Finite Group Databasebeta

← #27370164

Subgroups — #27370164

Order 13568.

672 conjugacy classes of subgroups.

The subgroup lattice has 482 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285653C211yesmaximal, characteristic
2C2×#16810823C211yesmaximal, characteristic
3#20283804C211yesmaximal, characteristic, complemented
4#20274285C212yesmaximal, complemented
5#20274189C212yesmaximal, complemented
6#20274161C212yesmaximal, complemented
7#20274065C212yesmaximal, complemented
8#20273501C211yesmaximal, characteristic, complemented
9#20273428C211yesmaximal, characteristic
10#20273415C211yesmaximal, characteristic, complemented
11C53×#3113C211yesmaximal, characteristic, complemented
12C22×#11719924—21no—
13C2×#13133214C2211yescharacteristic
14#16810997C2212yes—
15#16810971C2212yes—
16#16810823—22no—
17#16810823—21no—
18#16810823—21no—
19#16810823C2212yes—
20#16810823C2212yes—
21#16810773—21no—
22#16810773—21no—
23#16810754—22no—
24#16810754C2212yescomplemented
25Q8×#88322C2212yescomplemented
26#16810680—21no—
27#16810680C2211yescharacteristic
28#16810676—21no—
29#16810676C2211yescharacteristic
30C2×C4⋊#88322C2211yescharacteristic
31C2×#13133137—21no—
32#16810375C2212yescomplemented
33#16810373C2212yescomplemented
34#16810369C2212yescomplemented
35#16810369C2212yescomplemented
36#16810366C2212yes—
37#16810366C2212yes—
38C53×#533C2211yescharacteristic
39C2×C4×Q8×C53C2211yescharacteristic
40C53×#483—21no—
41C53×#475C2211yescharacteristic, complemented
42C53×#475C2211yescharacteristic, complemented
43C53×#462C2211yescharacteristic, complemented
44C53×#462C2211yescharacteristic, complemented
45C53×#451—21no—
46C53×#441—21no—
47C53×#441—21no—
48C53×#432C2211yescharacteristic
49C22×D212—21no—
50C22×D212—21no—
51C22×D212—21no—
52C2×#11719924—22no—
53C2×#11719924—22no—
54C2×#11719924—22no—
55C2×#11719924—22no—
56C2×#11719924—44no—
57C22×#88322—21no—
58#13133214—21no—
59#13133214—21no—
60#13133214—21no—
61#13133214C2312yes—
62C4⋊#88322—21no—
63C4⋊#88322—22no—
64C4⋊#88322—22no—
65C4⋊#88322D412yes—
66C4⋊#88322C2312yes—
67C4⋊#88322C2312yes—
68#13133137—22no—
69#13133137—22no—
70#13133137—22no—
71#13133137—22no—
72C22⋊#88322—21no—
73C22⋊#88322D411yescharacteristic
74C22⋊#88322D411yescharacteristic
75#13133128—21no—
76#13133128—22no—
77C4×#88322—22no—
78C4×#88322C2312yes—
79C22×Q8×C53D411yescharacteristic
80C2×Q16×C53—21no—
81C2×Q16×C53—21no—
82C4⋊Q8×C53C2311yescharacteristic
83C4⋊Q8×C53C2311yescharacteristic
84#125×C53—21no—
85C22⋊Q8×C53—21no—
86C22⋊Q8×C53—21no—
87C4×Q8×C53—21no—
88C4×Q8×C53—21no—
89C4×Q8×C53—21no—
90C4×Q8×C53—21no—
91C4×Q8×C53D411yescharacteristic
92C4×Q8×C53D411yescharacteristic
93C4×Q8×C53C2311yescharacteristic
94C4×Q8×C53C2311yescharacteristic
95#117×C53—21no—
96C2×C4⋊C4×C53—21no—
97C2×C4⋊C4×C53D411yescharacteristic
98C2×C42×C53—21no—
99C2×C42×C53C2311yescharacteristic
100#106×C53—21no—
101#106×C53—21no—
102C4⋊C8×C53C2311yescharacteristic
