Finite Group Databasebeta

← #27370104

Subgroups — #27370104

Order 13568.

658 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20284810C211yesmaximal, characteristic
2C2×#16810825C211yesmaximal, characteristic
3#20283749C211yesmaximal, characteristic
4#20274259C212yesmaximal, complemented
5#20274225C212yesmaximal, complemented
6#20274135C212yesmaximal, complemented
7#20274101C212yesmaximal, complemented
8#20273500C211yesmaximal, characteristic, complemented
9#20273432C211yesmaximal, characteristic, complemented
10#20273415C211yesmaximal, characteristic, complemented
11C53×#3230C211yesmaximal, characteristic, complemented
12C22×#11719924—21no—
13C2×C4×D212C2211yescharacteristic
14#16810956—21no—
15#16810900C2211yescharacteristic
16#16810900C2211yescharacteristic
17#16810825—21no—
18#16810825—21no—
19#16810825C2211yescharacteristic
20#16810825C2212yes—
21#16810825C2211yescharacteristic
22#16810825C2212yes—
23#16810773—21no—
24#16810721—21no—
25#16810721—21no—
26#16810702C2212yes—
27#16810693—22no—
28C8×D212C2212yes—
29#16810680C2211yescharacteristic
30#16810680C2211yescharacteristic
31C2×C4⋊#88322C2211yescharacteristic
32C2×#13133141—21no—
33#16810585—21no—
34C22⋊C8×D53—21no—
35#16810376C2212yescomplemented
36#16810374C2212yescomplemented
37#16810369C2212yescomplemented
38#16810369C2212yescomplemented
39#16810369C2212yescomplemented
40#16810369C2212yescomplemented
41C53×#533C2211yescharacteristic
42C2×C4⋊Q8×C53C2211yescharacteristic
43C53×#500C2212yescomplemented
44C53×#487C2212yescomplemented
45C53×#484—21no—
46C53×#484—21no—
47C53×#477—22no—
48C53×#431C2211yescharacteristic
49C22×D212D411yescharacteristic
50C22×C4×D53—21no—
51C2×#11719924—21no—
52C2×#11719924—21no—
53C2×#11719924—22no—
54C2×#11719924—22no—
55C2×#11719924—44no—
56C2×#11719912—22no—
57C22×#88322—21no—
58C2×C8×D53—22no—
59C4×D212—21no—
60C4×D212—21no—
61C4×D212D411yescharacteristic
62C4×D212D411yescharacteristic
63C4×D212C2311yescharacteristic
64C4×D212C2311yescharacteristic
65C2×C4⋊Dic53D411yescharacteristic
66C2×#11719921—21no—
67#13133150—21no—
68#13133150—21no—
69C4⋊#88322—21no—
70C4⋊#88322C2312yes—
71C4⋊#88322C2311yescharacteristic
72C4⋊#88322C2311yescharacteristic
73C4⋊#88322C2312yes—
74#13133141—22no—
75#13133141—22no—
76#13133141—21no—
77#13133141—21no—
78#13133137—21no—
79#13133137—21no—
80C22⋊#88322—21no—
81C22⋊#88322—21no—
82#13133129—22no—
83#13133129—22no—
84#13133128—21no—
85#13133128—21no—
86C22×Q8×C53—21no—
87C4⋊Q8×C53—21no—
88C4⋊Q8×C53—21no—
89C4⋊Q8×C53—21no—
90C4⋊Q8×C53C2311yescharacteristic
91C4⋊Q8×C53C2312yes—
92C4⋊Q8×C53C2311yescharacteristic
93#125×C53—21no—
94C22⋊Q8×C53—21no—
95C22⋊Q8×C53—21no—
96C4×Q8×C53—21no—
97C4×Q8×C53—21no—
98#117×C53—21no—
99C2×C4⋊C4×C53—21no—
100C2×C4⋊C4×C53—21no—
