Finite Group Databasebeta

← #27367614

Subgroups — #27367614

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#20286027C211yesmaximal, characteristic, complemented
2#20285633C211yesmaximal, characteristic
3#20283461C211yesmaximal, characteristic, complemented
4#20274273C212yesmaximal, complemented
5#20274245C212yesmaximal, complemented
6#20274149C212yesmaximal, complemented
7#20274121C212yesmaximal, complemented
8#20273467C211yesmaximal, characteristic, complemented
9#20273372C211yesmaximal, characteristic, complemented
10C2×#16810367C211yesmaximal, characteristic, complemented
11C53×#3138C211yesmaximal, characteristic, complemented
12C2×#13133225—21no—
13#16811173C2211yescharacteristic
14#16810999—21no—
15#16810969—21no—
16#16810910C2212yescomplemented
17#16810908—21no—
18#16810854—22no—
19#16810825C2212yescomplemented
20#16810823—21no—
21#16810815C2212yes—
22#16810815C2212yes—
23#16810757—22no—
24#16810702C2212yescomplemented
25#16810696—22no—
26C8×D212C2212yescomplemented
27#16810676C2211yescharacteristic
28C2×C4⋊#88322C2211yescharacteristic
29C2×C4⋊#88322C2211yescharacteristic
30#16810591—21no—
31#16810376C2212yescomplemented
32#16810374C2212yescomplemented
33#16810372C2211yescharacteristic, complemented
34#16810372C2211yescharacteristic, complemented
35#16810369C2211yescharacteristic, complemented
36#16810369C2211yescharacteristic, complemented
37#16810367C2212yes—
38#16810367C2212yes—
39C53×#533C2211yescharacteristic
40C2×#127×C53C2211yescharacteristic
41C53×#490—22no—
42C53×#486C2212yes—
43C53×#486C2212yes—
44C53×#418—21no—
45C53×#418—21no—
46C4×M16×C53C2211yescharacteristic
47C2×#11719945—21no—
48C2×#11719924—22no—
49C2×#11719922—22no—
50C2×#11719922—22no—
51#13133225—44no—
52C2×#11719912—22no—
53C22×#88322—21no—
54C22×#88322—21no—
55C2×C8×D53—22no—
56#13133217D411yescharacteristic
57#13133214D411yescharacteristic
58#13133205—21no—
59#13133191—21no—
60#13133191D412yes—
61C4×D212—21no—
62C4×D212C2312yes—
63#13133148—22no—
64C4⋊#88322—21no—
65C4⋊#88322—21no—
66C4⋊#88322C2311yescharacteristic
67C4⋊#88322C2311yescharacteristic
68C4⋊#88322C2312yes—
69C4⋊#88322C2311yescharacteristic
70C4⋊#88322C2311yescharacteristic
71#13133141—22no—
72#13133138—22no—
73#13133138—22no—
74#13133137—22no—
75C22⋊#88322—21no—
76#13133129—22no—
77#13133129—22no—
78C4×#88322—21no—
79C22×D4×C53—21no—
80C22×D4×C53—21no—
81C22×D4×C53—21no—
82C2×M16×C53D411yescharacteristic
83C2×M16×C53D411yescharacteristic
84C4⋊Q8×C53C2311yescharacteristic
85C4⋊Q8×C53C2311yescharacteristic
86#127×C53—21no—
87#127×C53—21no—
88#127×C53—21no—
89#127×C53C2312yes—
90#125×C53—21no—
91C22⋊Q8×C53—21no—
92C22⋊Q8×C53—21no—
93C4×Q8×C53—21no—
94C4×Q8×C53—21no—
