Finite Group Databasebeta

← #27370186

Subgroups — #27370186

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#20285630C211yesmaximal, characteristic
2C2×#16810823C211yesmaximal, characteristic, complemented
3#20283804C211yesmaximal, characteristic, complemented
4#20274257C212yesmaximal, complemented
5#20274197C212yesmaximal, complemented
6#20274133C212yesmaximal, complemented
7#20274073C212yesmaximal, complemented
8#20273501C211yesmaximal, characteristic, complemented
9#20273428C211yesmaximal, characteristic, complemented
10#20273410C211yesmaximal, characteristic, complemented
11C53×#3112C211yesmaximal, characteristic, complemented
12C22×#11719924—21no—
13C2×#13133214C2211yescharacteristic
14#16810999C2212yes—
15#16810971C2212yes—
16#16810823—21no—
17#16810823—21no—
18#16810823C2212yes—
19#16810823C2212yes—
20#16810818—22no—
21#16810756—21no—
22#16810756—21no—
23#16810754—22no—
24#16810754C2212yescomplemented
25Q8×#88322C2212yescomplemented
26#16810680—21no—
27#16810680C2211yescharacteristic
28#16810676—21no—
29#16810676C2211yescharacteristic
30C2×C4⋊#88322C2211yescharacteristic
31C2×#13133137—21no—
32#16810375C2212yescomplemented
33#16810373C2212yescomplemented
34#16810372C2212yescomplemented
35#16810367C2212yescomplemented
36#16810366C2212yes—
37#16810366C2212yes—
38C53×#532C2211yescharacteristic
39C2×C4×Q8×C53C2211yescharacteristic
40C53×#480—21no—
41C53×#477C2211yescharacteristic, complemented
42C53×#477C2211yescharacteristic, complemented
43C53×#460C2211yescharacteristic, complemented
44C53×#460C2211yescharacteristic, complemented
45C53×#448—21no—
46C53×#440—21no—
47C53×#440—21no—
48C53×#432C2211yescharacteristic
49C22×D212—21no—
50C22×D212—21no—
51C22×D212—21no—
52C2×#11719924—22no—
53C2×#11719924—22no—
54C2×#11719924—44no—
55C2×#11719922—22no—
56C2×#11719922—22no—
57C22×#88322—21no—
58#13133214—21no—
59#13133214—21no—
60#13133214—21no—
61#13133214C2312yes—
62C4⋊#88322—21no—
63C4⋊#88322—22no—
64C4⋊#88322—22no—
65C4⋊#88322D412yes—
66C4⋊#88322C2312yes—
67C4⋊#88322C2312yes—
68#13133137—22no—
69#13133137—22no—
70#13133137—22no—
71#13133137—22no—
72C22⋊#88322—21no—
73C22⋊#88322D411yescharacteristic
74C22⋊#88322D411yescharacteristic
75#13133128—21no—
76#13133128—22no—
77C4×#88322—22no—
78C4×#88322C2312yes—
79C2×#41×C53—21no—
80C22×Q8×C53D411yescharacteristic
81C2×SD16×C53—21no—
82C2×SD16×C53—21no—
83C4⋊Q8×C53C2311yescharacteristic
84#127×C53C2311yescharacteristic
85#124×C53—21no—
86#121×C53—21no—
87#121×C53—21no—
88C4×Q8×C53—21no—
89C4×Q8×C53—21no—
90C4×Q8×C53D411yescharacteristic
91C4×Q8×C53D411yescharacteristic
92C4×Q8×C53C2311yescharacteristic
93C4×Q8×C53C2311yescharacteristic
94C4×D4×C53—21no—
95C4×D4×C53—21no—
96C2×C4⋊C4×C53—21no—
97C2×C4⋊C4×C53D411yescharacteristic
98C2×C42×C53—21no—
99C2×C42×C53C2311yescharacteristic
100#107×C53—21no—
101#107×C53—21no—
