Finite Group Databasebeta

← #27367667

Subgroups — #27367667

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#20286031C211yesmaximal, characteristic, complemented
2#20285658C211yesmaximal, characteristic, complemented
3#20285168C211yesmaximal, characteristic
4#20283658C211yesmaximal, characteristic, complemented
5#20274198C212yesmaximal, complemented
6#20274190C212yesmaximal, complemented
7#20274074C212yesmaximal, complemented
8#20274066C212yesmaximal, complemented
9#20273468C211yesmaximal, characteristic, complemented
10C2×#16810366C211yesmaximal, characteristic, complemented
11C53×#1540C211yesmaximal, characteristic, complemented
12C2×#13133242—21no—
13C2×#13133242—21no—
14#16811180C2211yescharacteristic
15C2×#13133214C2211yescharacteristic
16#16811013—22no—
17#16810999C2212yes—
18#16810997C2212yes—
19#16810996—21no—
20#16810970—21no—
21#16810913—21no—
22#16810905C2212yescomplemented
23#16810850—22no—
24#16810828—21no—
25#16810824C2212yes—
26#16810824C2212yes—
27#16810820C2212yescomplemented
28#16810776—22no—
29#16810752—22no—
30#16810750C2212yescomplemented
31#16810747C2212yescomplemented
32#16810649—21no—
33C2×C4⋊#88322C2211yescharacteristic
34#16810578—21no—
35C4×#11719917C2211yescharacteristic
36#16810375C2212yescomplemented
37#16810373C2212yescomplemented
38#16810366C2212yes—
39#16810366C2212yes—
40C53×#532C2211yescharacteristic
41C53×#431C2211yescharacteristic
42C2×C4⋊C8×C53C2211yescharacteristic
43C53×#332C2211yescharacteristic, complemented
44C53×#332C2211yescharacteristic, complemented
45C53×#331C2211yescharacteristic, complemented
46C53×#331C2211yescharacteristic, complemented
47C22×D212—21no—
48C22×D212—21no—
49C22×D212—21no—
50#13133242—44no—
51#13133242—44no—
52C2×#11719913—22no—
53C2×#11719913—22no—
54C2×#11719923—22no—
55C2×#11719924—22no—
56C2×#11719924—22no—
57C2×#11719922—22no—
58C2×#11719922—22no—
59C2×#11719917D411yescharacteristic
60C2×#11719917D411yescharacteristic
61C22×#88322—21no—
62#13133217D411yescharacteristic
63#13133217D411yescharacteristic
64#13133214—21no—
65#13133214—21no—
66#13133214—21no—
67#13133214C2312yes—
68#13133209—21no—
69#13133196—21no—
70#13133196D412yes—
71C4×Dic106—21no—
72C4×Dic106C2312yes—
73#13133176—21no—
74#13133148—22no—
75C4⋊#88322—21no—
76C4⋊#88322C2312yes—
77#13133146—22no—
78#13133146—22no—
79#13133140—22no—
80#13133139—22no—
81#13133139—22no—
82#13133138—22no—
83#13133128D412yes—
84#13133127—22no—
85C4×#88322C2312yes—
86C8×Dic53—22no—
87C2×#41×C53—21no—
88C22×C8×C53—21no—
89C4⋊Q8×C53C2311yescharacteristic
90#127×C53C2311yescharacteristic
91#124×C53—21no—
92#121×C53—21no—
93#121×C53—21no—
94C4×D4×C53—21no—
95C4×D4×C53—21no—
96C2×C42×C53C2311yescharacteristic
97C4⋊C8×C53—21no—
98C4⋊C8×C53C2311yescharacteristic
99C4⋊C8×C53C2311yescharacteristic
100C4⋊C8×C53C2311yescharacteristic
101C4⋊C8×C53C2311yescharacteristic
102C22⋊C8×C53—21no—
103C4×C8×C53—21no—