95#117×C53—21no—
96C2×C42×C53C2311yescharacteristic
97#102×C53—22no—
98#102×C53—22no—
99#102×C53—22no—
100#102×C53—22no—
101#97×C53D412yes—
102C4×C8×C53C2312yes—
103C24×C53—41no—
104C2×Q8×C53—21no—
105C2×Q8×C53—21no—
106C2×D4×C53—22no—
107C2×D4×C53—22no—
108C2×D4×C53—22no—
109C2×D4×C53—41no—
110C2×D4×C53—41no—
111C2×D4×C53—41no—
112C2×D4×C53—42no—
113C2×D4×C53—42no—
114C2×D4×C53—81no—
115C2×D4×C53—81no—
116C2×D4×C53—82no—
117C22×C4×C53C2×D411yescharacteristic
118C22×C4×C53C2×D411yescharacteristic
119C22×C4×C53#4111yescharacteristic
120C2×C22⋊D53—22no—
121#11719945—42no—
122#11719945—42no—
123C2×D212—21no—
124C2×D212—21no—
125C2×D212C2×D412yes—
126C2×C4×D53—22no—
127C2×Dic106—21no—
128M16×C53—22no—
129M16×C53—22no—
130C2×C8×C53C2×D412yes—
131C2×C8×C53C2×D412yes—
132C4⋊C4×C53—21no—
133C4⋊C4×C53—21no—
134C4⋊C4×C53—21no—
135C4⋊C4×C53—21no—
136C4⋊C4×C53—41no—
137C4⋊C4×C53—41no—
138C22⋊C4×C53—41no—
139C42×C53—41no—
140C42×C53C2×D411yescharacteristic
141C42×C53C2×D411yescharacteristic
142C42×C53#4111yescharacteristic
143C42×C53C2411yescharacteristic
144#11719924—82no—
145#11719922—82no—
146#11719922—82no—
147#11719921—22no—
148#11719921—22no—
149C4⋊Dic53—21no—
150C4⋊Dic53C2×D412yes—
151C2×#88322—21no—
152C2×#88322—21no—
153C2×#88322—21no—
154C2×#88322—21no—
155C2×#88322—22no—
156C2×#88322—42no—
157C2×#88322—81no—
158#11719912—44no—
159C8×D53—44no—
160C23×C53—42no—
161C23×C53—82no—
162C23×C53—82no—
163C23×C53—82no—
164C23×C53#12111yescharacteristic
165C22×D53—22no—
166Q8×C53—41no—
167Q8×C53—41no—
168Q8×C53—81no—
169D4×C53—42no—
170D4×C53—42no—
171D4×C53—42no—
172D4×C53—42no—
173D4×C53—82no—
174D4×C53—82no—
175D4×C53—82no—
176D4×C53—82no—
177C2×C4×C53—21no—
178C2×C4×C53—21no—
179C2×C4×C53—41no—
180C2×C4×C53—41no—
181C2×C4×C53—41no—
182C2×C4×C53#12111yescharacteristic
183C2×C4×C53#12111yescharacteristic
184C2×C4×C53#12111yescharacteristic
185C2×C4×C53#13512yes—
186C2×C4×C53AGL(1,ℤ/8)12yes—
187C2×C4×C53C22×D411yescharacteristic
188C2×C4×C53C22×D411yescharacteristic
189C2×C4×C53C2×#4111yescharacteristic
190C22⋊D53—44no—
191C2×Dic53—22no—
192D212—22no—
193D212—22no—
194D212—42no—
195D212—81no—
196C4×D53—42no—
197C4×D53—42no—
198Dic106—42no—
199C8×C53—22no—
200C8×C53—22no—
201#88322—41no—
202#88322—41no—
203#88322—42no—
204#88322—41no—
205#88322—41no—
206#19234—531nomaximal, Sylow
207C22×C53—21no—
208C22×C53—21no—
209C22×C53—82no—
210C22×C53—82no—
211C22×C53—82no—
212C22×C53—164no—
213C22×C53#46312yes—
214C22×C53C2×#12111yescharacteristic
215C2×D53—42no—
216C2×D53—42no—
217C2×D53—42no—
218C4×C53—21no—
219C4×C53—21no—
220C4×C53—21no—
221C4×C53—81no—
222C4×C53—81no—
223C4×C53#46312yes—
224C4×C53C2×#13511yescharacteristic
