Finite Group Databasebeta

← #27367666

Subgroups — #27367666

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#20285653C211yesmaximal, characteristic, complemented
2#20285168C211yesmaximal, characteristic
3#20283664C211yesmaximal, characteristic, complemented
4#20274197C212yesmaximal, complemented
5#20274189C212yesmaximal, complemented
6#20274073C212yesmaximal, complemented
7#20274065C212yesmaximal, complemented
8#20273465C211yesmaximal, characteristic, complemented
9#20273364C211yesmaximal, characteristic, complemented
10C2×#16810366C211yesmaximal, characteristic, complemented
11C53×#3246C211yesmaximal, characteristic, complemented
12C2×#13133242—21no—
13C2×#13133242—21no—
14C2×#13133214C2211yescharacteristic
15#16811013—22no—
16#16810999C2212yes—
17#16810997—21no—
18#16810997C2212yes—
19#16810971—21no—
20#16810823C2212yes—
21#16810823C2212yes—
22#16810773—22no—
23#16810754—22no—
24#16810754C2212yescomplemented
25Q8×#88322C2212yescomplemented
26#16810676C2211yescharacteristic
27#16810650—21no—
28C2×C4⋊#88322C2211yescharacteristic
29#16810579—21no—
30C4×#11719917C2211yescharacteristic
31#16810375C2212yescomplemented
32#16810373C2212yescomplemented
33#16810369C2211yescharacteristic, complemented
34#16810369C2211yescharacteristic, complemented
35#16810367C2211yescharacteristic, complemented
36#16810367C2211yescharacteristic, complemented
37#16810366C2212yes—
38#16810366C2212yes—
39C53×#533C2211yescharacteristic
40C53×#532C2211yescharacteristic
41C53×#476—21no—
42C53×#475C2212yescomplemented
43C53×#465—22no—
44C53×#461—21no—
45C53×#460C2212yescomplemented
46C2×C4⋊C8×C53C2211yescharacteristic
47C22×D212—21no—
48C22×D212—21no—
49C22×D212—21no—
50#13133242—44no—
51#13133242—44no—
52C2×#11719924—22no—
53C2×#11719924—22no—
54C2×#11719924—22no—
55C2×#11719924—22no—
56C2×#11719922—22no—
57C2×#11719922—22no—
58C2×#11719917D411yescharacteristic
59C2×#11719917D411yescharacteristic
60C22×#88322—21no—
61#13133214—21no—
62#13133214—21no—
63#13133214—21no—
64#13133214C2312yes—
65C4⋊#88322—21no—
66C4⋊#88322—22no—
67C4⋊#88322—22no—
68C4⋊#88322C2311yescharacteristic
69C4⋊#88322C2311yescharacteristic
70C4⋊#88322C2311yescharacteristic
71C4⋊#88322C2311yescharacteristic
72#13133137—22no—
73#13133137—22no—
74C22⋊#88322—21no—
75#13133128—22no—
76#13133128D412yes—
77C4×#88322—21no—
78C4×#88322—22no—
79C4×#88322C2312yes—
80C2×#41×C53—21no—
81C2×SD16×C53—22no—
82C22×C8×C53—21no—
83C4⋊Q8×C53D411yescharacteristic
84C4⋊Q8×C53D411yescharacteristic
85C4⋊Q8×C53C2311yescharacteristic
86#127×C53C2311yescharacteristic
87#125×C53—21no—
88#124×C53—21no—
89C22⋊Q8×C53—21no—
90C22⋊Q8×C53D412yes—
91#121×C53—21no—
92#121×C53—21no—
93C4×Q8×C53—21no—
94C4×Q8×C53C2312yes—
95C4×D4×C53—21no—
96C4×D4×C53—21no—
97#117×C53—21no—
98C2×C42×C53C2311yescharacteristic
99#106×C53—22no—
100C4⋊C8×C53—21no—
101C4⋊C8×C53C2312yes—
102#103×C53—22no—
103#102×C53—22no—
104C23×D53—41no—
105#41×C53—42no—
106#41×C53—42no—
107C2×Q8×C53—21no—
108C2×Q8×C53—21no—
109C2×Q8×C53C2×D412yes—
110C2×D4×C53—21no—
111C2×D4×C53—21no—
112C2×D4×C53—21no—
113C2×D4×C53—41no—
114C22×C4×C53—41no—
115C22×C4×C53C2×D411yescharacteristic
116C22×C4×C53C2×D411yescharacteristic
117C22×C4×C53#4111yescharacteristic
118C2×D212—22no—
119C2×D212—22no—
120C2×D212—22no—
121C2×D212—42no—
122C2×D212—42no—
123C2×D212—41no—