101C2×C42×C53C2311yescharacteristic
102#107×C53—22no—
103#107×C53—22no—
104C4⋊C8×C53D412yes—
105#103×C53—22no—
106#103×C53—22no—
107C22⋊C8×C53D411yescharacteristic
108C22⋊C8×C53D411yescharacteristic
109C4×C8×C53C2312yes—
110C23×D53—21no—
111C2×Q8×C53—21no—
112C2×Q8×C53—21no—
113C2×Q8×C53—22no—
114C2×Q8×C53—42no—
115C2×Q8×C53—42no—
116C2×Q8×C53—41no—
117C22×C4×C53—41no—
118C22×C4×C53C2×D411yescharacteristic
119C22×C4×C53C2×D411yescharacteristic
120C22×C4×C53#4111yescharacteristic
121C22×Dic53—21no—
122C2×D212—22no—
123C2×D212—21no—
124C2×D212—41no—
125C2×D212C2×D411yescharacteristic
126C2×D212C2×D411yescharacteristic
127C2×C4×D53—21no—
128C2×C4×D53—21no—
129C2×C4×D53—21no—
130C2×C4×D53—21no—
131C2×C4×D53—82no—
132C2×C8×C53C2×D412yes—
133C2×C8×C53C2×D412yes—
134C4⋊C4×C53—22no—
135C4⋊C4×C53—21no—
136C4⋊C4×C53—21no—
137C4⋊C4×C53—22no—
138C4⋊C4×C53—21no—
139C4⋊C4×C53—21no—
140C4⋊C4×C53—41no—
141C4⋊C4×C53—41no—
142C4⋊C4×C53—41no—
143C4⋊C4×C53—41no—
144C22⋊C4×C53—41no—
145C42×C53—41no—
146C42×C53C2×D411yescharacteristic
147C42×C53C2×D411yescharacteristic
148C42×C53#4111yescharacteristic
149C42×C53C2411yescharacteristic
150#11719924—42no—
151#11719924—44no—
152#11719924—42no—
153#11719924—44no—
154#11719921—21no—
155#11719921—21no—
156#11719921—21no—
157#11719921—21no—
158C4⋊Dic53—21no—
159C4⋊Dic53C2×D411yescharacteristic
160C4⋊Dic53C2×D411yescharacteristic
161C2×#88322—22no—
162C2×#88322—21no—
163C2×#88322—21no—
164C2×#88322—22no—
165C2×#88322—42no—
166#11719912—82no—
167C8×D53—82no—
168C23×C53#12111yescharacteristic
169C22×D53—21no—
170C22×D53—21no—
171C22×D53—42no—
172C22×D53—42no—
173C22×D53—81no—
174Q8×C53—42no—
175Q8×C53—41no—
176Q8×C53—42no—
177Q8×C53—41no—
178Q8×C53—42no—
179Q8×C53—82no—
180C2×C4×C53—22no—
181C2×C4×C53—21no—
182C2×C4×C53—21no—
183C2×C4×C53—22no—
184C2×C4×C53—42no—
185C2×C4×C53—41no—
186C2×C4×C53—41no—
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—
201D212AGL(1,ℤ/8)12yes—
202C4×D53—42no—
203C4×D53—42no—
204C4×D53—81no—
205C4×D53—81no—
206C8×C53—22no—
207C8×C53—22no—
208#88322—42no—
209#88322—41no—
210#88322—41no—
211#88322—42no—
212#19464—531nomaximal, Sylow
213C22×C53—21no—
214C22×C53—21no—
215C22×C53#47312yes—
216C22×C53C2×#12111yescharacteristic
217C2×D53—22no—
218C2×D53—41no—
219C2×D53—41no—
220C2×D53—41no—
221C2×D53—41no—
222C2×D53—41no—
223C2×D53—84no—
224C4×C53—21no—
225C4×C53—21no—
226C4×C53—21no—
227C4×C53—21no—
228C4×C53—81no—
229C4×C53—82no—
230C4×C53—81no—
231C4×C53C2×AGL(1,ℤ/8)11yescharacteristic
232C4×C53#57511yescharacteristic
233C4×C53#57712yes—
234Dic53—41no—
235Dic53—41no—
236#3230D5311yescharacteristic, complemented
237#3110—531no—
238C2×#461—531no—
239#2974—531no—
240#1699—532no—
241#1693—532no—
242#1679—532no—
243#1673—532no—
