Finite Group Databasebeta

← #27283241

Subgroups — #27283241

Order 13568.

468 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285274C211yesmaximal, characteristic
2#20275659C211yesmaximal, characteristic, complemented
3#20275634C211yesmaximal, characteristic
4C4×#13133149C211yesmaximal, characteristic, complemented
5#20273215C211yesmaximal, characteristic, complemented
6#20273211C211yesmaximal, characteristic
7C8×M16×C53C211yesmaximal, characteristic, complemented
8#16811180C2211yescharacteristic
9#16811173C2211yescharacteristic
10#16811021C411yescharacteristic, complemented
11#16810996C411yescharacteristic, complemented
12#16810974C411yescharacteristic, complemented
13#16810970C411yescharacteristic, complemented
14#16810852—21no—
15#16810852—21no—
16C4×#11719913—21no—
17C4×#11719913—21no—
18C2×#13133149—21no—
19C2×#13133143—22no—
20C4×C4⋊Dic53C2211yescharacteristic
21C4×C4⋊Dic53C2211yescharacteristic
22#16810386—21no—
23#16810384—22no—
24#16810361—22no—
25#16810358—22no—
26C53×#431C2211yescharacteristic
27C4×M16×C53C2211yescharacteristic
28C2×C4×C8×C53C2211yescharacteristic
29C53×#322—21no—
30C82×C53—21no—
31C2×#11719945—21no—
32C2×#11719913—21no—
33C2×#11719913—21no—
34#13133217C2×C411yescharacteristic
35#13133217C2×C411yescharacteristic
36#13133217C2×C411yescharacteristic
37#13133214C2×C411yescharacteristic
38#13133209—21no—
39#13133205—21no—
40#13133196—21no—
41#13133196—21no—
42#13133191—21no—
43#13133191—21no—
44C4×Dic106—21no—
45C4×Dic106—21no—
46C4×D212—21no—
47C4×D212—21no—
48#13133176—21no—
49C2×C4⋊Dic53—21no—
50C2×C4⋊Dic53—21no—
51C2×C4×Dic53—21no—
52C2×C4×Dic53—21no—
53#13133149—24no—
54#13133149—21no—
55#13133149—21no—
56#13133143—44no—
57#13133142—21no—
58#13133142—21no—
59#13133139—21no—
60#13133139—21no—
61#13133124—21no—
62#13133124—21no—
63C2×M16×C53D412yes—
64C22×C8×C53D411yescharacteristic
65C22×C8×C53D411yescharacteristic
66C2×C42×C53C2311yescharacteristic
67C4⋊C8×C53—21no—
68C22⋊C8×C53D412yes—
69#97×C53—21no—
70C4×C8×C53—21no—
71C4×C8×C53—21no—
72C4×C8×C53C2×C411yescharacteristic
73C4×C8×C53C2×C411yescharacteristic
74C4×C8×C53D411yescharacteristic
75C4×C8×C53D411yescharacteristic
76C22×C4×C53C2×D411yescharacteristic
77C22×C4×C53C2×D411yescharacteristic
78C22×C4×C53C2×D411yescharacteristic
79C2×C22⋊D53—41no—
80C22×Dic53—41no—
81C22×Dic53—41no—
82#11719945—42no—
83#11719945—42no—
84C2×D212—21no—
85C2×D212—21no—
86C2×D212—21no—
87C2×C4×D53—41no—
88C2×Dic106—21no—
89C2×Dic106—21no—
90C2×Dic106—21no—
91M16×C53—22no—
92M16×C53SD1614yescomplemented
93C2×C8×C53—22no—
94C2×C8×C53—22no—
95C2×C8×C53—22no—
96C2×C8×C53—41no—
97C2×C8×C53C22⋊C411yescharacteristic
98C2×C8×C53C22⋊C411yescharacteristic
99C2×C8×C53C22⋊C411yescharacteristic
100C2×C8×C53C22⋊C411yescharacteristic
101C42×C53C22×C411yescharacteristic
102C42×C53C2×D411yescharacteristic
103C42×C53#4111yescharacteristic
104C42×C53#4111yescharacteristic
105C22⋊Dic53—41no—
106#11719921—41no—
107#11719921—41no—
108C4⋊Dic53—22no—
109C4⋊Dic53—22no—
110C4⋊Dic53—21no—
111C4⋊Dic53—21no—
112C4⋊Dic53—21no—
113C4⋊Dic53—21no—
114C4⋊Dic53—21no—
115C4⋊Dic53—21no—
116C4⋊Dic53—41no—
117C4⋊Dic53—41no—