124C2×D212—41no—
125C2×D212—41no—
126C2×D212—81no—
127C2×D212—81no—
128C2×D212—82no—
129SD16×C53—82no—
130C2×C8×C53—22no—
131C2×C8×C53—81no—
132C4⋊C4×C53—21no—
133C4⋊C4×C53—21no—
134C4⋊C4×C53—21no—
135C4⋊C4×C53—22no—
136C4⋊C4×C53—22no—
137C4⋊C4×C53—21no—
138C4⋊C4×C53—21no—
139C4⋊C4×C53C2×D412yes—
140C22⋊C4×C53—22no—
141C22⋊C4×C53—41no—
142C22⋊C4×C53—41no—
143C42×C53—22no—
144C42×C53C2×D411yescharacteristic
145C42×C53C2×D411yescharacteristic
146C42×C53#4111yescharacteristic
147C42×C53C2411yescharacteristic
148#11719924—44no—
149#11719924—44no—
150#11719924—82no—
151#11719924—82no—
152#11719922—44no—
153#11719922—44no—
154#11719917—22no—
155#11719917SD1614yescomplemented
156C2×#88322—21no—
157C2×#88322—21no—
158C2×#88322—21no—
159C2×#88322—21no—
160C2×#88322—42no—
161C2×#88322C2×D412yes—
162C2×#88322C2×D412yes—
163C23×C53—41no—
164C23×C53#12111yescharacteristic
165C22×D53—42no—
166C22×D53—82no—
167C22×D53—82no—
168C22×D53—82no—
169Q8×C53—22no—
170Q8×C53—22no—
171Q8×C53—41no—
172Q8×C53—41no—
173Q8×C53—81no—
174D4×C53—41no—
175D4×C53—41no—
176D4×C53—41no—
177D4×C53—41no—
178D4×C53—81no—
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—41no—
186C2×C4×C53—82no—
187C2×C4×C53#12111yescharacteristic
188C2×C4×C53#12111yescharacteristic
189C2×C4×C53#12111yescharacteristic
190C2×C4×C53C2×SD1612yes—
191C2×C4×C53AGL(1,ℤ/8)12yes—
192C2×C4×C53C22×D411yescharacteristic
193C2×C4×C53C22×D411yescharacteristic
194C2×C4×C53C2×#4111yescharacteristic
195D212—42no—
196D212—42no—
197D212—42no—
198D212—42no—
199D212—82no—
200D212—82no—
201D212—82no—
202D212—82no—
203C8×C53—42no—
204#88322—22no—
205#88322—41no—
206#88322—41no—
207#88322—41no—
208#88322—41no—
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—42no—
226C4×C53—42no—
227C4×C53—81no—
228C4×C53#46012yes—
229C4×C53C22×SD1611yescharacteristic
230C4×C53C2×AGL(1,ℤ/8)11yescharacteristic
231C4×C53#57611yescharacteristic
232C4×C53#57711yescharacteristic
233#3246D5311yescharacteristic, complemented
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#1540—531no—
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#533C2×D5311yescharacteristic
253#532C2×D5311yescharacteristic
254C2×#127—531no—
255#496—1062no—
256#493—532no—
257#492—532no—
258#492—1061no—
259#486—1061no—
260#476—21no—
261#475C2×D5312yescomplemented
262#465—22no—
263#461—21no—
264#460—532no—
265#460—532no—
266#460C2×D5312yescomplemented
267#450—1062no—
268#446—532no—
269#446—1062no—
270C8×Q8—532no—
271#431—531no—
272#424—1061no—
273C2×C4⋊C8—531no—
274C2×C4⋊C8C2×D5311yescharacteristic
275#405—1061no—
276C4×M16—531no—
277#335—532no—
278#334—532no—
279#332—531no—
280#332—531no—
281#331—531no—
282#331—531no—
283#331—532no—
284#331—532no—
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—22no—
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⋊Q8C22⋊D5311yescharacteristic
304C4⋊Q8C22⋊D5311yescharacteristic
305C4⋊Q8C22×D5311yescharacteristic
306#127—532no—
307#127—1061no—
308#127—1061no—
309#127—1061no—
310#127C22×D5311yescharacteristic
311#125—21no—
312#124—21no—
313C22⋊Q8—21no—
314C22⋊Q8C22⋊D5312yes—
315#121—21no—
316#121—21no—
317C4×Q8—21no—
318C4×Q8C22×D5312yes—
319C4×D4—21no—
320C4×D4—21no—
321#117—21no—
322C2×C42C22×D5311yescharacteristic
323#106—22no—
324C4⋊C8—21no—
325C4⋊C8—531no—
326C4⋊C8—531no—
327C4⋊C8—531no—
328C4⋊C8—531no—
