Finite Group Databasebeta

← #27368492

Subgroups — #27368492

Order 13568.

682 conjugacy classes of subgroups.

The subgroup lattice has 488 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285091C211yesmaximal, characteristic, complemented
2#20284195C211yesmaximal, characteristic
3#20283729C211yesmaximal, characteristic, complemented
4#20274285C212yesmaximal, complemented
5#20274257C212yesmaximal, complemented
6#20274154C212yesmaximal, complemented
7#20274126C212yesmaximal, complemented
8#20273483C211yesmaximal, characteristic, complemented
9#20273445C211yesmaximal, characteristic, complemented
10#20273428C211yesmaximal, characteristic, complemented
11C53×#3116C211yesmaximal, characteristic, complemented
12C2×#13133242—21no—
13C2×#13133214C2211yescharacteristic
14#16810989—21no—
15#16810989—21no—
16#16810971C2212yes—
17#16810971C2212yes—
18#16810823—21no—
19#16810823C2212yes—
20#16810818—21no—
21#16810818C2212yes—
22#16810754C2212yescomplemented
23#16810695—22no—
24D4×#88322C2212yescomplemented
25#16810680—21no—
26#16810680C2211yescharacteristic
27#16810647C2211yescharacteristic
28#16810627—22no—
29#16810627—21no—
30C4×#11719917C2211yescharacteristic
31C2×#13133128—21no—
32#16810375C2212yescomplemented
33#16810373C2212yescomplemented
34#16810372C2212yescomplemented
35#16810369C2212yescomplemented
36#16810366C2212yes—
37#16810366C2212yes—
38C53×#533C2211yescharacteristic
39C4×#41×C53C2211yescharacteristic
40C53×#485—21no—
41C53×#477C2211yescharacteristic, complemented
42C53×#476C2211yescharacteristic, complemented
43C53×#462C2211yescharacteristic, complemented
44C53×#461C2211yescharacteristic, complemented
45C53×#452—21no—
46C53×#441—21no—
47C53×#440—21no—
48C53×#432C2211yescharacteristic
49C22×D212—21no—
50C22×D212—21no—
51C22×D212—21no—
52#13133242—44no—
53C2×#11719924—22no—
54C2×#11719922—22no—
55C2×#11719917—21no—
56C2×#11719917D412yes—
57C22×#88322—22no—
58#13133214—21no—
59#13133214—21no—
60#13133214—21no—
61#13133214C2312yes—
62C4⋊#88322—21no—
63C4⋊#88322—22no—
64C4⋊#88322C2312yes—
65C4⋊#88322C2312yes—
66#13133137—22no—
67#13133137—22no—
68#13133137—22no—
69#13133137—22no—
70#13133137—22no—
71#13133137—22no—
72C22⋊#88322—22no—
73C22⋊#88322—21no—
74#13133128—21no—
75#13133128—22no—
76#13133128D411yescharacteristic
77#13133128D411yescharacteristic
78C4×#88322C2312yes—
79C2×#41×C53D411yescharacteristic
80C2×Q16×C53—21no—
81C2×SD16×C53—21no—
82C4⋊Q8×C53C2311yescharacteristic
83C4⋊Q8×C53C2311yescharacteristic
84#125×C53—21no—
85C22⋊Q8×C53—21no—
86C22⋊Q8×C53—21no—
87C4×Q8×C53—21no—
88C4×Q8×C53—21no—
89C4×Q8×C53D411yescharacteristic
90C4×Q8×C53C2311yescharacteristic
91C4×D4×C53—21no—
92C4×D4×C53—21no—
93C4×D4×C53D411yescharacteristic
94C4×D4×C53C2311yescharacteristic
95#117×C53—21no—
96#117×C53—21no—
97#117×C53D411yescharacteristic
98C2×C42×C53—21no—
99C2×C42×C53C2311yescharacteristic
100#107×C53—21no—
101#106×C53—21no—
102C4⋊C8×C53C2311yescharacteristic
103C4⋊C8×C53C2311yescharacteristic
104#103×C53—21no—
105#103×C53—21no—
106#103×C53—21no—
107#102×C53—21no—
108C22⋊C8×C53—21no—
109#97×C53—21no—
110C23×D53—41no—
111#41×C53—22no—
112#41×C53—41no—
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—44no—
130C2×D212—41no—
131C2×D212—41no—
132C2×D212—41no—
133C2×D212—82no—
134Q16×C53—42no—
135SD16×C53—42no—
136C2×C8×C53—21no—
137C2×C8×C53—21no—
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#11719922—44no—
162#11719917—42no—
163#11719917—42no—
164C2×#88322—22no—
165C2×#88322—22no—
166C2×#88322—44no—
167C2×#88322C2×D412yes—
168C2×#88322C2×D412yes—
169C23×C53—21no—
170C23×C53#12111yescharacteristic
171C22×D53—42no—
172C22×D53—82no—
