Finite Group Databasebeta

← #27367664

Subgroups — #27367664

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#20285653C211yesmaximal, characteristic, complemented
3#20285167C211yesmaximal, characteristic
4#20274199C212yesmaximal, complemented
5#20274187C212yesmaximal, complemented
6#20274075C212yesmaximal, complemented
7#20274063C212yesmaximal, complemented
8#20273467C211yesmaximal, characteristic, complemented
9#20273365C211yesmaximal, characteristic, complemented
10C2×#16810366C211yesmaximal, characteristic, complemented
11C53×#2962C211yesmaximal, characteristic, complemented
12C2×#13133245—21no—
13C2×#13133245—21no—
14#16811173C2211yescharacteristic
15C2×#13133214C2211yescharacteristic
16#16811012—22no—
17#16811000C2212yes—
18#16810997—21no—
19#16810996C2212yes—
20#16810971—21no—
21#16810910—21no—
22#16810908C2212yescomplemented
23#16810854—22no—
24#16810825—21no—
25#16810823C2212yes—
26#16810823C2212yes—
27#16810823C2212yescomplemented
28#16810773—22no—
29#16810676C2211yescharacteristic
30C2×C4⋊#88322C2211yescharacteristic
31#16810376C2212yescomplemented
32#16810374C2212yescomplemented
33#16810370C2211yescharacteristic, complemented
34#16810370C2211yescharacteristic, complemented
35#16810366C2212yes—
36#16810366C2212yes—
37#16810366C2211yescharacteristic, complemented
38#16810366C2211yescharacteristic, complemented
39C53×#533C2211yescharacteristic
40C53×#446—22no—
41C53×#446C2212yescomplemented
42C8×Q8×C53C2212yescomplemented
43C53×#424—21no—
44C2×C4⋊C8×C53C2211yescharacteristic
45C53×#405—21no—
46C4×M16×C53C2211yescharacteristic
47C2×#11719945—21no—
48C22×D212—21no—
49C22×D212—21no—
50C22×D212—21no—
51#13133245—44no—
52#13133245—44no—
53C2×#11719913—22no—
54C2×#11719913—22no—
55C2×#11719924—22no—
56C2×#11719924—22no—
57C2×#11719924—22no—
58C2×D424—22no—
59C2×D424—22no—
60C22×#88322—21no—
61#13133217C2311yescharacteristic
62#13133214—21no—
63#13133214—21no—
64#13133214—21no—
65#13133214C2312yes—
66#13133214C2311yescharacteristic
67#13133205—21no—
68#13133191—21no—
69#13133191—21no—
70C4×D212—21no—
71C4×D212—21no—
72#13133148—22no—
73C4⋊#88322—21no—
74C4⋊#88322C2311yescharacteristic
75C4⋊#88322C2311yescharacteristic
76C4⋊#88322C2312yes—
77#13133141—22no—
78#13133137—22no—
79#13133137—22no—
80#13133137—22no—
81C22⋊#88322—21no—
82C4×#88322—21no—
83C2×M16×C53D411yescharacteristic
84C2×M16×C53D411yescharacteristic
85C22×C8×C53—21no—
86C4⋊Q8×C53D411yescharacteristic
87C4⋊Q8×C53D411yescharacteristic
88#125×C53—21no—
89C22⋊Q8×C53—21no—
90C22⋊Q8×C53D412yes—
91C4×Q8×C53—21no—
92C4×Q8×C53C2312yes—
93#117×C53—21no—
94C2×C42×C53C2311yescharacteristic
95C4⋊C8×C53—22no—
96C4⋊C8×C53—22no—
97C4⋊C8×C53—21no—
98C4⋊C8×C53C2311yescharacteristic
99C4⋊C8×C53C2311yescharacteristic
100#97×C53—22no—
101#97×C53D412yes—
102C4×C8×C53—22no—
103C4×C8×C53C2312yes—
104C23×D53—41no—
105C2×Q8×C53—21no—
106C2×Q8×C53C2×D412yes—
107C22×C4×C53C2×D411yescharacteristic
108C22×C4×C53C2×D411yescharacteristic
109C22×C4×C53#4111yescharacteristic
110C2×C22⋊D53—41no—
111#11719945—42no—
112#11719945—42no—
113C2×D212—21no—
114C2×D212—22no—
115C2×D212—21no—
116C2×D212—22no—
117C2×D212—21no—
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×C4×D53—41no—
128C2×Dic106—21no—
129M16×C53—22no—
130M16×C53SD1614yescomplemented
131C2×C8×C53—21no—
132C2×C8×C53—21no—
133C2×C8×C53—42no—
134C2×C8×C53C2×D412yes—
135C2×C8×C53C2×D412yes—
136C4⋊C4×C53—21no—
137C4⋊C4×C53—21no—
138C4⋊C4×C53—21no—
139C4⋊C4×C53—22no—
