Finite Group Databasebeta

← #27369873

Subgroups — #27369873

Order 13568.

682 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285653C211yesmaximal, characteristic, complemented
2#20284195C211yesmaximal, characteristic
3#20283832C211yesmaximal, characteristic, complemented
4#20274257C212yesmaximal, complemented
5#20274250C212yesmaximal, complemented
6#20274189C212yesmaximal, complemented
7#20274182C212yesmaximal, complemented
8#20273489C211yesmaximal, characteristic, complemented
9#20273405C211yesmaximal, characteristic, complemented
10C2×#16810366C211yesmaximal, characteristic, complemented
11C53×#3076C211yesmaximal, characteristic, complemented
12C2×#13133242—21no—
13C2×#13133214C2211yescharacteristic
14#16810997C2212yes—
15#16810971C2212yes—
16#16810823—21no—
17#16810823—21no—
18#16810823—21no—
19#16810823C2212yes—
20#16810818—21no—
21#16810818C2212yes—
22#16810773—22no—
23#16810754C2212yescomplemented
24#16810701C2212yescomplemented
25#16810695—22no—
26#16810680—21no—
27#16810676C2211yescharacteristic
28#16810647—21no—
29#16810647C2211yescharacteristic
30C2×C4⋊#88322C2211yescharacteristic
31#16810627—21no—
32#16810375C2212yescomplemented
33#16810375C2212yescomplemented
34#16810372C2212yescomplemented
35#16810369C2212yescomplemented
36#16810366C2212yes—
37#16810366C2212yes—
38C53×#533C2211yescharacteristic
39C4×#41×C53C2211yescharacteristic
40C53×#479—21no—
41C53×#478—21no—
42C53×#477C2211yescharacteristic, complemented
43C53×#476C2211yescharacteristic, complemented
44C53×#475C2211yescharacteristic, complemented
45C53×#474C2211yescharacteristic, complemented
46C53×#427—21no—
47C2×C4⋊C8×C53C2211yescharacteristic
48C53×#416—21no—
49C22×D212—21no—
50C22×D212—21no—
51C22×D212—21no—
52#13133242—44no—
53C2×#11719924—22no—
54C2×#11719924—22no—
55C2×#11719924—22no—
56C2×#11719922—22no—
57C2×#11719917—22no—
58C2×#11719917—21no—
59C22×#88322—21no—
60#13133214—21no—
61#13133214—21no—
62#13133214—21no—
63#13133214C2312yes—
64C4⋊#88322—21no—
65C4⋊#88322—21no—
66C4⋊#88322—22no—
67C4⋊#88322D411yescharacteristic
68C4⋊#88322D411yescharacteristic
69C4⋊#88322C2312yes—
70C4⋊#88322C2312yes—
71#13133137—22no—
72#13133137—22no—
73#13133137—22no—
74#13133137—22no—
75C22⋊#88322—22no—
76C22⋊#88322D412yes—
77#13133128—22no—
78C4×#88322C2312yes—
79C2×#41×C53D411yescharacteristic
80C22×C8×C53—21no—
81C4⋊Q8×C53C2311yescharacteristic
82C4⋊Q8×C53C2311yescharacteristic
83#125×C53—21no—
84C22⋊Q8×C53—21no—
85C22⋊Q8×C53—21no—
86C4×Q8×C53—21no—
87C4×Q8×C53—21no—
88C4×Q8×C53D411yescharacteristic
89C4×Q8×C53C2311yescharacteristic
90C4×D4×C53—21no—
91C4×D4×C53—21no—
92C4×D4×C53D411yescharacteristic
93C4×D4×C53C2311yescharacteristic
94#117×C53—21no—
95#117×C53—21no—
96#117×C53D411yescharacteristic
97C2×C42×C53—21no—
98C2×C42×C53C2311yescharacteristic
99#107×C53—21no—
100#107×C53—21no—
101#106×C53—21no—
102#106×C53—21no—
103C4⋊C8×C53—21no—
104C4⋊C8×C53C2311yescharacteristic
105C4⋊C8×C53C2311yescharacteristic