173C22×D53—82no—
174C22×D53—82no—
175Q8×C53—21no—
176Q8×C53—21no—
177Q8×C53—41no—
178Q8×C53—41no—
179Q8×C53—81no—
180D4×C53—21no—
181D4×C53—42no—
182D4×C53AGL(1,ℤ/8)12yes—
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—41no—
190C2×C4×C53—42no—
191C2×C4×C53—42no—
192C2×C4×C53—41no—
193C2×C4×C53—41no—
194C2×C4×C53—81no—
195C2×C4×C53#12111yescharacteristic
196C2×C4×C53#12111yescharacteristic
197C2×C4×C53#12111yescharacteristic
198C2×C4×C53C22×D411yescharacteristic
199C2×C4×C53C22×D411yescharacteristic
200C2×C4×C53C2×#4111yescharacteristic
201D212—42no—
202D212—42no—
203D212—44no—
204D212—82no—
205D212—82no—
206D212—82no—
207D212—82no—
208C8×C53—41no—
209C8×C53—41no—
210#88322—22no—
211#88322—22no—
212#88322—42no—
213#88322—42no—
214#19316—531nomaximal, Sylow
215C22×C53—21no—
216C22×C53—21no—
217C22×C53—22no—
218C22×C53—21no—
219C22×C53—41no—
220C22×C53—41no—
221C22×C53C2×#12111yescharacteristic
222C2×D53—82no—
223C2×D53—82no—
224C2×D53—82no—
225C2×D53—164no—
226C4×C53—21no—
227C4×C53—21no—
228C4×C53—21no—
229C4×C53—21no—
230C4×C53—41no—
231C4×C53—41no—
232C4×C53—41no—
233C4×C53—81no—
234C4×C53—81no—
235C4×C53C2×AGL(1,ℤ/8)11yescharacteristic
236C4×C53#57511yescharacteristic
237C4×C53#57512yes—
238#3138—531no—
239#3116D5311yescharacteristic, complemented
240#3038—531no—
241#2970—531no—
242#1702—532no—
243#1698—532no—
244#1681—532no—
245#1677—532no—
246#1569—531no—
247#1559—531no—
248#1555—531no—
249C2×C53—41no—
250C2×C53—42no—
251C2×C53#304011yescharacteristic
252C2×C53#305511yescharacteristic
253C2×C53#306811yescharacteristic
254D53—162no—
255C2×AGL(1,ℤ/8)—1061no—
256#533C2×D5311yescharacteristic
257C2×#127—531no—
258C4×#41C2×D5311yescharacteristic
259#490—1061no—
260#490—1061no—
261#486—532no—
262#486—532no—
263#485—21no—
264#477C2×D5311yescharacteristic, complemented
265#476C2×D5311yescharacteristic, complemented
266#462C2×D5311yescharacteristic, complemented
267#461C2×D5311yescharacteristic, complemented
268#460—532no—
269#460—1061no—
270#459—532no—
271#459—1061no—
272#452—21no—
273#446—532no—
274#441—21no—
275#440—21no—
276#435—1062no—
277C8×D4—532no—
278#432—531no—
279#432—1061no—
280#432C2×D5311yescharacteristic
281#423—531no—
282#418—1062no—
283#418—1061no—
284C4×M16—531no—
285C2×#97—1061no—
286#335—532no—
287#334—532no—
288#333—532no—
289#332—532no—
290#331—532no—
291#331—532no—
292C53#1931611yescharacteristic, complemented, Sylow
293C2×#41C22⋊D5311yescharacteristic
294C22×D4—1061no—
295C22×D4—1061no—
296C22×D4—1061no—
297AGL(1,ℤ/8)—2124no—
298C2×Q16—21no—
299C2×SD16—21no—
300C2×SD16—1062no—
301C2×D8—1062no—
302C2×M16—532no—
303C2×M16—1061no—
304C22×C8—1062no—
305C4⋊Q8C22×D5311yescharacteristic
306C4⋊Q8C22×D5311yescharacteristic
307#127—532no—
308#127—1061no—
309#127—1061no—
310#127—1061no—
311#125—21no—
312C22⋊Q8—21no—
313C22⋊Q8—21no—
314C4×Q8—21no—
315C4×Q8—21no—
316C4×Q8C22⋊D5311yescharacteristic
317C4×Q8C22×D5311yescharacteristic
318C4×D4—21no—
319C4×D4—21no—
320C4×D4C22⋊D5311yescharacteristic
321C4×D4C22×D5311yescharacteristic
322#117—21no—
323#117—21no—
324#117C22⋊D5311yescharacteristic
325C2×C42—21no—
326C2×C42C22×D5311yescharacteristic
327#107—21no—
328#106—21no—
329C4⋊C8—532no—
330C4⋊C8—532no—
331C4⋊C8—1062no—
332C4⋊C8—1061no—
333C4⋊C8C22×D5311yescharacteristic
334C4⋊C8C22×D5311yescharacteristic
335#103—21no—
336#103—21no—
337#103—21no—
338#102—21no—
339#102—1062no—
340#102—1062no—
341#102—1062no—
342#102—1062no—
343#102—1062no—
344#102—1062no—
345C22⋊C8—21no—
346C22⋊C8—1062no—
347C22⋊C8—1061no—
348#97—21no—
349#97—531no—
350#97—531no—
351#97—1062no—
352#97—1061no—
353C4×C8—532no—
354C24—2121no—
355#41—22no—
356#41—41no—
