Finite Group Databasebeta

← #27367616

Subgroups — #27367616

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#20286031C211yesmaximal, characteristic, complemented
2#20285089C211yesmaximal, characteristic, complemented
3#20283459C211yesmaximal, characteristic, complemented
4#20274274C212yesmaximal, complemented
5#20274246C212yesmaximal, complemented
6#20274150C212yesmaximal, complemented
7#20274122C212yesmaximal, complemented
8#20273468C211yesmaximal, characteristic, complemented
9#20273370C211yesmaximal, characteristic, complemented
10C2×#16810367C211yesmaximal, characteristic, complemented
11C53×#3205C211yesmaximal, characteristic
12C2×#13133224—21no—
13#16811180C2211yescharacteristic
14#16810987—22no—
15#16810969C2212yes—
16#16810969C2212yes—
17#16810913C2212yescomplemented
18#16810905—21no—
19#16810850—22no—
20#16810828C2212yescomplemented
21#16810820—21no—
22#16810701C2212yescomplemented
23#16810695—22no—
24D4×#88322C2212yescomplemented
25C2×C4⋊#88322C2211yescharacteristic
26C2×C4⋊#88322C2211yescharacteristic
27#16810628—21no—
28#16810628—21no—
29#16810590—21no—
30C4×#11719917C2211yescharacteristic
31#16810375C2212yescomplemented
32#16810373C2212yescomplemented
33#16810371C2211yescharacteristic, complemented
34#16810371C2211yescharacteristic, complemented
35#16810370C2211yescharacteristic, complemented
36#16810370C2211yescharacteristic, complemented
37#16810367C2212yes—
38#16810367C2212yes—
39C53×#532C2211yescharacteristic
40C2×#127×C53C2211yescharacteristic
41C53×#493—21no—
42C53×#486—21no—
43C53×#459C2212yes—
44C53×#459C2212yes—
45C53×#447—22no—
46C53×#431C2211yescharacteristic
47C2×#11719923—22no—
48#13133224—44no—
49C2×#11719917—22no—
50C2×#11719917D411yescharacteristic
51C2×#11719917D411yescharacteristic
52C22×#88322—22no—
53C22×#88322—21no—
54C22×#88322—21no—
55#13133217C2311yescharacteristic
56#13133217C2311yescharacteristic
57#13133209—21no—
58#13133196—21no—
59#13133196—21no—
60C4×Dic106—21no—
61C4×Dic106—21no—
62#13133176—21no—
63#13133148—22no—
64C4⋊#88322—21no—
65C4⋊#88322—21no—
66C4⋊#88322C2312yes—
67C4⋊#88322C2311yescharacteristic
68C4⋊#88322C2311yescharacteristic
69#13133140—22no—
70#13133138—22no—
71#13133138—22no—
72#13133138—22no—
73#13133138—22no—
74#13133138—22no—
75C22⋊#88322—22no—
76C22⋊#88322—22no—
77#13133128D412yes—
78C4×#88322C2312yes—
79C2×#41×C53—21no—
80C22×D4×C53—21no—
81C22×D4×C53—21no—
82C22×D4×C53—21no—
83C2×D8×C53—22no—
84C2×D8×C53—22no—
85C4⋊Q8×C53D411yescharacteristic
86#127×C53—21no—
87#127×C53—21no—
88#127×C53—21no—
89#127×C53D411yescharacteristic
90#127×C53C2312yes—
91#124×C53—21no—
92#121×C53—21no—
93#121×C53D412yes—
94C4×D4×C53—21no—
95C4×D4×C53C2312yes—
