Finite Group Databasebeta

← #27370027

Subgroups — #27370027

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#20285637C211yesmaximal, characteristic, complemented
2C2×#16810903C211yesmaximal, characteristic, complemented
3#20283750C211yesmaximal, characteristic, complemented
4#20274292C212yesmaximal, complemented
5#20274178C212yesmaximal, complemented
6#20274168C212yesmaximal, complemented
7#20274054C212yesmaximal, complemented
8#20273501C211yesmaximal, characteristic, complemented
9#20273431C211yesmaximal, characteristic, complemented
10#20273410C211yesmaximal, characteristic, complemented
11C53×#3109C211yesmaximal, characteristic
12C2×#13133217C2211yescharacteristic
13#16810995C2212yescomplemented
14#16810975C2212yescomplemented
15#16810903—21no—
16#16810903—21no—
17#16810903C2212yes—
18#16810903C2212yes—
19#16810828—22no—
20#16810761—21no—
21#16810761—21no—
22#16810701C2212yes—
23#16810695—22no—
24D4×#88322C2212yes—
25#16810680C2211yescharacteristic
26#16810676C2211yescharacteristic
27C2×#13133150—21no—
28C2×C4⋊#88322C2211yescharacteristic
29C2×#13133138—21no—
30#16810587—21no—
31C2×C22⋊#88322—21no—
32#16810375C2212yescomplemented
33#16810373C2212yescomplemented
34#16810372C2212yescomplemented
35#16810368C2212yescomplemented
36#16810368C2212yescomplemented
37#16810367C2212yescomplemented
38C53×#532C2211yescharacteristic
39C2×C4×D4×C53C2211yescharacteristic
40C53×#481—21no—
41C53×#476C2211yescharacteristic
42C53×#476C2211yescharacteristic
43C53×#459C2211yescharacteristic
44C53×#459C2211yescharacteristic
45C53×#447—21no—
46C53×#442—21no—
47C53×#442—21no—
48C53×#432C2211yescharacteristic
49C22×Dic106—21no—
50C2×#11719923—22no—
51C2×#11719923—22no—
52C2×#11719917—22no—
53C22×#88322—22no—
54C22×#88322—21no—
55#13133217—21no—
56#13133217—21no—
57#13133217—21no—
58#13133217C2312yes—
59C2×C4⋊Dic53—21no—
60C2×C4⋊Dic53—21no—
61#13133150—22no—
62#13133150—22no—
63C4⋊#88322—21no—
64C4⋊#88322D412yes—
65C4⋊#88322C2312yes—
66C4⋊#88322C2312yes—
67#13133140—22no—
68#13133140—22no—
69#13133138—22no—
70#13133138—22no—
71C22⋊#88322—22no—
72C22⋊#88322—22no—
73C22⋊#88322—21no—
74C22⋊#88322D411yescharacteristic
75C22⋊#88322D411yescharacteristic
76#13133128—21no—
77C4×#88322C2312yes—
78C2×#41×C53—21no—
79C22×D4×C53D411yescharacteristic
80C23×C4×C53—21no—
81C2×D8×C53—21no—
82C2×D8×C53—21no—
83C4⋊Q8×C53C2311yescharacteristic
84#127×C53C2311yescharacteristic
85#124×C53—21no—
86#121×C53—21no—
87#121×C53—21no—
88C4×D4×C53—21no—
89C4×D4×C53—21no—
90C4×D4×C53—21no—
91C4×D4×C53—21no—
92C4×D4×C53D411yescharacteristic
93C4×D4×C53D411yescharacteristic
94C4×D4×C53C2311yescharacteristic
95C4×D4×C53C2311yescharacteristic
96C2×C4⋊C4×C53D411yescharacteristic
97C2×C22⋊C4×C53—21no—
98C2×C42×C53C2311yescharacteristic
99#106×C53—21no—
100#106×C53—21no—
101C4⋊C8×C53C2311yescharacteristic