106#103×C53—21no—
107#103×C53—21no—
108#102×C53—21no—
109#102×C53—21no—
110C23×D53—41no—
111#41×C53—22no—
112#41×C53—22no—
113C2×Q8×C53—21no—
114C2×Q8×C53—21no—
115C2×Q8×C53C2×D411yescharacteristic
116C2×D4×C53—21no—
117C2×D4×C53C2×D411yescharacteristic
118C22×C4×C53—21no—
119C22×C4×C53—21no—
120C22×C4×C53—21no—
121C22×C4×C53C2×D411yescharacteristic
122C22×C4×C53C2×D411yescharacteristic
123C22×C4×C53#4111yescharacteristic
124C2×D212—22no—
125C2×D212—22no—
126C2×D212—22no—
127C2×D212—42no—
128C2×D212—42no—
129C2×D212—41no—
130C2×D212—41no—
131C2×D212—41no—
132C2×D212—81no—
133C2×D212—81no—
134C2×D212—82no—
135C2×C8×C53—21no—
136C2×C8×C53—21no—
137C2×C8×C53—42no—
138C4⋊C4×C53—21no—
139C4⋊C4×C53—21no—
140C4⋊C4×C53—21no—
141C4⋊C4×C53—21no—
142C4⋊C4×C53—21no—
143C4⋊C4×C53—21no—
144C4⋊C4×C53—41no—
145C4⋊C4×C53—41no—
146C4⋊C4×C53C2×D411yescharacteristic
147C4⋊C4×C53C2×D411yescharacteristic
148C22⋊C4×C53—21no—
149C22⋊C4×C53—21no—
150C22⋊C4×C53—21no—
151C22⋊C4×C53—41no—
152C42×C53—21no—
153C42×C53—21no—
154C42×C53—21no—
155C42×C53—41no—
156C42×C53C2×D411yescharacteristic
157C42×C53C2×D411yescharacteristic
158C42×C53#4111yescharacteristic
159C42×C53C2411yescharacteristic
160#11719924—44no—
161#11719924—82no—
162#11719924—82no—
163#11719922—44no—
164#11719917—42no—
165#11719917—82no—
166C2×#88322—22no—
167C2×#88322—22no—
168C2×#88322—81no—
169C2×#88322C2×D412yes—
170C2×#88322C2×D412yes—
171C23×C53—21no—
172C23×C53#12111yescharacteristic
173C22×D53—42no—
174C22×D53—82no—
175C22×D53—82no—
176C22×D53—82no—
177Q8×C53—21no—
178Q8×C53—21no—
179Q8×C53—41no—
180Q8×C53—41no—
181Q8×C53—81no—
182D4×C53—21no—
183D4×C53—21no—
184D4×C53—42no—
185C2×C4×C53—21no—
186C2×C4×C53—21no—
187C2×C4×C53—21no—
188C2×C4×C53—21no—
189C2×C4×C53—21no—
190C2×C4×C53—41no—
191C2×C4×C53—41no—
192C2×C4×C53—41no—
193C2×C4×C53—42no—
194C2×C4×C53—41no—
195C2×C4×C53—41no—
196C2×C4×C53—81no—
197C2×C4×C53#12111yescharacteristic
198C2×C4×C53#12111yescharacteristic
199C2×C4×C53#12111yescharacteristic
200C2×C4×C53AGL(1,ℤ/8)12yes—
201C2×C4×C53#13712yes—
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#19445—531nomaximal, Sylow
220C22×C53—21no—
221C22×C53—21no—
222C22×C53—41no—
223C22×C53—41no—
224C22×C53—41no—
225C22×C53#46812yes—
226C22×C53C2×#12111yescharacteristic
227C2×D53—82no—
228C2×D53—82no—
229C2×D53—82no—
230C2×D53—164no—
231C4×C53—21no—
232C4×C53—21no—
233C4×C53—21no—
234C4×C53—41no—
235C4×C53—41no—
236C4×C53—41no—
237C4×C53—81no—
238C4×C53—81no—
239C4×C53#46812yes—
240C4×C53C2×AGL(1,ℤ/8)11yescharacteristic
241C4×C53C2×#13711yescharacteristic
242C4×C53#57611yescharacteristic
243C4×C53#57711yescharacteristic
244#3208—531no—
245#3076D5311yescharacteristic, complemented
246#3038—531no—
247#2989—531no—
248#1698—532no—
249#1697—532no—
250#1686—532no—
251#1685—532no—
252#1571—531no—
253#1550—531no—
254C2×#331—531no—
255C2×C53—22no—
256C2×C53—81no—
257C2×C53#304511yescharacteristic
258C2×C53C2×#46811yescharacteristic
