Finite Group Databasebeta

← #27370184

Subgroups — #27370184

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#20285635C211yesmaximal, characteristic
2#20284837C211yesmaximal, characteristic, complemented
3C2×#16810823C211yesmaximal, characteristic, complemented
4#20274255C212yesmaximal, complemented
5#20274199C212yesmaximal, complemented
6#20274131C212yesmaximal, complemented
7#20274075C212yesmaximal, complemented
8#20273500C211yesmaximal, characteristic, complemented
9#20273428C211yesmaximal, characteristic, complemented
10#20273411C211yesmaximal, characteristic, complemented
11C53×#2984C211yesmaximal, characteristic, complemented
12C22×#11719924—21no—
13#16811173C2211yescharacteristic
14C2×#13133214C2211yescharacteristic
15#16811000C2212yes—
16#16810970C2212yes—
17#16810936—21no—
18#16810918C2211yescharacteristic, complemented
19#16810918C2211yescharacteristic, complemented
20#16810823—21no—
21#16810823—21no—
22#16810823C2212yes—
23#16810823C2212yes—
24#16810823C2211yescharacteristic, complemented
25#16810823C2211yescharacteristic, complemented
26#16810819—22no—
27#16810762—21no—
28#16810759—21no—
29#16810759—21no—
30#16810721—21no—
31#16810721—21no—
32#16810680C2211yescharacteristic
33C2×C4⋊#88322C2211yescharacteristic
34C2×#13133137—21no—
35#16810376C2212yescomplemented
36#16810374C2212yescomplemented
37#16810371C2212yescomplemented
38#16810366C2212yes—
39#16810366C2212yes—
40#16810366C2212yescomplemented
41C2×C4×Q8×C53C2211yescharacteristic
42C53×#446—22no—
43C53×#446C2212yescomplemented
44C8×Q8×C53C2212yescomplemented
45C53×#432—21no—
46C53×#432C2211yescharacteristic
47C53×#431—21no—
48C53×#431C2211yescharacteristic
49C2×#11719945—21no—
50C22×D212—21no—
51C22×D212—21no—
52C22×D212—21no—
53C2×#11719924—21no—
54C2×#11719924—21no—
55C2×#11719924—22no—
56C2×#11719924—22no—
57C2×#11719924—44no—
58C2×D424—22no—
59C2×D424—22no—
60C22×#88322—21no—
61#13133217C2311yescharacteristic
62#13133214—21no—
63#13133214—21no—
64#13133214—21no—
65#13133214C2312yes—
66#13133214C2311yescharacteristic
67#13133205—21no—
68#13133191—21no—
69#13133191—21no—
70C4×D212—21no—
71C4×D212—21no—
72#13133150—21no—
73#13133150—21no—
74C4⋊#88322—21no—
75C4⋊#88322C2312yes—
76C4⋊#88322C2311yescharacteristic
77C4⋊#88322C2311yescharacteristic
78#13133141—21no—
79#13133141—21no—
80#13133139—22no—
81#13133139—22no—
82#13133137—22no—
83#13133137—22no—
84#13133137—21no—
85#13133137—21no—
86C22⋊#88322—21no—
87#13133128—21no—
88C22×Q8×C53D411yescharacteristic
89C4×Q8×C53—21no—
90C4×Q8×C53—21no—
91C4×Q8×C53D411yescharacteristic
92C4×Q8×C53D411yescharacteristic
93C4×Q8×C53C2311yescharacteristic
94C4×Q8×C53C2311yescharacteristic
95C2×C4⋊C4×C53—21no—
96C2×C4⋊C4×C53D411yescharacteristic
97C2×C42×C53—21no—
98C2×C42×C53C2311yescharacteristic
99C4⋊C8×C53—22no—
100C4⋊C8×C53—22no—