102C4⋊C8×C53C2311yescharacteristic
103#102×C53—21no—
104#102×C53—21no—
105#102×C53—21no—
106#102×C53—21no—
107C22⋊C8×C53—21no—
108#97×C53—21no—
109C24×C53—21no—
110#41×C53—82no—
111C2×Q8×C53—21no—
112C2×D4×C53—22no—
113C2×D4×C53—21no—
114C2×D4×C53—21no—
115C2×D4×C53—21no—
116C2×D4×C53—21no—
117C2×D4×C53—41no—
118C2×D4×C53—41no—
119C2×D4×C53C2×D411yescharacteristic
120C2×D4×C53C2×D411yescharacteristic
121C22×C4×C53—21no—
122C22×C4×C53—21no—
123C22×C4×C53—21no—
124C22×C4×C53—21no—
125C22×C4×C53—21no—
126C22×C4×C53—41no—
127C22×C4×C53—82no—
128C22×C4×C53C2×D411yescharacteristic
129C22×C4×C53C2×D411yescharacteristic
130C22×C4×C53#4111yescharacteristic
131C22×Dic53—41no—
132C2×Dic106—22no—
133C2×Dic106—41no—
134C2×Dic106—82no—
135D8×C53—42no—
136D8×C53—42no—
137C2×C8×C53—21no—
138C2×C8×C53—21no—
139C4⋊C4×C53—21no—
140C4⋊C4×C53—21no—
141C4⋊C4×C53—21no—
142C4⋊C4×C53C2×D411yescharacteristic
143C4⋊C4×C53C2×D411yescharacteristic
144C22⋊C4×C53—21no—
145C22⋊C4×C53—21no—
146C22⋊C4×C53—21no—
147C22⋊C4×C53—21no—
148C22⋊C4×C53—41no—
149C22⋊C4×C53—41no—
150C42×C53C2×D411yescharacteristic
151C42×C53C2×D411yescharacteristic
152C42×C53#4111yescharacteristic
153C42×C53C2411yescharacteristic
154#11719923—82no—
155#11719923—82no—
156C4⋊Dic53—22no—
157C4⋊Dic53—22no—
158C4⋊Dic53—41no—
159C4⋊Dic53—41no—
160#11719917—82no—
161C2×#88322—22no—
162C2×#88322—22no—
163C2×#88322—42no—
164C2×#88322—82no—
165C2×#88322C2×D412yes—
166C2×#88322C2×D412yes—
167C23×C53—21no—
168C23×C53—21no—
169C23×C53—42no—
170C23×C53—42no—
171C23×C53—41no—
172C23×C53—81no—
173C23×C53#12111yescharacteristic
174Q8×C53—81no—
175D4×C53—21no—
176D4×C53—21no—
177D4×C53—21no—
178D4×C53—42no—
179D4×C53—41no—
180D4×C53—41no—
181D4×C53—42no—
182D4×C53—81no—
183D4×C53—82no—
184D4×C53#13712yes—
185C2×C4×C53—22no—
186C2×C4×C53—21no—
187C2×C4×C53—21no—
188C2×C4×C53—22no—
189C2×C4×C53—21no—
190C2×C4×C53—21no—
191C2×C4×C53—42no—
192C2×C4×C53—42no—
193C2×C4×C53—41no—
194C2×C4×C53—81no—
195C2×C4×C53—81no—
196C2×C4×C53—82no—
197C2×C4×C53—81no—
198C2×C4×C53#12111yescharacteristic
199C2×C4×C53#12111yescharacteristic
200C2×C4×C53#12111yescharacteristic
201C2×C4×C53C22×D411yescharacteristic
202C2×C4×C53C22×D411yescharacteristic
203C2×C4×C53C2×#4111yescharacteristic
204C2×Dic53—42no—
205C2×Dic53—82no—
206Dic106—42no—
207Dic106—42no—
208Dic106—82no—
209C8×C53—41no—
210C8×C53—41no—
211#88322—22no—
212#88322—22no—
213#88322—42no—
214#88322—42no—
215#19457—531nomaximal, Sylow
216C22×C53—21no—
217C22×C53—21no—
218C22×C53—22no—
219C22×C53—41no—
220C22×C53—41no—
221C22×C53—41no—
222C22×C53—41no—
223C22×C53—41no—
224C22×C53—84no—
225C22×C53—81no—
226C22×C53—81no—
227C22×C53—81no—
228C22×C53#47112yes—
229C22×C53C2×#12111yescharacteristic
230C4×C53—21no—
231C4×C53—21no—
232C4×C53—21no—
233C4×C53—21no—
234C4×C53—41no—
235C4×C53—41no—
236C4×C53—81no—