259C2×C53#306811yescharacteristic
260D53—162no—
261C2×AGL(1,ℤ/8)—1061no—
262#533C2×D5311yescharacteristic
263C2×#127—531no—
264C4×#41C2×D5311yescharacteristic
265#492—532no—
266#486—532no—
267#479—21no—
268#478—21no—
269#477C2×D5311yescharacteristic, complemented
270#476C2×D5311yescharacteristic, complemented
271#475C2×D5311yescharacteristic, complemented
272#474C2×D5311yescharacteristic, complemented
273#460—532no—
274#460—1061no—
275#460—1061no—
276#460—1061no—
277#459—532no—
278#459—1061no—
279#450—1062no—
280#446—532no—
281#436—532no—
282#435—1062no—
283#432—1061no—
284#431—531no—
285#427—21no—
286#423—531no—
287#423—1061no—
288C2×C4⋊C8—531no—
289C2×C4⋊C8C2×D5311yescharacteristic
290#418—1061no—
291#416—21no—
292#335—532no—
293#335—532no—
294#333—532no—
295#332—532no—
296#331—532no—
297#331—532no—
298C53#1944511yescharacteristic, complemented, Sylow
299C2×#41C22⋊D5311yescharacteristic
300C22×D4—1061no—
301C22×D4—1061no—
302C22×D4—1061no—
303AGL(1,ℤ/8)—2124no—
304C2×SD16—1062no—
305C2×SD16—1062no—
306C2×SD16—1062no—
307C2×D8—1062no—
308C2×M16—1062no—
309C2×M16—1061no—
310C22×C8—21no—
311C22×C8—1061no—
312C4⋊Q8C22×D5311yescharacteristic
313C4⋊Q8C22×D5311yescharacteristic
314#127—532no—
315#127—1061no—
316#127—1061no—
317#127—1061no—
318#125—21no—
319C22⋊Q8—21no—
320C22⋊Q8—21no—
321C4×Q8—21no—
322C4×Q8—21no—
323C4×Q8C22⋊D5311yescharacteristic
324C4×Q8C22×D5311yescharacteristic
325C4×D4—21no—
326C4×D4—21no—
327C4×D4C22⋊D5311yescharacteristic
328C4×D4C22×D5311yescharacteristic
329#117—21no—
330#117—21no—
331#117C22⋊D5311yescharacteristic
332C2×C42—21no—
333C2×C42C22×D5311yescharacteristic
334#107—21no—
335#107—21no—
336#106—21no—
337#106—21no—
338C4⋊C8—21no—
339C4⋊C8—531no—
340C4⋊C8—531no—
341C4⋊C8—532no—
342C4⋊C8—532no—
343C4⋊C8—1062no—
344C4⋊C8—1061no—
345C4⋊C8—1061no—
346C4⋊C8C22×D5311yescharacteristic
347C4⋊C8C22×D5311yescharacteristic
348#103—21no—
349#103—21no—
350#102—21no—
351#102—21no—
352#102—1062no—
353#102—1062no—
354#102—1062no—
355#102—1062no—
356C22⋊C8—532no—
357C22⋊C8—1062no—
358#97—1062no—
359C4×C8—532no—
360C24—2121no—
361#41—22no—
362#41—22no—
363C2×Q8—21no—
364C2×Q8—21no—
365C2×Q8C2×C22⋊D5311yescharacteristic
366C2×D4—21no—
367C2×D4—1062no—
368C2×D4—1062no—
369C2×D4—1062no—
370C2×D4—4241no—
371C2×D4—4241no—
372C2×D4—4242no—
373C2×D4—2121no—
374C2×D4—2121no—
375C2×D4—2121no—
376C2×D4—2122no—
377C2×D4—2122no—
378C2×D4C2×C22⋊D5311yescharacteristic
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4D4×D5311yescharacteristic
383C22×C4#1171994711yescharacteristic
384C22×C4C2×C22⋊D5311yescharacteristic
385SD16—4242no—
386SD16—4242no—
387SD16—2124no—
388D8—2124no—
389M16—4242no—
390M16—2122no—
391C2×C8—21no—
392C2×C8—21no—
393C2×C8—42no—
394C2×C8—532no—
395C2×C8—532no—
396C2×C8—1062no—
397C2×C8—1062no—
398C2×C8—4241no—
399C4⋊C4—21no—
400C4⋊C4—21no—
401C4⋊C4—21no—
402C4⋊C4—21no—
403C4⋊C4—21no—
404C4⋊C4—21no—
405C4⋊C4—41no—
406C4⋊C4—41no—