101C4⋊C8×C53D412yes—
102C4⋊C8×C53C2312yes—
103C22⋊C8×C53—21no—
104C22⋊C8×C53D411yescharacteristic
105C22⋊C8×C53D411yescharacteristic
106#97×C53—22no—
107#97×C53—21no—
108C4×C8×C53—22no—
109C4×C8×C53C2312yes—
110C23×D53—41no—
111C2×Q8×C53—21no—
112C2×Q8×C53—22no—
113C2×Q8×C53—41no—
114C2×Q8×C53C2×D411yescharacteristic
115C2×Q8×C53C2×D411yescharacteristic
116C22×C4×C53—21no—
117C22×C4×C53—21no—
118C22×C4×C53C2×D411yescharacteristic
119C22×C4×C53C2×D411yescharacteristic
120C22×C4×C53#4111yescharacteristic
121C2×C22⋊D53—41no—
122#11719945—82no—
123C2×D212—21no—
124C2×D212—22no—
125C2×D212—22no—
126C2×D212—21no—
127C2×D212—21no—
128C2×D212—22no—
129C2×D212—42no—
130C2×D212—42no—
131C2×D212—41no—
132C2×D212—41no—
133C2×D212—41no—
134C2×D212—82no—
135C2×D212—82no—
136C2×C4×D53—41no—
137C2×Dic106—21no—
138C2×C8×C53—22no—
139C2×C8×C53C2×D412yes—
140C2×C8×C53C2×D412yes—
141C4⋊C4×C53—21no—
142C4⋊C4×C53—21no—
143C4⋊C4×C53—21no—
144C4⋊C4×C53—21no—
145C4⋊C4×C53—21no—
146C4⋊C4×C53C2×D411yescharacteristic
147C4⋊C4×C53C2×D411yescharacteristic
148C42×C53—21no—
149C42×C53—21no—
150C42×C53—21no—
151C42×C53—21no—
152C42×C53C2×D411yescharacteristic
153C42×C53C2×D411yescharacteristic
154C42×C53#4111yescharacteristic
155C42×C53C2411yescharacteristic
156#11719924—42no—
157#11719924—44no—
158#11719924—42no—
159#11719924—44no—
160#11719921—41no—
161#11719921—41no—
162C4⋊Dic53—21no—
163C4⋊Dic53—21no—
164C2×#88322—21no—
165C2×#88322—21no—
166C2×#88322—22no—
167C2×#88322—42no—
168D424—82no—
169D424—82no—
170C23×C53#12111yescharacteristic
171C22×D53—41no—
172C22×D53—42no—
173C22×D53—82no—
174C22×D53—82no—
175C22×D53—82no—
176Q8×C53—21no—
177Q8×C53—21no—
178Q8×C53—21no—
179Q8×C53—42no—
180Q8×C53AGL(1,ℤ/8)12yes—
181C2×C4×C53—21no—
182C2×C4×C53—22no—
183C2×C4×C53—21no—
184C2×C4×C53—21no—
185C2×C4×C53—22no—
186C2×C4×C53—21no—
187C2×C4×C53—21no—
188C2×C4×C53—21no—
189C2×C4×C53—42no—
190C2×C4×C53—42no—
191C2×C4×C53#12111yescharacteristic
192C2×C4×C53#12111yescharacteristic
193C2×C4×C53#12111yescharacteristic
194C2×C4×C53C22×D411yescharacteristic
195C2×C4×C53C22×D411yescharacteristic
196C2×C4×C53C2×#4111yescharacteristic
197C22⋊D53—82no—
198C2×Dic53—41no—
199D212—42no—
200D212—41no—
201D212—42no—
202D212—41no—
203D212—42no—
204D212—42no—
205D212—42no—
206D212—82no—
207D212—81no—
208D212—82no—
209D212—82no—
210D212—82no—
211C4×D53—82no—
212Dic106—81no—
213C8×C53—22no—
214C8×C53—22no—
215C8×C53—42no—
216#88322—42no—
217#88322—41no—
218#88322—41no—
219#19470—531nomaximal, Sylow
220C22×C53—21no—
221C22×C53—21no—
222C22×C53#47212yes—
223C22×C53C2×#12111yescharacteristic
224C2×D53—82no—
225C2×D53—81no—
226C2×D53—82no—
227C2×D53—81no—
228C2×D53—81no—
229C2×D53—82no—
230C2×D53—164no—
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×#460—531no—
246#2984D5311yescharacteristic, complemented