103C4⋊C8×C53C2311yescharacteristic
104#103×C53—21no—
105#103×C53—21no—
106#103×C53—21no—
107#103×C53—21no—
108C22⋊C8×C53—21no—
109#97×C53—21no—
110C23×D53—41no—
111C2×Q8×C53—21no—
112C2×Q8×C53—22no—
113C2×Q8×C53—21no—
114C2×Q8×C53—21no—
115C2×Q8×C53—41no—
116C2×Q8×C53C2×D411yescharacteristic
117C2×Q8×C53C2×D411yescharacteristic
118C22×C4×C53—21no—
119C22×C4×C53—21no—
120C22×C4×C53C2×D411yescharacteristic
121C22×C4×C53C2×D411yescharacteristic
122C22×C4×C53#4111yescharacteristic
123C2×D212—22no—
124C2×D212—22no—
125C2×D212—22no—
126C2×D212—42no—
127C2×D212—42no—
128C2×D212—41no—
129C2×D212—41no—
130C2×D212—41no—
131C2×D212—82no—
132C2×D212—82no—
133Q16×C53—42no—
134Q16×C53—42no—
135C2×C8×C53—21no—
136C2×C8×C53—21no—
137C4⋊C4×C53—21no—
138C4⋊C4×C53—21no—
139C4⋊C4×C53—21no—
140C4⋊C4×C53—21no—
141C4⋊C4×C53—21no—
142C4⋊C4×C53—21no—
143C4⋊C4×C53—21no—
144C4⋊C4×C53—21no—
145C4⋊C4×C53—21no—
146C4⋊C4×C53—41no—
147C4⋊C4×C53—41no—
148C4⋊C4×C53C2×D411yescharacteristic
149C4⋊C4×C53C2×D411yescharacteristic
150C22⋊C4×C53—41no—
151C42×C53—21no—
152C42×C53—21no—
153C42×C53—21no—
154C42×C53—21no—
155C42×C53—41no—
156C42×C53C2×D411yescharacteristic
157C42×C53C2×D411yescharacteristic
158C42×C53#4111yescharacteristic
159C42×C53C2411yescharacteristic
160#11719924—44no—
161#11719924—44no—
162#11719924—82no—
163#11719924—82no—
164C2×#88322—22no—
165C2×#88322—22no—
166C2×#88322—42no—
167C2×#88322C2×D412yes—
168C2×#88322C2×D412yes—
169C23×C53#12111yescharacteristic
170C22×D53—42no—
171C22×D53—82no—
172C22×D53—82no—
173C22×D53—82no—
174Q8×C53—21no—
175Q8×C53—21no—
176Q8×C53—21no—
177Q8×C53—42no—
178Q8×C53—41no—
179Q8×C53—41no—
180Q8×C53—42no—
181Q8×C53AGL(1,ℤ/8)12yes—
182C2×C4×C53—21no—
183C2×C4×C53—22no—
184C2×C4×C53—21no—
185C2×C4×C53—21no—
186C2×C4×C53—22no—
187C2×C4×C53—21no—
188C2×C4×C53—21no—
189C2×C4×C53—21no—
190C2×C4×C53—42no—
191C2×C4×C53—42no—
192C2×C4×C53—41no—
193C2×C4×C53—41no—
194C2×C4×C53#12111yescharacteristic
195C2×C4×C53#12111yescharacteristic
196C2×C4×C53#12111yescharacteristic
197C2×C4×C53C22×D411yescharacteristic
198C2×C4×C53C22×D411yescharacteristic
199C2×C4×C53C2×#4111yescharacteristic
200D212—42no—
201D212—42no—
202D212—42no—
203D212—42no—
204D212—82no—
205D212—82no—
206D212—82no—
207D212—82no—
208C8×C53—41no—
209C8×C53—41no—
210#88322—22no—
211#88322—22no—
212#88322—42no—
213#88322—42no—
214#19468—531nomaximal, Sylow
215C22×C53—21no—
216C22×C53—21no—
217C22×C53#47112yes—
218C22×C53C2×#12111yescharacteristic
219C2×D53—82no—
220C2×D53—82no—
221C2×D53—82no—
222C2×D53—164no—
223C4×C53—21no—
224C4×C53—21no—
225C4×C53—22no—
226C4×C53—21no—
227C4×C53—21no—
228C4×C53—41no—
229C4×C53—41no—
230C4×C53—41no—
231C4×C53—81no—
232C4×C53—81no—
233C4×C53C2×AGL(1,ℤ/8)11yescharacteristic
234C4×C53#57511yescharacteristic
235C4×C53#57712yes—
236#3208—531no—
237#3113D5311yescharacteristic, complemented
238C2×#460—531no—
239#2984—531no—
240#1702—532no—
241#1686—532no—
