Finite Group Databasebeta

← #27370115

Subgroups — #27370115

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#20284802C211yesmaximal, characteristic
2C2×#16810910C211yesmaximal, characteristic, complemented
3#20283749C211yesmaximal, characteristic, complemented
4#20274245C212yesmaximal, complemented
5#20274235C212yesmaximal, complemented
6#20274121C212yesmaximal, complemented
7#20274111C212yesmaximal, complemented
8#20273500C211yesmaximal, characteristic, complemented
9#20273441C211yesmaximal, characteristic, complemented
10#20273410C211yesmaximal, characteristic, complemented
11C53×#3226C211yesmaximal, characteristic, complemented
12C2×C4×D212C2211yescharacteristic
13#16810939—21no—
14#16810910—21no—
15#16810910—21no—
16#16810910C2212yes—
17#16810910C2212yes—
18#16810910C2211yescharacteristic
19#16810910C2211yescharacteristic
20#16810815C2211yescharacteristic
21#16810815C2211yescharacteristic
22#16810756—21no—
23#16810733—21no—
24#16810733—21no—
25#16810702C2212yes—
26#16810693—22no—
27C8×D212C2212yes—
28#16810680C2211yescharacteristic
29#16810680C2211yescharacteristic
30C2×#13133148—21no—
31C2×C4⋊#88322C2211yescharacteristic
32C2×#13133137—21no—
33#16810585—21no—
34C22⋊C8×D53—21no—
35#16810376C2212yescomplemented
36#16810374C2212yescomplemented
37#16810372C2212yescomplemented
38#16810372C2212yescomplemented
39#16810372C2212yescomplemented
40#16810367C2212yescomplemented
41C53×#532C2211yescharacteristic
42C2×C4⋊Q8×C53C2211yescharacteristic
43C53×#500C2212yescomplemented
44C53×#486C2212yescomplemented
45C53×#480—21no—
46C53×#480—21no—
47C53×#474—22no—
48C53×#431C2211yescharacteristic
49C22×D212D411yescharacteristic
50C22×C4×D53—21no—
51C2×#11719922—21no—
52C2×#11719922—21no—
53C2×#11719912—22no—
54C22×#88322—21no—
55C2×C8×D53—22no—
56C4×D212—21no—
57C4×D212—21no—
58C4×D212D411yescharacteristic
59C4×D212D411yescharacteristic
60C4×D212C2311yescharacteristic
61C4×D212C2311yescharacteristic
62C2×C4⋊Dic53D411yescharacteristic
63C2×#11719921—21no—
64#13133148—22no—
65#13133148—22no—
66#13133148—21no—
67#13133148—21no—
68C4⋊#88322—21no—
69C4⋊#88322C2311yescharacteristic
70C4⋊#88322C2311yescharacteristic
71C4⋊#88322C2312yes—
72C4⋊#88322C2312yes—
73#13133138—21no—
74#13133138—21no—
75#13133137—21no—
76#13133137—21no—
77#13133137—22no—
78#13133137—22no—
79C22⋊#88322—21no—
80C22⋊#88322—21no—
81#13133129—22no—
82#13133129—22no—
83#13133128—21no—
84#13133128—21no—
85C2×#41×C53—21no—
86C22×Q8×C53—21no—
87C4⋊Q8×C53—21no—
88C4⋊Q8×C53—21no—
89C4⋊Q8×C53—21no—
90C4⋊Q8×C53C2312yes—
91C4⋊Q8×C53C2311yescharacteristic
92#127×C53C2311yescharacteristic
93#124×C53—21no—
94#121×C53—21no—
95#121×C53—21no—
96C4×D4×C53—21no—
97C4×D4×C53—21no—
98C2×C4⋊C4×C53—21no—
99C2×C4⋊C4×C53—21no—
100C2×C42×C53C2311yescharacteristic
101#107×C53—22no—
102#107×C53—22no—
103C4⋊C8×C53D412yes—
104#102×C53—22no—
