Finite Group Databasebeta

← #27367660

Subgroups — #27367660

Order 13568.

722 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20286027C211yesmaximal, characteristic, complemented
2#20285165C211yesmaximal, characteristic
3#20283664C211yesmaximal, characteristic, complemented
4#20274197C212yesmaximal, complemented
5#20274185C212yesmaximal, complemented
6#20274073C212yesmaximal, complemented
7#20274061C212yesmaximal, complemented
8#20273468C211yesmaximal, characteristic, complemented
9#20273362C211yesmaximal, characteristic, complemented
10C2×#16810367C211yesmaximal, characteristic, complemented
11C53×#3208C211yesmaximal, characteristic, complemented
12C2×#13133243—21no—
13C2×#13133243—21no—
14#16811173C2211yescharacteristic
15#16811010—22no—
16#16810999C2212yes—
17#16810995C2212yes—
18#16810910—21no—
19#16810908C2212yescomplemented
20#16810854—22no—
21#16810825—21no—
22#16810823C2212yescomplemented
23#16810754—22no—
24#16810754C2212yescomplemented
25Q8×#88322C2212yescomplemented
26#16810650—21no—
27C2×C4⋊#88322C2211yescharacteristic
28C2×C4⋊#88322C2211yescharacteristic
29#16810579—21no—
30C4×#11719917C2211yescharacteristic
31#16810375C2212yescomplemented
32#16810373C2212yescomplemented
33#16810368C2211yescharacteristic, complemented
34#16810368C2211yescharacteristic, complemented
35#16810367C2212yes—
36#16810367C2212yes—
37#16810366C2211yescharacteristic, complemented
38#16810366C2211yescharacteristic, complemented
39C53×#533C2211yescharacteristic
40C2×#127×C53C2211yescharacteristic
41C53×#492—21no—
42C53×#486—21no—
43C53×#460C2212yes—
44C53×#460C2212yes—
45C53×#450—22no—
46C53×#431C2211yescharacteristic
47C2×#11719945—21no—
48#13133243—44no—
49#13133243—44no—
50C2×#11719923—22no—
51C2×#11719923—22no—
52C2×#11719924—22no—
53C2×#11719922—22no—
54C2×#11719922—22no—
55C2×#11719917D411yescharacteristic
56C2×#11719917D411yescharacteristic
57C22×#88322—21no—
58C22×#88322—21no—
59#13133217C2311yescharacteristic
60#13133214C2311yescharacteristic
61#13133205—21no—
62#13133191—21no—
63#13133191—21no—
64C4×D212—21no—
65C4×D212—21no—
66#13133148—22no—
67C4⋊#88322—21no—
68C4⋊#88322—21no—
69C4⋊#88322—22no—
70C4⋊#88322—22no—
71C4⋊#88322C2311yescharacteristic
72C4⋊#88322C2312yes—
73C4⋊#88322C2311yescharacteristic
74#13133141—22no—
75#13133137—22no—
76#13133128—22no—
77#13133128D412yes—
78C4×#88322—22no—
79C4×#88322C2312yes—
80C22×D4×C53—21no—
81C22×D4×C53—21no—
82C22×D4×C53—21no—
83C2×SD16×C53—22no—
84C2×SD16×C53—22no—
85C4⋊Q8×C53D411yescharacteristic
86C4⋊Q8×C53D411yescharacteristic
87#127×C53—21no—
88#127×C53—21no—
89#127×C53—21no—
90#127×C53C2312yes—
91#125×C53—21no—
92C22⋊Q8×C53—21no—
93C22⋊Q8×C53D412yes—
94C4×Q8×C53—21no—
95C4×Q8×C53C2312yes—
96#117×C53—21no—
97C2×C42×C53C2311yescharacteristic
98C4⋊C8×C53C2311yescharacteristic
99C4⋊C8×C53C2311yescharacteristic