102C4⋊C8×C53C2311yescharacteristic
103C4⋊C8×C53C2311yescharacteristic
104#103×C53—21no—
105#103×C53—21no—
106#102×C53—21no—
107#102×C53—21no—
108C22⋊C8×C53—21no—
109#97×C53—21no—
110C23×D53—41no—
111#41×C53—82no—
112C2×Q8×C53—21no—
113C2×Q8×C53—22no—
114C2×Q8×C53—21no—
115C2×Q8×C53—41no—
116C2×Q8×C53C2×D411yescharacteristic
117C2×Q8×C53C2×D411yescharacteristic
118C2×D4×C53—21no—
119C2×D4×C53—21no—
120C2×D4×C53—21no—
121C2×D4×C53—41no—
122C22×C4×C53—21no—
123C22×C4×C53—21no—
124C22×C4×C53—41no—
125C22×C4×C53C2×D411yescharacteristic
126C22×C4×C53C2×D411yescharacteristic
127C22×C4×C53#4111yescharacteristic
128C2×D212—22no—
129C2×D212—22no—
130C2×D212—22no—
131C2×D212—42no—
132C2×D212—42no—
133C2×D212—41no—
134C2×D212—41no—
135C2×D212—41no—
136C2×D212—82no—
137C2×D212—82no—
138SD16×C53—42no—
139SD16×C53—42no—
140C2×C8×C53—21no—
141C2×C8×C53—21no—
142C4⋊C4×C53—21no—
143C4⋊C4×C53—21no—
144C4⋊C4×C53—21no—
145C4⋊C4×C53—21no—
146C4⋊C4×C53—21no—
147C4⋊C4×C53—21no—
148C4⋊C4×C53—21no—
149C4⋊C4×C53C2×D411yescharacteristic
150C4⋊C4×C53C2×D411yescharacteristic
151C22⋊C4×C53—41no—
152C22⋊C4×C53—41no—
153C42×C53—21no—
154C42×C53—21no—
155C42×C53—21no—
156C42×C53—21no—
157C42×C53C2×D411yescharacteristic
158C42×C53C2×D411yescharacteristic
159C42×C53#4111yescharacteristic
160C42×C53C2411yescharacteristic
161#11719924—44no—
162#11719924—44no—
163#11719922—82no—
164#11719922—82no—
165C2×#88322—22no—
166C2×#88322—22no—
167C2×#88322—42no—
168C2×#88322C2×D412yes—
169C2×#88322C2×D412yes—
170C23×C53—41no—
171C23×C53#12111yescharacteristic
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×C53—81no—
181Q8×C53AGL(1,ℤ/8)12yes—
182D4×C53—41no—
183D4×C53—41no—
184D4×C53—42no—
185D4×C53—81no—
186D4×C53—82no—
187C2×C4×C53—21no—
188C2×C4×C53—22no—
189C2×C4×C53—21no—
190C2×C4×C53—21no—
191C2×C4×C53—22no—
192C2×C4×C53—21no—
193C2×C4×C53—21no—
194C2×C4×C53—21no—
195C2×C4×C53—42no—
196C2×C4×C53—42no—
197C2×C4×C53—41no—
198C2×C4×C53—82no—
199C2×C4×C53#12111yescharacteristic
200C2×C4×C53#12111yescharacteristic
201C2×C4×C53#12111yescharacteristic
202C2×C4×C53C22×D411yescharacteristic
203C2×C4×C53C22×D411yescharacteristic
204C2×C4×C53C2×#4111yescharacteristic
205D212—42no—
206D212—42no—
207D212—42no—
208D212—42no—
209D212—82no—
210D212—82no—
211D212—82no—
212D212—82no—
213C8×C53—41no—
214C8×C53—41no—
215#88322—22no—
216#88322—22no—
217#88322—42no—
218#88322—42no—
219#19470—531nomaximal, Sylow
220C22×C53—21no—
221C22×C53—21no—
222C22×C53—81no—
223C22×C53—81no—
224C22×C53—81no—
225C22×C53#47212yes—
226C22×C53C2×#12111yescharacteristic
227C2×D53—82no—
228C2×D53—82no—
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×C53—81no—
240C4×C53C2×AGL(1,ℤ/8)11yescharacteristic
241C4×C53#57511yescharacteristic
242C4×C53#57712yes—
243#3205—531no—
244#3112D5311yescharacteristic, complemented
245C2×#460—531no—