237C4×C53C2×#13711yescharacteristic
238C4×C53#57511yescharacteristic
239C4×C53#57612yes—
240Dic53—82no—
241#3206—531no—
242#3109D5311yescharacteristic
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—42no—
254C2×C53—81no—
255C2×C53—161no—
256C2×C53#304111yescharacteristic
257C2×C53#305611yescharacteristic
258C2×C53C2×#47111yescharacteristic
259#532C2×D5311yescharacteristic
260C2×C4⋊Q8—531no—
261C2×C4×D4C2×D5311yescharacteristic
262#492—532no—
263#487—532no—
264#481—21no—
265#476C2×D5311yescharacteristic
266#476C2×D5311yescharacteristic
267#474—532no—
268#474—532no—
269#474—1061no—
270#474—1061no—
271#461—1062no—
272#459C2×D5311yescharacteristic
273#459C2×D5311yescharacteristic
274#448—1061no—
275#448—1061no—
276#447—21no—
277#442—21no—
278#442—21no—
279#436—532no—
280#435—1062no—
281C8×D4—532no—
282#432—531no—
283#432C2×D5311yescharacteristic
284#431—531no—
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—1061no—
299C22×D4C22⋊D5311yescharacteristic
300C23×C4—21no—
301C2×SD16—1062no—
302C2×SD16—1062no—
303C2×D8—21no—
304C2×D8—21no—
305C2×M16—1062no—
306C22×C8—1062no—
307C22×C8—1061no—
308C4⋊Q8—532no—
309C4⋊Q8—1061no—
310C4⋊Q8—1061no—
311C4⋊Q8—1061no—
312C4⋊Q8C22×D5311yescharacteristic
313#127C22×D5311yescharacteristic
314#124—21no—
315#121—21no—
316#121—21no—
317C4×D4—21no—
318C4×D4—21no—
319C4×D4—21no—
320C4×D4—21no—
321C4×D4C22⋊D5311yescharacteristic
322C4×D4C22⋊D5311yescharacteristic
323C4×D4C22×D5311yescharacteristic
324C4×D4C22×D5311yescharacteristic
325C2×C4⋊C4—1061no—
326C2×C4⋊C4—1061no—
327C2×C4⋊C4C22⋊D5311yescharacteristic
328C2×C22⋊C4—21no—
329C2×C42C22×D5311yescharacteristic
330#107—1062no—
331#107—1062no—
332#106—21no—
333#106—21no—
334C4⋊C8—532no—
335C4⋊C8—532no—
336C4⋊C8—532no—
337C4⋊C8—1061no—
338C4⋊C8C22×D5311yescharacteristic
339C4⋊C8C22×D5311yescharacteristic
340#103—1062no—
341#103—1062no—
342#102—21no—
343#102—21no—
344#102—21no—
345#102—21no—
346#102—1062no—
347#102—1062no—
348C22⋊C8—21no—
349C22⋊C8—531no—
350C22⋊C8—531no—
351C22⋊C8—1062no—
352C22⋊C8—1062no—
353C22⋊C8—1061no—
354#97—21no—
355#97—1061no—
356C4×C8—532no—
357C24—21no—
358#41—82no—
359C2×Q8—21no—
360C2×Q8—1062no—
361C2×Q8—4242no—
362C2×Q8—2121no—
363C2×D4—22no—
364C2×D4—21no—
365C2×D4—21no—
366C2×D4—21no—
367C2×D4—21no—
368C2×D4—41no—
369C2×D4—41no—
370C2×D4C2×C22⋊D5311yescharacteristic
371C2×D4C2×C22⋊D5311yescharacteristic
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—41no—
378C22×C4—82no—
379C22×C4—2121no—
380C22×C4D4×D5311yescharacteristic
381C22×C4#1171994711yescharacteristic
382C22×C4C2×C22⋊D5311yescharacteristic
383SD16—4242no—
384SD16—4242no—
385D8—42no—
386D8—42no—
387M16—4242no—
388C2×C8—21no—
389C2×C8—21no—
390C2×C8—532no—
391C2×C8—532no—
392C2×C8—1062no—
393C2×C8—1062no—
394C2×C8—4242no—
395C2×C8—2122no—
396C4⋊C4—21no—
397C4⋊C4—21no—
398C4⋊C4—21no—