140C4⋊C4×C53—22no—
141C4⋊C4×C53C2×D412yes—
142C22⋊C4×C53—22no—
143C42×C53—22no—
144C42×C53C2×D411yescharacteristic
145C42×C53C2×D411yescharacteristic
146C42×C53#4111yescharacteristic
147C42×C53C2411yescharacteristic
148#11719924—82no—
149#11719924—82no—
150#11719924—82no—
151#11719921—41no—
152#11719921—41no—
153C4⋊Dic53—21no—
154C4⋊Dic53—21no—
155C2×#88322—21no—
156C2×#88322—21no—
157C2×#88322—22no—
158C2×#88322—81no—
159D424—44no—
160D424—44no—
161#11719913—44no—
162#11719913—44no—
163C23×C53#12111yescharacteristic
164C22×D53—41no—
165C22×D53—42no—
166C22×D53—82no—
167C22×D53—82no—
168C22×D53—82no—
169Q8×C53—22no—
170Q8×C53—22no—
171Q8×C53—81no—
172C2×C4×C53—21no—
173C2×C4×C53—21no—
174C2×C4×C53—22no—
175C2×C4×C53—22no—
176C2×C4×C53—41no—
177C2×C4×C53#12111yescharacteristic
178C2×C4×C53#12111yescharacteristic
179C2×C4×C53#12111yescharacteristic
180C2×C4×C53C2×SD1612yes—
181C2×C4×C53AGL(1,ℤ/8)12yes—
182C2×C4×C53C22×D411yescharacteristic
183C2×C4×C53C22×D411yescharacteristic
184C2×C4×C53C2×#4111yescharacteristic
185C22⋊D53—82no—
186C2×Dic53—41no—
187D212—42no—
188D212—41no—
189D212—42no—
190D212—41no—
191D212—42no—
192D212—41no—
193D212—41no—
194D212—42no—
195D212—81no—
196D212—82no—
197D212—82no—
198D212—82no—
199D212—82no—
200C4×D53—82no—
201Dic106—41no—
202Dic106—41no—
203C8×C53—22no—
204C8×C53—41no—
205C8×C53—41no—
206C8×C53C2×SD1614yes—
207#88322—42no—
208#88322—41no—
209#88322—41no—
210#19238—531nomaximal, Sylow
211C22×C53—21no—
212C22×C53—21no—
213C22×C53#46012yes—
214C22×C53C2×#12111yescharacteristic
215C2×D53—81no—
216C2×D53—82no—
217C2×D53—82no—
218C2×D53—81no—
219C2×D53—82no—
220C2×D53—81no—
221C2×D53—164no—
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#3208—531no—
235#3151—531no—
236#2962D5311yescharacteristic, complemented
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×C53C2×#46011yescharacteristic
246C2×C53#306211yescharacteristic
247C2×C53#306811yescharacteristic
248D53—161no—
249D53—162no—
250C2×AGL(1,ℤ/8)—1061no—
251C2×AGL(1,ℤ/8)—1061no—
252#533C2×D5311yescharacteristic
253#532—531no—
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—22no—
269#446C2×D5312yescomplemented
270C8×Q8C2×D5312yescomplemented
271#431—531no—
272#424—21no—
273C2×C4⋊C8—531no—
274C2×C4⋊C8C2×D5311yescharacteristic
275#405—21no—
276C4×M16C2×D5311yescharacteristic
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—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×M16C22⋊D5311yescharacteristic
300C2×M16C22⋊D5311yescharacteristic
301C22×C8—21no—
302C22×C8—1061no—
303C4⋊Q8—531no—
304C4⋊Q8D21211yescharacteristic
305C4⋊Q8D21211yescharacteristic
306#127—532no—
307#127—531no—
308#127—1061no—
309#127—1061no—
310#127—1061no—
311#125—21no—
312#124—1061no—
313C22⋊Q8—21no—
314C22⋊Q8D21212yes—
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—22no—
325C4⋊C8—22no—
326C4⋊C8—21no—
327C4⋊C8—532no—
328C4⋊C8—531no—
329C4⋊C8—531no—
330C4⋊C8—1061no—
331C4⋊C8C22×D5311yescharacteristic
332C4⋊C8C22×D5311yescharacteristic
333#103—1062no—
334#102—1062no—
335#102—1062no—
336#102—1062no—
337C22⋊C8—1061no—
338#97—22no—
339#97C22⋊D5312yes—
340C4×C8—22no—
341C4×C8—1061no—
342C4×C8C22×D5312yes—
343C24—2121no—
344#41—2122no—
345#41—2122no—
346C2×Q8—21no—
347C2×Q8—1061no—
348C2×Q8C2×D21212yes—
349C2×D4—1061no—
350C2×D4—1062no—
351C2×D4—1061no—
352C2×D4—1062no—
353C2×D4—1061no—
354C2×D4—1062no—
355C2×D4—4241no—
356C2×D4—4241no—
357C2×D4—4242no—
358C2×D4—2121no—
359C2×D4—2121no—
