Finite Group Databasebeta

← #27370121

Subgroups — #27370121

Order 13568.

648 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285838C211yesmaximal, characteristic, complemented
2#20284806C211yesmaximal, characteristic
3C2×#16810913C211yesmaximal, characteristic, complemented
4#20274248C212yesmaximal, complemented
5#20274238C212yesmaximal, complemented
6#20274124C212yesmaximal, complemented
7#20274114C212yesmaximal, complemented
8#20273500C211yesmaximal, characteristic, complemented
9#20273440C211yesmaximal, characteristic, complemented
10#20273411C211yesmaximal, characteristic, complemented
11C53×#2974C211yesmaximal, characteristic, complemented
12#16811173C2211yescharacteristic
13C2×#13133217C2211yescharacteristic
14#16811028C2212yescomplemented
15#16810970C2212yescomplemented
16#16810941—21no—
17#16810937—21no—
18#16810937—21no—
19#16810913—21no—
20#16810913—21no—
21#16810913C2212yes—
22#16810913C2212yes—
23#16810913C2211yescharacteristic
24#16810913C2211yescharacteristic
25#16810904—22no—
26#16810818C2211yescharacteristic
27#16810818C2211yescharacteristic
28#16810757—21no—
29#16810733—21no—
30#16810733—21no—
31#16810680C2211yescharacteristic
32C2×#13133148—21no—
33C2×C4⋊#88322C2211yescharacteristic
34C2×#13133138—21no—
35#16810376C2212yescomplemented
36#16810374C2212yescomplemented
37#16810371C2212yescomplemented
38#16810371C2212yescomplemented
39#16810371C2212yescomplemented
40#16810366C2212yescomplemented
41C2×C4×D4×C53C2211yescharacteristic
42C53×#436C2212yes—
43C53×#435—22no—
44C8×D4×C53C2212yes—
45C53×#432C2211yescharacteristic
46C53×#431C2211yescharacteristic
47C53×#407—21no—
48C2×C22⋊C8×C53—21no—
49C2×#11719945—21no—
50C22×Dic106—21no—
51C2×#11719922—21no—
52C2×#11719922—21no—
53C22×#88322—21no—
54#13133217—21no—
55#13133217—21no—
56#13133217—21no—
57#13133217C2311yescharacteristic
58#13133217C2312yes—
59#13133214C2311yescharacteristic
60#13133205—21no—
61#13133191—21no—
62#13133191—21no—
63C4×D212—21no—
64C4×D212—21no—
65C2×C4⋊Dic53—21no—
66C2×C4⋊Dic53—21no—
67#13133151—22no—
68#13133151—22no—
69#13133148—22no—
70#13133148—22no—
71#13133148—21no—
72#13133148—21no—
73C4⋊#88322—21no—
74C4⋊#88322C2311yescharacteristic
75C4⋊#88322C2311yescharacteristic
76C4⋊#88322C2312yes—
77#13133139—22no—
78#13133139—22no—
79#13133138—21no—
80#13133138—21no—
81#13133138—22no—
82#13133138—22no—
83#13133137—21no—
84#13133137—21no—
85C22⋊#88322—21no—
86#13133128—21no—
87C22×D4×C53D411yescharacteristic
88C23×C4×C53—21no—
89C2×M16×C53—22no—
90C22×C8×C53—22no—
91C4×D4×C53—21no—
92C4×D4×C53—21no—
93C4×D4×C53D411yescharacteristic
94C4×D4×C53D411yescharacteristic
95C4×D4×C53C2311yescharacteristic
96C4×D4×C53C2311yescharacteristic
97C2×C4⋊C4×C53D411yescharacteristic
98C2×C22⋊C4×C53—21no—
99C2×C42×C53C2311yescharacteristic
100C4⋊C8×C53D412yes—
101C4⋊C8×C53C2312yes—
102C22⋊C8×C53—22no—
103C22⋊C8×C53—22no—
104C22⋊C8×C53—21no—