105#102×C53—22no—
106C22⋊C8×C53D411yescharacteristic
107C22⋊C8×C53D411yescharacteristic
108C4×C8×C53C2312yes—
109C23×D53—21no—
110#41×C53—82no—
111C2×Q8×C53—21no—
112C2×Q8×C53—22no—
113C2×Q8×C53—41no—
114C2×Q8×C53—82no—
115C2×D4×C53—21no—
116C2×D4×C53—21no—
117C2×D4×C53—21no—
118C2×D4×C53—41no—
119C22×C4×C53—41no—
120C22×C4×C53—41no—
121C22×C4×C53C2×D411yescharacteristic
122C22×C4×C53C2×D411yescharacteristic
123C22×C4×C53#4111yescharacteristic
124C22×Dic53—21no—
125C2×D212—22no—
126C2×D212—21no—
127C2×D212—41no—
128C2×D212C2×D411yescharacteristic
129C2×D212C2×D411yescharacteristic
130C2×C4×D53—21no—
131C2×C4×D53—21no—
132C2×C4×D53—21no—
133C2×C4×D53—21no—
134C2×C4×D53—82no—
135C2×C8×C53C2×D412yes—
136C2×C8×C53C2×D412yes—
137C4⋊C4×C53—22no—
138C4⋊C4×C53—21no—
139C4⋊C4×C53—22no—
140C4⋊C4×C53—21no—
141C4⋊C4×C53—41no—
142C4⋊C4×C53—41no—
143C22⋊C4×C53—41no—
144C22⋊C4×C53—41no—
145C42×C53C2×D411yescharacteristic
146C42×C53C2×D411yescharacteristic
147C42×C53#4111yescharacteristic
148C42×C53C2411yescharacteristic
149#11719922—42no—
150#11719922—42no—
151#11719921—21no—
152#11719921—21no—
153#11719921—21no—
154#11719921—21no—
155C4⋊Dic53—21no—
156C4⋊Dic53C2×D411yescharacteristic
157C4⋊Dic53C2×D411yescharacteristic
158C2×#88322—22no—
159C2×#88322—21no—
160C2×#88322—21no—
161C2×#88322—22no—
162C2×#88322—42no—
163#11719912—82no—
164C8×D53—82no—
165C23×C53—41no—
166C23×C53#12111yescharacteristic
167C22×D53—21no—
168C22×D53—21no—
169C22×D53—42no—
170C22×D53—42no—
171C22×D53—81no—
172Q8×C53—82no—
173Q8×C53—82no—
174Q8×C53—81no—
175D4×C53—41no—
176D4×C53—41no—
177D4×C53—42no—
178D4×C53—81no—
179D4×C53—82no—
180C2×C4×C53—22no—
181C2×C4×C53—21no—
182C2×C4×C53—21no—
183C2×C4×C53—22no—
184C2×C4×C53—41no—
185C2×C4×C53—42no—
186C2×C4×C53—82no—
187C2×C4×C53—82no—
188C2×C4×C53#12111yescharacteristic
189C2×C4×C53#12111yescharacteristic
190C2×C4×C53#12111yescharacteristic
191C2×C4×C53C22×D411yescharacteristic
192C2×C4×C53C22×D411yescharacteristic
193C2×C4×C53C2×#4111yescharacteristic
194C2×Dic53—21no—
195C2×Dic53—21no—
196C2×Dic53—81no—
197D212—21no—
198D212—21no—
199D212—21no—
200D212—42no—
201D212#13712yes—
202C4×D53—42no—
203C4×D53—42no—
204C4×D53—81no—
205C4×D53—81no—
206C8×C53—22no—
207C8×C53—22no—
208#88322—41no—
209#88322—42no—
210#88322—41no—
211#88322—42no—
212#19465—531nomaximal, Sylow
213C22×C53—21no—
214C22×C53—21no—
215C22×C53—81no—
216C22×C53—81no—
217C22×C53—81no—
218C22×C53#47212yes—
219C22×C53C2×#12111yescharacteristic
220C2×D53—22no—
221C2×D53—41no—
222C2×D53—41no—
223C2×D53—41no—
224C2×D53—41no—
225C2×D53—41no—
226C2×D53—84no—
227C4×C53—21no—
228C4×C53—21no—
229C4×C53—21no—
230C4×C53—21no—
231C4×C53—81no—
232C4×C53—82no—
233C4×C53C2×#13711yescharacteristic
234C4×C53#57511yescharacteristic
235C4×C53#57712yes—
236Dic53—41no—
237Dic53—41no—
238#3226D5311yescharacteristic, complemented
