Finite Group Databasebeta

← #27360166

Subgroups — #27360166

Order 13568.

1048 conjugacy classes of subgroups.

The subgroup lattice has 488 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285172C211yesmaximal, characteristic, complemented
2#20284837C211yesmaximal, characteristic, complemented
3#20284806C211yesmaximal, characteristic
4#20273849C214yesmaximal, complemented
5#20273787C214yesmaximal, complemented
6#20273441C211yesmaximal, characteristic, complemented
7#20273162C211yesmaximal, characteristic, complemented
8#20273158C211yesmaximal, characteristic, complemented
9C2×C53×#554C211yesmaximal, characteristic, complemented
10C2×#13133248—22no—
11#16811173C2211yescharacteristic
12#16811015—22no—
13#16811002C2212yescomplemented
14#16810997C2212yescomplemented
15#16810941—21no—
16#16810936—21no—
17#16810918C2212yescomplemented
18#16810913C2212yes—
19#16810823C2212yescomplemented
20#16810818C2212yes—
21#16810762—21no—
22#16810757—21no—
23#16810733—22no—
24#16810721—22no—
25#16810680C2211yescharacteristic
26#16810680C2211yescharacteristic
27C4×#11719917C2211yescharacteristic
28#16810372C2214yescomplemented
29Q8⋊#88322C2214yescomplemented
30D4⋊#88322C2214yes—
31C53×#554—22no—
32C53×#554—22no—
33C53×#554—22no—
34C53×#554C2214yes—
35C2×C4⋊Q8×C53C2211yescharacteristic
36C2×#125×C53—21no—
37C2×C22⋊Q8×C53—21no—
38C2×C22⋊Q8×C53—22no—
39C2×C4×Q8×C53C2211yescharacteristic
40C2×C4×D4×C53—21no—
41C2×C4×D4×C53C2211yescharacteristic
42C22×C4⋊C4×C53—21no—
43C2×#11719945—21no—
44#13133248—48no—
45C2×#11719925—24no—
46C2×#11719924—24no—
47C2×#11719924—22no—
48C2×#11719922—22no—
49C2×#11719917—21no—
50#13133217C2311yescharacteristic
51#13133214C2311yescharacteristic
52#13133205—21no—
53#13133191—21no—
54#13133191—21no—
55C4×D212—21no—
56C4×D212—21no—
57#13133150—22no—
58#13133148—22no—
59C4⋊#88322C2312yes—
60C4⋊#88322C2312yes—
61#13133141—22no—
62#13133138—22no—
63#13133137—22no—
64#13133137—22no—
65C22⋊#88322—21no—
66C22⋊#88322—21no—
67#13133128—21no—
68#13133128—21no—
69#13133128—21no—
70C4×#88322C2312yes—
71C22×Q8×C53D411yescharacteristic
72C22×Q8×C53D412yes—
73C22×D4×C53D411yescharacteristic
74C23×C4×C53—22no—
75C23×C4×C53—21no—
76C4⋊Q8×C53—21no—
77C4⋊Q8×C53—21no—
78C4⋊Q8×C53D412yes—
79C4⋊Q8×C53C2312yes—
80#125×C53—24no—
81#125×C53—24no—
82C22⋊Q8×C53—28no—
83C22⋊Q8×C53—24no—
84C22⋊Q8×C53—28no—
85C22⋊Q8×C53—24no—
86C4×Q8×C53—21no—
87C4×Q8×C53—21no—
88C4×Q8×C53D412yes—
89C4×Q8×C53C2312yes—
90C4×D4×C53—24no—
91C4×D4×C53—21no—
92C4×D4×C53—21no—
93C4×D4×C53—24no—
94C4×D4×C53D412yes—
95C4×D4×C53C2312yes—
96C2×C4⋊C4×C53—22no—
97C2×C4⋊C4×C53—24no—
98C2×C4⋊C4×C53—22no—
99C2×C4⋊C4×C53—24no—
100C2×C4⋊C4×C53—21no—
101C2×C4⋊C4×C53—21no—
102C2×C4⋊C4×C53—21no—
103C2×C4⋊C4×C53D411yescharacteristic
104C2×C4⋊C4×C53D411yescharacteristic
105C2×C22⋊C4×C53—22no—