105C22⋊C8×C53D411yescharacteristic
106C22⋊C8×C53D411yescharacteristic
107#97×C53—21no—
108C4×C8×C53C2312yes—
109C24×C53—21no—
110C2×D4×C53—22no—
111C2×D4×C53—21no—
112C2×D4×C53—41no—
113C2×D4×C53C2×D411yescharacteristic
114C2×D4×C53C2×D411yescharacteristic
115C22×C4×C53—21no—
116C22×C4×C53—21no—
117C22×C4×C53—21no—
118C22×C4×C53—21no—
119C22×C4×C53—21no—
120C22×C4×C53—82no—
121C22×C4×C53C2×D411yescharacteristic
122C22×C4×C53C2×D411yescharacteristic
123C22×C4×C53#4111yescharacteristic
124C2×C22⋊D53—41no—
125C22×Dic53—41no—
126#11719945—82no—
127C2×D212—21no—
128C2×D212—21no—
129C2×D212—21no—
130C2×C4×D53—41no—
131C2×Dic106—21no—
132C2×Dic106—22no—
133C2×Dic106—41no—
134C2×Dic106—82no—
135M16×C53—82no—
136C2×C8×C53—22no—
137C2×C8×C53—82no—
138C2×C8×C53C2×D412yes—
139C2×C8×C53C2×D412yes—
140C4⋊C4×C53—21no—
141C4⋊C4×C53C2×D411yescharacteristic
142C4⋊C4×C53C2×D411yescharacteristic
143C22⋊C4×C53—21no—
144C22⋊C4×C53—21no—
145C22⋊C4×C53—21no—
146C22⋊C4×C53—21no—
147C42×C53C2×D411yescharacteristic
148C42×C53C2×D411yescharacteristic
149C42×C53#4111yescharacteristic
150C42×C53C2411yescharacteristic
151#11719922—42no—
152#11719922—42no—
153#11719921—41no—
154#11719921—41no—
155C4⋊Dic53—22no—
156C4⋊Dic53—21no—
157C4⋊Dic53—22no—
158C4⋊Dic53—21no—
159C4⋊Dic53—41no—
160C4⋊Dic53—41no—
161C2×#88322—21no—
162C2×#88322—21no—
163C2×#88322—22no—
164C2×#88322—42no—
165C23×C53—21no—
166C23×C53—21no—
167C23×C53—42no—
168C23×C53—42no—
169C23×C53—81no—
170C23×C53#12111yescharacteristic
171C22×D53—41no—
172D4×C53—21no—
173D4×C53—21no—
174D4×C53—21no—
175D4×C53—42no—
176D4×C53#13712yes—
177C2×C4×C53—22no—
178C2×C4×C53—21no—
179C2×C4×C53—21no—
180C2×C4×C53—22no—
181C2×C4×C53—21no—
182C2×C4×C53—21no—
183C2×C4×C53—42no—
184C2×C4×C53—42no—
185C2×C4×C53—81no—
186C2×C4×C53—81no—
187C2×C4×C53—81no—
188C2×C4×C53#12111yescharacteristic
189C2×C4×C53#12111yescharacteristic
190C2×C4×C53#12111yescharacteristic
191C2×C4×C53C22×D411yescharacteristic
192C2×C4×C53C22×D411yescharacteristic
193C2×C4×C53C2×#4111yescharacteristic
194C22⋊D53—82no—
195C2×Dic53—42no—
196C2×Dic53—41no—
197C2×Dic53—82no—
198D212—41no—
199D212—41no—
200D212—42no—
201D212—81no—
202C4×D53—82no—
203Dic106—82no—
204Dic106—82no—
205Dic106—81no—
206C8×C53—22no—
207C8×C53—22no—
208C8×C53—42no—
209#88322—41no—
210#88322—42no—
211#88322—41no—
212#19465—531nomaximal, Sylow
213C22×C53—21no—
214C22×C53—21no—
215C22×C53—22no—
216C22×C53—41no—
217C22×C53—41no—
218C22×C53—41no—
219C22×C53—41no—
220C22×C53—41no—
221C22×C53—84no—
222C22×C53#47212yes—
223C22×C53C2×#12111yescharacteristic
224C2×D53—81no—
225C2×D53—81no—
226C2×D53—81no—
227C4×C53—21no—
228C4×C53—21no—
229C4×C53—21no—
230C4×C53—21no—
231C4×C53—41no—
232C4×C53—41no—
233C4×C53C2×#13711yescharacteristic
234C4×C53#57511yescharacteristic
235C4×C53#57712yes—
236Dic53—82no—
237Dic53—81no—
238#3226—531no—
239#3109—531no—
240C2×#476—531no—