360C2×D4—2121no—
361C2×D4—2121no—
362C2×D4—2122no—
363C2×D4—2122no—
364C22×C4—2121no—
365C22×C4C2×D21211yescharacteristic
366C22×C4#1171994511yescharacteristic
367C22×C4C2×C22⋊D5311yescharacteristic
368SD16—4242no—
369SD16—4242no—
370SD16—4242no—
371SD16—2124no—
372SD16—2124no—
373D8—2124no—
374D8—2124no—
375M16—22no—
376M16#1171992414yescomplemented
377C2×C8—21no—
378C2×C8—21no—
379C2×C8—42no—
380C2×C8—1061no—
381C2×C8—1061no—
382C2×C8—1062no—
383C2×C8—4241no—
384C2×C8C2×C22⋊D5312yes—
385C2×C8C2×C22⋊D5312yes—
386C4⋊C4—21no—
387C4⋊C4—21no—
388C4⋊C4—21no—
389C4⋊C4—22no—
390C4⋊C4—22no—
391C4⋊C4—1061no—
392C4⋊C4—1061no—
393C4⋊C4C2×D21212yes—
394C22⋊C4—22no—
395C22⋊C4—2121no—
396C22⋊C4—2121no—
397C42—22no—
398C42C2×D21211yescharacteristic
399C42#1171994511yescharacteristic
400C42C2×C22⋊D5311yescharacteristic
401C42C23×D5311yescharacteristic
402C23—4242no—
403C23—4242no—
404C23—4242no—
405C23—2121no—
406C23—2122no—
407C23#1313319111yescharacteristic
408Q8—22no—
409Q8—22no—
410Q8—81no—
411Q8—2121no—
412Q8—2121no—
413D4—4241no—
414D4—4242no—
415D4—4242no—
416D4—4242no—
417D4—4242no—
418D4—4242no—
419D4—2121no—
420D4—2122no—
421D4—2121no—
422D4—2122no—
423D4—2121no—
424D4—2122no—
425D4—2122no—
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#1313319111yescharacteristic
435C2×C4#1313319111yescharacteristic
436C2×C4#1313319111yescharacteristic
437C2×C4C2×#1171992412yes—
438C2×C4#1313324212yes—
439C2×C4C22×C22⋊D5311yescharacteristic
440C2×C4C22×D21211yescharacteristic
441C2×C4C2×#1171994511yescharacteristic
442C8—22no—
443C8—41no—
444C8—41no—
445C8—2122no—
446C8—2121no—
447C8—2121no—
448C8C2×#1171992414yes—
449C22—21no—
450C22—21no—
451C22—8484no—
452C22—4242no—
453C22—4241no—
454C22—4242no—
455C22—4241no—
456C22—4241no—
457C22—4242no—
458C22#1681082312yes—
459C22C2×#1313319111yescharacteristic
460C4—21no—
461C4—21no—
462C4—21no—
463C4—42no—
464C4—42no—
465C4—4241no—
466C4#1681082312yes—
467C4C22×#1171992411yescharacteristic
468C4C2×#1313324211yescharacteristic
469C4#1681142811yescharacteristic
470C4#1681143511yescharacteristic
471C2—22no—
472C2—8482no—
473C2—8481no—
474C2C2×#1681082311yescharacteristic
475C2#2028439511yescharacteristic
476C2#2028444311yescharacteristic
subgroupG/H
centerC22C2×#13133191
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#2962C2
socleC22×C53C2×#121
prime radical#20285167C2
hypercenter#2962D53
hypocenterC53#19238
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2962

Complements

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

NH ≅ G/Ntype
1. #20286027C2semidirect
2. #20285653C2semidirect
4. #20274199C2semidirect
5. #20274187C2semidirect
6. #20274075C2semidirect
7. #20274063C2semidirect
8. #20273467C2semidirect
9. #20273365C2semidirect
10. C2×#16810366C2semidirect
11. C53×#2962C2semidirect
22. #16810908C22semidirect
27. #16810823C22semidirect
31. #16810376C22semidirect
32. #16810374C22semidirect
33. #16810370C22semidirect
34. #16810370C22semidirect
37. #16810366C22semidirect
38. #16810366C22semidirect
41. C53×#446C22semidirect
42. C8×Q8×C53C22semidirect
130. M16×C53SD16semidirect
236. #2962D53semidirect
269. #446C2×D53semidirect
270. C8×Q8C2×D53semidirect
285. C53#19238semidirect
376. M16#11719924semidirect

Maximal subgroups

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

subgroupindex
#202860272
#202856532
#202851672
#202741992
#202741872
#202740752
#202740632
#202734672
#202733652
C2×#168103662
C53×#29622
#1923853