Finite Group Databasebeta

← #27367622

Subgroups — #27367622

Order 13568.

704 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285630C211yesmaximal, characteristic
2#20285091C211yesmaximal, characteristic, complemented
3#20283459C211yesmaximal, characteristic, complemented
4#20274278C212yesmaximal, complemented
5#20274250C212yesmaximal, complemented
6#20274154C212yesmaximal, complemented
7#20274126C212yesmaximal, complemented
8#20273465C211yesmaximal, characteristic, complemented
9#20273372C211yesmaximal, characteristic, complemented
10C2×#16810366C211yesmaximal, characteristic, complemented
11C53×#3246C211yesmaximal, characteristic, complemented
12C2×#13133224—21no—
13C2×#13133214C2211yescharacteristic
14#16810999—21no—
15#16810989—22no—
16#16810971—21no—
17#16810971C2212yes—
18#16810971C2212yes—
19#16810818C2212yes—
20#16810818C2212yes—
21#16810756—22no—
22#16810701C2212yescomplemented
23#16810695—22no—
24D4×#88322C2212yescomplemented
25#16810676C2211yescharacteristic
26C2×C4⋊#88322C2211yescharacteristic
27#16810627—21no—
28#16810627—21no—
29#16810590—21no—
30C4×#11719917C2211yescharacteristic
31#16810375C2212yescomplemented
32#16810373C2212yescomplemented
33#16810372C2211yescharacteristic, complemented
34#16810372C2211yescharacteristic, complemented
35#16810369C2211yescharacteristic, complemented
36#16810369C2211yescharacteristic, complemented
37#16810366C2212yes—
38#16810366C2212yes—
39C53×#533C2211yescharacteristic
40C53×#532C2211yescharacteristic
41C53×#476C2212yescomplemented
42C53×#475—21no—
43C53×#465—22no—
44C53×#461C2212yescomplemented
45C53×#460—21no—
46C2×C4⋊C8×C53C2211yescharacteristic
47C22×D212—21no—
48C22×D212—21no—
49C22×D212—21no—
50C2×#11719922—22no—
51C2×#11719922—22no—
52#13133224—44no—
53C2×#11719917—22no—
54C2×#11719917D411yescharacteristic
55C2×#11719917D411yescharacteristic
56C22×#88322—22no—
57C22×#88322—21no—
58#13133214—21no—
59#13133214—21no—
60#13133214—21no—
61#13133214C2312yes—
62C4⋊#88322—21no—
63C4⋊#88322C2311yescharacteristic
64C4⋊#88322C2311yescharacteristic
65C4⋊#88322C2311yescharacteristic
66C4⋊#88322C2311yescharacteristic
67#13133137—22no—
68#13133137—22no—
69#13133137—22no—
70#13133137—22no—
71#13133137—22no—
72#13133137—22no—
73C22⋊#88322—22no—
74C22⋊#88322—22no—
75C22⋊#88322—21no—
76#13133128D412yes—
77C4×#88322—21no—
78C4×#88322C2312yes—
79C2×#41×C53—21no—
80C2×SD16×C53—22no—
81C22×C8×C53—21no—
82C4⋊Q8×C53D411yescharacteristic
83C4⋊Q8×C53C2311yescharacteristic
84C4⋊Q8×C53C2311yescharacteristic
85#127×C53D411yescharacteristic
86#125×C53—21no—
87#124×C53—21no—
88C22⋊Q8×C53—21no—
89C22⋊Q8×C53—21no—
90#121×C53—21no—
91#121×C53D412yes—
92C4×Q8×C53—21no—
93C4×Q8×C53—21no—
94C4×D4×C53—21no—
95C4×D4×C53C2312yes—
96#117×C53—21no—
97C2×C42×C53C2311yescharacteristic
98#106×C53—22no—