106C2×C22⋊C4×C53—21no—
107C2×C42×C53—21no—
108C2×C42×C53C2311yescharacteristic
109C24×C53—21no—
110C2×Q8×C53—21no—
111C2×Q8×C53—22no—
112C2×Q8×C53—42no—
113C2×Q8×C53—44no—
114C2×Q8×C53C2×D412yes—
115C2×Q8×C53C2×D414yes—
116C2×D4×C53—24no—
117C2×D4×C53—21no—
118C2×D4×C53C2×D412yes—
119C22×C4×C53—21no—
120C22×C4×C53—21no—
121C22×C4×C53—21no—
122C22×C4×C53—22no—
123C22×C4×C53—28no—
124C22×C4×C53—22no—
125C22×C4×C53—22no—
126C22×C4×C53—44no—
127C22×C4×C53—84no—
128C22×C4×C53C2×D411yescharacteristic
129C22×C4×C53C2×D411yescharacteristic
130C22×C4×C53C2×D411yescharacteristic
131C2×C22⋊D53—41no—
132#11719945—44no—
133C2×D212—21no—
134C2×D212—21no—
135C2×D212—21no—
136C2×C4×D53—41no—
137C2×Dic106—21no—
138C4⋊C4×C53—28no—
139C4⋊C4×C53—28no—
140C4⋊C4×C53—21no—
141C4⋊C4×C53—22no—
142C4⋊C4×C53—22no—
143C4⋊C4×C53—22no—
144C4⋊C4×C53—21no—
145C4⋊C4×C53—22no—
146C4⋊C4×C53—22no—
147C4⋊C4×C53—22no—
148C4⋊C4×C53C2×D412yes—
149C4⋊C4×C53C2×D412yes—
150C22⋊C4×C53—28no—
151C22⋊C4×C53—22no—
152C22⋊C4×C53—22no—
153C42×C53—22no—
154C42×C53—22no—
155C42×C53C2×D411yescharacteristic
156C42×C53C2×D411yescharacteristic
157C42×C53C2×D411yescharacteristic
158C42×C53C2411yescharacteristic
159#11719925—48no—
160#11719924—48no—
161#11719924—44no—
162#11719922—44no—
163#11719921—41no—
164#11719921—41no—
165C4⋊Dic53—21no—
166C4⋊Dic53—21no—
167#11719917—24no—
168C2×#88322—22no—
169C2×#88322—22no—
170C2×#88322—22no—
171C23×C53—24no—
172C23×C53—22no—
173C23×C53—42no—
174C23×C53—42no—
175C23×C53#12011yescharacteristic
176C22×D53—41no—
177Q8×C53—22no—
178Q8×C53—22no—
179Q8×C53—42no—
180Q8×C53—44no—
181Q8×C53—82no—
182D4×C53—24no—
183D4×C53—22no—
184D4×C53#13714yescomplemented
185C2×C4×C53—22no—
186C2×C4×C53—22no—
187C2×C4×C53—22no—
188C2×C4×C53—22no—
189C2×C4×C53—24no—
190C2×C4×C53—22no—
191C2×C4×C53—22no—
192C2×C4×C53—44no—
193C2×C4×C53—42no—
194C2×C4×C53—42no—
195C2×C4×C53—42no—
196C2×C4×C53—44no—
197C2×C4×C53—44no—
198C2×C4×C53—88no—
199C2×C4×C53#12011yescharacteristic
200C2×C4×C53#12011yescharacteristic
201C2×C4×C53#12011yescharacteristic
202C2×C4×C53C22×D411yescharacteristic
203C2×C4×C53C22×D411yescharacteristic
204C2×C4×C53C22×D411yescharacteristic
205C22⋊D53—82no—
206C2×Dic53—41no—
207D212—42no—
208D212—42no—
209D212—42no—
210C4×D53—82no—
211Dic106—42no—
212#88322—24no—
213#88322—42no—
214#88322—42no—
215#18525—531nomaximal, Sylow
216C22×C53—21no—
217C22×C53—21no—
218C22×C53—24no—
219C22×C53—48no—
220C22×C53—42no—
221C22×C53—42no—
222C22×C53#45312yes—
223C22×C53C2×#12011yescharacteristic
224C2×D53—81no—
225C2×D53—81no—
226C2×D53—81no—
227C4×C53—21no—
228C4×C53—21no—
229C4×C53—21no—
230C4×C53—21no—
231C4×C53—42no—
232C4×C53—42no—
233C4×C53—42no—
234C4×C53—44no—
235C4×C53C2×#13712yes—
236C4×C53#57512yes—
237Dic53—81no—
238C2×#554D5311yescharacteristic, complemented