239#3109—531no—
240C2×#476—531no—
241#2974—531no—
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—161no—
251C2×C53#304211yescharacteristic
252C2×C53#305611yescharacteristic
253C2×C53C2×#47211yescharacteristic
254D53—42no—
255D53—81no—
256#532C2×D5311yescharacteristic
257C2×C4⋊Q8C2×D5311yescharacteristic
258C2×C4×D4—531no—
259#500C2×D5312yescomplemented
260#486C2×D5312yescomplemented
261#481—1061no—
262#480—21no—
263#480—21no—
264#476—531no—
265#476—531no—
266#476—532no—
267#476—532no—
268#476—1061no—
269#476—1061no—
270#474—22no—
271#459—531no—
272#459—531no—
273#447—1061no—
274#442—1061no—
275#442—1061no—
276#436—532no—
277#435—1062no—
278C8×D4—532no—
279#432—531no—
280#432—531no—
281#431C2×D5311yescharacteristic
282C2×#106—1061no—
283C2×C4⋊C8—531no—
284C2×#102—1061no—
285#407—1061no—
286C2×C22⋊C8—1061no—
287#335—532no—
288#334—532no—
289#333—532no—
290#333—532no—
291#333—532no—
292#331—532no—
293C53#1946511yescharacteristic, complemented, Sylow
294C2×#41—21no—
295C22×Q8—21no—
296C22×D4—531no—
297C23×C4—1061no—
298C2×D8—1061no—
299C2×D8—1061no—
300C2×M16—1062no—
301C22×C8—1061no—
302C22×C8—1062no—
303C4⋊Q8—21no—
304C4⋊Q8—21no—
305C4⋊Q8—21no—
306C4⋊Q8C22×D5312yes—
307C4⋊Q8C22×D5311yescharacteristic
308#127C22×D5311yescharacteristic
309#124—21no—
310#121—21no—
311#121—21no—
312C4×D4—21no—
313C4×D4—21no—
314C4×D4—531no—
315C4×D4—531no—
316C4×D4—531no—
317C4×D4—531no—
318C4×D4—1061no—
319C4×D4—1061no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—531no—
323C2×C22⋊C4—1061no—
324C2×C42C22×D5311yescharacteristic
325#107—22no—
326#107—22no—
327#106—1062no—
328#106—1062no—
329#106—1061no—
330#106—1061no—
331C4⋊C8—532no—
332C4⋊C8—532no—
333C4⋊C8—531no—
334C4⋊C8—531no—
335C4⋊C8—1061no—
336C4⋊C8C22⋊D5312yes—
337#102—22no—
338#102—22no—
339#102—1062no—
340#102—1062no—
341#102—1061no—
342#102—1061no—
343#102—1061no—
344#102—1061no—
345C22⋊C8—1061no—
346C22⋊C8—1061no—
347C22⋊C8—1062no—
348C22⋊C8—1062no—
349C22⋊C8C22⋊D5311yescharacteristic
350C22⋊C8C22⋊D5311yescharacteristic
351#97—1061no—
352#97—1061no—
353C4×C8C22×D5312yes—
354C24—1061no—
355#41—82no—
356C2×Q8—21no—
357C2×Q8—22no—
358C2×Q8—41no—
359C2×Q8—82no—
360C2×D4—21no—
361C2×D4—21no—
362C2×D4—21no—
363C2×D4—41no—
364C2×D4—531no—
365C2×D4—531no—
366C2×D4—1062no—
367C2×D4—1061no—
368C2×D4—2121no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—1061no—
372C22×C4—1061no—
373C22×C4—1061no—
374C22×C4—1061no—
375C22×C4—1061no—
376C22×C4—4242no—
377C22×C4D4×D5311yescharacteristic
378C22×C4#1171994711yescharacteristic
379C22×C4C2×C22⋊D5311yescharacteristic
380D8—2122no—
381D8—2122no—
382M16—4242no—
383C2×C8—1061no—
384C2×C8—1061no—
385C2×C8—1062no—
386C2×C8—1062no—
387C2×C8—4242no—
388C2×C8—2122no—
389C2×C8C2×C22⋊D5312yes—
390C2×C8C2×C22⋊D5312yes—
391C4⋊C4—22no—
392C4⋊C4—21no—
393C4⋊C4—22no—
394C4⋊C4—21no—
395C4⋊C4—41no—
396C4⋊C4—41no—
