Finite Group Databasebeta

← #27370019

Subgroups — #27370019

Order 13568.

660 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20284802C211yesmaximal, characteristic
2C2×#16810900C211yesmaximal, characteristic, complemented
3#20283749C211yesmaximal, characteristic, complemented
4#20274287C212yesmaximal, complemented
5#20274177C212yesmaximal, complemented
6#20274163C212yesmaximal, complemented
7#20274053C212yesmaximal, complemented
8#20273500C211yesmaximal, characteristic, complemented
9#20273432C211yesmaximal, characteristic, complemented
10#20273410C211yesmaximal, characteristic, complemented
11C53×#3206C211yesmaximal, characteristic, complemented
12C2×C4×D212C2211yescharacteristic
13#16810939—21no—
14#16810910C2211yescharacteristic
15#16810910C2211yescharacteristic
16#16810900—21no—
17#16810900—21no—
18#16810900C2212yes—
19#16810900C2212yes—
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×#13133150—21no—
31C2×C4⋊#88322C2211yescharacteristic
32C2×#13133137—21no—
33#16810585—21no—
34C22⋊C8×D53—21no—
35#16810376C2212yescomplemented
36#16810374C2212yescomplemented
37#16810372C2212yescomplemented
38#16810369C2212yescomplemented
39#16810369C2212yescomplemented
40#16810367C2212yescomplemented
41C53×#532C2211yescharacteristic
42C2×C4⋊Q8×C53C2211yescharacteristic
43C53×#492C2212yescomplemented
44C53×#487C2212yescomplemented
45C53×#461—22no—
46C53×#448—21no—
47C53×#448—21no—
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#13133150—22no—
65#13133150—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—
87C2×SD16×C53—22no—
88C2×SD16×C53—22no—
89C4⋊Q8×C53—21no—
90C4⋊Q8×C53—21no—
91C4⋊Q8×C53—21no—
92C4⋊Q8×C53C2311yescharacteristic
93C4⋊Q8×C53C2312yes—
94#127×C53C2311yescharacteristic
95#124×C53—21no—
96#121×C53—21no—
97#121×C53—21no—
98C4×D4×C53—21no—
99C4×D4×C53—21no—
100C2×C4⋊C4×C53—21no—
101C2×C4⋊C4×C53—21no—
102C2×C42×C53C2311yescharacteristic
103C4⋊C8×C53D412yes—
104#103×C53—22no—
105#103×C53—22no—
106C22⋊C8×C53D411yescharacteristic
107C22⋊C8×C53D411yescharacteristic
108C4×C8×C53C2312yes—
109C23×D53—21no—
110#41×C53—82no—
111C2×Q8×C53—22no—
112C2×Q8×C53—21no—
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—
135SD16×C53—82no—
136SD16×C53—82no—
137C2×C8×C53C2×D412yes—
138C2×C8×C53C2×D412yes—
139C4⋊C4×C53—21no—
140C4⋊C4×C53—22no—
141C4⋊C4×C53—21no—
142C4⋊C4×C53—22no—
143C4⋊C4×C53—41no—
144C4⋊C4×C53—41no—
145C22⋊C4×C53—41no—
146C22⋊C4×C53—41no—
147C42×C53C2×D411yescharacteristic
148C42×C53C2×D411yescharacteristic
149C42×C53#4111yescharacteristic
150C42×C53C2411yescharacteristic
151#11719922—42no—
152#11719922—42no—
153#11719921—21no—
154#11719921—21no—
155#11719921—21no—
156#11719921—21no—
157C4⋊Dic53—21no—
158C4⋊Dic53C2×D411yescharacteristic
159C4⋊Dic53C2×D411yescharacteristic
160C2×#88322—22no—
161C2×#88322—21no—
162C2×#88322—21no—
163C2×#88322—22no—
164C2×#88322—42no—
165#11719912—82no—
166C8×D53—82no—
167C23×C53—41no—
168C23×C53#12111yescharacteristic
169C22×D53—21no—
170C22×D53—21no—
171C22×D53—42no—
172C22×D53—42no—
173C22×D53—81no—
174Q8×C53—42no—
175Q8×C53—42no—
176Q8×C53—82no—
177Q8×C53—81no—
178D4×C53—41no—
179D4×C53—41no—
180D4×C53—42no—
181D4×C53—81no—
182D4×C53—82no—
183C2×C4×C53—22no—
184C2×C4×C53—21no—
185C2×C4×C53—21no—
186C2×C4×C53—22no—
187C2×C4×C53—41no—
188C2×C4×C53—42no—
189C2×C4×C53—82no—
190C2×C4×C53—82no—
191C2×C4×C53#12111yescharacteristic
192C2×C4×C53#12111yescharacteristic