239#3152—531no—
240#3112—531no—
241#3109—531no—
242#1632—534no—
243#1622—534no—
244#1558—531no—
245#1491—531no—
246#1490—531no—
247C2×C53—22no—
248C2×C53—44no—
249C2×C53#300411yescharacteristic
250C2×C53#300511yescharacteristic
251C2×C53C2×#45311yescharacteristic
252D53—161no—
253C2×#137—1062no—
254#554—22no—
255#554—22no—
256#554—22no—
257#554C2×D5314yes—
258#532—531no—
259C2×C4⋊Q8C2×D5311yescharacteristic
260C2×#125—21no—
261C2×C22⋊Q8—21no—
262C2×C22⋊Q8—22no—
263C2×C4×Q8C2×D5311yescharacteristic
264C2×C4×D4—21no—
265C2×C4×D4C2×D5311yescharacteristic
266C22×C4⋊C4—21no—
267#497—1062no—
268#494—532no—
269#492—532no—
270#481—1061no—
271#480—1061no—
272#477—532no—
273#476—532no—
274#460—532no—
275#459—532no—
276#448—1061no—
277#447—1061no—
278#442—1062no—
279#440—1062no—
280#432—531no—
281#432—531no—
282C4×M16—531no—
283#333—534no—
284Q8⋊C8—534no—
285D4⋊C8—534no—
286C53#1852511yescharacteristic, complemented, Sylow
287C2×#41—1061no—
288C22×Q8C22⋊D5311yescharacteristic
289C22×Q8C22⋊D5312yes—
290C22×D4C22⋊D5311yescharacteristic
291C23×C4—22no—
292C23×C4—21no—
293#137—2128no—
294C2×Q16—1064no—
295C2×SD16—1064no—
296C2×SD16—1062no—
297C2×D8—1062no—
298C2×M16—1061no—
299C4⋊Q8—21no—
300C4⋊Q8—21no—
301C4⋊Q8—531no—
302C4⋊Q8C22⋊D5312yes—
303C4⋊Q8C22×D5312yes—
304#127—531no—
305#125—24no—
306#125—24no—
307#124—1061no—
308C22⋊Q8—24no—
309C22⋊Q8—28no—
310C22⋊Q8—24no—
311C22⋊Q8—28no—
312#121—1061no—
313#121—1061no—
314C4×Q8—21no—
315C4×Q8—21no—
316C4×Q8C22⋊D5312yes—
317C4×Q8C22×D5312yes—
318C4×D4—24no—
319C4×D4—24no—
320C4×D4—21no—
321C4×D4—21no—
322C4×D4—1061no—
323C4×D4—1061no—
324C4×D4C22⋊D5312yes—
325C4×D4C22×D5312yes—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—24no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—24no—
333C2×C4⋊C4C22⋊D5311yescharacteristic
334C2×C4⋊C4C22⋊D5311yescharacteristic
335C2×C22⋊C4—22no—
336C2×C22⋊C4—21no—
337C2×C42—21no—
338C2×C42C22×D5311yescharacteristic
339#107—1062no—
340#106—1062no—
341C4⋊C8—532no—
342C4⋊C8—532no—
343#103—1062no—
344#102—1062no—
345#102—1062no—
346#102—1062no—
347C22⋊C8—1061no—
348C22⋊C8—1061no—
349#97—1061no—
350#97—1061no—
351#97—1061no—
352C4×C8—532no—
353C24—21no—
354#41—2124no—
355C2×Q8—21no—
356C2×Q8—22no—
357C2×Q8—42no—
358C2×Q8—44no—
359C2×Q8—1061no—
360C2×Q8C2×C22⋊D5312yes—
361C2×Q8C2×C22⋊D5314yes—
362C2×D4—24no—
363C2×D4—21no—
364C2×D4—1061no—
365C2×D4—1061no—
366C2×D4—1061no—
367C2×D4—2121no—
368C2×D4C2×C22⋊D5312yes—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—22no—
373C22×C4—28no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—44no—
377C22×C4—84no—
378C22×C4—2121no—
379C22×C4C2×C22⋊D5311yescharacteristic
380C22×C4C2×C22⋊D5311yescharacteristic
381C22×C4C2×C22⋊D5311yescharacteristic
382Q16—2128no—
383SD16—2128no—
384SD16—2124no—
385D8—2124no—
