—
73#13133137—21no—
74#13133137—21no—
75C22⋊#88322—21no—
76C22⋊#88322—21no—
77C22⋊#88322D411yescharacteristic
78C22⋊#88322D411yescharacteristic
79#13133128—21no—
80#13133128—21no—
81#13133128—22no—
82C4×#88322—22no—
83C4×#88322C2312yes—
84C22×Q8×C53D411yescharacteristic
85C22×D4×C53—21no—
86C22×D4×C53—21no—
87C22×D4×C53—21no—
88C2×SD16×C53—22no—
89C2×SD16×C53—22no—
90C2×SD16×C53—44no—
91C22×C8×C53—21no—
92#127×C53—21no—
93#127×C53—21no—
94#127×C53—21no—
95#127×C53C2312yes—
96C4×Q8×C53—21no—
97C4×Q8×C53—21no—
98C4×Q8×C53D411yescharacteristic
99C4×Q8×C53D411yescharacteristic
100C4×Q8×C53C2311yescharacteristic
101C4×Q8×C53C2311yescharacteristic
102C2×C4⋊C4×C53—21no—
103C2×C4⋊C4×C53D411yescharacteristic
104C2×C42×C53—21no—
105C2×C42×C53C2311yescharacteristic
106C4⋊C8×C53—21no—
107C4⋊C8×C53C2312yes—
108#102×C53—22no—
109#102×C53—22no—
110C24×C53—41no—
111C2×Q8×C53—21no—
112C2×Q8×C53—22no—
113C2×Q8×C53—41no—
114C2×Q8×C53C2×D411yescharacteristic
115C2×Q8×C53C2×D411yescharacteristic
116C2×D4×C53—22no—
117C2×D4×C53—22no—
118C2×D4×C53—22no—
119C2×D4×C53—42no—
120C2×D4×C53—42no—
121C2×D4×C53—41no—
122C2×D4×C53—41no—
123C2×D4×C53—41no—
124C2×D4×C53—82no—
125C2×D4×C53—82no—
126C22×C4×C53—21no—
127C22×C4×C53—21no—
128C22×C4×C53C2×D411yescharacteristic
129C22×C4×C53C2×D411yescharacteristic
130C22×C4×C53#4111yescharacteristic
131C2×C22⋊D53—41no—
132#11719945—82no—
133C2×D212—21no—
134C2×D212—21no—
135C2×D212—21no—
136C2×C4×D53—41no—
137C2×Dic106—21no—
138SD16×C53—44no—
139SD16×C53—44no—
140C2×C8×C53—22no—
141C2×C8×C53—42no—
142C4⋊C4×C53—21no—
143C4⋊C4×C53—21no—
144C4⋊C4×C53—21no—
145C4⋊C4×C53—21no—
146C4⋊C4×C53—21no—
147C4⋊C4×C53C2×D411yescharacteristic
148C4⋊C4×C53C2×D411yescharacteristic
149C42×C53—21no—
150C42×C53—21no—
151C42×C53—21no—
152C42×C53—21no—
153C42×C53C2×D411yescharacteristic
154C42×C53C2×D411yescharacteristic
155C42×C53#4111yescharacteristic
156C42×C53C2411yescharacteristic
157#11719924—42no—
158#11719924—42no—
159#11719922—82no—
160#11719922—82no—
161#11719921—41no—
162#11719921—41no—
163C4⋊Dic53—21no—
164C4⋊Dic53—21no—
165C2×#88322—22no—
166C2×#88322—21no—
167C2×#88322—21no—
168C2×#88322C2×D412yes—
169C2×#88322C2×D412yes—
170C23×C53—42no—
171C23×C53—82no—
172C23×C53—82no—
173C23×C53—82no—
174C23×C53#12111yescharacteristic
175C22×D53—41no—
176Q8×C53—21no—
177Q8×C53—21no—
178Q8×C53—21no—
179Q8×C53—42no—
180Q8×C53AGL(1,ℤ/8)12yes—
181D4×C53—42no—
182D4×C53—42no—
183D4×C53—42no—
184D4×C53—42no—
185D4×C53—82no—
186D4×C53—82no—
187D4×C53—82no—
188D4×C53—82no—
189C2×C4×C53—21no—
190C2×C4×C53—22no—
191C2×C4×C53—21no—
192C2×C4×C53—21no—
193C2×C4×C53—22no—
194C2×C4×C53—21no—
195C2×C4×C53—21no—
196C2×C4×C53—21no—
197C2×C4×C53—42no—
198C2×C4×C53—42no—
199C2×C4×C53#12111yescharacteristic
200C2×C4×C53#12111yescharacteristic
201C2×C4×C53#12111yescharacteristic
202C2×C4×C53C22×D411yescharacteristic
203C2×C4×C53C22×D411yescharacteristic
204C2×C4×C53C2×#4111yescharacteristic
205C22⋊D53—82no—
206C2×Dic53—41no—
207D212—41no—
208D212—41no—
209D212—42no—
210D212—81no—
211C4×D53—82no—
212Dic106—81no—
213C8×C53—42no—
214#88322—22no—
215#88322—22no—
216#88322—41no—
217#88322—41no—
218#88322—42no—
219#19470—531nomaximal, Sylow