96C2×C42×C53C2311yescharacteristic
97C4⋊C8×C53C2311yescharacteristic
98C4⋊C8×C53C2311yescharacteristic
99#102×C53—22no—
100#102×C53—22no—
101C22⋊C8×C53—21no—
102C4×C8×C53—21no—
103C24×C53—41no—
104#41×C53—42no—
105#41×C53—42no—
106C2×Q8×C53—21no—
107C2×D4×C53—22no—
108C2×D4×C53—21no—
109C2×D4×C53—21no—
110C2×D4×C53—22no—
111C2×D4×C53—22no—
112C2×D4×C53—22no—
113C2×D4×C53—41no—
114C2×D4×C53—41no—
115C2×D4×C53—41no—
116C2×D4×C53—42no—
117C2×D4×C53—42no—
118C2×D4×C53—81no—
119C2×D4×C53—81no—
120C2×D4×C53—82no—
121C2×D4×C53C2×D412yes—
122C22×C4×C53—22no—
123C22×C4×C53C2×D411yescharacteristic
124C22×C4×C53C2×D411yescharacteristic
125C22×C4×C53#4111yescharacteristic
126C2×Dic106—21no—
127C2×Dic106—21no—
128D8×C53—82no—
129D8×C53—82no—
130C2×C8×C53—21no—
131C2×C8×C53—21no—
132C4⋊C4×C53—21no—
133C4⋊C4×C53C2×D412yes—
134C22⋊C4×C53—22no—
135C22⋊C4×C53—22no—
136C42×C53C2×D411yescharacteristic
137C42×C53C2×D411yescharacteristic
138C42×C53#4111yescharacteristic
139C42×C53C2411yescharacteristic
140C22⋊Dic53—41no—
141#11719923—82no—
142C4⋊Dic53—21no—
143C4⋊Dic53—21no—
144C4⋊Dic53—21no—
145C4⋊Dic53—21no—
146#11719919—41no—
147#11719919—41no—
148C4×Dic53—41no—
149#11719917—22no—
150#11719917—22no—
151#11719917—44no—
152C2×#88322—21no—
153C2×#88322—21no—
154C2×#88322—22no—
155C2×#88322—44no—
156C2×#88322—42no—
157C2×#88322—81no—
158C2×#88322C2×D412yes—
159C2×#88322C2×D412yes—
160C23×C53—22no—
161C23×C53—42no—
162C23×C53—82no—
163C23×C53—82no—
164C23×C53—82no—
165C23×C53#12111yescharacteristic
166Q8×C53—42no—
167D4×C53—22no—
168D4×C53—22no—
169D4×C53—44no—
170D4×C53—42no—
171D4×C53—42no—
172D4×C53—42no—
173D4×C53—42no—
174D4×C53—42no—
175D4×C53—81no—
176D4×C53—82no—
177D4×C53—82no—
178D4×C53—82no—
179D4×C53—82no—
180C2×C4×C53—21no—
181C2×C4×C53—21no—
182C2×C4×C53—22no—
183C2×C4×C53—41no—
184C2×C4×C53—42no—
185C2×C4×C53—42no—
186C2×C4×C53#12111yescharacteristic
187C2×C4×C53#12111yescharacteristic
188C2×C4×C53#12111yescharacteristic
189C2×C4×C53#13512yes—
190C2×C4×C53AGL(1,ℤ/8)12yes—
191C2×C4×C53C22×D411yescharacteristic
192C2×C4×C53C22×D411yescharacteristic
193C2×C4×C53C2×#4111yescharacteristic
194C2×Dic53—41no—
195C2×Dic53—41no—
196Dic106—41no—
197Dic106—41no—
198Dic106—81no—
199C8×C53—41no—
200C8×C53—41no—
201#88322—22no—
202#88322—22no—
203#88322—41no—
204#88322—41no—
205#88322—42no—
206#19234—531nomaximal, Sylow
207C22×C53—21no—
208C22×C53—21no—
209C22×C53—42no—
210C22×C53—42no—
211C22×C53—42no—
212C22×C53—82no—
213C22×C53—82no—
214C22×C53—82no—
215C22×C53—164no—
216C22×C53#46312yes—
217C22×C53C2×#12111yescharacteristic
218C4×C53—21no—
219C4×C53—21no—
220C4×C53—21no—
221C4×C53—42no—
222C4×C53#46312yes—
223C4×C53C2×#13511yescharacteristic