99C4⋊C8×C53—21no—
100C4⋊C8×C53C2312yes—
101#103×C53—22no—
102#102×C53—22no—
103C23×D53—41no—
104#41×C53—42no—
105#41×C53—42no—
106C2×Q8×C53—21no—
107C2×Q8×C53—21no—
108C2×Q8×C53—21no—
109C2×D4×C53—22no—
110C2×D4×C53—21no—
111C2×D4×C53—21no—
112C2×D4×C53C2×D412yes—
113C22×C4×C53—22no—
114C22×C4×C53C2×D411yescharacteristic
115C22×C4×C53C2×D411yescharacteristic
116C22×C4×C53#4111yescharacteristic
117C2×D212—22no—
118C2×D212—22no—
119C2×D212—22no—
120C2×D212—42no—
121C2×D212—42no—
122C2×D212—41no—
123C2×D212—41no—
124C2×D212—41no—
125C2×D212—81no—
126C2×D212—81no—
127C2×D212—82no—
128SD16×C53—82no—
129C2×C8×C53—22no—
130C2×C8×C53—81no—
131C4⋊C4×C53—21no—
132C4⋊C4×C53—21no—
133C4⋊C4×C53—21no—
134C4⋊C4×C53—21no—
135C4⋊C4×C53—21no—
136C4⋊C4×C53—41no—
137C4⋊C4×C53—41no—
138C4⋊C4×C53C2×D412yes—
139C22⋊C4×C53—22no—
140C22⋊C4×C53—22no—
141C22⋊C4×C53—41no—
142C42×C53—41no—
143C42×C53C2×D411yescharacteristic
144C42×C53C2×D411yescharacteristic
145C42×C53#4111yescharacteristic
146C42×C53C2411yescharacteristic
147#11719922—82no—
148#11719922—82no—
149#11719917—22no—
150#11719917—22no—
151#11719917—44no—
152C2×#88322—21no—
153C2×#88322—21no—
154C2×#88322—21no—
155C2×#88322—21no—
156C2×#88322—44no—
157C2×#88322—42no—
158C2×#88322C2×D412yes—
159C2×#88322C2×D412yes—
160C23×C53—22no—
161C23×C53#12111yescharacteristic
162C22×D53—42no—
163C22×D53—82no—
164C22×D53—82no—
165C22×D53—82no—
166Q8×C53—42no—
167Q8×C53—41no—
168Q8×C53—41no—
169Q8×C53—81no—
170D4×C53—22no—
171D4×C53—22no—
172D4×C53—44no—
173D4×C53—42no—
174D4×C53—81no—
175C2×C4×C53—21no—
176C2×C4×C53—21no—
177C2×C4×C53—22no—
178C2×C4×C53—41no—
179C2×C4×C53—42no—
180C2×C4×C53—42no—
181C2×C4×C53—41no—
182C2×C4×C53—41no—
183C2×C4×C53#12111yescharacteristic
184C2×C4×C53#12111yescharacteristic
185C2×C4×C53#12111yescharacteristic
186C2×C4×C53#13512yes—
187C2×C4×C53AGL(1,ℤ/8)12yes—
188C2×C4×C53C22×D411yescharacteristic
189C2×C4×C53C22×D411yescharacteristic
190C2×C4×C53C2×#4111yescharacteristic
191D212—42no—
192D212—42no—
193D212—42no—
194D212—42no—
195D212—82no—
196D212—82no—
197D212—82no—
198D212—82no—
199C8×C53—42no—
200#88322—22no—
201#88322—22no—
202#88322—41no—
203#88322—41no—
204#88322—41no—
205#88322—41no—
206#19234—531nomaximal, Sylow
207C22×C53—21no—
208C22×C53—21no—
209C22×C53—42no—
210C22×C53—42no—
211C22×C53—42no—
212C22×C53#46312yes—
213C22×C53C2×#12111yescharacteristic
214C2×D53—82no—
215C2×D53—82no—
216C2×D53—82no—
217C2×D53—164no—
218C4×C53—21no—
219C4×C53—21no—
220C4×C53—21no—
221C4×C53—42no—
222C4×C53—81no—
223C4×C53—81no—
224C4×C53#46312yes—
225C4×C53C2×#13511yescharacteristic
226C4×C53C2×AGL(1,ℤ/8)11yescharacteristic
227C4×C53#57611yescharacteristic
228C4×C53#57711yescharacteristic
229#3246D5311yescharacteristic, complemented
230#3205—531no—
231#3138—531no—
232#2935—531no—