241#2974D5311yescharacteristic, complemented
242#1697—532no—
243#1695—532no—
244#1677—532no—
245#1675—532no—
246#1573—531no—
247#1558—531no—
248#1551—531no—
249C2×C53—22no—
250C2×C53—42no—
251C2×C53—81no—
252C2×C53#304211yescharacteristic
253C2×C53#305611yescharacteristic
254C2×C53C2×#47211yescharacteristic
255D53—161no—
256#532—531no—
257C2×C4⋊Q8—531no—
258C2×C4×D4C2×D5311yescharacteristic
259#500—532no—
260#486—532no—
261#481—1061no—
262#480—1061no—
263#480—1061no—
264#476—531no—
265#476—531no—
266#476—532no—
267#476—532no—
268#476—1061no—
269#476—1061no—
270#474—1062no—
271#459—531no—
272#459—531no—
273#447—1061no—
274#442—1061no—
275#442—1061no—
276#436C2×D5312yes—
277#435—22no—
278C8×D4C2×D5312yes—
279#432—531no—
280#432C2×D5311yescharacteristic
281#431C2×D5311yescharacteristic
282C2×#106—1061no—
283C2×C4⋊C8—531no—
284C2×#102—1061no—
285#407—21no—
286C2×C22⋊C8—21no—
287#335—532no—
288#334—532no—
289#333—532no—
290#333—532no—
291#333—532no—
292#331—532no—
293C53#1946511yescharacteristic, complemented, Sylow
294C2×#41—1061no—
295C22×Q8—1061no—
296C22×D4D21211yescharacteristic
297C23×C4—21no—
298C2×D8—1061no—
299C2×D8—1061no—
300C2×M16—22no—
301C22×C8—22no—
302C22×C8—1061no—
303C4⋊Q8—532no—
304C4⋊Q8—531no—
305C4⋊Q8—1061no—
306C4⋊Q8—1061no—
307C4⋊Q8—1061no—
308#127—531no—
309#124—1061no—
310#121—1061no—
311#121—1061no—
312C4×D4—21no—
313C4×D4—21no—
314C4×D4—1061no—
315C4×D4—1061no—
316C4×D4D21211yescharacteristic
317C4×D4D21211yescharacteristic
318C4×D4C22×D5311yescharacteristic
319C4×D4C22×D5311yescharacteristic
320C2×C4⋊C4—1061no—
321C2×C4⋊C4—1061no—
322C2×C4⋊C4D21211yescharacteristic
323C2×C22⋊C4—21no—
324C2×C42C22×D5311yescharacteristic
325#107—1062no—
326#107—1062no—
327#106—1062no—
328#106—1062no—
329#106—1061no—
330#106—1061no—
331C4⋊C8—532no—
332C4⋊C8—531no—
333C4⋊C8—531no—
334C4⋊C8—1061no—
335C4⋊C8C22⋊D5312yes—
336C4⋊C8C22×D5312yes—
337#102—1062no—
338#102—1062no—
339#102—1061no—
340#102—1061no—
341#102—1062no—
342#102—1062no—
343#102—1061no—
344#102—1061no—
345C22⋊C8—21no—
346C22⋊C8—22no—
347C22⋊C8—22no—
348C22⋊C8—1061no—
349C22⋊C8C22⋊D5311yescharacteristic
350C22⋊C8C22⋊D5311yescharacteristic
351#97—21no—
352#97—1061no—
353C4×C8C22×D5312yes—
354C24—21no—
355#41—4242no—
356C2×Q8—1061no—
357C2×Q8—1062no—
358C2×Q8—4242no—
359C2×Q8—2121no—
360C2×D4—22no—
361C2×D4—21no—
362C2×D4—41no—
363C2×D4—1061no—
364C2×D4—1061no—
365C2×D4—1061no—
366C2×D4—2121no—
367C2×D4C2×D21211yescharacteristic
368C2×D4C2×D21211yescharacteristic
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—82no—
375C22×C4—2121no—
376C22×C4—2121no—
377C22×C4C2×D21211yescharacteristic
378C22×C4#1171994511yescharacteristic
379C22×C4C2×C22⋊D5311yescharacteristic
380D8—2122no—
381D8—2122no—
382M16—82no—
383C2×C8—22no—
384C2×C8—82no—
385C2×C8—1061no—
386C2×C8—1061no—
387C2×C8—1062no—
388C2×C8—2122no—
389C2×C8C2×C22⋊D5312yes—
390C2×C8C2×C22⋊D5312yes—
391C4⋊C4—21no—