225C4×C53C2×AGL(1,ℤ/8)11yescharacteristic
226C4×C53#57611yescharacteristic
227C4×C53#57711yescharacteristic
228Dic53—42no—
229#3246—531no—
230#3205—531no—
231#3138D5311yescharacteristic, complemented
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—162no—
242C2×C53C2×#46311yescharacteristic
243C2×C53#306211yescharacteristic
244C2×C53#306811yescharacteristic
245D53—82no—
246C2×#131—1061no—
247#533C2×D5311yescharacteristic
248#532—531no—
249C2×#127C2×D5311yescharacteristic
250#493—1061no—
251#490—22no—
252#486—1061no—
253#486C2×D5312yes—
254#486C2×D5312yes—
255#476—532no—
256#475—1061no—
257#465—1062no—
258#461—532no—
259#460—1061no—
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⋊C8—531no—
269#418—21no—
270#418—21no—
271#408—1061no—
272C4×M16C2×D5311yescharacteristic
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—1061no—
283C22×D4—21no—
284C22×D4—21no—
285C22×D4—21no—
286C2×SD16—1062no—
287C2×D8—1062no—
288C2×D8—1062no—
289#131—2124no—
290C2×M16—1062no—
291C2×M16C22⋊D5311yescharacteristic
292C2×M16C22⋊D5311yescharacteristic
293C22×C8—1061no—
294C22×C8—1061no—
295C22×C8—1062no—
296C4⋊Q8—531no—
297C4⋊Q8C22×D5311yescharacteristic
298C4⋊Q8C22×D5311yescharacteristic
299#127—21no—
300#127—21no—
301#127—21no—
302#127—531no—
303#127C22×D5312yes—
304#125—21no—
305#124—1061no—
306C22⋊Q8—21no—
307C22⋊Q8—21no—
308#121—532no—
309#121—1061no—
310C4×Q8—21no—
311C4×Q8—21no—
312C4×D4—532no—
313C4×D4—1061no—
314#117—21no—
315C2×C42C22×D5311yescharacteristic
316#106—1062no—
317C4⋊C8—531no—
318C4⋊C8—531no—
319C4⋊C8—532no—
320C4⋊C8—531no—
321C4⋊C8—531no—
322C4⋊C8—1061no—
323C4⋊C8—1061no—
324#103—1062no—
325#102—22no—
326#102—22no—
327#102—22no—
328#102—22no—
329#102—1062no—
330#102—1062no—
331#102—1062no—
332C22⋊C8—1061no—
333C22⋊C8—1062no—
334C22⋊C8—1062no—
335#97C22⋊D5312yes—
336C4×C8—1061no—
337C4×C8C22×D5312yes—
338C24—41no—
339#41—2122no—
340#41—2122no—
341C2×Q8—21no—
342C2×Q8—21no—
343C2×Q8—1061no—
344C2×D4—22no—
345C2×D4—22no—
346C2×D4—22no—
347C2×D4—41no—
348C2×D4—41no—
349C2×D4—41no—
350C2×D4—42no—
351C2×D4—42no—
352C2×D4—81no—
353C2×D4—81no—
354C2×D4—82no—
355C2×D4—532no—
356C2×D4—1062no—
357C2×D4—1061no—
358C2×D4—1061no—
359C22×C4—1062no—
360C22×C4D4×D5311yescharacteristic
361C22×C4#1171994711yescharacteristic
362C22×C4C2×C22⋊D5311yescharacteristic
363SD16—4242no—
364D8—4242no—
365D8—4242no—
366M16—22no—
367M16—22no—
368M16—2124no—
369C2×C8—1061no—
370C2×C8—1061no—
371C2×C8—1062no—
372C2×C8—1061no—
373C2×C8—1061no—
374C2×C8—4241no—
375C2×C8—2124no—
376C2×C8—2122no—
377C2×C8C2×C22⋊D5312yes—
378C2×C8C2×C22⋊D5312yes—
379C4⋊C4—21no—
380C4⋊C4—21no—
381C4⋊C4—21no—
382C4⋊C4—21no—
383C4⋊C4—41no—
384C4⋊C4—41no—
385C4⋊C4—532no—
386C4⋊C4—1061no—