100#102×C53—22no—
101#102×C53—22no—
102C22⋊C8×C53—21no—
103C4×C8×C53—21no—
104C24×C53—41no—
105C2×Q8×C53—21no—
106C2×Q8×C53C2×D412yes—
107C2×D4×C53—22no—
108C2×D4×C53—22no—
109C2×D4×C53—22no—
110C2×D4×C53—41no—
111C2×D4×C53—41no—
112C2×D4×C53—41no—
113C2×D4×C53—42no—
114C2×D4×C53—42no—
115C2×D4×C53—81no—
116C2×D4×C53—81no—
117C2×D4×C53—82no—
118C22×C4×C53C2×D411yescharacteristic
119C22×C4×C53C2×D411yescharacteristic
120C22×C4×C53#4111yescharacteristic
121C2×C22⋊D53—41no—
122#11719945—42no—
123#11719945—42no—
124C2×D212—21no—
125C2×D212—21no—
126C2×D212—21no—
127C2×C4×D53—41no—
128C2×Dic106—21no—
129SD16×C53—82no—
130SD16×C53—82no—
131C2×C8×C53—21no—
132C2×C8×C53—21no—
133C4⋊C4×C53—21no—
134C4⋊C4×C53—21no—
135C4⋊C4×C53—21no—
136C4⋊C4×C53—22no—
137C4⋊C4×C53—22no—
138C4⋊C4×C53C2×D412yes—
139C22⋊C4×C53—22no—
140C42×C53—22no—
141C42×C53C2×D411yescharacteristic
142C42×C53C2×D411yescharacteristic
143C42×C53#4111yescharacteristic
144C42×C53C2411yescharacteristic
145#11719924—82no—
146#11719923—44no—
147#11719923—44no—
148#11719922—44no—
149#11719922—44no—
150#11719921—41no—
151#11719921—41no—
152C4⋊Dic53—21no—
153C4⋊Dic53—21no—
154#11719917—22no—
155#11719917SD1614yescomplemented
156C2×#88322—21no—
157C2×#88322—21no—
158C2×#88322—22no—
159C2×#88322—42no—
160C2×#88322—81no—
161C2×#88322C2×D412yes—
162C2×#88322C2×D412yes—
163C23×C53—42no—
164C23×C53—82no—
165C23×C53—82no—
166C23×C53—82no—
167C23×C53#12111yescharacteristic
168C22×D53—41no—
169Q8×C53—22no—
170Q8×C53—22no—
171Q8×C53—81no—
172D4×C53—42no—
173D4×C53—42no—
174D4×C53—42no—
175D4×C53—42no—
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—22no—
184C2×C4×C53—41no—
185C2×C4×C53#12111yescharacteristic
186C2×C4×C53#12111yescharacteristic
187C2×C4×C53#12111yescharacteristic
188C2×C4×C53C2×SD1612yes—
189C2×C4×C53AGL(1,ℤ/8)12yes—
190C2×C4×C53C22×D411yescharacteristic
191C2×C4×C53C22×D411yescharacteristic
192C2×C4×C53C2×#4111yescharacteristic
193C22⋊D53—82no—
194C2×Dic53—41no—
195D212—41no—
196D212—41no—
197D212—41no—
198D212—41no—
199D212—81no—
200C4×D53—82no—
201Dic106—41no—
202Dic106—41no—
203C8×C53—41no—
204C8×C53—41no—
205#88322—22no—
206#88322—41no—
207#88322—41no—
208#88322—42no—
209#88322C2×SD1614yes—
210#19238—531nomaximal, Sylow
211C22×C53—21no—
212C22×C53—21no—
213C22×C53—82no—
214C22×C53—82no—
215C22×C53—82no—
216C22×C53—164no—
217C22×C53#46012yes—
218C22×C53C2×#12111yescharacteristic
219C2×D53—81no—
220C2×D53—81no—
221C2×D53—81no—
222C4×C53—21no—
223C4×C53—21no—
224C4×C53—21no—
225C4×C53—42no—
226C4×C53—42no—
227C4×C53#46012yes—
228C4×C53C22×SD1611yescharacteristic
229C4×C53C2×AGL(1,ℤ/8)11yescharacteristic
230C4×C53#57611yescharacteristic
231C4×C53#57711yescharacteristic
232Dic53—81no—
233#3246—531no—
234#3208D5311yescharacteristic, complemented
235#3151—531no—
236#2962—531no—
