Finite Group Databasebeta

← #27366405

Subgroups — #27366405

Order 13568.

716 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285040C211yesmaximal, characteristic, complemented
2#20284715C211yesmaximal, characteristic
3C2×D4×#88322C211yesmaximal, characteristic, complemented
4#20274168C212yesmaximal, complemented
5#20274154C212yesmaximal, complemented
6#20274140C212yesmaximal, complemented
7#20274126C212yesmaximal, complemented
8#20273461C211yesmaximal, characteristic, complemented
9#20273372C211yesmaximal, characteristic, complemented
10#20273362C211yesmaximal, characteristic, complemented
11C53×#3101C211yesmaximal, characteristic, complemented
12C23×#88322—21no—
13#16811173C2211yescharacteristic
14#16810980—21no—
15#16810980—21no—
16#16810975C2211yescharacteristic, complemented
17#16810975C2211yescharacteristic, complemented
18#16810971C2211yescharacteristic, complemented
19#16810971C2211yescharacteristic, complemented
20#16810936—21no—
21#16810903C2212yes—
22#16810818C2212yes—
23#16810757—21no—
24C4×#11719922—22no—
25D4×#88322—21no—
26D4×#88322—21no—
27D4×#88322C2212yes—
28D4×#88322C2212yes—
29#16810676C2211yescharacteristic
30C2×C4⋊#88322C2211yescharacteristic
31#16810617—21no—
32#16810617—21no—
33C2×C22⋊#88322—21no—
34C2×C4×#88322C2211yescharacteristic
35#16810375C2212yescomplemented
36#16810373C2212yescomplemented
37#16810372C2212yescomplemented
38#16810369C2212yescomplemented
39#16810368C2212yescomplemented
40#16810366C2212yescomplemented
41C53×#533C2211yescharacteristic
42C2×C4×D4×C53C2211yescharacteristic
43C53×#483—21no—
44C53×#476C2212yes—
45C53×#461C2212yes—
46C53×#450—21no—
47C4×SD16×C53—22no—
48C53×#431C2211yescharacteristic
49C2×#11719945—21no—
50C2×#11719922—22no—
51C22×#88322—22no—
52C22×#88322—22no—
53C22×#88322—21no—
54C22×#88322—44no—
55C22×#88322D411yescharacteristic
56C22×#88322D411yescharacteristic
57#13133217C2311yescharacteristic
58#13133214C2311yescharacteristic
59#13133205—21no—
60#13133191—21no—
61#13133191—21no—
62C4×D212—21no—
63C4×D212—21no—
64#13133150—22no—
65C4⋊#88322—21no—
66C4⋊#88322C2312yes—
67C4⋊#88322C2312yes—
68#13133140—21no—
69#13133140—21no—
70#13133140—21no—
71#13133140—21no—
72#13133138—22no—
73#13133137—22no—
74#13133137—21no—
75#13133137—21no—
76#13133137—21no—
77#13133137—21no—
78C22⋊#88322—22no—
79C22⋊#88322—22no—
80C22⋊#88322—21no—
81C4×#88322—21no—
82C4×#88322D411yescharacteristic
83C4×#88322D411yescharacteristic
84C4×#88322C2311yescharacteristic
85C4×#88322C2311yescharacteristic
86C22×D4×C53D411yescharacteristic
87C23×C4×C53—21no—
88C2×SD16×C53—22no—
89C4⋊Q8×C53C2311yescharacteristic
90C4⋊Q8×C53C2311yescharacteristic
91#125×C53—21no—
92C22⋊Q8×C53—21no—
93C22⋊Q8×C53—21no—
94C4×Q8×C53—21no—
95C4×Q8×C53—21no—
96C4×D4×C53—21no—
97C4×D4×C53—21no—
98C4×D4×C53D412yes—
99C4×D4×C53C2312yes—
100#117×C53—21no—