329C4⋊C8—1062no—
330C4⋊C8—1062no—
331C4⋊C8—1061no—
332C4⋊C8C22×D5312yes—
333#103—22no—
334#102—22no—
335#102—1062no—
336#102—1062no—
337C22⋊C8—1061no—
338#97—532no—
339#97—1062no—
340C4×C8—532no—
341C4×C8—1062no—
342C4×C8—1061no—
343C24—2121no—
344#41—42no—
345#41—42no—
346C2×Q8—21no—
347C2×Q8—21no—
348C2×Q8C2×C22⋊D5312yes—
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×C4D4×D5311yescharacteristic
366C22×C4#1171994711yescharacteristic
367C22×C4C2×C22⋊D5311yescharacteristic
368SD16—82no—
369SD16—4242no—
370SD16—4242no—
371SD16—2124no—
372SD16—2124no—
373D8—2124no—
374D8—2124no—
375M16—534no—
376M16—1062no—
377C2×C8—22no—
378C2×C8—81no—
379C2×C8—532no—
380C2×C8—532no—
381C2×C8—1061no—
382C2×C8—1061no—
383C2×C8—1061no—
384C2×C8—1061no—
385C2×C8—2122no—
386C4⋊C4—21no—
387C4⋊C4—21no—
388C4⋊C4—21no—
389C4⋊C4—22no—
390C4⋊C4—22no—
391C4⋊C4—21no—
392C4⋊C4—21no—
393C4⋊C4C2×C22⋊D5312yes—
394C22⋊C4—22no—
395C22⋊C4—41no—
396C22⋊C4—41no—
397C42—22no—
398C42D4×D5311yescharacteristic
399C42#1171994711yescharacteristic
400C42C2×C22⋊D5311yescharacteristic
401C42C23×D5311yescharacteristic
402C23—41no—
403C23—4242no—
404C23—4242no—
405C23—4242no—
406C23—2122no—
407C23#1313318911yescharacteristic
408Q8—22no—
409Q8—22no—
410Q8—41no—
411Q8—41no—
412Q8—81no—
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—22no—
430C2×C4—22no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—82no—
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—42no—
443C8—534no—
444C8—1062no—
445C8—2121no—
446C8—2121no—
447C8—2121no—
448C8—2121no—
449C22—21no—
450C22—21no—
451C22—81no—
452C22—81no—
453C22—81no—
454C22—8484no—
455C22—4242no—
456C22—4242no—
457C22—4242no—
458C22#1681082112yes—
459C22C2×#1313318911yescharacteristic
460C4—21no—
461C4—21no—
462C4—21no—
463C4—42no—
464C4—42no—
465C4—81no—
466C4#1681082112yes—
467C4C2×SD16×D5311yescharacteristic
468C4C2×#1313324411yescharacteristic
469C4#1681142611yescharacteristic
470C4#1681143111yescharacteristic
471C2—22no—
472C2—161no—
473C2—8482no—
474C2C2×#1681082111yescharacteristic
475C2#2028439711yescharacteristic
476C2#2028444511yescharacteristic
subgroupG/H
centerC22C2×#13133189
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#3246C2
socleC22×C53C2×#121
prime radical#20285168C2
hypercenter#3246D53
hypocenterC53#19238
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3246

Complements

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

NH ≅ G/Ntype
1. #20285653C2semidirect
3. #20283664C2semidirect
4. #20274197C2semidirect
5. #20274189C2semidirect
6. #20274073C2semidirect
7. #20274065C2semidirect
8. #20273465C2semidirect
9. #20273364C2semidirect
10. C2×#16810366C2semidirect
11. C53×#3246C2semidirect
24. #16810754C22semidirect
25. Q8×#88322C22semidirect
31. #16810375C22semidirect
32. #16810373C22semidirect
33. #16810369C22semidirect
34. #16810369C22semidirect
35. #16810367C22semidirect
36. #16810367C22semidirect
42. C53×#475C22semidirect
45. C53×#460C22semidirect
155. #11719917SD16semidirect
233. #3246D53semidirect
261. #475C2×D53semidirect
266. #460C2×D53semidirect
285. C53#19238semidirect

Maximal subgroups

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

subgroupindex
#202856532
#202851682
#202836642
#202741972
#202741892
#202740732
#202740652
#202734652
#202733642
C2×#168103662
C53×#32462
#1923853