Finite Group Databasebeta

← #27370162

Subgroups — #27370162

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#20285658C211yesmaximal, characteristic
2#20284845C211yesmaximal, characteristic, complemented
3C2×#16810823C211yesmaximal, characteristic
4#20274283C212yesmaximal, complemented
5#20274187C212yesmaximal, complemented
6#20274159C212yesmaximal, complemented
7#20274063C212yesmaximal, complemented
8#20273500C211yesmaximal, characteristic, complemented
9#20273428C211yesmaximal, characteristic
10#20273416C211yesmaximal, characteristic, complemented
11C53×#2984C211yesmaximal, characteristic, complemented
12C22×#11719924—21no—
13#16811180C2211yescharacteristic
14C2×#13133214C2211yescharacteristic
15#16810996C2212yes—
16#16810970C2212yes—
17#16810953—21no—
18#16810908C2211yescharacteristic, complemented
19#16810908C2211yescharacteristic, complemented
20#16810833C2211yescharacteristic, complemented
21#16810833C2211yescharacteristic, complemented
22#16810824—22no—
23#16810823—21no—
24#16810823—21no—
25#16810823C2212yes—
26#16810823C2212yes—
27#16810779—21no—
28#16810776—21no—
29#16810776—21no—
30#16810728—21no—
31#16810728—21no—
32#16810680C2211yescharacteristic
33C2×C4⋊#88322C2211yescharacteristic
34C2×#13133137—21no—
35#16810376C2212yescomplemented
36#16810374C2212yescomplemented
37#16810370C2212yescomplemented
38#16810368C2212yescomplemented
39#16810366C2212yes—
40#16810366C2212yes—
41C2×C4×Q8×C53C2211yescharacteristic
42C53×#446—22no—
43C53×#446C2212yescomplemented
44C8×Q8×C53C2212yescomplemented
45C53×#432—21no—
46C53×#432C2211yescharacteristic
47C53×#431—21no—
48C53×#431C2211yescharacteristic
49C22×D212—21no—
50C22×D212—21no—
51C22×D212—21no—
52C2×#11719925—21no—
53C2×#11719925—21no—
54C2×#11719913—22no—
55C2×#11719913—22no—
56C2×#11719924—22no—
57C2×#11719924—22no—
58C2×#11719924—44no—
59C22×#88322—21no—
60#13133217C2311yescharacteristic
61#13133217C2311yescharacteristic
62#13133214—21no—
63#13133214—21no—
64#13133214—21no—
65#13133214C2312yes—
66#13133209—21no—
67#13133196—21no—
68#13133196—21no—
69C4×Dic106—21no—
70C4×Dic106—21no—
71#13133176—21no—
72#13133148—21no—
73#13133148—21no—
74C4⋊#88322—21no—
75C4⋊#88322C2312yes—
76C4⋊#88322C2311yescharacteristic
77C4⋊#88322C2311yescharacteristic
78#13133141—21no—
79#13133141—21no—
80#13133140—21no—
81#13133140—21no—
82#13133139—22no—
83#13133139—22no—
84#13133137—22no—
85#13133137—22no—
86C22⋊#88322—21no—
87#13133128—21no—
88C22×Q8×C53D411yescharacteristic
89C4×Q8×C53—21no—
90C4×Q8×C53—21no—
91C4×Q8×C53D411yescharacteristic
92C4×Q8×C53D411yescharacteristic
93C4×Q8×C53C2311yescharacteristic
94C4×Q8×C53C2311yescharacteristic
95C2×C4⋊C4×C53—21no—
96C2×C4⋊C4×C53D411yescharacteristic
97C2×C42×C53—21no—
98C2×C42×C53C2311yescharacteristic
99C4⋊C8×C53—22no—
100C4⋊C8×C53—22no—
101C4⋊C8×C53D412yes—
102C4⋊C8×C53C2312yes—
103C22⋊C8×C53—21no—
104C22⋊C8×C53D411yescharacteristic
105C22⋊C8×C53D411yescharacteristic
106#97×C53—22no—
107#97×C53—21no—
108C4×C8×C53—22no—
109C4×C8×C53C2312yes—
110C23×D53—41no—
111C2×Q8×C53—21no—
112C2×Q8×C53—22no—
113C2×Q8×C53—41no—
114C2×Q8×C53C2×D411yescharacteristic
115C2×Q8×C53C2×D411yescharacteristic
116C22×C4×C53—21no—
117C22×C4×C53—21no—
118C22×C4×C53C2×D411yescharacteristic
119C22×C4×C53C2×D411yescharacteristic
120C22×C4×C53#4111yescharacteristic
121C2×D212—22no—
122C2×D212—22no—
123C2×D212—22no—
124C2×D212—42no—
125C2×D212—42no—
126C2×D212—41no—
127C2×D212—41no—
128C2×D212—41no—
129C2×D212—82no—
130C2×D212—82no—