242#1682—532no—
243#1666—532no—
244#1573—531no—
245#1555—531no—
246#1552—531no—
247C2×C53—22no—
248C2×C53#304311yescharacteristic
249C2×C53#305511yescharacteristic
250C2×C53C2×#47111yescharacteristic
251D53—162no—
252C22×SD16—1061no—
253#533C2×D5311yescharacteristic
254C2×#127—531no—
255C2×C4×Q8C2×D5311yescharacteristic
256#492—532no—
257#486—532no—
258#483—21no—
259#475C2×D5311yescharacteristic, complemented
260#475C2×D5311yescharacteristic, complemented
261#462C2×D5311yescharacteristic, complemented
262#462C2×D5311yescharacteristic, complemented
263#460—532no—
264#460—532no—
265#460—1062no—
266#460—1061no—
267#460—1061no—
268#451—21no—
269#450—1061no—
270#450—1061no—
271#446—532no—
272#446—1062no—
273C8×Q8—532no—
274#441—21no—
275#441—21no—
276#432—531no—
277#432—1061no—
278#432C2×D5311yescharacteristic
279#431—531no—
280#431—1061no—
281C2×C4⋊C8—531no—
282C2×#102—1061no—
283#335—532no—
284#334—532no—
285#332—532no—
286#332—532no—
287#331—532no—
288#331—532no—
289C53#1946811yescharacteristic, complemented, Sylow
290C22×Q8C22⋊D5311yescharacteristic
291C22×D4—1061no—
292C22×D4—1061no—
293C22×D4—1061no—
294C2×Q16—21no—
295C2×Q16—21no—
296C2×SD16—1062no—
297C2×SD16—1062no—
298C2×SD16—1062no—
299C2×SD16—1062no—
300C2×SD16—2124no—
301C22×C8—1061no—
302C4⋊Q8C22×D5311yescharacteristic
303C4⋊Q8C22×D5311yescharacteristic
304#127—532no—
305#127—1061no—
306#127—1061no—
307#127—1061no—
308#125—21no—
309C22⋊Q8—21no—
310C22⋊Q8—21no—
311C4×Q8—21no—
312C4×Q8—21no—
313C4×Q8—21no—
314C4×Q8—21no—
315C4×Q8C22⋊D5311yescharacteristic
316C4×Q8C22⋊D5311yescharacteristic
317C4×Q8C22×D5311yescharacteristic
318C4×Q8C22×D5311yescharacteristic
319#117—21no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4C22⋊D5311yescharacteristic
322C2×C42—21no—
323C2×C42C22×D5311yescharacteristic
324#106—21no—
325#106—21no—
326C4⋊C8—532no—
327C4⋊C8—532no—
328C4⋊C8—532no—
329C4⋊C8—1062no—
330C4⋊C8—1062no—
331C4⋊C8—1061no—
332C4⋊C8C22×D5311yescharacteristic
333C4⋊C8C22×D5311yescharacteristic
334#103—21no—
335#103—21no—
336#103—21no—
337#103—21no—
338#102—1062no—
339#102—1062no—
340#102—1062no—
341#102—1062no—
342C22⋊C8—21no—
343C22⋊C8—531no—
344C22⋊C8—531no—
345C22⋊C8—1061no—
346#97—21no—
347#97—1062no—
348#97—1061no—
349C4×C8—532no—
350C4×C8—1062no—
351C24—2121no—
352C2×Q8—21no—
353C2×Q8—22no—
354C2×Q8—21no—
355C2×Q8—21no—
356C2×Q8—41no—
357C2×Q8C2×C22⋊D5311yescharacteristic
358C2×Q8C2×C22⋊D5311yescharacteristic
359C2×D4—1062no—
360C2×D4—1062no—
361C2×D4—1062no—
362C2×D4—4242no—
363C2×D4—4242no—
364C2×D4—2122no—
365C2×D4—2122no—
366C2×D4—2121no—
367C2×D4—2121no—
368C2×D4—2121no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4D4×D5311yescharacteristic
372C22×C4#1171994711yescharacteristic
373C22×C4C2×C22⋊D5311yescharacteristic
374Q16—42no—
375Q16—42no—
376SD16—4242no—
377SD16—4242no—
378SD16—2124no—
379SD16—2124no—
380C2×C8—21no—
381C2×C8—21no—
382C2×C8—532no—
383C2×C8—532no—
384C2×C8—1062no—