118#11719919—41no—
119#11719919—41no—
120C4×Dic53—41no—
121C4×Dic53—42no—
122C4×Dic53—42no—
123#11719913—42no—
124#11719913—42no—
125C23×C53#12711yescharacteristic
126C22×D53—41no—
127C2×C4×C53—21no—
128C2×C4×C53—21no—
129C2×C4×C53—41no—
130C2×C4×C53C2×C22⋊C411yescharacteristic
131C2×C4×C53C4×D411yescharacteristic
132C2×C4×C53C4×D411yescharacteristic
133C2×C4×C53#12111yescharacteristic
134C2×C4×C53#12111yescharacteristic
135C2×C4×C53#12411yescharacteristic
136C2×C4×C53C2×SD1612yes—
137C2×C4×C53#13512yes—
138C22⋊D53—81no—
139C22⋊D53—81no—
140C2×Dic53—42no—
141C2×Dic53—42no—
142C2×Dic53—41no—
143C2×Dic53—41no—
144C2×Dic53—41no—
145C2×Dic53—81no—
146C2×Dic53—81no—
147C2×Dic53—81no—
148C2×Dic53—81no—
149D212—41no—
150D212—41no—
151D212—42no—
152D212—81no—
153C4×D53—82no—
154Dic106—41no—
155Dic106—41no—
156Dic106—42no—
157Dic106—81no—
158C8×C53—21no—
159C8×C53—21no—
160C8×C53—21no—
161C8×C53—24no—
162C8×C53—42no—
163C8×C53#10412yescomplemented
164#9717—531nomaximal, Sylow
165C22×C53—21no—
166C22×C53—21no—
167C22×C53#39011yescharacteristic
168C22×C53#49211yescharacteristic
169C22×C53#49511yescharacteristic
170C2×D53—81no—
171C2×D53—81no—
172C2×D53—81no—
173C4×C53—21no—
174C4×C53—21no—
175C4×C53—21no—
176C4×C53C2×#10411yescharacteristic
177C4×C53#42111yescharacteristic
178C4×C53C4×SD1612yes—
179C4×C53#46512yes—
180Dic53—82no—
181Dic53—82no—
182Dic53—81no—
183Dic53—81no—
184Dic53—81no—
185#3167—531no—
186#1986—531no—
187#1980—531no—
188C4×#106—531no—
189#1503—531no—
190#1502—531no—
191C8×M16D5311yescharacteristic, complemented
192C2×C53—21no—
193C2×C53—21no—
194C2×C53#196811yescharacteristic
195C2×C53#197611yescharacteristic
196C2×C53#198011yescharacteristic
197D53—161no—
198#533—531no—
199#532—531no—
200#498—531no—
201#492—531no—
202#487—531no—
203#486—531no—
204#465—1061no—
205#465—1061no—
206C4×SD16—1061no—
207C4×SD16—1061no—
208#431C2×D5311yescharacteristic
209C2×#106—1061no—
210C2×#104—1062no—
211C4×M16C2×D5311yescharacteristic
212C2×C4×C8C2×D5311yescharacteristic
213C4×C4⋊C4—531no—
214C4×C4⋊C4—531no—
215#340—1061no—
216#339—1062no—
217#328—1062no—
218#327—1062no—
219#322—21no—
220C82—21no—
221C53#971711yescharacteristic, complemented, Sylow
222C2×#41—1061no—
223C2×SD16—1061no—
224C2×SD16—1061no—
225C2×M16D21212yes—
226C22×C8D21211yescharacteristic
227C22×C8D21211yescharacteristic
228C4⋊Q8—531no—
229C4⋊Q8—531no—
230C4⋊Q8—531no—
231#127—531no—
232#125—1061no—
233#124—1061no—
234C22⋊Q8—1061no—
235C22⋊Q8—1061no—
236#121—1061no—
237#121—1061no—
238C4×Q8—1061no—
239C4×Q8—1061no—
240C4×D4—1061no—
241C4×D4—1061no—
242#117—1061no—
243C2×C4⋊C4—1061no—
244C2×C4⋊C4—1061no—
245C2×C42—1061no—
246C2×C42—1061no—
247C2×C42C22×D5311yescharacteristic
248#106—1064no—
249#106—1061no—
250#106—1061no—
251C4⋊C8—21no—
252#104—2124no—
253#103—1061no—
254#103—1061no—
255#102—1061no—
256#102—1061no—
257C22⋊C8D21212yes—
258#97—21no—
259C4×C8—21no—
260C4×C8—21no—
261C4×C8C4×D5311yescharacteristic
262C4×C8C4×D5311yescharacteristic
263C4×C8C22⋊D5311yescharacteristic
264C4×C8C22⋊D5311yescharacteristic
265#95—1061no—
266#95—1061no—
267#41—2122no—
268#41—2122no—
269C2×Q8—1061no—