131C2×Dic106—21no—
132C2×Dic106—21no—
133C2×C8×C53—22no—
134C2×C8×C53C2×D412yes—
135C2×C8×C53C2×D412yes—
136C4⋊C4×C53—21no—
137C4⋊C4×C53—21no—
138C4⋊C4×C53—21no—
139C4⋊C4×C53—21no—
140C4⋊C4×C53—21no—
141C4⋊C4×C53C2×D411yescharacteristic
142C4⋊C4×C53C2×D411yescharacteristic
143C42×C53—21no—
144C42×C53—21no—
145C42×C53—21no—
146C42×C53—21no—
147C42×C53C2×D411yescharacteristic
148C42×C53C2×D411yescharacteristic
149C42×C53#4111yescharacteristic
150C42×C53C2411yescharacteristic
151C22⋊Dic53—41no—
152#11719925—42no—
153#11719925—42no—
154#11719924—44no—
155#11719924—44no—
156C4⋊Dic53—21no—
157C4⋊Dic53—21no—
158C4⋊Dic53—21no—
159C4⋊Dic53—21no—
160#11719919—41no—
161#11719919—41no—
162C4×Dic53—41no—
163C2×#88322—21no—
164C2×#88322—21no—
165C2×#88322—22no—
166C2×#88322—42no—
167#11719913—82no—
168#11719913—82no—
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×C53AGL(1,ℤ/8)12yes—
179C2×C4×C53—21no—
180C2×C4×C53—22no—
181C2×C4×C53—21no—
182C2×C4×C53—21no—
183C2×C4×C53—22no—
184C2×C4×C53—21no—
185C2×C4×C53—21no—
186C2×C4×C53—21no—
187C2×C4×C53—42no—
188C2×C4×C53—42no—
189C2×C4×C53#12111yescharacteristic
190C2×C4×C53#12111yescharacteristic
191C2×C4×C53#12111yescharacteristic
192C2×C4×C53C22×D411yescharacteristic
193C2×C4×C53C22×D411yescharacteristic
194C2×C4×C53C2×#4111yescharacteristic
195C2×Dic53—41no—
196C2×Dic53—41no—
197D212—42no—
198D212—42no—
199D212—42no—
200D212—42no—
201D212—82no—
202D212—82no—
203D212—82no—
204D212—82no—
205Dic106—41no—
206Dic106—41no—
207Dic106—42no—
208C8×C53—22no—
209C8×C53—22no—
210C8×C53—42no—
211#88322—42no—
212#88322—41no—
213#88322—41no—
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×C53C2×AGL(1,ℤ/8)11yescharacteristic
232C4×C53#57511yescharacteristic
233C4×C53#57712yes—
234Dic53—81no—
235Dic53—81no—
236#3208—531no—
237#3113—531no—
238C2×#460—531no—
239#2984D5311yescharacteristic, complemented
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#533—531no—
254C2×#127—531no—
255C2×C4×Q8C2×D5311yescharacteristic
256#492—532no—
257#486—532no—
258#483—1061no—
259#475—531no—
260#475—531no—
261#462—531no—
262#462—531no—
263#460—532no—
264#460—532no—
265#460—1061no—
266#460—1061no—
267#460—1062no—
268#451—1061no—
269#450—1061no—
270#450—1061no—
271#446—22no—
272#446C2×D5312yescomplemented
273C8×Q8C2×D5312yescomplemented
274#441—1061no—
275#441—1061no—
276#432—21no—
277#432—531no—
278#432C2×D5311yescharacteristic
279#431—21no—
280#431C2×D5311yescharacteristic
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×Q8D21211yescharacteristic
291C22×D4—1061no—
292C22×D4—1061no—
293C22×D4—1061no—
294C2×Q16—1061no—
295C2×Q16—1061no—
296C2×SD16—1062no—
297C2×SD16—1062no—
298C2×SD16—1062no—
299C2×SD16—1062no—
300C2×SD16—2124no—
301C22×C8—1061no—
302C4⋊Q8—531no—
303C4⋊Q8—531no—
304#127—532no—
305#127—1061no—
306#127—1061no—
307#127—1061no—
308#125—1061no—
309C22⋊Q8—1061no—
310C22⋊Q8—1061no—
311C4×Q8—21no—
312C4×Q8—21no—
313C4×Q8—1061no—
314C4×Q8—1061no—
315C4×Q8D21211yescharacteristic
316C4×Q8D21211yescharacteristic
317C4×Q8C22×D5311yescharacteristic
318C4×Q8C22×D5311yescharacteristic
319#117—1061no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4D21211yescharacteristic
322C2×C42—21no—
323C2×C42C22×D5311yescharacteristic
324#106—1061no—
325#106—1061no—
326C4⋊C8—22no—
327C4⋊C8—22no—
328C4⋊C8—531no—
329C4⋊C8—531no—
330C4⋊C8—532no—
331C4⋊C8—1061no—
332C4⋊C8C22⋊D5312yes—
333C4⋊C8C22×D5312yes—
334#103—1061no—
335#103—1061no—
336#103—1061no—
337#103—1061no—
338#102—1062no—