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#304311yescharacteristic
256C2×C53#305511yescharacteristic
257C2×C53C2×#47211yescharacteristic
258D53—161no—
259D53—162no—
260C22×SD16—1061no—
261#532—531no—
262C2×#127—531no—
263C2×C4×Q8C2×D5311yescharacteristic
264#493—532no—
265#486—532no—
266#480—1061no—
267#477—531no—
268#477—531no—
269#460—531no—
270#460—531no—
271#460—532no—
272#460—532no—
273#460—1061no—
274#460—1061no—
275#459—1062no—
276#448—1061no—
277#447—1061no—
278#447—1061no—
279#446—22no—
280#446C2×D5312yescomplemented
281C8×Q8C2×D5312yescomplemented
282#440—1061no—
283#440—1061no—
284#432—21no—
285#432—531no—
286#432C2×D5311yescharacteristic
287#431—21no—
288#431C2×D5311yescharacteristic
289C2×C4⋊C8—531no—
290C2×#102—1061no—
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×Q8D21211yescharacteristic
300C22×D4—1061no—
301C22×D4—1061no—
302C22×D4—1061no—
303C2×SD16—1062no—
304C2×SD16—1062no—
305C2×SD16—1061no—
306C2×SD16—1061no—
307C2×SD16—2124no—
308C2×D8—1062no—
309C2×D8—1062no—
310C22×C8—1061no—
311C4⋊Q8—531no—
312#127—532no—
313#127—531no—
314#127—1061no—
315#127—1061no—
316#127—1061no—
317#124—1061no—
318#121—1061no—
319#121—1061no—
320C4×Q8—21no—
321C4×Q8—21no—
322C4×Q8D21211yescharacteristic
323C4×Q8D21211yescharacteristic
324C4×Q8C22×D5311yescharacteristic
325C4×Q8C22×D5311yescharacteristic
326C4×D4—1061no—
327C4×D4—1061no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4D21211yescharacteristic
330C2×C42—21no—
331C2×C42C22×D5311yescharacteristic
332#107—1061no—
333#107—1061no—
334C4⋊C8—22no—
335C4⋊C8—22no—
336C4⋊C8—531no—
337C4⋊C8—531no—
338C4⋊C8—532no—
339C4⋊C8—1061no—
340C4⋊C8C22⋊D5312yes—
341C4⋊C8C22×D5312yes—
342#103—1061no—
343#103—1061no—
344#102—1062no—
345#102—1062no—
346#102—1062no—
347#102—1062no—
348#102—1061no—
349#102—1061no—
350C22⋊C8—21no—
351C22⋊C8—1061no—
352C22⋊C8C22⋊D5311yescharacteristic
353C22⋊C8C22⋊D5311yescharacteristic
354#97—22no—
355#97—21no—
356#97—1061no—
357C4×C8—22no—
358C4×C8C22×D5312yes—
359C24—2121no—
360#41—4242no—
361C2×Q8—21no—
362C2×Q8—22no—
363C2×Q8—41no—
364C2×Q8—1061no—
365C2×Q8C2×D21211yescharacteristic
366C2×Q8C2×D21211yescharacteristic
367C2×D4—1062no—
368C2×D4—1061no—
369C2×D4—1061no—
370C2×D4—1062no—
371C2×D4—1061no—
372C2×D4—1062no—
373C2×D4—4242no—
374C2×D4—4242no—
375C2×D4—2122no—
376C2×D4—2122no—
377C2×D4—2121no—
378C2×D4—2121no—
379C2×D4—2121no—
380C2×D4—2121no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—2121no—
384C22×C4C2×D21211yescharacteristic
385C22×C4#1171994511yescharacteristic
386C22×C4C2×C22⋊D5311yescharacteristic
387SD16—2124no—
388SD16—2122no—
389SD16—2124no—
390SD16—2122no—
391D8—4242no—
392D8—4242no—
393C2×C8—22no—
394C2×C8—1061no—
395C2×C8—1061no—
396C2×C8—1062no—
397C2×C8—2122no—
398C2×C8C2×C22⋊D5312yes—
399C2×C8C2×C22⋊D5312yes—
400C4⋊C4—21no—