220C22×C53—21no—
221C22×C53—21no—
222C22×C53—82no—
223C22×C53—82no—
224C22×C53—82no—
225C22×C53—164no—
226C22×C53#47212yes—
227C22×C53C2×#12111yescharacteristic
228C2×D53—81no—
229C2×D53—81no—
230C2×D53—81no—
231C4×C53—21no—
232C4×C53—21no—
233C4×C53—22no—
234C4×C53—21no—
235C4×C53—21no—
236C4×C53—41no—
237C4×C53—41no—
238C4×C53—41no—
239C4×C53C2×AGL(1,ℤ/8)11yescharacteristic
240C4×C53#57511yescharacteristic
241C4×C53#57712yes—
242Dic53—81no—
243#3205—531no—
244#3112—531no—
245C2×#460D5311yescharacteristic, complemented
246#2984—531no—
247#1698—532no—
248#1688—532no—
249#1678—532no—
250#1668—532no—
251#1573—531no—
252#1555—531no—
253#1551—531no—
254C2×C53—22no—
255C2×C53—162no—
256C2×C53#304311yescharacteristic
257C2×C53#305511yescharacteristic
258C2×C53C2×#47211yescharacteristic
259D53—161no—
260C22×SD16—21no—
261#532—531no—
262C2×#127C2×D5311yescharacteristic
263C2×C4×Q8C2×D5311yescharacteristic
264#493—532no—
265#486—532no—
266#480—1061no—
267#477—531no—
268#477—531no—
269#460—21no—
270#460—21no—
271#460—531no—
272#460—531no—
273#460C2×D5312yes—
274#460C2×D5312yes—
275#459—1062no—
276#448—1061no—
277#447—1061no—
278#447—1061no—
279#446—532no—
280#446—1062no—
281C8×Q8—532no—
282#440—1061no—
283#440—1061no—
284#432—531no—
285#432—531no—
286#432—1061no—
287#431—531no—
288#431—1061no—
289C2×C4⋊C8C2×D5311yescharacteristic
290C2×#102—21no—
291#335—532no—
292#334—532no—
293#333—532no—
294#331—532no—
295#331—532no—
296#331—532no—
297C53#1947011yescharacteristic, complemented, Sylow
298C2×#41—1061no—
299C22×Q8C22⋊D5311yescharacteristic
300C22×D4—21no—
301C22×D4—21no—
302C22×D4—21no—
303C2×SD16—22no—
304C2×SD16—22no—
305C2×SD16—44no—
306C2×SD16—1061no—
307C2×SD16—1061no—
308C2×D8—1062no—
309C2×D8—1062no—
310C22×C8—21no—
311C4⋊Q8—531no—
312#127—21no—
313#127—21no—
314#127—21no—
315#127—531no—
316#127C22×D5312yes—
317#124—1061no—
318#121—1061no—
319#121—1061no—
320C4×Q8—21no—
321C4×Q8—21no—
322C4×Q8C22⋊D5311yescharacteristic
323C4×Q8C22⋊D5311yescharacteristic
324C4×Q8C22×D5311yescharacteristic
325C4×Q8C22×D5311yescharacteristic
326C4×D4—1061no—
327C4×D4—1061no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4C22⋊D5311yescharacteristic
330C2×C42—21no—
331C2×C42C22×D5311yescharacteristic
332#107—1061no—
333#107—1061no—
334C4⋊C8—21no—
335C4⋊C8—532no—
336C4⋊C8—532no—
337C4⋊C8—531no—
338C4⋊C8—531no—
339C4⋊C8—1062no—
340C4⋊C8—1062no—
341C4⋊C8C22×D5312yes—
342#103—1061no—
343#103—1061no—
344#102—22no—
345#102—22no—
346#102—1062no—
347#102—1062no—
348#102—1061no—
349#102—1061no—
350C22⋊C8—531no—
351C22⋊C8—531no—
352C22⋊C8—1061no—
353C22⋊C8—1061no—
354#97—1062no—
355#97—1061no—
356#97—1061no—
357C4×C8—532no—
358C4×C8—1062no—
359C24—41no—
360#41—4242no—
361C2×Q8—21no—
362C2×Q8—22no—
363C2×Q8—41no—
364C2×Q8—1061no—
365C2×Q8C2×C22⋊D5311yescharacteristic
366C2×Q8C2×C22⋊D5311yescharacteristic
367C2×D4—22no—
368C2×D4—22no—
369C2×D4—22no—
370C2×D4—42no—
371C2×D4—42no—
372C2×D4—41no—
373C2×D4—41no—
374C2×D4—41no—
375C2×D4—82no—
376C2×D4—82no—
377C2×D4—1061no—
378C2×D4—1061no—
379C2×D4—1061no—
380C2×D4—2121no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—2121no—
384C22×C4D4×D5311yescharacteristic