397C4⋊C4—531no—
398C4⋊C4—531no—
399C4⋊C4—1061no—
400C22⋊C4—41no—
401C22⋊C4—41no—
402C22⋊C4—1061no—
403C22⋊C4—1061no—
404C22⋊C4—1061no—
405C22⋊C4—1061no—
406C42D4×D5311yescharacteristic
407C42#1171994711yescharacteristic
408C42C2×C22⋊D5311yescharacteristic
409C42C23×D5311yescharacteristic
410C23—41no—
411C23—1061no—
412C23—1061no—
413C23—4241no—
414C23—2122no—
415C23—2122no—
416C23#1313318811yescharacteristic
417Q8—82no—
418Q8—81no—
419Q8—82no—
420D4—41no—
421D4—41no—
422D4—42no—
423D4—81no—
424D4—82no—
425D4—532no—
426D4—1061no—
427D4—1061no—
428D4—1061no—
429D4—2122no—
430C2×C4—22no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—22no—
434C2×C4—41no—
435C2×C4—42no—
436C2×C4—82no—
437C2×C4—82no—
438C2×C4—1061no—
439C2×C4—1061no—
440C2×C4—4241no—
441C2×C4—4241no—
442C2×C4—4241no—
443C2×C4—2122no—
444C2×C4—2122no—
445C2×C4#1313318811yescharacteristic
446C2×C4#1313318811yescharacteristic
447C2×C4#1313318811yescharacteristic
448C2×C4C2×D4×D5311yescharacteristic
449C2×C4C22×C22⋊D5311yescharacteristic
450C2×C4C2×#1171994711yescharacteristic
451C8—22no—
452C8—22no—
453C8—2121no—
454C8—2122no—
455C8—2121no—
456C8—2122no—
457C22—21no—
458C22—21no—
459C22—81no—
460C22—81no—
461C22—81no—
462C22—1062no—
463C22—4244no—
464C22—2121no—
465C22—2121no—
466C22—2121no—
467C22—2121no—
468C22—2121no—
469C22#1681089012yes—
470C22C2×#1313318811yescharacteristic
471C4—21no—
472C4—21no—
473C4—21no—
474C4—21no—
475C4—81no—
476C4—82no—
477C4—2121no—
478C4—2121no—
479C4#137×D5311yescharacteristic
480C4#1681142211yescharacteristic
481C4#1681143112yes—
482C2—22no—
483C2—161no—
484C2—4241no—
485C2—2122no—
486C2#2028422311yescharacteristic
487C2#2028433911yescharacteristic
488C2C2×#1681089011yescharacteristic
subgroupG/H
centerC22C2×#13133188
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#3226C2
socleC22×C53C2×#121
prime radical#20284802C2
hypercenter#3226D53
hypocenterC53#19465
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3226

Complements

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

NH ≅ G/Ntype
2. C2×#16810910C2semidirect
3. #20283749C2semidirect
4. #20274245C2semidirect
5. #20274235C2semidirect
6. #20274121C2semidirect
7. #20274111C2semidirect
8. #20273500C2semidirect
9. #20273441C2semidirect
10. #20273410C2semidirect
11. C53×#3226C2semidirect
35. #16810376C22semidirect
36. #16810374C22semidirect
37. #16810372C22semidirect
38. #16810372C22semidirect
39. #16810372C22semidirect
40. #16810367C22semidirect
43. C53×#500C22semidirect
44. C53×#486C22semidirect
238. #3226D53semidirect
259. #500C2×D53semidirect
260. #486C2×D53semidirect
293. C53#19465semidirect

Maximal subgroups

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

subgroupindex
#202848022
C2×#168109102
#202837492
#202742452
#202742352
#202741212
#202741112
#202735002
#202734412
#202734102
C53×#32262
#1946553