224C4×C53C2×AGL(1,ℤ/8)11yescharacteristic
225C4×C53#57611yescharacteristic
226C4×C53#57711yescharacteristic
227Dic53—81no—
228Dic53—81no—
229#3246—531no—
230#3205D5311yescharacteristic
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×C53—162no—
243C2×C53C2×#46311yescharacteristic
244C2×C53#306211yescharacteristic
245C2×C53#306811yescharacteristic
246C2×#131—1061no—
247#533—531no—
248#532C2×D5311yescharacteristic
249C2×#127C2×D5311yescharacteristic
250#493—21no—
251#490—1062no—
252#486—21no—
253#486—532no—
254#486—532no—
255#476—532no—
256#475—1061no—
257#465—1062no—
258#461—532no—
259#460—1061no—
260#459C2×D5312yes—
261#459C2×D5312yes—
262#447—22no—
263#436—532no—
264#435—1062no—
265C8×D4—532no—
266#431C2×D5311yescharacteristic
267C2×C4⋊C8—531no—
268C2×C4⋊C8—531no—
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—21no—
284C22×D4—21no—
285C22×D4—21no—
286C2×SD16—1062no—
287C2×D8—22no—
288C2×D8—22no—
289#131—2124no—
290C2×M16—531no—
291C2×M16—531no—
292C2×M16—1062no—
293C22×C8—1062no—
294C22×C8—1061no—
295C22×C8—1061no—
296C4⋊Q8—531no—
297C4⋊Q8—531no—
298C4⋊Q8C22⋊D5311yescharacteristic
299#127—21no—
300#127—21no—
301#127—21no—
302#127C22⋊D5311yescharacteristic
303#127C22×D5312yes—
304#125—1061no—
305#124—21no—
306C22⋊Q8—1061no—
307C22⋊Q8—1061no—
308#121—21no—
309#121C22⋊D5312yes—
310C4×Q8—1061no—
311C4×Q8—1061no—
312C4×D4—21no—
313C4×D4C22×D5312yes—
314#117—1061no—
315C2×C42C22×D5311yescharacteristic
316#106—1062no—
317C4⋊C8—531no—
318C4⋊C8—531no—
319C4⋊C8—532no—
320C4⋊C8—1061no—
321C4⋊C8—1061no—
322C4⋊C8C22×D5311yescharacteristic
323C4⋊C8C22×D5311yescharacteristic
324#103—1062no—
325#102—22no—
326#102—22no—
327#102—1062no—
328#102—1062no—
329#102—1062no—
330#102—1062no—
331#102—1062no—
332C22⋊C8—21no—
333C22⋊C8—1062no—
334C22⋊C8—1062no—
335#97—532no—
336C4×C8—21no—
337C4×C8—532no—
338C24—41no—
339#41—42no—
340#41—42no—
341C2×Q8—21no—
342C2×Q8—1061no—
343C2×Q8—1061no—
344C2×D4—22no—
345C2×D4—21no—
346C2×D4—21no—
347C2×D4—22no—
348C2×D4—22no—
349C2×D4—22no—
350C2×D4—41no—
351C2×D4—41no—
352C2×D4—41no—
353C2×D4—42no—
354C2×D4—42no—
355C2×D4—81no—
356C2×D4—81no—
357C2×D4—82no—
358C2×D4C2×C22⋊D5312yes—
359C22×C4—22no—
360C22×C4D4×D5311yescharacteristic
361C22×C4#1171994711yescharacteristic
362C22×C4C2×C22⋊D5311yescharacteristic
363SD16—4242no—
364D8—82no—
365D8—82no—
366M16—1062no—
367M16—1062no—
368M16—2124no—
369C2×C8—21no—
370C2×C8—21no—
371C2×C8—532no—
372C2×C8—532no—
373C2×C8—1061no—
374C2×C8—1061no—
375C2×C8—1062no—
376C2×C8—4241no—
377C2×C8—2124no—
378C2×C8—2122no—
379C4⋊C4—21no—
380C4⋊C4—1061no—
381C4⋊C4—1061no—
382C4⋊C4—1061no—
383C4⋊C4—1061no—
384C4⋊C4—2121no—
385C4⋊C4—2121no—
386C4⋊C4C2×C22⋊D5312yes—
387C22⋊C4—22no—