101C2×C4⋊C4×C53D411yescharacteristic
102C2×C22⋊C4×C53—21no—
103C2×C42×C53C2311yescharacteristic
104#106×C53—22no—
105C4⋊C8×C53C2312yes—
106#103×C53—22no—
107#102×C53—22no—
108C22⋊C8×C53—21no—
109C4×C8×C53—21no—
110C24×C53—21no—
111C2×Q8×C53—21no—
112C2×Q8×C53—21no—
113C2×D4×C53—24no—
114C2×D4×C53—21no—
115C2×D4×C53C2×D412yes—
116C22×C4×C53—21no—
117C22×C4×C53—22no—
118C22×C4×C53—22no—
119C22×C4×C53—42no—
120C22×C4×C53—42no—
121C22×C4×C53C2×D411yescharacteristic
122C22×C4×C53C2×D411yescharacteristic
123C22×C4×C53#4111yescharacteristic
124C2×C22⋊D53—41no—
125#11719945—42no—
126#11719945—42no—
127C2×D212—21no—
128C2×D212—21no—
129C2×D212—21no—
130C2×C4×D53—41no—
131C2×Dic106—21no—
132SD16×C53—44no—
133C2×C8×C53—22no—
134C4⋊C4×C53—21no—
135C4⋊C4×C53—21no—
136C4⋊C4×C53—21no—
137C4⋊C4×C53—21no—
138C4⋊C4×C53—21no—
139C4⋊C4×C53—41no—
140C4⋊C4×C53—41no—
141C4⋊C4×C53C2×D412yes—
142C22⋊C4×C53—22no—
143C22⋊C4×C53—22no—
144C22⋊C4×C53—41no—
145C42×C53—41no—
146C42×C53C2×D411yescharacteristic
147C42×C53C2×D411yescharacteristic
148C42×C53#4111yescharacteristic
149C42×C53C2411yescharacteristic
150#11719922—44no—
151#11719921—41no—
152#11719921—41no—
153C4⋊Dic53—21no—
154C4⋊Dic53—21no—
155C2×#88322—22no—
156C2×#88322—22no—
157C2×#88322—22no—
158C2×#88322—22no—
159C2×#88322—44no—
160C2×#88322—44no—
161C2×#88322—42no—
162C2×#88322C2×D411yescharacteristic
163C2×#88322C2×D411yescharacteristic
164C2×#88322C2×D411yescharacteristic
165C2×#88322C2×D411yescharacteristic
166C23×C53—24no—
167C23×C53—22no—
168C23×C53—42no—
169C23×C53—42no—
170C23×C53#12111yescharacteristic
171C22×D53—41no—
172Q8×C53—42no—
173Q8×C53—81no—
174D4×C53—24no—
175D4×C53—22no—
176D4×C53#13514yes—
177C2×C4×C53—22no—
178C2×C4×C53—22no—
179C2×C4×C53—41no—
180C2×C4×C53—42no—
181C2×C4×C53—42no—
182C2×C4×C53—42no—
183C2×C4×C53—44no—
184C2×C4×C53—41no—
185C2×C4×C53—41no—
186C2×C4×C53#12111yescharacteristic
187C2×C4×C53#12111yescharacteristic
188C2×C4×C53#12111yescharacteristic
189C2×C4×C53#13512yes—
190C2×C4×C53AGL(1,ℤ/8)12yes—
191C2×C4×C53C22×D411yescharacteristic
192C2×C4×C53C22×D411yescharacteristic
193C2×C4×C53C2×#4111yescharacteristic
194C22⋊D53—82no—
195C2×Dic53—41no—
196D212—41no—
197D212—42no—
198D212—41no—
199D212—81no—
200C4×D53—82no—
201Dic106—41no—
202Dic106—41no—
203C8×C53—42no—
204#88322—21no—
205#88322—21no—
206#88322—21no—
207#88322—21no—
208#88322—42no—
209#88322—42no—
210#19110—531nomaximal, Sylow
211C22×C53—21no—
212C22×C53—21no—
213C22×C53—24no—
214C22×C53—48no—
215C22×C53—42no—
216C22×C53—42no—
217C22×C53#46312yes—
218C22×C53C2×#12111yescharacteristic
219C2×D53—81no—
220C2×D53—81no—
221C2×D53—81no—
222C4×C53—21no—
223C4×C53—21no—
224C4×C53—21no—
225C4×C53—42no—
226C4×C53—81no—
227C4×C53—81no—
228C4×C53#46312yes—
229C4×C53C2×#13512yes—
230C4×C53C2×#13511yescharacteristic
231C4×C53C2×AGL(1,ℤ/8)11yescharacteristic
232Dic53—81no—