385C2×C8—1062no—
386C2×C8—2122no—
387C4⋊C4—21no—
388C4⋊C4—21no—
389C4⋊C4—21no—
390C4⋊C4—21no—
391C4⋊C4—21no—
392C4⋊C4—21no—
393C4⋊C4—21no—
394C4⋊C4—21no—
395C4⋊C4—21no—
396C4⋊C4—41no—
397C4⋊C4—41no—
398C4⋊C4C2×C22⋊D5311yescharacteristic
399C4⋊C4C2×C22⋊D5311yescharacteristic
400C22⋊C4—41no—
401C42—21no—
402C42—21no—
403C42—21no—
404C42—21no—
405C42—41no—
406C42D4×D5311yescharacteristic
407C42#1171994711yescharacteristic
408C42C2×C22⋊D5311yescharacteristic
409C42C23×D5311yescharacteristic
410C23—4242no—
411C23—4242no—
412C23—4242no—
413C23—2122no—
414C23#1313318911yescharacteristic
415Q8—21no—
416Q8—21no—
417Q8—21no—
418Q8—42no—
419Q8—41no—
420Q8—41no—
421Q8—42no—
422Q8#1313324212yes—
423D4—4242no—
424D4—4242no—
425D4—4242no—
426D4—4242no—
427D4—2122no—
428D4—2122no—
429D4—2122no—
430D4—2122no—
431C2×C4—21no—
432C2×C4—22no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—22no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4#1313318911yescharacteristic
444C2×C4#1313318911yescharacteristic
445C2×C4#1313318911yescharacteristic
446C2×C4C2×D4×D5311yescharacteristic
447C2×C4C22×C22⋊D5311yescharacteristic
448C2×C4C2×#1171994711yescharacteristic
449C8—41no—
450C8—41no—
451C8—1062no—
452C8—1062no—
453C8—2122no—
454C8—2122no—
455C22—21no—
456C22—21no—
457C22—8484no—
458C22—4242no—
459C22—4242no—
460C22—4242no—
461C22#1681088812yes—
462C22C2×#1313318911yescharacteristic
463C4—21no—
464C4—21no—
465C4—21no—
466C4—21no—
467C4—22no—
468C4—41no—
469C4—41no—
470C4—41no—
471C4—81no—
472C4—81no—
473C4C2×#1313324211yescharacteristic
474C4#1681142111yescharacteristic
475C4#1681143212yes—
476C2—22no—
477C2—8482no—
478C2#2028423811yescharacteristic
479C2#2028433611yescharacteristic
480C2C2×#1681088811yescharacteristic
subgroupG/H
centerC22C2×#13133189
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#3113C2
socleC22×C53C2×#121
prime radicalC2×#13133214C22
hypercenter#3113D53
hypocenterC53#19468
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3113

Complements

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

NH ≅ G/Ntype
3. #20283804C2semidirect
4. #20274285C2semidirect
5. #20274189C2semidirect
6. #20274161C2semidirect
7. #20274065C2semidirect
8. #20273501C2semidirect
10. #20273415C2semidirect
11. C53×#3113C2semidirect
24. #16810754C22semidirect
25. Q8×#88322C22semidirect
32. #16810375C22semidirect
33. #16810373C22semidirect
34. #16810369C22semidirect
35. #16810369C22semidirect
41. C53×#475C22semidirect
42. C53×#475C22semidirect
43. C53×#462C22semidirect
44. C53×#462C22semidirect
237. #3113D53semidirect
259. #475C2×D53semidirect
260. #475C2×D53semidirect
261. #462C2×D53semidirect
262. #462C2×D53semidirect
289. C53#19468semidirect

Maximal subgroups

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

subgroupindex
#202856532
C2×#168108232
#202838042
#202742852
#202741892
#202741612
#202740652
#202735012
#202734282
#202734152
C53×#31132
#1946853