237#1688—532no—
238#1686—532no—
239#1668—532no—
240#1666—532no—
241#1565—531no—
242#1540—531no—
243C2×#331—531no—
244C2×C53—22no—
245C2×C53—162no—
246C2×C53C2×#46011yescharacteristic
247C2×C53#306211yescharacteristic
248C2×C53#306811yescharacteristic
249D53—161no—
250C2×AGL(1,ℤ/8)—1061no—
251C2×AGL(1,ℤ/8)—1061no—
252#533C2×D5311yescharacteristic
253#532—531no—
254C2×#127C2×D5311yescharacteristic
255#496—1062no—
256#493—532no—
257#492—21no—
258#492—532no—
259#486—21no—
260#476—1061no—
261#475—532no—
262#465—1062no—
263#461—1061no—
264#460—532no—
265#460C2×D5312yes—
266#460C2×D5312yes—
267#450—22no—
268#446—532no—
269#446—1062no—
270C8×Q8—532no—
271#431C2×D5311yescharacteristic
272#424—1061no—
273C2×C4⋊C8—531no—
274C2×C4⋊C8—531no—
275#405—1061no—
276C4×M16—531no—
277#335—532no—
278#334—532no—
279#332—531no—
280#332—531no—
281#331—532no—
282#331—532no—
283#331—531no—
284#331—531no—
285C53#1923811yescharacteristic, complemented, Sylow
286C2×#41—1061no—
287C22×D4—21no—
288C22×D4—21no—
289C22×D4—21no—
290AGL(1,ℤ/8)—2124no—
291AGL(1,ℤ/8)—2124no—
292C2×SD16—22no—
293C2×SD16—22no—
294C2×SD16—1062no—
295C2×SD16—1062no—
296C2×SD16—1062no—
297C2×D8—1062no—
298C2×D8—1062no—
299C2×M16—531no—
300C2×M16—531no—
301C22×C8—1061no—
302C22×C8—1061no—
303C4⋊Q8—531no—
304C4⋊Q8C22⋊D5311yescharacteristic
305C4⋊Q8C22⋊D5311yescharacteristic
306#127—21no—
307#127—21no—
308#127—21no—
309#127—531no—
310#127C22×D5312yes—
311#125—21no—
312#124—1061no—
313C22⋊Q8—21no—
314C22⋊Q8C22⋊D5312yes—
315#121—1061no—
316#121—1061no—
317C4×Q8—21no—
318C4×Q8C22×D5312yes—
319C4×D4—1061no—
320C4×D4—1061no—
321#117—21no—
322C2×C42C22×D5311yescharacteristic
323#106—1062no—
324C4⋊C8—531no—
325C4⋊C8—531no—
326C4⋊C8—532no—
327C4⋊C8—1062no—
328C4⋊C8—1062no—
329C4⋊C8—1061no—
330C4⋊C8—1061no—
331C4⋊C8C22×D5311yescharacteristic
332C4⋊C8C22×D5311yescharacteristic
333#103—1062no—
334#102—22no—
335#102—22no—
336#102—1062no—
337C22⋊C8—21no—
338#97—532no—
339#97—1062no—
340C4×C8—21no—
341C4×C8—532no—
342C4×C8—1062no—
343C24—41no—
344#41—2122no—
345#41—2122no—
346C2×Q8—21no—
347C2×Q8—1061no—
348C2×Q8C2×C22⋊D5312yes—
349C2×D4—22no—
350C2×D4—22no—
351C2×D4—22no—
352C2×D4—41no—
353C2×D4—41no—
354C2×D4—41no—
355C2×D4—42no—
356C2×D4—42no—
357C2×D4—81no—
358C2×D4—81no—
359C2×D4—82no—
360C2×D4—1061no—
361C2×D4—1061no—
362C2×D4—1061no—
363C2×D4—2121no—
364C22×C4—2121no—
365C22×C4D4×D5311yescharacteristic
366C22×C4#1171994711yescharacteristic
367C22×C4C2×C22⋊D5311yescharacteristic
368SD16—82no—
369SD16—82no—
370SD16—4242no—
371SD16—2124no—
372SD16—2124no—
373D8—2124no—
374D8—2124no—
375M16—534no—
376M16—1062no—
377C2×C8—21no—
378C2×C8—21no—
379C2×C8—532no—
380C2×C8—532no—
381C2×C8—1061no—
382C2×C8—1061no—
383C2×C8—1062no—
384C2×C8—4241no—
385C2×C8—2122no—
386C4⋊C4—21no—
387C4⋊C4—21no—
388C4⋊C4—21no—
389C4⋊C4—22no—