104C23×D53—41no—
105#41×C53—42no—
106#41×C53—42no—
107C2×Q8×C53—21no—
108C2×D4×C53—21no—
109C2×D4×C53—21no—
110C2×D4×C53—21no—
111C2×D4×C53—41no—
112C22×C4×C53—41no—
113C22×C4×C53C2×D411yescharacteristic
114C22×C4×C53C2×D411yescharacteristic
115C22×C4×C53#4111yescharacteristic
116C2×D212—22no—
117C2×D212—22no—
118C2×D212—22no—
119C2×D212—42no—
120C2×D212—42no—
121C2×D212—41no—
122C2×D212—41no—
123C2×D212—41no—
124C2×D212—81no—
125C2×D212—81no—
126C2×D212—82no—
127C2×Dic106—21no—
128C2×Dic106C2×D412yes—
129C2×C8×C53—21no—
130C2×C8×C53—21no—
131C2×C8×C53—21no—
132C2×C8×C53—21no—
133C2×C8×C53—42no—
134C4⋊C4×C53—21no—
135C4⋊C4×C53—21no—
136C22⋊C4×C53—41no—
137C22⋊C4×C53—41no—
138C42×C53C2×D411yescharacteristic
139C42×C53C2×D411yescharacteristic
140C42×C53#4111yescharacteristic
141C42×C53C2411yescharacteristic
142C22⋊Dic53—22no—
143#11719924—44no—
144#11719924—44no—
145#11719923—82no—
146#11719922—44no—
147#11719922—44no—
148C4⋊Dic53—21no—
149C4⋊Dic53—21no—
150C4⋊Dic53—21no—
151C4⋊Dic53C2×D412yes—
152#11719919—22no—
153#11719919—22no—
154C4×Dic53—22no—
155#11719917—22no—
156#11719917SD1614yescomplemented
157C2×#88322—22no—
158C2×#88322—81no—
159C2×#88322C2×D412yes—
160C2×#88322C2×D412yes—
161#11719913—82no—
162#11719913—82no—
163C23×C53—41no—
164C23×C53#12111yescharacteristic
165C22×D53—42no—
166C22×D53—82no—
167C22×D53—82no—
168C22×D53—82no—
169Q8×C53—41no—
170Q8×C53—41no—
171D4×C53—41no—
172D4×C53—41no—
173D4×C53—41no—
174D4×C53—41no—
175D4×C53—81no—
176D4×C53—82no—
177C2×C4×C53—21no—
178C2×C4×C53—21no—
179C2×C4×C53—41no—
180C2×C4×C53—41no—
181C2×C4×C53—82no—
182C2×C4×C53#12111yescharacteristic
183C2×C4×C53#12111yescharacteristic
184C2×C4×C53#12111yescharacteristic
185C2×C4×C53C2×SD1612yes—
186C2×C4×C53AGL(1,ℤ/8)12yes—
187C2×C4×C53C22×D411yescharacteristic
188C2×C4×C53C22×D411yescharacteristic
189C2×C4×C53C2×#4111yescharacteristic
190C2×Dic53—22no—
191C2×Dic53—22no—
192D212—42no—
193D212—42no—
194D212—42no—
195D212—42no—
196D212—82no—
197D212—82no—
198D212—82no—
199D212—82no—
200Dic106—22no—
201Dic106—22no—
202Dic106—81no—
203C8×C53—41no—
204C8×C53—41no—
205C8×C53—41no—
206C8×C53—41no—
207#88322—22no—
208#88322—42no—
209#88322C2×SD1614yes—
210#19238—531nomaximal, Sylow
211C22×C53—21no—
212C22×C53—21no—
213C22×C53—81no—
214C22×C53—81no—
215C22×C53—81no—
216C22×C53#46012yes—
217C22×C53C2×#12111yescharacteristic
218C2×D53—82no—
219C2×D53—82no—
220C2×D53—82no—
221C2×D53—164no—
222C4×C53—21no—
223C4×C53—21no—
224C4×C53—21no—
225C4×C53—81no—
226C4×C53#46012yes—
227C4×C53C22×SD1611yescharacteristic
228C4×C53C2×AGL(1,ℤ/8)11yescharacteristic
229C4×C53#57611yescharacteristic
230C4×C53#57711yescharacteristic
231Dic53—42no—
232Dic53—42no—
233#3246—531no—
234#3208—531no—
235#3151—531no—
236#2962—531no—
237#1688—532no—
238#1686—532no—
239#1668—532no—
240#1666—532no—
241#1565—531no—
242#1540D5311yescharacteristic, complemented