233#1701—532no—
234#1697—532no—
235#1681—532no—
236#1677—532no—
237#1565—531no—
238#1542—531no—
239C2×#331—531no—
240C2×C53—22no—
241C2×C53—82no—
242C2×C53C2×#46311yescharacteristic
243C2×C53#306211yescharacteristic
244C2×C53#306811yescharacteristic
245D53—162no—
246C2×#131—1061no—
247#533C2×D5311yescharacteristic
248#532C2×D5311yescharacteristic
249C2×#127—531no—
250#493—1061no—
251#490—1062no—
252#486—532no—
253#486—532no—
254#486—1061no—
255#476C2×D5312yescomplemented
256#475—21no—
257#465—22no—
258#461C2×D5312yescomplemented
259#460—21no—
260#459—532no—
261#459—532no—
262#447—1062no—
263#436—532no—
264#435—1062no—
265C8×D4—532no—
266#431—531no—
267C2×C4⋊C8—531no—
268C2×C4⋊C8C2×D5311yescharacteristic
269#418—1061no—
270#418—1061no—
271#408—1061no—
272C4×M16—531no—
273#335—532no—
274#334—532no—
275#333—531no—
276#333—531no—
277#332—531no—
278#332—531no—
279#331—532no—
280#331—532no—
281C53#1923411yescharacteristic, complemented, Sylow
282C2×#41—21no—
283C22×D4—1061no—
284C22×D4—1061no—
285C22×D4—1061no—
286C2×SD16—22no—
287C2×D8—1062no—
288C2×D8—1062no—
289#131—2124no—
290C2×M16—531no—
291C2×M16—531no—
292C2×M16—1062no—
293C22×C8—21no—
294C22×C8—1062no—
295C22×C8—1061no—
296C4⋊Q8C22⋊D5311yescharacteristic
297C4⋊Q8C22×D5311yescharacteristic
298C4⋊Q8C22×D5311yescharacteristic
299#127—532no—
300#127—1061no—
301#127—1061no—
302#127—1061no—
303#127C22⋊D5311yescharacteristic
304#125—21no—
305#124—21no—
306C22⋊Q8—21no—
307C22⋊Q8—21no—
308#121—21no—
309#121C22⋊D5312yes—
310C4×Q8—21no—
311C4×Q8—21no—
312C4×D4—21no—
313C4×D4C22×D5312yes—
314#117—21no—
315C2×C42C22×D5311yescharacteristic
316#106—22no—
317C4⋊C8—21no—
318C4⋊C8—531no—
319C4⋊C8—531no—
320C4⋊C8—531no—
321C4⋊C8—531no—
322C4⋊C8—1061no—
323C4⋊C8C22×D5312yes—
324#103—22no—
325#102—22no—
326#102—1062no—
327#102—1062no—
328#102—1062no—
329#102—1062no—
330#102—1062no—
331#102—1062no—
332C22⋊C8—1062no—
333C22⋊C8—1062no—
334C22⋊C8—1061no—
335#97—532no—
336C4×C8—532no—
337C4×C8—1061no—
338C24—2121no—
339#41—42no—
340#41—42no—
341C2×Q8—21no—
342C2×Q8—21no—
343C2×Q8—21no—
344C2×D4—22no—
345C2×D4—21no—
346C2×D4—21no—
347C2×D4—1062no—
348C2×D4—1062no—
349C2×D4—1062no—
350C2×D4—4241no—
351C2×D4—4241no—
352C2×D4—4242no—
353C2×D4—2121no—
354C2×D4—2121no—
355C2×D4—2121no—
356C2×D4—2122no—
357C2×D4—2122no—
358C2×D4C2×C22⋊D5312yes—
359C22×C4—22no—
360C22×C4D4×D5311yescharacteristic
361C22×C4#1171994711yescharacteristic
362C22×C4C2×C22⋊D5311yescharacteristic
363SD16—82no—
364D8—4242no—
365D8—4242no—
366M16—1062no—
367M16—1062no—
368M16—2124no—
369C2×C8—22no—
370C2×C8—81no—
371C2×C8—532no—
372C2×C8—532no—
373C2×C8—1061no—
374C2×C8—1061no—
375C2×C8—1061no—
376C2×C8—1061no—
377C2×C8—2124no—
378C2×C8—2122no—
379C4⋊C4—21no—
380C4⋊C4—21no—
381C4⋊C4—21no—
382C4⋊C4—21no—
383C4⋊C4—21no—
384C4⋊C4—41no—