390C4⋊C4—22no—
391C4⋊C4—1061no—
392C4⋊C4—1061no—
393C4⋊C4C2×C22⋊D5312yes—
394C22⋊C4—22no—
395C22⋊C4—2121no—
396C22⋊C4—2121no—
397C42—22no—
398C42D4×D5311yescharacteristic
399C42#1171994711yescharacteristic
400C42C2×C22⋊D5311yescharacteristic
401C42C23×D5311yescharacteristic
402C23—42no—
403C23—82no—
404C23—82no—
405C23—82no—
406C23—2121no—
407C23#1313318911yescharacteristic
408Q8—22no—
409Q8—22no—
410Q8—81no—
411Q8—2121no—
412Q8—2121no—
413D4—42no—
414D4—42no—
415D4—42no—
416D4—42no—
417D4—82no—
418D4—82no—
419D4—82no—
420D4—82no—
421D4—4241no—
422D4—4242no—
423D4—2121no—
424D4—2121no—
425D4—2121no—
426D4—2121no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—41no—
432C2×C4—4242no—
433C2×C4—2121no—
434C2×C4#1313318911yescharacteristic
435C2×C4#1313318911yescharacteristic
436C2×C4#1313318911yescharacteristic
437C2×C4SD16×D5312yes—
438C2×C4#1313324412yes—
439C2×C4C2×D4×D5311yescharacteristic
440C2×C4C22×C22⋊D5311yescharacteristic
441C2×C4C2×#1171994711yescharacteristic
442C8—41no—
443C8—41no—
444C8—534no—
445C8—1062no—
446C8—2121no—
447C8—2121no—
448C8—2122no—
449C22—21no—
450C22—21no—
451C22—82no—
452C22—82no—
453C22—82no—
454C22—164no—
455C22—4241no—
456C22—4241no—
457C22—4241no—
458C22#1681082112yes—
459C22C2×#1313318911yescharacteristic
460C4—21no—
461C4—21no—
462C4—21no—
463C4—42no—
464C4—42no—
465C4—4241no—
466C4#1681082112yes—
467C4C2×SD16×D5311yescharacteristic
468C4C2×#1313324411yescharacteristic
469C4#1681142611yescharacteristic
470C4#1681143411yescharacteristic
471C2—22no—
472C2—162no—
473C2—8481no—
474C2C2×#1681082111yescharacteristic
475C2#2028439111yescharacteristic
476C2#2028444011yescharacteristic
subgroupG/H
centerC22C2×#13133189
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#3208C2
socleC22×C53C2×#121
prime radical#20285165C2
hypercenter#3208D53
hypocenterC53#19238
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3208

Complements

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

NH ≅ G/Ntype
1. #20286027C2semidirect
3. #20283664C2semidirect
4. #20274197C2semidirect
5. #20274185C2semidirect
6. #20274073C2semidirect
7. #20274061C2semidirect
8. #20273468C2semidirect
9. #20273362C2semidirect
10. C2×#16810367C2semidirect
11. C53×#3208C2semidirect
19. #16810908C22semidirect
22. #16810823C22semidirect
24. #16810754C22semidirect
25. Q8×#88322C22semidirect
31. #16810375C22semidirect
32. #16810373C22semidirect
33. #16810368C22semidirect
34. #16810368C22semidirect
37. #16810366C22semidirect
38. #16810366C22semidirect
155. #11719917SD16semidirect
234. #3208D53semidirect
285. C53#19238semidirect

Maximal subgroups

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

subgroupindex
#202860272
#202851652
#202836642
#202741972
#202741852
#202740732
#202740612
#202734682
#202733622
C2×#168103672
C53×#32082
#1923853