385C22×C4#1171994711yescharacteristic
386C22×C4C2×C22⋊D5311yescharacteristic
387SD16—44no—
388SD16—44no—
389SD16—2122no—
390SD16—2122no—
391D8—4242no—
392D8—4242no—
393C2×C8—22no—
394C2×C8—42no—
395C2×C8—532no—
396C2×C8—532no—
397C2×C8—1062no—
398C2×C8—1061no—
399C2×C8—1061no—
400C4⋊C4—21no—
401C4⋊C4—21no—
402C4⋊C4—21no—
403C4⋊C4—21no—
404C4⋊C4—21no—
405C4⋊C4—1061no—
406C4⋊C4—1061no—
407C4⋊C4C2×C22⋊D5311yescharacteristic
408C4⋊C4C2×C22⋊D5311yescharacteristic
409C22⋊C4—2121no—
410C22⋊C4—2121no—
411C42—21no—
412C42—21no—
413C42—21no—
414C42—21no—
415C42D4×D5311yescharacteristic
416C42#1171994711yescharacteristic
417C42C2×C22⋊D5311yescharacteristic
418C42C23×D5311yescharacteristic
419C23—42no—
420C23—82no—
421C23—82no—
422C23—82no—
423C23—2121no—
424C23#1313318911yescharacteristic
425Q8—21no—
426Q8—21no—
427Q8—21no—
428Q8—42no—
429Q8—4241no—
430Q8#1313324312yes—
431D4—42no—
432D4—42no—
433D4—42no—
434D4—42no—
435D4—82no—
436D4—82no—
437D4—82no—
438D4—82no—
439D4—4241no—
440D4—4242no—
441D4—2121no—
442D4—2121no—
443D4—2122no—
444C2×C4—21no—
445C2×C4—22no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—22no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—4242no—
455C2×C4—2121no—
456C2×C4#1313318911yescharacteristic
457C2×C4#1313318911yescharacteristic
458C2×C4#1313318911yescharacteristic
459C2×C4C2×D4×D5311yescharacteristic
460C2×C4C22×C22⋊D5311yescharacteristic
461C2×C4C2×#1171994711yescharacteristic
462C8—42no—
463C8—1062no—
464C8—1062no—
465C8—2121no—
466C8—2121no—
467C8—2122no—
468C22—21no—
469C22—21no—
470C22—82no—
471C22—82no—
472C22—82no—
473C22—164no—
474C22—4241no—
475C22—4241no—
476C22—4241no—
477C22#1681089112yes—
478C22C2×#1313318911yescharacteristic
479C4—21no—
480C4—21no—
481C4—21no—
482C4—21no—
483C4—22no—
484C4—41no—
485C4—41no—
486C4—41no—
487C4—4241no—
488C4C2×#1313324311yescharacteristic
489C4#1681142111yescharacteristic
490C4#1681142912yes—
491C2—22no—
492C2—162no—
493C2—8481no—
494C2#2028423311yescharacteristic
495C2#2028433011yescharacteristic
496C2C2×#1681089111yescharacteristic
subgroupG/H
centerC22C2×#13133189
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC2×C53×#460C2
socleC22×C53C2×#121
prime radical#20285633C2
hypercenterC2×#460D53
hypocenterC53#19470
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#460

Complements

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

NH ≅ G/Ntype
2. #20284837C2semidirect
3. #20283804C2semidirect
4. #20274253C2semidirect
5. #20274197C2semidirect
6. #20274129C2semidirect
7. #20274073C2semidirect
8. #20273498C2semidirect
9. #20273429C2semidirect
10. #20273408C2semidirect
11. C2×C53×#460C2semidirect
16. #16810918C22semidirect
17. #16810918C22semidirect
18. #16810823C22semidirect
19. #16810823C22semidirect
25. #16810754C22semidirect
26. Q8×#88322C22semidirect
34. #16810375C22semidirect
35. #16810373C22semidirect
36. #16810371C22semidirect
39. #16810366C22semidirect
245. C2×#460D53semidirect
297. C53#19470semidirect

Maximal subgroups

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

subgroupindex
#202856332
#202848372
#202838042
#202742532
#202741972
#202741292
#202740732
#202734982
#202734292
#202734082
C2×C53×#4602
#1947053