385C4⋊C4—41no—
386C4⋊C4C2×C22⋊D5312yes—
387C22⋊C4—22no—
388C22⋊C4—22no—
389C22⋊C4—41no—
390C42—41no—
391C42D4×D5311yescharacteristic
392C42#1171994711yescharacteristic
393C42C2×C22⋊D5311yescharacteristic
394C42C23×D5311yescharacteristic
395C23—22no—
396C23—4242no—
397C23—4242no—
398C23—4242no—
399C23—2122no—
400C23#1313318911yescharacteristic
401Q8—42no—
402Q8—41no—
403Q8—41no—
404Q8—81no—
405D4—22no—
406D4—22no—
407D4—44no—
408D4—42no—
409D4—81no—
410D4—4242no—
411D4—4242no—
412D4—4242no—
413D4—4242no—
414D4—2122no—
415D4—2122no—
416D4—2122no—
417D4—2122no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—22no—
421C2×C4—41no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4#1313318911yescharacteristic
427C2×C4#1313318911yescharacteristic
428C2×C4#1313318911yescharacteristic
429C2×C4#1313323812yes—
430C2×C4#1313324412yes—
431C2×C4C2×D4×D5311yescharacteristic
432C2×C4C22×C22⋊D5311yescharacteristic
433C2×C4C2×#1171994711yescharacteristic
434C8—42no—
435C8—1062no—
436C8—1062no—
437C8—2121no—
438C8—2121no—
439C8—2121no—
440C8—2121no—
441C22—21no—
442C22—21no—
443C22—42no—
444C22—42no—
445C22—42no—
446C22—8484no—
447C22—4242no—
448C22—4242no—
449C22—4242no—
450C22#1681084012yes—
451C22C2×#1313318911yescharacteristic
452C4—21no—
453C4—21no—
454C4—21no—
455C4—42no—
456C4—81no—
457C4—81no—
458C4#1681084012yes—
459C4C2×#1313323811yescharacteristic
460C4C2×#1313324411yescharacteristic
461C4#1681142611yescharacteristic
462C4#1681143111yescharacteristic
463C2—22no—
464C2—82no—
465C2—8482no—
466C2C2×#1681084011yescharacteristic
467C2#2028439711yescharacteristic
468C2#2028444511yescharacteristic
subgroupG/H
centerC22C2×#13133189
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#3246C2
socleC22×C53C2×#121
prime radical#20285630C2
hypercenter#3246D53
hypocenterC53#19234
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3246

Complements

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

NH ≅ G/Ntype
2. #20285091C2semidirect
3. #20283459C2semidirect
4. #20274278C2semidirect
5. #20274250C2semidirect
6. #20274154C2semidirect
7. #20274126C2semidirect
8. #20273465C2semidirect
9. #20273372C2semidirect
10. C2×#16810366C2semidirect
11. C53×#3246C2semidirect
22. #16810701C22semidirect
24. D4×#88322C22semidirect
31. #16810375C22semidirect
32. #16810373C22semidirect
33. #16810372C22semidirect
34. #16810372C22semidirect
35. #16810369C22semidirect
36. #16810369C22semidirect
41. C53×#476C22semidirect
44. C53×#461C22semidirect
229. #3246D53semidirect
255. #476C2×D53semidirect
258. #461C2×D53semidirect
281. C53#19234semidirect

Maximal subgroups

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

subgroupindex
#202856302
#202850912
#202834592
#202742782
#202742502
#202741542
#202741262
#202734652
#202733722
C2×#168103662
C53×#32462
#1923453