233#3131—531no—
234#3101D5311yescharacteristic, complemented
235#3100—531no—
236C2×C8×D4—531no—
237#1683—532no—
238#1681—532no—
239#1679—532no—
240#1677—532no—
241#1564—531no—
242#1542—531no—
243#1540—531no—
244C2×C53—22no—
245C2×C53—44no—
246C2×C53C2×#46311yescharacteristic
247C2×C53#305011yescharacteristic
248C2×C53#306711yescharacteristic
249D53—161no—
250C23×C8—1061no—
251#533C2×D5311yescharacteristic
252#532—531no—
253C2×C4×D4C2×D5311yescharacteristic
254#488—1061no—
255#488—1061no—
256#487—531no—
257#487—531no—
258#486—531no—
259#486—531no—
260#483—21no—
261#480—1061no—
262#476C2×D5312yes—
263#474—532no—
264#461C2×D5312yes—
265#459—532no—
266#450—21no—
267#447—1061no—
268C4×SD16—22no—
269C4×D8—1062no—
270C8×D4—532no—
271C8×D4—532no—
272C8×D4—1061no—
273C8×D4—1061no—
274#431—531no—
275#431C2×D5311yescharacteristic
276C2×C4⋊C8—531no—
277#416—1061no—
278#416—1061no—
279C2×C22⋊C8—1061no—
280C2×C4×C8—531no—
281#335—532no—
282#334—532no—
283#333—532no—
284#332—532no—
285#332—532no—
286#331—532no—
287C53#1911011yescharacteristic, complemented, Sylow
288C2×#41—1061no—
289C22×D4C22⋊D5311yescharacteristic
290C23×C4—21no—
291C2×SD16—22no—
292C2×D8—1062no—
293C22×C8—531no—
294C22×C8—531no—
295C22×C8—1062no—
296C22×C8—1062no—
297C22×C8—1061no—
298C22×C8—2124no—
299C4⋊Q8—531no—
300C4⋊Q8C22×D5311yescharacteristic
301C4⋊Q8C22×D5311yescharacteristic
302#127—531no—
303#125—21no—
304#124—1061no—
305C22⋊Q8—21no—
306C22⋊Q8—21no—
307#121—1061no—
308#121—1061no—
309C4×Q8—21no—
310C4×Q8—21no—
311C4×D4—21no—
312C4×D4—21no—
313C4×D4—1061no—
314C4×D4—1061no—
315C4×D4C22⋊D5312yes—
316C4×D4C22×D5312yes—
317#117—21no—
318C2×C4⋊C4C22⋊D5311yescharacteristic
319C2×C22⋊C4—21no—
320C2×C42C22×D5311yescharacteristic
321#107—1062no—
322#106—22no—
323C4⋊C8—532no—
324C4⋊C8—532no—
325C4⋊C8—1061no—
326C4⋊C8C22×D5312yes—
327#103—22no—
328#103—1061no—
329#103—1061no—
330#103—1061no—
331#103—1061no—
332#102—22no—
333#102—1061no—
334#102—1061no—
335#102—1061no—
336#102—1061no—
337#102—1062no—
338#102—1062no—
339C22⋊C8—21no—
340C22⋊C8—1062no—
341C22⋊C8—1062no—
342C22⋊C8—1061no—
343C4×C8—21no—
344C4×C8—531no—
345C4×C8—531no—
346C4×C8—531no—
347C4×C8—531no—
348C4×C8—1061no—
349C24—21no—
350#41—2122no—
351#41—2122no—
352C2×Q8—21no—
353C2×Q8—21no—
354C2×Q8—1061no—
355C2×D4—24no—
356C2×D4—21no—
357C2×D4—1061no—
358C2×D4—1061no—
359C2×D4—1061no—
360C2×D4—2121no—
361C2×D4C2×C22⋊D5312yes—
362C22×C4—21no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—2121no—
368C22×C4D4×D5311yescharacteristic
369C22×C4#1171994711yescharacteristic
370C22×C4C2×C22⋊D5311yescharacteristic
371SD16—44no—
372D8—2124no—
373C2×C8—22no—
374C2×C8—531no—
375C2×C8—531no—
376C2×C8—531no—
377C2×C8—531no—
378C2×C8—1062no—
379C2×C8—1062no—
380C2×C8—1062no—
381C2×C8—1062no—
382C2×C8—2124no—
383C2×C8—2124no—
384C2×C8—2122no—
385C4⋊C4—21no—
386C4⋊C4—21no—