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—161no—
256C2×C53#304311yescharacteristic
257C2×C53#305511yescharacteristic
258C2×C53C2×#47211yescharacteristic
259D53—162no—
260C22×SD16—1061no—
261#532C2×D5311yescharacteristic
262C2×#127—531no—
263C2×C4×Q8C2×D5311yescharacteristic
264#493—532no—
265#486—532no—
266#480—21no—
267#477C2×D5311yescharacteristic, complemented
268#477C2×D5311yescharacteristic, complemented
269#460—532no—
270#460—532no—
271#460—1061no—
272#460—1061no—
273#460C2×D5311yescharacteristic, complemented
274#460C2×D5311yescharacteristic, complemented
275#459—1062no—
276#448—21no—
277#447—1061no—
278#447—1061no—
279#446—532no—
280#446—1062no—
281C8×Q8—532no—
282#440—21no—
283#440—21no—
284#432—531no—
285#432—1061no—
286#432C2×D5311yescharacteristic
287#431—531no—
288#431—1061no—
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—21no—
299C22×Q8C22⋊D5311yescharacteristic
300C22×D4—1061no—
301C22×D4—1061no—
302C22×D4—1061no—
303C2×SD16—21no—
304C2×SD16—21no—
305C2×SD16—1062no—
306C2×SD16—1062no—
307C2×SD16—2124no—
308C2×D8—1062no—
309C2×D8—1062no—
310C22×C8—1061no—
311C4⋊Q8C22×D5311yescharacteristic
312#127—532no—
313#127—1061no—
314#127—1061no—
315#127—1061no—
316#127C22×D5311yescharacteristic
317#124—21no—
318#121—21no—
319#121—21no—
320C4×Q8—21no—
321C4×Q8—21no—
322C4×Q8C22⋊D5311yescharacteristic
323C4×Q8C22⋊D5311yescharacteristic
324C4×Q8C22×D5311yescharacteristic
325C4×Q8C22×D5311yescharacteristic
326C4×D4—21no—
327C4×D4—21no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4C22⋊D5311yescharacteristic
330C2×C42—21no—
331C2×C42C22×D5311yescharacteristic
332#107—21no—
333#107—21no—
334C4⋊C8—532no—
335C4⋊C8—532no—
336C4⋊C8—532no—
337C4⋊C8—1062no—
338C4⋊C8—1062no—
339C4⋊C8—1061no—
340C4⋊C8C22×D5311yescharacteristic
341C4⋊C8C22×D5311yescharacteristic
342#103—21no—
343#103—21no—
344#102—21no—
345#102—21no—
346#102—1062no—
347#102—1062no—
348#102—1062no—
349#102—1062no—
350C22⋊C8—21no—
351C22⋊C8—531no—
352C22⋊C8—531no—
353C22⋊C8—1061no—
354#97—21no—
355#97—1062no—
356#97—1061no—
357C4×C8—532no—
358C4×C8—1062no—
359C24—2121no—
360#41—82no—
361C2×Q8—21no—
362C2×Q8—22no—
363C2×Q8—21no—
364C2×Q8—41no—
365C2×Q8C2×C22⋊D5311yescharacteristic
366C2×Q8C2×C22⋊D5311yescharacteristic
367C2×D4—21no—
368C2×D4—21no—
369C2×D4—21no—
370C2×D4—41no—
371C2×D4—1062no—
372C2×D4—1062no—
373C2×D4—1062no—
374C2×D4—4242no—
375C2×D4—4242no—
376C2×D4—2122no—
377C2×D4—2122no—
378C2×D4—2121no—
379C2×D4—2121no—
380C2×D4—2121no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—41no—
384C22×C4D4×D5311yescharacteristic
385C22×C4#1171994711yescharacteristic
386C22×C4C2×C22⋊D5311yescharacteristic
387SD16—42no—
388SD16—42no—
389SD16—2124no—
390SD16—2124no—
391D8—4242no—
392D8—4242no—
393C2×C8—21no—
394C2×C8—21no—
395C2×C8—532no—
396C2×C8—532no—
397C2×C8—1062no—
398C2×C8—1062no—
399C2×C8—2122no—
400C4⋊C4—21no—