193C2×C4×C53#12111yescharacteristic
194C2×C4×C53C22×D411yescharacteristic
195C2×C4×C53C22×D411yescharacteristic
196C2×C4×C53C2×#4111yescharacteristic
197C2×Dic53—21no—
198C2×Dic53—21no—
199C2×Dic53—81no—
200D212—21no—
201D212—21no—
202D212—21no—
203D212—42no—
204D212#13712yes—
205C4×D53—42no—
206C4×D53—42no—
207C4×D53—81no—
208C4×D53—81no—
209C8×C53—22no—
210C8×C53—22no—
211#88322—41no—
212#88322—42no—
213#88322—41no—
214#88322—42no—
215#19457—531nomaximal, Sylow
216C22×C53—21no—
217C22×C53—21no—
218C22×C53—81no—
219C22×C53—81no—
220C22×C53—81no—
221C22×C53#47112yes—
222C22×C53C2×#12111yescharacteristic
223C2×D53—22no—
224C2×D53—41no—
225C2×D53—41no—
226C2×D53—41no—
227C2×D53—41no—
228C2×D53—41no—
229C2×D53—84no—
230C4×C53—21no—
231C4×C53—21no—
232C4×C53—21no—
233C4×C53—21no—
234C4×C53—82no—
235C4×C53—81no—
236C4×C53C2×#13711yescharacteristic
237C4×C53#57511yescharacteristic
238C4×C53#57612yes—
239Dic53—41no—
240Dic53—41no—
241#3206D5311yescharacteristic, complemented
242#3109—531no—
243C2×#474—531no—
244#2974—531no—
245#1703—532no—
246#1685—532no—
247#1683—532no—
248#1665—532no—
249#1573—531no—
250#1556—531no—
251#1551—531no—
252C2×C53—22no—
253C2×C53—161no—
254C2×C53#304111yescharacteristic
255C2×C53#305611yescharacteristic
256C2×C53C2×#47111yescharacteristic
257D53—42no—
258D53—81no—
259#532C2×D5311yescharacteristic
260C2×C4⋊Q8C2×D5311yescharacteristic
261C2×C4×D4—531no—
262#492C2×D5312yescomplemented
263#487C2×D5312yescomplemented
264#481—1061no—
265#476—531no—
266#476—531no—
267#474—532no—
268#474—532no—
269#474—1061no—
270#474—1061no—
271#461—22no—
272#459—531no—
273#459—531no—
274#448—21no—
275#448—21no—
276#447—1061no—
277#442—1061no—
278#442—1061no—
279#436—532no—
280#435—1062no—
281C8×D4—532no—
282#432—531no—
283#432—531no—
284#431C2×D5311yescharacteristic
285C2×#107—1061no—
286C2×C4⋊C8—531no—
287C2×#102—1061no—
288#407—1061no—
289C2×C22⋊C8—1061no—
290#335—532no—
291#334—532no—
292#333—532no—
293#332—532no—
294#332—532no—
295#331—532no—
296C53#1945711yescharacteristic, complemented, Sylow
297C2×#41—21no—
298C22×Q8—21no—
299C22×D4—531no—
300C23×C4—1061no—
301C2×SD16—22no—
302C2×SD16—22no—
303C2×D8—1061no—
304C2×D8—1061no—
305C2×M16—1062no—
306C22×C8—1061no—
307C22×C8—1062no—
308C4⋊Q8—21no—
309C4⋊Q8—21no—
310C4⋊Q8—21no—
311C4⋊Q8C22×D5312yes—
312C4⋊Q8C22×D5311yescharacteristic
313#127C22×D5311yescharacteristic
314#124—21no—
315#121—21no—
316#121—21no—
317C4×D4—21no—
318C4×D4—21no—
319C4×D4—531no—
320C4×D4—531no—
321C4×D4—531no—
322C4×D4—531no—
323C4×D4—1061no—
324C4×D4—1061no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—531no—
328C2×C22⋊C4—1061no—
329C2×C42C22×D5311yescharacteristic
330#107—1062no—
331#107—1062no—
332#106—1061no—
333#106—1061no—
334C4⋊C8—532no—
335C4⋊C8—532no—
336C4⋊C8—531no—
337C4⋊C8—531no—
338C4⋊C8—1061no—
339C4⋊C8C22⋊D5312yes—
340#103—22no—
341#103—22no—
342#102—1062no—
343#102—1062no—
344#102—1061no—
345#102—1061no—
346#102—1061no—
347#102—1061no—
348C22⋊C8—1061no—
349C22⋊C8—1061no—
350C22⋊C8—1062no—
351C22⋊C8—1062no—
352C22⋊C8C22⋊D5311yescharacteristic
353C22⋊C8C22⋊D5311yescharacteristic
354#97—1061no—
355#97—1061no—
356C4×C8C22×D5312yes—
357C24—1061no—
358#41—82no—
359C2×Q8—22no—
360C2×Q8—21no—
361C2×Q8—41no—
362C2×Q8—82no—
363C2×D4—21no—
364C2×D4—21no—
365C2×D4—21no—
366C2×D4—41no—
367C2×D4—531no—
368C2×D4—531no—
369C2×D4—1062no—
370C2×D4—1061no—
371C2×D4—2121no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—1061no—