244#1573—531no—
245#1556—531no—
246#1552—531no—
247C2×C53—22no—
248C2×C53#304211yescharacteristic
249C2×C53#305511yescharacteristic
250C2×C53C2×#47311yescharacteristic
251D53—42no—
252D53—81no—
253C22×SD16—1061no—
254#533C2×D5311yescharacteristic
255C2×C4⋊Q8C2×D5311yescharacteristic
256C2×C4×D4—531no—
257#500C2×D5312yescomplemented
258#487C2×D5312yescomplemented
259#484—21no—
260#484—21no—
261#484—1061no—
262#477—22no—
263#474—531no—
264#474—531no—
265#461—531no—
266#461—531no—
267#461—532no—
268#461—532no—
269#461—1061no—
270#461—1061no—
271#450—1061no—
272#440—1061no—
273#440—1061no—
274#436—532no—
275#435—1062no—
276C8×D4—532no—
277#432—531no—
278#432—531no—
279#431C2×D5311yescharacteristic
280C2×C4⋊C8—531no—
281C2×#103—1061no—
282#407—1061no—
283C2×C22⋊C8—1061no—
284#335—532no—
285#334—532no—
286#332—532no—
287#332—532no—
288#332—532no—
289#332—532no—
290C53#1946411yescharacteristic, complemented, Sylow
291C22×Q8—21no—
292C22×D4—531no—
293C23×C4—1061no—
294C2×SD16—1062no—
295C2×SD16—1062no—
296C2×SD16—1061no—
297C2×SD16—1061no—
298C2×SD16—2124no—
299C2×M16—1062no—
300C22×C8—1061no—
301C22×C8—1062no—
302C4⋊Q8—21no—
303C4⋊Q8—21no—
304C4⋊Q8—21no—
305C4⋊Q8C22×D5312yes—
306C4⋊Q8C22×D5311yescharacteristic
307C4⋊Q8C22×D5311yescharacteristic
308#125—21no—
309C22⋊Q8—21no—
310C22⋊Q8—21no—
311C4×Q8—21no—
312C4×Q8—21no—
313C4×D4—531no—
314C4×D4—531no—
315C4×D4—531no—
316C4×D4—531no—
317C4×D4—1061no—
318C4×D4—1061no—
319#117—21no—
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#107—1061no—
328#107—1061no—
329C4⋊C8—532no—
330C4⋊C8—531no—
331C4⋊C8—532no—
332C4⋊C8—531no—
333C4⋊C8—1061no—
334C4⋊C8C22⋊D5312yes—
335#103—22no—
336#103—22no—
337#103—1062no—
338#103—1062no—
339#103—1061no—
340#103—1061no—
341#102—1061no—
342#102—1061no—
343C22⋊C8—1061no—
344C22⋊C8—1061no—
345C22⋊C8—1062no—
346C22⋊C8—1062no—
347C22⋊C8C22⋊D5311yescharacteristic
348C22⋊C8C22⋊D5311yescharacteristic
349#97—1061no—
350#97—1061no—
351C4×C8C22×D5312yes—
352C24—1061no—
353C2×Q8—21no—
354C2×Q8—22no—
355C2×Q8—21no—
356C2×Q8—42no—
357C2×Q8—42no—
358C2×Q8—41no—
359C2×D4—531no—
360C2×D4—531no—
361C2×D4—1062no—
362C2×D4—1061no—
363C2×D4—2121no—
364C22×C4—41no—
365C22×C4—1061no—
366C22×C4—1061no—
367C22×C4—1061no—
368C22×C4—1061no—
369C22×C4—1061no—
370C22×C4—4242no—
371C22×C4D4×D5311yescharacteristic
372C22×C4#1171994711yescharacteristic
373C22×C4C2×C22⋊D5311yescharacteristic
374SD16—2124no—
375SD16—2122no—
376SD16—2124no—
377SD16—2122no—
378M16—4242no—
379C2×C8—1061no—
380C2×C8—1061no—
381C2×C8—1062no—
382C2×C8—1062no—
383C2×C8—4242no—
384C2×C8—2122no—
385C2×C8C2×C22⋊D5312yes—
386C2×C8C2×C22⋊D5312yes—
387C4⋊C4—21no—
388C4⋊C4—22no—
389C4⋊C4—21no—
390C4⋊C4—22no—
391C4⋊C4—21no—
392C4⋊C4—21no—
393C4⋊C4—41no—
394C4⋊C4—41no—
395C4⋊C4—41no—