401C4⋊C4—21no—
402C4⋊C4—21no—
403C4⋊C4—21no—
404C4⋊C4—21no—
405C4⋊C4—21no—
406C4⋊C4—21no—
407C4⋊C4C2×C22⋊D5311yescharacteristic
408C4⋊C4C2×C22⋊D5311yescharacteristic
409C22⋊C4—41no—
410C22⋊C4—41no—
411C42—21no—
412C42—21no—
413C42—21no—
414C42—21no—
415C42D4×D5311yescharacteristic
416C42#1171994711yescharacteristic
417C42C2×C22⋊D5311yescharacteristic
418C42C23×D5311yescharacteristic
419C23—41no—
420C23—4242no—
421C23—4242no—
422C23—4242no—
423C23—2122no—
424C23#1313318911yescharacteristic
425Q8—21no—
426Q8—21no—
427Q8—21no—
428Q8—42no—
429Q8—81no—
430Q8#1313324212yes—
431D4—41no—
432D4—41no—
433D4—42no—
434D4—81no—
435D4—82no—
436D4—4242no—
437D4—4242no—
438D4—4242no—
439D4—4242no—
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—41no—
455C2×C4—82no—
456C2×C4#1313318911yescharacteristic
457C2×C4#1313318911yescharacteristic
458C2×C4#1313318911yescharacteristic
459C2×C4C2×D4×D5311yescharacteristic
460C2×C4C22×C22⋊D5311yescharacteristic
461C2×C4C2×#1171994711yescharacteristic
462C8—41no—
463C8—41no—
464C8—1062no—
465C8—1062no—
466C8—2122no—
467C8—2122no—
468C22—21no—
469C22—21no—
470C22—81no—
471C22—81no—
472C22—81no—
473C22—8484no—
474C22—4242no—
475C22—4242no—
476C22—4242no—
477C22#1681089112yes—
478C22C2×#1313318911yescharacteristic
479C4—21no—
480C4—21no—
481C4—21no—
482C4—21no—
483C4—22no—
484C4—41no—
485C4—41no—
486C4—41no—
487C4—81no—
488C4C2×#1313324211yescharacteristic
489C4#1681142111yescharacteristic
490C4#1681143212yes—
491C2—22no—
492C2—161no—
493C2—8482no—
494C2#2028423811yescharacteristic
495C2#2028433611yescharacteristic
496C2C2×#1681089111yescharacteristic
subgroupG/H
centerC22C2×#13133189
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#3112C2
socleC22×C53C2×#121
prime radical#20285630C2
hypercenter#3112D53
hypocenterC53#19470
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3112

Complements

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

NH ≅ G/Ntype
2. C2×#16810823C2semidirect
3. #20283804C2semidirect
4. #20274257C2semidirect
5. #20274197C2semidirect
6. #20274133C2semidirect
7. #20274073C2semidirect
8. #20273501C2semidirect
9. #20273428C2semidirect
10. #20273410C2semidirect
11. C53×#3112C2semidirect
24. #16810754C22semidirect
25. Q8×#88322C22semidirect
32. #16810375C22semidirect
33. #16810373C22semidirect
34. #16810372C22semidirect
35. #16810367C22semidirect
41. C53×#477C22semidirect
42. C53×#477C22semidirect
43. C53×#460C22semidirect
44. C53×#460C22semidirect
244. #3112D53semidirect
267. #477C2×D53semidirect
268. #477C2×D53semidirect
273. #460C2×D53semidirect
274. #460C2×D53semidirect
297. C53#19470semidirect

Maximal subgroups

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

subgroupindex
#202856302
C2×#168108232
#202838042
#202742572
#202741972
#202741332
#202740732
#202735012
#202734282
#202734102
C53×#31122
#1947053