392C4⋊C4—1062no—
393C4⋊C4—1061no—
394C4⋊C4—1062no—
395C4⋊C4—1061no—
396C4⋊C4—2121no—
397C4⋊C4—2121no—
398C4⋊C4C2×D21211yescharacteristic
399C4⋊C4C2×D21211yescharacteristic
400C22⋊C4—21no—
401C22⋊C4—21no—
402C22⋊C4—21no—
403C22⋊C4—21no—
404C22⋊C4—2121no—
405C22⋊C4—2121no—
406C42C2×D21211yescharacteristic
407C42#1171994511yescharacteristic
408C42C2×C22⋊D5311yescharacteristic
409C42C23×D5311yescharacteristic
410C23—21no—
411C23—21no—
412C23—42no—
413C23—42no—
414C23—81no—
415C23—2121no—
416C23#1313319111yescharacteristic
417Q8—4242no—
418Q8—4241no—
419Q8—4242no—
420D4—21no—
421D4—21no—
422D4—21no—
423D4—42no—
424D4—4241no—
425D4—4242no—
426D4—2121no—
427D4—2121no—
428D4—2122no—
429D4#1313325012yes—
430C2×C4—22no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—22no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—81no—
439C2×C4—81no—
440C2×C4—81no—
441C2×C4—4242no—
442C2×C4—4242no—
443C2×C4—2121no—
444C2×C4—2122no—
445C2×C4#1313319111yescharacteristic
446C2×C4#1313319111yescharacteristic
447C2×C4#1313319111yescharacteristic
448C2×C4C22×C22⋊D5311yescharacteristic
449C2×C4C22×D21211yescharacteristic
450C2×C4C2×#1171994511yescharacteristic
451C8—22no—
452C8—22no—
453C8—42no—
454C8—2121no—
455C8—2122no—
456C8—2121no—
457C22—21no—
458C22—21no—
459C22—22no—
460C22—41no—
461C22—41no—
462C22—41no—
463C22—41no—
464C22—41no—
465C22—84no—
466C22—4241no—
467C22—4241no—
468C22—4241no—
469C22#1681089312yes—
470C22C2×#1313319111yescharacteristic
471C4—21no—
472C4—21no—
473C4—21no—
474C4—21no—
475C4—41no—
476C4—41no—
477C4—4241no—
478C4—4242no—
479C4C2×#1313325011yescharacteristic
480C4#1681142311yescharacteristic
481C4#1681143412yes—
482C2—22no—
483C2—42no—
484C2—81no—
485C2—8481no—
486C2#2028422711yescharacteristic
487C2#2028434511yescharacteristic
488C2C2×#1681089311yescharacteristic
subgroupG/H
centerC22C2×#13133191
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#2974C2
socleC22×C53C2×#121
prime radical#20284806C2
hypercenter#2974D53
hypocenterC53#19465
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2974

Complements

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

NH ≅ G/Ntype
1. #20285838C2semidirect
3. C2×#16810913C2semidirect
4. #20274248C2semidirect
5. #20274238C2semidirect
6. #20274124C2semidirect
7. #20274114C2semidirect
8. #20273500C2semidirect
9. #20273440C2semidirect
10. #20273411C2semidirect
11. C53×#2974C2semidirect
14. #16811028C22semidirect
15. #16810970C22semidirect
35. #16810376C22semidirect
36. #16810374C22semidirect
37. #16810371C22semidirect
38. #16810371C22semidirect
39. #16810371C22semidirect
40. #16810366C22semidirect
241. #2974D53semidirect
293. C53#19465semidirect

Maximal subgroups

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

subgroupindex
#202858382
#202848062
C2×#168109132
#202742482
#202742382
#202741242
#202741142
#202735002
#202734402
#202734112
C53×#29742
#1946553