388C22⋊C4—22no—
389C22⋊C4—2121no—
390C42—2121no—
391C42D4×D5311yescharacteristic
392C42#1171994711yescharacteristic
393C42C2×C22⋊D5311yescharacteristic
394C42C23×D5311yescharacteristic
395C23—22no—
396C23—42no—
397C23—82no—
398C23—82no—
399C23—82no—
400C23#1313318911yescharacteristic
401Q8—42no—
402Q8—4241no—
403Q8—2121no—
404Q8—2121no—
405D4—22no—
406D4—22no—
407D4—44no—
408D4—42no—
409D4—42no—
410D4—42no—
411D4—42no—
412D4—42no—
413D4—81no—
414D4—82no—
415D4—82no—
416D4—82no—
417D4—82no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—22no—
421C2×C4—41no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—2121no—
425C2×C4—2121no—
426C2×C4#1313318911yescharacteristic
427C2×C4#1313318911yescharacteristic
428C2×C4#1313318911yescharacteristic
429C2×C4#1313323612yes—
430C2×C4#1313324112yes—
431C2×C4C2×D4×D5311yescharacteristic
432C2×C4C22×C22⋊D5311yescharacteristic
433C2×C4C2×#1171994711yescharacteristic
434C8—41no—
435C8—41no—
436C8—1062no—
437C8—1062no—
438C8—2121no—
439C8—2121no—
440C8—2122no—
441C22—21no—
442C22—21no—
443C22—42no—
444C22—42no—
445C22—42no—
446C22—82no—
447C22—82no—
448C22—82no—
449C22—164no—
450C22#1681083612yes—
451C22C2×#1313318911yescharacteristic
452C4—21no—
453C4—21no—
454C4—21no—
455C4—42no—
456C4—4241no—
457C4—4241no—
458C4#1681083612yes—
459C4C2×#1313323611yescharacteristic
460C4C2×#1313324111yescharacteristic
461C4#1681142611yescharacteristic
462C4#1681143411yescharacteristic
463C2—22no—
464C2—82no—
465C2—162no—
466C2C2×#1681083611yescharacteristic
467C2#2028439111yescharacteristic
468C2#2028444011yescharacteristic
subgroupG/H
centerC22C2×#13133189
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
Fitting, prime radicalC53×#3205C2
socleC22×C53C2×#121
hypercenter#3205D53
hypocenterC53#19234
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3205

Complements

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

NH ≅ G/Ntype
1. #20286031C2semidirect
2. #20285089C2semidirect
3. #20283459C2semidirect
4. #20274274C2semidirect
5. #20274246C2semidirect
6. #20274150C2semidirect
7. #20274122C2semidirect
8. #20273468C2semidirect
9. #20273370C2semidirect
10. C2×#16810367C2semidirect
17. #16810913C22semidirect
20. #16810828C22semidirect
22. #16810701C22semidirect
24. D4×#88322C22semidirect
31. #16810375C22semidirect
32. #16810373C22semidirect
33. #16810371C22semidirect
34. #16810371C22semidirect
35. #16810370C22semidirect
36. #16810370C22semidirect
281. C53#19234semidirect

Maximal subgroups

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

subgroupindex
#202860312
#202850892
#202834592
#202742742
#202742462
#202741502
#202741222
#202734682
#202733702
C2×#168103672
C53×#32052
#1923453