399C4⋊C4—1062no—
400C4⋊C4—1062no—
401C4⋊C4—2121no—
402C4⋊C4—2121no—
403C4⋊C4C2×C22⋊D5311yescharacteristic
404C4⋊C4C2×C22⋊D5311yescharacteristic
405C22⋊C4—21no—
406C22⋊C4—21no—
407C22⋊C4—21no—
408C22⋊C4—21no—
409C22⋊C4—41no—
410C22⋊C4—41no—
411C42D4×D5311yescharacteristic
412C42#1171994711yescharacteristic
413C42C2×C22⋊D5311yescharacteristic
414C42C23×D5311yescharacteristic
415C23—21no—
416C23—21no—
417C23—42no—
418C23—42no—
419C23—41no—
420C23—81no—
421C23#1313318911yescharacteristic
422Q8—81no—
423Q8—4242no—
424Q8—2122no—
425Q8—2122no—
426D4—21no—
427D4—21no—
428D4—21no—
429D4—42no—
430D4—41no—
431D4—41no—
432D4—42no—
433D4—81no—
434D4—82no—
435D4#1313324712yes—
436C2×C4—22no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—22no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—41no—
445C2×C4—81no—
446C2×C4—81no—
447C2×C4—82no—
448C2×C4—81no—
449C2×C4—4242no—
450C2×C4—2122no—
451C2×C4#1313318911yescharacteristic
452C2×C4#1313318911yescharacteristic
453C2×C4#1313318911yescharacteristic
454C2×C4C2×D4×D5311yescharacteristic
455C2×C4C22×C22⋊D5311yescharacteristic
456C2×C4C2×#1171994711yescharacteristic
457C8—41no—
458C8—41no—
459C8—1062no—
460C8—1062no—
461C8—2122no—
462C8—2122no—
463C22—21no—
464C22—21no—
465C22—22no—
466C22—41no—
467C22—41no—
468C22—41no—
469C22—41no—
470C22—41no—
471C22—84no—
472C22—81no—
473C22—81no—
474C22—81no—
475C22#1681088412yes—
476C22C2×#1313318911yescharacteristic
477C4—21no—
478C4—21no—
479C4—21no—
480C4—21no—
481C4—41no—
482C4—41no—
483C4—81no—
484C4—4242no—
485C4C2×#1313324711yescharacteristic
486C4#1681142111yescharacteristic
487C4#1681142712yes—
488C2—22no—
489C2—42no—
490C2—81no—
491C2—161no—
492C2#2028422011yescharacteristic
493C2#2028434711yescharacteristic
494C2C2×#1681088411yescharacteristic
subgroupG/H
centerC22C2×#13133189
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
Fitting, prime radicalC53×#3109C2
socleC22×C53C2×#121
hypercenter#3109D53
hypocenterC53#19457
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3109

Complements

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

NH ≅ G/Ntype
1. #20285637C2semidirect
2. C2×#16810903C2semidirect
3. #20283750C2semidirect
4. #20274292C2semidirect
5. #20274178C2semidirect
6. #20274168C2semidirect
7. #20274054C2semidirect
8. #20273501C2semidirect
9. #20273431C2semidirect
10. #20273410C2semidirect
13. #16810995C22semidirect
14. #16810975C22semidirect
32. #16810375C22semidirect
33. #16810373C22semidirect
34. #16810372C22semidirect
35. #16810368C22semidirect
36. #16810368C22semidirect
37. #16810367C22semidirect
296. C53#19457semidirect

Maximal subgroups

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

subgroupindex
#202856372
C2×#168109032
#202837502
#202742922
#202741782
#202741682
#202740542
#202735012
#202734312
#202734102
C53×#31092
#1945753