357C2×Q8—21no—
358C2×Q8—21no—
359C2×Q8C2×C22⋊D5311yescharacteristic
360C2×D4—21no—
361C2×D4—1062no—
362C2×D4—1062no—
363C2×D4—1062no—
364C2×D4—4242no—
365C2×D4—2122no—
366C2×D4—2122no—
367C2×D4—2124no—
368C2×D4—2121no—
369C2×D4—2121no—
370C2×D4—2121no—
371C2×D4C2×C22⋊D5311yescharacteristic
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4D4×D5311yescharacteristic
376C22×C4#1171994711yescharacteristic
377C22×C4C2×C22⋊D5311yescharacteristic
378Q16—42no—
379SD16—42no—
380SD16—2124no—
381D8—2124no—
382M16—2122no—
383M16—2122no—
384C2×C8—21no—
385C2×C8—21no—
386C2×C8—532no—
387C2×C8—532no—
388C2×C8—1062no—
389C2×C8—1062no—
390C2×C8—2124no—
391C4⋊C4—21no—
392C4⋊C4—21no—
393C4⋊C4—21no—
394C4⋊C4—21no—
395C4⋊C4—21no—
396C4⋊C4—21no—
397C4⋊C4—41no—
398C4⋊C4—41no—
399C4⋊C4C2×C22⋊D5311yescharacteristic
400C4⋊C4C2×C22⋊D5311yescharacteristic
401C22⋊C4—21no—
402C22⋊C4—21no—
403C22⋊C4—21no—
404C22⋊C4—41no—
405C42—21no—
406C42—21no—
407C42—21no—
408C42—41no—
409C42D4×D5311yescharacteristic
410C42#1171994711yescharacteristic
411C42C2×C22⋊D5311yescharacteristic
412C42C23×D5311yescharacteristic
413C23—21no—
414C23—4242no—
415C23—4242no—
416C23—4242no—
417C23—2122no—
418C23#1313318911yescharacteristic
419Q8—21no—
420Q8—21no—
421Q8—41no—
422Q8—41no—
423Q8—81no—
424D4—21no—
425D4—42no—
426D4—4242no—
427D4—4242no—
428D4—4242no—
429D4—4242no—
430D4—2122no—
431D4—2122no—
432D4—2124no—
433D4#1313324212yes—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—22no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—41no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—81no—
446C2×C4#1313318911yescharacteristic
447C2×C4#1313318911yescharacteristic
448C2×C4#1313318911yescharacteristic
449C2×C4C2×D4×D5311yescharacteristic
450C2×C4C22×C22⋊D5311yescharacteristic
451C2×C4C2×#1171994711yescharacteristic
452C8—41no—
453C8—41no—
454C8—1062no—
455C8—1062no—
456C8—2122no—
457C8—2122no—
458C22—21no—
459C22—21no—
460C22—21no—
461C22—22no—
462C22—41no—
463C22—41no—
464C22—8484no—
465C22—4242no—
466C22—4242no—
467C22—4242no—
468C22C2×#1313318911yescharacteristic
469C4—21no—
470C4—21no—
471C4—21no—
472C4—21no—
473C4—41no—
474C4—41no—
475C4—41no—
476C4—81no—
477C4—81no—
478C4C2×#1313324211yescharacteristic
479C4#1681142111yescharacteristic
480C4#1681142212yes—
481C2—41no—
482C2—42no—
483C2—8482no—
484C2#2028421111yescharacteristic
485C2#2028433611yescharacteristic
486C2#2028444511yescharacteristic
subgroupG/H
centerC22C2×#13133189
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#3116C2
socleC22×C53C2×#121
prime radical#20284195C2
hypercenter#3116D53
hypocenterC53#19316
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3116

Complements

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

NH ≅ G/Ntype
1. #20285091C2semidirect
3. #20283729C2semidirect
4. #20274285C2semidirect
5. #20274257C2semidirect
6. #20274154C2semidirect
7. #20274126C2semidirect
8. #20273483C2semidirect
9. #20273445C2semidirect
10. #20273428C2semidirect
11. C53×#3116C2semidirect
22. #16810754C22semidirect
24. D4×#88322C22semidirect
32. #16810375C22semidirect
33. #16810373C22semidirect
34. #16810372C22semidirect
35. #16810369C22semidirect
41. C53×#477C22semidirect
42. C53×#476C22semidirect
43. C53×#462C22semidirect
44. C53×#461C22semidirect
239. #3116D53semidirect
264. #477C2×D53semidirect
265. #476C2×D53semidirect
266. #462C2×D53semidirect
267. #461C2×D53semidirect
292. C53#19316semidirect

Maximal subgroups

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

subgroupindex
#202850912
#202841952
#202837292
#202742852
#202742572
#202741542
#202741262
#202734832
#202734452
#202734282
C53×#31162
#1931653