401C4⋊C4—21no—
402C4⋊C4—21no—
403C4⋊C4—21no—
404C4⋊C4—21no—
405C4⋊C4—1061no—
406C4⋊C4—1061no—
407C4⋊C4C2×D21211yescharacteristic
408C4⋊C4C2×D21211yescharacteristic
409C22⋊C4—2121no—
410C22⋊C4—2121no—
411C42—21no—
412C42—21no—
413C42—21no—
414C42—21no—
415C42C2×D21211yescharacteristic
416C42#1171994511yescharacteristic
417C42C2×C22⋊D5311yescharacteristic
418C42C23×D5311yescharacteristic
419C23—4242no—
420C23—4242no—
421C23—4242no—
422C23—2121no—
423C23—2122no—
424C23#1313319111yescharacteristic
425Q8—21no—
426Q8—21no—
427Q8—21no—
428Q8—42no—
429Q8—4241no—
430Q8#1313324512yes—
431D4—4242no—
432D4—4241no—
433D4—4242no—
434D4—4242no—
435D4—4242no—
436D4—4242no—
437D4—2122no—
438D4—2121no—
439D4—2121no—
440D4—2122no—
441D4—2122no—
442D4—2122no—
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#1313319111yescharacteristic
457C2×C4#1313319111yescharacteristic
458C2×C4#1313319111yescharacteristic
459C2×C4C22×C22⋊D5311yescharacteristic
460C2×C4C22×D21211yescharacteristic
461C2×C4C2×#1171994511yescharacteristic
462C8—22no—
463C8—22no—
464C8—42no—
465C8—2122no—
466C8—2121no—
467C8—2121no—
468C22—21no—
469C22—21no—
470C22—8484no—
471C22—4241no—
472C22—4242no—
473C22—4242no—
474C22—4241no—
475C22—4242no—
476C22—4241no—
477C22#1681089312yes—
478C22C2×#1313319111yescharacteristic
479C4—21no—
480C4—21no—
481C4—21no—
482C4—21no—
483C4—22no—
484C4—41no—
485C4—41no—
486C4—41no—
487C4—4241no—
488C4C2×#1313324511yescharacteristic
489C4#1681142311yescharacteristic
490C4#1681143112yes—
491C2—22no—
492C2—8482no—
493C2—8481no—
494C2#2028423611yescharacteristic
495C2#2028433411yescharacteristic
496C2C2×#1681089311yescharacteristic
subgroupG/H
centerC22C2×#13133191
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#2984C2
socleC22×C53C2×#121
prime radical#20285635C2
hypercenter#2984D53
hypocenterC53#19470
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2984

Complements

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

NH ≅ G/Ntype
2. #20284837C2semidirect
3. C2×#16810823C2semidirect
4. #20274255C2semidirect
5. #20274199C2semidirect
6. #20274131C2semidirect
7. #20274075C2semidirect
8. #20273500C2semidirect
9. #20273428C2semidirect
10. #20273411C2semidirect
11. C53×#2984C2semidirect
18. #16810918C22semidirect
19. #16810918C22semidirect
24. #16810823C22semidirect
25. #16810823C22semidirect
35. #16810376C22semidirect
36. #16810374C22semidirect
37. #16810371C22semidirect
40. #16810366C22semidirect
43. C53×#446C22semidirect
44. C8×Q8×C53C22semidirect
246. #2984D53semidirect
280. #446C2×D53semidirect
281. C8×Q8C2×D53semidirect
297. C53#19470semidirect

Maximal subgroups

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

subgroupindex
#202856352
#202848372
C2×#168108232
#202742552
#202741992
#202741312
#202740752
#202735002
#202734282
#202734112
C53×#29842
#1947053