387C4⋊C4—21no—
388C4⋊C4—21no—
389C4⋊C4—21no—
390C4⋊C4—41no—
391C4⋊C4—41no—
392C4⋊C4—1061no—
393C4⋊C4—1061no—
394C4⋊C4C2×C22⋊D5312yes—
395C22⋊C4—22no—
396C22⋊C4—22no—
397C22⋊C4—41no—
398C22⋊C4—2121no—
399C22⋊C4—2121no—
400C42—41no—
401C42D4×D5311yescharacteristic
402C42#1171994711yescharacteristic
403C42C2×C22⋊D5311yescharacteristic
404C42C23×D5311yescharacteristic
405C23—24no—
406C23—22no—
407C23—42no—
408C23—42no—
409C23—2121no—
410C23#1313318911yescharacteristic
411Q8—42no—
412Q8—81no—
413Q8—2121no—
414Q8—2121no—
415D4—24no—
416D4—22no—
417D4—4241no—
418D4—4242no—
419D4—2122no—
420D4—2121no—
421D4—2121no—
422D4#1313323714yes—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—41no—
426C2×C4—42no—
427C2×C4—44no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—4242no—
433C2×C4—2121no—
434C2×C4#1313318911yescharacteristic
435C2×C4#1313318911yescharacteristic
436C2×C4#1313318911yescharacteristic
437C2×C4#1313323812yes—
438C2×C4#1313324412yes—
439C2×C4C2×D4×D5311yescharacteristic
440C2×C4C22×C22⋊D5311yescharacteristic
441C2×C4C2×#1171994711yescharacteristic
442C8—42no—
443C8—1061no—
444C8—1061no—
445C8—1061no—
446C8—1061no—
447C8—2122no—
448C8—2122no—
449C22—21no—
450C22—21no—
451C22—24no—
452C22—48no—
453C22—42no—
454C22—42no—
455C22—4241no—
456C22—4241no—
457C22—4241no—
458C22#1681084012yes—
459C22C2×#1313318911yescharacteristic
460C4—21no—
461C4—21no—
462C4—21no—
463C4—42no—
464C4—81no—
465C4—81no—
466C4—4241no—
467C4#1681084012yes—
468C4C2×#1313323712yes—
469C4C2×#1313323811yescharacteristic
470C4C2×#1313324411yescharacteristic
471C2—22no—
472C2—44no—
473C2—8481no—
474C2C2×#1681084011yescharacteristic
475C2#2028429311yescharacteristic
476C2#2028443211yescharacteristic
subgroupG/H
centerC2×C4#13133189
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#3101C2
socleC22×C53C2×#121
prime radical#20284715C2
hypercenter#3101D53
hypocenterC53#19110
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3101

Complements

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

NH ≅ G/Ntype
1. #20285040C2semidirect
3. C2×D4×#88322C2semidirect
4. #20274168C2semidirect
5. #20274154C2semidirect
6. #20274140C2semidirect
7. #20274126C2semidirect
8. #20273461C2semidirect
9. #20273372C2semidirect
10. #20273362C2semidirect
11. C53×#3101C2semidirect
16. #16810975C22semidirect
17. #16810975C22semidirect
18. #16810971C22semidirect
19. #16810971C22semidirect
35. #16810375C22semidirect
36. #16810373C22semidirect
37. #16810372C22semidirect
38. #16810369C22semidirect
39. #16810368C22semidirect
40. #16810366C22semidirect
234. #3101D53semidirect
287. C53#19110semidirect

Maximal subgroups

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

subgroupindex
#202850402
#202847152
C2×D4×#883222
#202741682
#202741542
#202741402
#202741262
#202734612
#202733722
#202733622
C53×#31012
#1911053