386M16—1064no—
387C2×C8—1062no—
388C2×C8—1062no—
389C2×C8—1062no—
390C4⋊C4—28no—
391C4⋊C4—28no—
392C4⋊C4—21no—
393C4⋊C4—22no—
394C4⋊C4—22no—
395C4⋊C4—22no—
396C4⋊C4—21no—
397C4⋊C4—22no—
398C4⋊C4—22no—
399C4⋊C4—22no—
400C4⋊C4—1061no—
401C4⋊C4—1061no—
402C4⋊C4C2×C22⋊D5312yes—
403C4⋊C4C2×C22⋊D5312yes—
404C22⋊C4—28no—
405C22⋊C4—22no—
406C22⋊C4—22no—
407C22⋊C4—2121no—
408C22⋊C4—2121no—
409C42—22no—
410C42—22no—
411C42C2×C22⋊D5311yescharacteristic
412C42C2×C22⋊D5311yescharacteristic
413C42C2×C22⋊D5311yescharacteristic
414C42C23×D5311yescharacteristic
415C23—24no—
416C23—22no—
417C23—42no—
418C23—42no—
419C23—2121no—
420C23#1313318611yescharacteristic
421Q8—22no—
422Q8—22no—
423Q8—44no—
424Q8—42no—
425Q8—82no—
426Q8—2122no—
427D4—24no—
428D4—22no—
429D4—4242no—
430D4—2122no—
431D4—2122no—
432D4—2122no—
433D4#1313324814yescomplemented
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—24no—
441C2×C4—44no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—44no—
446C2×C4—44no—
447C2×C4—88no—
448C2×C4—4242no—
449C2×C4—2121no—
450C2×C4#1313318611yescharacteristic
451C2×C4#1313318611yescharacteristic
452C2×C4#1313318611yescharacteristic
453C2×C4C22×C22⋊D5311yescharacteristic
454C2×C4C22×C22⋊D5311yescharacteristic
455C2×C4C22×C22⋊D5311yescharacteristic
456C8—1064no—
457C8—2122no—
458C8—2122no—
459C22—21no—
460C22—21no—
461C22—24no—
462C22—48no—
463C22—42no—
464C22—42no—
465C22—4241no—
466C22—4241no—
467C22—4241no—
468C22#1681079112yes—
469C22C2×#1313318611yescharacteristic
470C4—21no—
471C4—21no—
472C4—21no—
473C4—21no—
474C4—42no—
475C4—42no—
476C4—42no—
477C4—44no—
478C4—4241no—
479C4C2×#1313324812yes—
480C4#1681142112yes—
481C2—22no—
482C2—44no—
483C2—8481no—
484C2#2028394911yescharacteristic
485C2#2028395711yescharacteristic
486C2C2×#1681079111yescharacteristic
subgroupG/H
centerC22C2×#13133186
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC2×C53×#554C2
socleC22×C53C2×#120
prime radical#20284806C2
hypercenterC2×#554D53
hypocenterC53#18525
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#554

Complements

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

NH ≅ G/Ntype
1. #20285172C2semidirect
2. #20284837C2semidirect
4. #20273849C2semidirect
5. #20273787C2semidirect
6. #20273441C2semidirect
7. #20273162C2semidirect
8. #20273158C2semidirect
9. C2×C53×#554C2semidirect
13. #16811002C22semidirect
14. #16810997C22semidirect
17. #16810918C22semidirect
19. #16810823C22semidirect
28. #16810372C22semidirect
29. Q8⋊#88322C22semidirect
184. D4×C53#137semidirect
238. C2×#554D53semidirect
286. C53#18525semidirect
433. D4#13133248semidirect

Maximal subgroups

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

subgroupindex
#202851722
#202848372
#202848062
#202738492
#202737872
#202734412
#202731622
#202731582
C2×C53×#5542
#1852553