396C4⋊C4—41no—
397C4⋊C4—531no—
398C4⋊C4—531no—
399C4⋊C4—1061no—
400C22⋊C4—41no—
401C22⋊C4—1061no—
402C22⋊C4—1061no—
403C22⋊C4—1061no—
404C22⋊C4—1061no—
405C42—41no—
406C42D4×D5311yescharacteristic
407C42#1171994711yescharacteristic
408C42C2×C22⋊D5311yescharacteristic
409C42C23×D5311yescharacteristic
410C23—1061no—
411C23—1061no—
412C23—4241no—
413C23—2122no—
414C23—2122no—
415C23#1313318811yescharacteristic
416Q8—42no—
417Q8—41no—
418Q8—41no—
419Q8—42no—
420Q8—42no—
421Q8—82no—
422D4—532no—
423D4—1061no—
424D4—1061no—
425D4—1061no—
426D4—2122no—
427C2×C4—22no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—22no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—42no—
434C2×C4—82no—
435C2×C4—1061no—
436C2×C4—1061no—
437C2×C4—4241no—
438C2×C4—4241no—
439C2×C4—4241no—
440C2×C4—2122no—
441C2×C4—2122no—
442C2×C4#1313318811yescharacteristic
443C2×C4#1313318811yescharacteristic
444C2×C4#1313318811yescharacteristic
445C2×C4C2×D4×D5311yescharacteristic
446C2×C4C22×C22⋊D5311yescharacteristic
447C2×C4C2×#1171994711yescharacteristic
448C8—22no—
449C8—22no—
450C8—2122no—
451C8—2121no—
452C8—2121no—
453C8—2122no—
454C22—21no—
455C22—21no—
456C22—1062no—
457C22—4244no—
458C22—2121no—
459C22—2121no—
460C22—2121no—
461C22—2121no—
462C22—2121no—
463C22#1681089512yes—
464C22C2×#1313318811yescharacteristic
465C4—21no—
466C4—21no—
467C4—21no—
468C4—21no—
469C4—81no—
470C4—82no—
471C4—81no—
472C4—2121no—
473C4—2121no—
474C4AGL(1,ℤ/8)×D5311yescharacteristic
475C4#1681141911yescharacteristic
476C4#1681143112yes—
477C2—22no—
478C2—4241no—
479C2—2122no—
480C2#2028422311yescharacteristic
481C2#2028432811yescharacteristic
482C2C2×#1681089511yescharacteristic
subgroupG/H
centerC22C2×#13133188
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#3230C2
socleC22×C53C2×#121
prime radicalC22×D212D4
hypercenter#3230D53
hypocenterC53#19464
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3230

Complements

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

NH ≅ G/Ntype
4. #20274259C2semidirect
5. #20274225C2semidirect
6. #20274135C2semidirect
7. #20274101C2semidirect
8. #20273500C2semidirect
9. #20273432C2semidirect
10. #20273415C2semidirect
11. C53×#3230C2semidirect
35. #16810376C22semidirect
36. #16810374C22semidirect
37. #16810369C22semidirect
38. #16810369C22semidirect
39. #16810369C22semidirect
40. #16810369C22semidirect
43. C53×#500C22semidirect
44. C53×#487C22semidirect
236. #3230D53semidirect
257. #500C2×D53semidirect
258. #487C2×D53semidirect
290. C53#19464semidirect

Maximal subgroups

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

subgroupindex
#202848102
C2×#168108252
#202837492
#202742592
#202742252
#202741352
#202741012
#202735002
#202734322
#202734152
C53×#32302
#1946453