375C22×C4—1061no—
376C22×C4—1061no—
377C22×C4—1061no—
378C22×C4—1061no—
379C22×C4—4242no—
380C22×C4D4×D5311yescharacteristic
381C22×C4#1171994711yescharacteristic
382C22×C4C2×C22⋊D5311yescharacteristic
383SD16—82no—
384SD16—82no—
385D8—2122no—
386D8—2122no—
387M16—4242no—
388C2×C8—1061no—
389C2×C8—1061no—
390C2×C8—1062no—
391C2×C8—1062no—
392C2×C8—4242no—
393C2×C8—2122no—
394C2×C8C2×C22⋊D5312yes—
395C2×C8C2×C22⋊D5312yes—
396C4⋊C4—21no—
397C4⋊C4—22no—
398C4⋊C4—21no—
399C4⋊C4—22no—
400C4⋊C4—41no—
401C4⋊C4—41no—
402C4⋊C4—531no—
403C4⋊C4—531no—
404C4⋊C4—1061no—
405C22⋊C4—41no—
406C22⋊C4—41no—
407C22⋊C4—1061no—
408C22⋊C4—1061no—
409C22⋊C4—1061no—
410C22⋊C4—1061no—
411C42D4×D5311yescharacteristic
412C42#1171994711yescharacteristic
413C42C2×C22⋊D5311yescharacteristic
414C42C23×D5311yescharacteristic
415C23—41no—
416C23—1061no—
417C23—1061no—
418C23—4241no—
419C23—2122no—
420C23—2122no—
421C23#1313318811yescharacteristic
422Q8—42no—
423Q8—42no—
424Q8—81no—
425Q8—82no—
426D4—41no—
427D4—41no—
428D4—42no—
429D4—81no—
430D4—82no—
431D4—532no—
432D4—1061no—
433D4—1061no—
434D4—1061no—
435D4—2122no—
436C2×C4—22no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—22no—
440C2×C4—41no—
441C2×C4—42no—
442C2×C4—82no—
443C2×C4—82no—
444C2×C4—1061no—
445C2×C4—1061no—
446C2×C4—4241no—
447C2×C4—4241no—
448C2×C4—4241no—
449C2×C4—2122no—
450C2×C4—2122no—
451C2×C4#1313318811yescharacteristic
452C2×C4#1313318811yescharacteristic
453C2×C4#1313318811yescharacteristic
454C2×C4C2×D4×D5311yescharacteristic
455C2×C4C22×C22⋊D5311yescharacteristic
456C2×C4C2×#1171994711yescharacteristic
457C8—22no—
458C8—22no—
459C8—2121no—
460C8—2122no—
461C8—2121no—
462C8—2122no—
463C22—21no—
464C22—21no—
465C22—81no—
466C22—81no—
467C22—81no—
468C22—1062no—
469C22—4244no—
470C22—2121no—
471C22—2121no—
472C22—2121no—
473C22—2121no—
474C22—2121no—
475C22#1681088312yes—
476C22C2×#1313318811yescharacteristic
477C4—21no—
478C4—21no—
479C4—21no—
480C4—21no—
481C4—82no—
482C4—81no—
483C4—2121no—
484C4—2121no—
485C4#137×D5311yescharacteristic
486C4#1681142211yescharacteristic
487C4#1681142612yes—
488C2—22no—
489C2—161no—
490C2—4241no—
491C2—2122no—
492C2#2028421411yescharacteristic
493C2#2028433911yescharacteristic
494C2C2×#1681088311yescharacteristic
subgroupG/H
centerC22C2×#13133188
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#3206C2
socleC22×C53C2×#121
prime radical#20284802C2
hypercenter#3206D53
hypocenterC53#19457
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3206

Complements

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

NH ≅ G/Ntype
2. C2×#16810900C2semidirect
3. #20283749C2semidirect
4. #20274287C2semidirect
5. #20274177C2semidirect
6. #20274163C2semidirect
7. #20274053C2semidirect
8. #20273500C2semidirect
9. #20273432C2semidirect
10. #20273410C2semidirect
11. C53×#3206C2semidirect
35. #16810376C22semidirect
36. #16810374C22semidirect
37. #16810372C22semidirect
38. #16810369C22semidirect
39. #16810369C22semidirect
40. #16810367C22semidirect
43. C53×#492C22semidirect
44. C53×#487C22semidirect
241. #3206D53semidirect
262. #492C2×D53semidirect
263. #487C2×D53semidirect
296. C53#19457semidirect

Maximal subgroups

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

subgroupindex
#202848022
C2×#168109002
#202837492
#202742872
#202741772
#202741632
#202740532
#202735002
#202734322
#202734102
C53×#32062
#1945753