243C2×#331—531no—
244C2×C53—22no—
245C2×C53—161no—
246C2×C53C2×#46011yescharacteristic
247C2×C53#306211yescharacteristic
248C2×C53#306811yescharacteristic
249D53—162no—
250C2×AGL(1,ℤ/8)—1061no—
251C2×AGL(1,ℤ/8)—1061no—
252#533—531no—
253#532C2×D5311yescharacteristic
254C2×#127—531no—
255#496—1062no—
256#493—532no—
257#492—532no—
258#492—1061no—
259#486—1061no—
260#476—1061no—
261#475—532no—
262#465—1062no—
263#461—1061no—
264#460—532no—
265#460—532no—
266#460—532no—
267#450—1062no—
268#446—532no—
269#446—1062no—
270C8×Q8—532no—
271#431C2×D5311yescharacteristic
272#424—1061no—
273C2×C4⋊C8—531no—
274C2×C4⋊C8C2×D5311yescharacteristic
275#405—1061no—
276C4×M16—531no—
277#335—532no—
278#334—532no—
279#332C2×D5311yescharacteristic, complemented
280#332C2×D5311yescharacteristic, complemented
281#331—532no—
282#331—532no—
283#331C2×D5311yescharacteristic, complemented
284#331C2×D5311yescharacteristic, complemented
285C53#1923811yescharacteristic, complemented, Sylow
286C2×#41—21no—
287C22×D4—1061no—
288C22×D4—1061no—
289C22×D4—1061no—
290AGL(1,ℤ/8)—2124no—
291AGL(1,ℤ/8)—2124no—
292C2×SD16—1062no—
293C2×SD16—1062no—
294C2×SD16—1062no—
295C2×SD16—1062no—
296C2×SD16—1062no—
297C2×D8—1062no—
298C2×D8—1062no—
299C2×M16—531no—
300C2×M16—531no—
301C22×C8—21no—
302C22×C8—1061no—
303C4⋊Q8—531no—
304C4⋊Q8—531no—
305C4⋊Q8C22×D5311yescharacteristic
306#127—532no—
307#127—1061no—
308#127—1061no—
309#127—1061no—
310#127C22×D5311yescharacteristic
311#125—1061no—
312#124—21no—
313C22⋊Q8—532no—
314C22⋊Q8—1061no—
315#121—21no—
316#121—21no—
317C4×Q8—532no—
318C4×Q8—1061no—
319C4×D4—21no—
320C4×D4—21no—
321#117—1061no—
322C2×C42C22×D5311yescharacteristic
323#106—1062no—
324C4⋊C8—21no—
325C4⋊C8—532no—
326C4⋊C8—1062no—
327C4⋊C8—1062no—
328C4⋊C8—1061no—
329C4⋊C8C22×D5311yescharacteristic
330C4⋊C8C22×D5311yescharacteristic
331C4⋊C8C22×D5311yescharacteristic
332C4⋊C8C22×D5311yescharacteristic
333#103—1062no—
334#102—1062no—
335#102—1062no—
336#102—1062no—
337C22⋊C8—21no—
338#97—532no—
339#97—1062no—
340C4×C8—21no—
341C4×C8—532no—
342C4×C8—1062no—
343C24—2121no—
344#41—42no—
345#41—42no—
346C2×Q8—21no—
347C2×Q8—532no—
348C2×Q8—1061no—
349C2×D4—21no—
350C2×D4—21no—
351C2×D4—21no—
352C2×D4—41no—
353C2×D4—1062no—
354C2×D4—1062no—
355C2×D4—1062no—
356C2×D4—4241no—
357C2×D4—4241no—
358C2×D4—4242no—
359C2×D4—2121no—
360C2×D4—2121no—
361C2×D4—2121no—
362C2×D4—2122no—
363C2×D4—2122no—
364C22×C4—41no—
365C22×C4#1171994511yescharacteristic
366C22×C4D4×D5311yescharacteristic
367C22×C4D4×D5311yescharacteristic
368SD16—4242no—
369SD16—4242no—
370SD16—4242no—
371SD16—2124no—
372SD16—2124no—
373D8—2124no—
374D8—2124no—
375M16—534no—
376M16—1062no—
377C2×C8—21no—
378C2×C8—21no—
379C2×C8—21no—
380C2×C8—21no—
381C2×C8—42no—
382C2×C8—532no—
383C2×C8—532no—
384C2×C8—1062no—
385C2×C8—4241no—
386C4⋊C4—21no—
387C4⋊C4—21no—
388C4⋊C4—532no—
389C4⋊C4—1061no—
390C4⋊C4—1061no—