339#102—1062no—
340#102—1062no—
341#102—1062no—
342C22⋊C8—21no—
343C22⋊C8—1061no—
344C22⋊C8C22⋊D5311yescharacteristic
345C22⋊C8C22⋊D5311yescharacteristic
346#97—22no—
347#97—21no—
348#97—1061no—
349C4×C8—22no—
350C4×C8C22×D5312yes—
351C24—2121no—
352C2×Q8—21no—
353C2×Q8—22no—
354C2×Q8—41no—
355C2×Q8—1061no—
356C2×Q8—1061no—
357C2×Q8C2×D21211yescharacteristic
358C2×Q8C2×D21211yescharacteristic
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×C4C2×D21211yescharacteristic
372C22×C4#1171994511yescharacteristic
373C22×C4C2×C22⋊D5311yescharacteristic
374Q16—2122no—
375Q16—2122no—
376SD16—4242no—
377SD16—4242no—
378SD16—2124no—
379SD16—2124no—
380C2×C8—22no—
381C2×C8—1061no—
382C2×C8—1061no—
383C2×C8—1062no—
384C2×C8—2122no—
385C2×C8C2×C22⋊D5312yes—
386C2×C8C2×C22⋊D5312yes—
387C4⋊C4—21no—
388C4⋊C4—21no—
389C4⋊C4—21no—
390C4⋊C4—21no—
391C4⋊C4—21no—
392C4⋊C4—1061no—
393C4⋊C4—1061no—
394C4⋊C4—1061no—
395C4⋊C4—1061no—
396C4⋊C4—2121no—
397C4⋊C4—2121no—
398C4⋊C4C2×D21211yescharacteristic
399C4⋊C4C2×D21211yescharacteristic
400C22⋊C4—2121no—
401C42—21no—
402C42—21no—
403C42—21no—
404C42—21no—
405C42—2121no—
406C42C2×D21211yescharacteristic
407C42#1171994511yescharacteristic
408C42C2×C22⋊D5311yescharacteristic
409C42C23×D5311yescharacteristic
410C23—4242no—
411C23—4242no—
412C23—4242no—
413C23—2122no—
414C23#1313319111yescharacteristic
415Q8—21no—
416Q8—21no—
417Q8—21no—
418Q8—42no—
419Q8—2121no—
420Q8—2121no—
421Q8—2122no—
422Q8#1313324512yes—
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—2121no—
442C2×C4—2121no—
443C2×C4#1313319111yescharacteristic
444C2×C4#1313319111yescharacteristic
445C2×C4#1313319111yescharacteristic
446C2×C4C22×C22⋊D5311yescharacteristic
447C2×C4C22×D21211yescharacteristic
448C2×C4C2×#1171994511yescharacteristic
449C8—22no—
450C8—22no—
451C8—42no—
452C8—2122no—
453C8—2121no—
454C8—2121no—
455C22—21no—
456C22—21no—
457C22—8484no—
458C22—4242no—
459C22—4242no—
460C22—4242no—
461C22#1681088612yes—
462C22C2×#1313319111yescharacteristic
463C4—21no—
464C4—21no—
465C4—22no—
466C4—21no—
467C4—21no—
468C4—41no—
469C4—41no—
470C4—41no—
471C4—4241no—
472C4—4241no—
473C4C2×#1313324511yescharacteristic
474C4#1681142311yescharacteristic
475C4#1681143112yes—
476C2—22no—
477C2—8482no—
478C2#2028423611yescharacteristic
479C2#2028433411yescharacteristic
480C2C2×#1681088611yescharacteristic
subgroupG/H
centerC22C2×#13133191
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#2984C2
socleC22×C53C2×#121
prime radicalC2×#13133214C22
hypercenter#2984D53
hypocenterC53#19468
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2984

Complements

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

NH ≅ G/Ntype
2. #20284845C2semidirect
4. #20274283C2semidirect
5. #20274187C2semidirect
6. #20274159C2semidirect
7. #20274063C2semidirect
8. #20273500C2semidirect
10. #20273416C2semidirect
11. C53×#2984C2semidirect
18. #16810908C22semidirect
19. #16810908C22semidirect
20. #16810833C22semidirect
21. #16810833C22semidirect
35. #16810376C22semidirect
36. #16810374C22semidirect
37. #16810370C22semidirect
38. #16810368C22semidirect
43. C53×#446C22semidirect
44. C8×Q8×C53C22semidirect
239. #2984D53semidirect
272. #446C2×D53semidirect
273. C8×Q8C2×D53semidirect
289. C53#19468semidirect

Maximal subgroups

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

subgroupindex
#202856582
#202848452
C2×#168108232
#202742832
#202741872
#202741592
#202740632
#202735002
#202734282
#202734162
C53×#29842
#1946853