387C22⋊C4—41no—
388C22⋊C4—1062no—
389C22⋊C4—1062no—
390C42—41no—
391C42D4×D5311yescharacteristic
392C42#1171994711yescharacteristic
393C42C2×C22⋊D5311yescharacteristic
394C42C23×D5311yescharacteristic
395C23—42no—
396C23—82no—
397C23—82no—
398C23—82no—
399C23—1062no—
400C23#1313318811yescharacteristic
401Q8—41no—
402Q8—41no—
403Q8—81no—
404Q8—2122no—
405D4—42no—
406D4—42no—
407D4—42no—
408D4—42no—
409D4—82no—
410D4—82no—
411D4—82no—
412D4—82no—
413D4—1062no—
414D4—1062no—
415D4—4241no—
416D4—2124no—
417D4—2122no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—1062no—
424C2×C4—2122no—
425C2×C4—2122no—
426C2×C4#1313318811yescharacteristic
427C2×C4#1313318811yescharacteristic
428C2×C4#1313318811yescharacteristic
429C2×C4#1313323712yes—
430C2×C4#1313324212yes—
431C2×C4C2×D4×D5311yescharacteristic
432C2×C4C22×C22⋊D5311yescharacteristic
433C2×C4C2×#1171994711yescharacteristic
434C8—22no—
435C8—22no—
436C8—2121no—
437C8—2121no—
438C8—2122no—
439C8—2121no—
440C8—2121no—
441C22—21no—
442C22—21no—
443C22—82no—
444C22—82no—
445C22—82no—
446C22—164no—
447C22—2122no—
448C22—2122no—
449C22—2122no—
450C22#1681083512yes—
451C22C2×#1313318811yescharacteristic
452C4—21no—
453C4—21no—
454C4—21no—
455C4—81no—
456C4—81no—
457C4—2122no—
458C4#1681083512yes—
459C4C2×#1313323711yescharacteristic
460C4C2×#1313324211yescharacteristic
461C4#1681142711yescharacteristic
462C4#1681143311yescharacteristic
463C2—22no—
464C2—162no—
465C2—4242no—
466C2C2×#1681083511yescharacteristic
467C2#2028438911yescharacteristic
468C2#2028443911yescharacteristic
subgroupG/H
centerC22C2×#13133188
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#3138C2
socleC22×C53C2×#121
prime radical#20285633C2
hypercenter#3138D53
hypocenterC53#19234
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3138

Complements

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

NH ≅ G/Ntype
1. #20286027C2semidirect
3. #20283461C2semidirect
4. #20274273C2semidirect
5. #20274245C2semidirect
6. #20274149C2semidirect
7. #20274121C2semidirect
8. #20273467C2semidirect
9. #20273372C2semidirect
10. C2×#16810367C2semidirect
11. C53×#3138C2semidirect
16. #16810910C22semidirect
19. #16810825C22semidirect
24. #16810702C22semidirect
26. C8×D212C22semidirect
31. #16810376C22semidirect
32. #16810374C22semidirect
33. #16810372C22semidirect
34. #16810372C22semidirect
35. #16810369C22semidirect
36. #16810369C22semidirect
231. #3138D53semidirect
281. C53#19234semidirect

Maximal subgroups

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

subgroupindex
#202860272
#202856332
#202834612
#202742732
#202742452
#202741492
#202741212
#202734672
#202733722
C2×#168103672
C53×#31382
#1923453