407C4⋊C4C2×C22⋊D5311yescharacteristic
408C4⋊C4C2×C22⋊D5311yescharacteristic
409C22⋊C4—21no—
410C22⋊C4—21no—
411C22⋊C4—21no—
412C22⋊C4—41no—
413C42—21no—
414C42—21no—
415C42—21no—
416C42—41no—
417C42D4×D5311yescharacteristic
418C42#1171994711yescharacteristic
419C42C2×C22⋊D5311yescharacteristic
420C42C23×D5311yescharacteristic
421C23—21no—
422C23—4242no—
423C23—4242no—
424C23—4242no—
425C23—2122no—
426C23#1313318911yescharacteristic
427Q8—21no—
428Q8—21no—
429Q8—41no—
430Q8—41no—
431Q8—81no—
432D4—21no—
433D4—21no—
434D4—42no—
435D4—4242no—
436D4—4242no—
437D4—4242no—
438D4—4242no—
439D4—2122no—
440D4—2122no—
441D4—2122no—
442D4—2122no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—41no—
449C2×C4—42no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—81no—
455C2×C4#1313318911yescharacteristic
456C2×C4#1313318911yescharacteristic
457C2×C4#1313318911yescharacteristic
458C2×C4#1313324212yes—
459C2×C4#1313324812yes—
460C2×C4C2×D4×D5311yescharacteristic
461C2×C4C22×C22⋊D5311yescharacteristic
462C2×C4C2×#1171994711yescharacteristic
463C8—41no—
464C8—41no—
465C8—1062no—
466C8—1062no—
467C8—2122no—
468C8—2122no—
469C22—21no—
470C22—21no—
471C22—41no—
472C22—41no—
473C22—41no—
474C22—8484no—
475C22—4242no—
476C22—4242no—
477C22—4242no—
478C22#1681087112yes—
479C22C2×#1313318911yescharacteristic
480C4—21no—
481C4—21no—
482C4—21no—
483C4—41no—
484C4—41no—
485C4—41no—
486C4—81no—
487C4—81no—
488C4#1681087112yes—
489C4C2×#1313324211yescharacteristic
490C4C2×#1313324811yescharacteristic
491C4#1681142511yescharacteristic
492C4#1681143011yescharacteristic
493C2—22no—
494C2—81no—
495C2—8482no—
496C2#2028425211yescharacteristic
497C2C2×#1681087111yescharacteristic
498C2#2028444511yescharacteristic
subgroupG/H
centerC22C2×#13133189
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#3076C2
socleC22×C53C2×#121
prime radical#20284195C2
hypercenter#3076D53
hypocenterC53#19445
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3076

Complements

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

NH ≅ G/Ntype
1. #20285653C2semidirect
3. #20283832C2semidirect
4. #20274257C2semidirect
5. #20274250C2semidirect
6. #20274189C2semidirect
7. #20274182C2semidirect
8. #20273489C2semidirect
9. #20273405C2semidirect
10. C2×#16810366C2semidirect
11. C53×#3076C2semidirect
23. #16810754C22semidirect
24. #16810701C22semidirect
32. #16810375C22semidirect
33. #16810375C22semidirect
34. #16810372C22semidirect
35. #16810369C22semidirect
42. C53×#477C22semidirect
43. C53×#476C22semidirect
44. C53×#475C22semidirect
45. C53×#474C22semidirect
245. #3076D53semidirect
269. #477C2×D53semidirect
270. #476C2×D53semidirect
271. #475C2×D53semidirect
272. #474C2×D53semidirect
298. C53#19445semidirect

Maximal subgroups

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

subgroupindex
#202856532
#202841952
#202838322
#202742572
#202742502
#202741892
#202741822
#202734892
#202734052
C2×#168103662
C53×#30762
#1944553