391C4⋊C4—1062no—
392C4⋊C4—1061no—
393C4⋊C4—1062no—
394C22⋊C4—41no—
395C22⋊C4—41no—
396C22⋊C4—1062no—
397C42—1062no—
398C42#1171994511yescharacteristic
399C42D4×D5311yescharacteristic
400C42D4×D5311yescharacteristic
401C42C23×D5311yescharacteristic
402C23—41no—
403C23—4242no—
404C23—4242no—
405C23—4242no—
406C23—2122no—
407C23#1313319011yescharacteristic
408Q8—41no—
409Q8—41no—
410Q8—1062no—
411Q8—1062no—
412Q8—4241no—
413D4—41no—
414D4—41no—
415D4—41no—
416D4—41no—
417D4—81no—
418D4—82no—
419D4—4242no—
420D4—4242no—
421D4—4242no—
422D4—4242no—
423D4—2122no—
424D4—2122no—
425D4—2122no—
426D4—2122no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—82no—
432C2×C4—1062no—
433C2×C4—1062no—
434C2×C4#1313319011yescharacteristic
435C2×C4#1313319011yescharacteristic
436C2×C4#1313319011yescharacteristic
437C2×C4SD16×D5312yes—
438C2×C4#1313324112yes—
439C2×C4C2×D4×D5311yescharacteristic
440C2×C4C2×D4×D5311yescharacteristic
441C2×C4C2×#1171994511yescharacteristic
442C8—41no—
443C8—41no—
444C8—41no—
445C8—41no—
446C8—534no—
447C8—1062no—
448C8—2122no—
449C22—21no—
450C22—21no—
451C22—81no—
452C22—81no—
453C22—81no—
454C22—8484no—
455C22—4242no—
456C22—4242no—
457C22—4242no—
458C22#1681082212yes—
459C22C2×#1313319011yescharacteristic
460C4—21no—
461C4—21no—
462C4—21no—
463C4—81no—
464C4—2122no—
465C4—2122no—
466C4#1681082212yes—
467C4C2×SD16×D5311yescharacteristic
468C4C2×#1313324111yescharacteristic
469C4#1681142511yescharacteristic
470C4#1681143011yescharacteristic
471C2—22no—
472C2—161no—
473C2—8482no—
474C2C2×#1681082211yescharacteristic
475C2#2028439811yescharacteristic
476C2#2028444611yescharacteristic
subgroupG/H
centerC22C2×#13133190
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#1540C2
socleC22×C53C2×#121
prime radical#20285168C2
hypercenter#1540D53
hypocenterC53#19238
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1540

Complements

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

NH ≅ G/Ntype
1. #20286031C2semidirect
2. #20285658C2semidirect
4. #20283658C2semidirect
5. #20274198C2semidirect
6. #20274190C2semidirect
7. #20274074C2semidirect
8. #20274066C2semidirect
9. #20273468C2semidirect
10. C2×#16810366C2semidirect
11. C53×#1540C2semidirect
22. #16810905C22semidirect
27. #16810820C22semidirect
30. #16810750C22semidirect
31. #16810747C22semidirect
36. #16810375C22semidirect
37. #16810373C22semidirect
43. C53×#332C22semidirect
44. C53×#332C22semidirect
45. C53×#331C22semidirect
46. C53×#331C22semidirect
156. #11719917SD16semidirect
242. #1540D53semidirect
279. #332C2×D53semidirect
280. #332C2×D53semidirect
283. #331C2×D53semidirect
284. #331C2×D53semidirect
285. C53#19238semidirect

Maximal subgroups

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

subgroupindex
#202860312
#202856582
#202851682
#202836582
#202741982
#202741902
#202740742
#202740662
#202734682
C2×#168103662
C53×#15402
#1923853