270C2×Q8—1061no—
271C2×Q8—1061no—
272C2×D4—1061no—
273C2×D4—1061no—
274C2×D4—1061no—
275C2×D4—2121no—
276C22×C4—2121no—
277C22×C4—2121no—
278C22×C4—2121no—
279C22×C4C2×D21211yescharacteristic
280C22×C4C2×D21211yescharacteristic
281C22×C4C2×D21211yescharacteristic
282SD16—2122no—
283SD16—2122no—
284M16—22no—
285M16#1171991314yescomplemented
286C2×C8—22no—
287C2×C8—22no—
288C2×C8—22no—
289C2×C8—41no—
290C2×C8#1171992111yescharacteristic
291C2×C8#1171992111yescharacteristic
292C2×C8#1171992111yescharacteristic
293C2×C8#1171992111yescharacteristic
294C4⋊C4—1062no—
295C4⋊C4—1062no—
296C4⋊C4—1061no—
297C4⋊C4—1061no—
298C4⋊C4—1061no—
299C4⋊C4—1061no—
300C4⋊C4—1061no—
301C4⋊C4—1061no—
302C4⋊C4—2121no—
303C4⋊C4—2121no—
304C4⋊C4—2121no—
305C4⋊C4—2121no—
306C22⋊C4—2121no—
307C22⋊C4—2121no—
308C22⋊C4—2121no—
309C42—2122no—
310C42—2122no—
311C42—2121no—
312C42C2×C4×D5311yescharacteristic
313C42#1171994511yescharacteristic
314C42#1171994511yescharacteristic
315C42C2×C22⋊D5311yescharacteristic
316C23—2121no—
317C23#1313321411yescharacteristic
318Q8—4241no—
319Q8—2121no—
320Q8—2121no—
321Q8—2122no—
322D4—4241no—
323D4—4241no—
324D4—4241no—
325D4—2121no—
326D4—2121no—
327D4—2122no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—41no—
331C2×C4—4241no—
332C2×C4—4241no—
333C2×C4—4241no—
334C2×C4—4241no—
335C2×C4—4242no—
336C2×C4—2122no—
337C2×C4—2122no—
338C2×C4—2121no—
339C2×C4—2121no—
340C2×C4—2121no—
341C2×C4C2×#1171992111yescharacteristic
342C2×C4C4×D21211yescharacteristic
343C2×C4C4×D21211yescharacteristic
344C2×C4#1313319111yescharacteristic
345C2×C4#1313319111yescharacteristic
346C2×C4#1313320511yescharacteristic
347C2×C4C2×#1171991312yes—
348C2×C4#1313323912yes—
349C8—24no—
350C8—21no—
351C8—21no—
352C8—21no—
353C8—42no—
354C8#1313314312yescomplemented
355C22—21no—
356C22—21no—
357C22—4241no—
358C22—4241no—
359C22—4241no—
360C22#1681052511yescharacteristic
361C22#1681099611yescharacteristic
362C22#1681100711yescharacteristic
363C4—21no—
364C4—21no—
365C4—21no—
366C4—4242no—
367C4—4242no—
368C4—4241no—
369C4—4241no—
370C4—4241no—
371C4C2×#1313314311yescharacteristic
372C4#1681063911yescharacteristic
373C4C4×#1171991312yes—
374C4#1681085212yes—
375C2—21no—
376C2—21no—
377C2—8481no—
378C2#2027556911yescharacteristic
379C2#2027561311yescharacteristic
380C2#2027563411yescharacteristic
subgroupG/H
centerC2×C4#13133191
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC8×M16×C53C2
socleC22×C53#390
prime radical#16811173C22
cosocleC2×C42×C53C23
hypercenterC8×M16D53
hypocenterC53#9717
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C8×M16

Complements

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

NH ≅ G/Ntype
2. #20275659C2semidirect
4. C4×#13133149C2semidirect
5. #20273215C2semidirect
7. C8×M16×C53C2semidirect
10. #16811021C4semidirect
11. #16810996C4semidirect
12. #16810974C4semidirect
13. #16810970C4semidirect
92. M16×C53SD16semidirect
163. C8×C53#104semidirect
191. C8×M16D53semidirect
221. C53#9717semidirect
285. M16#11719913semidirect
354. C8#13133143semidirect

Maximal subgroups

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

subgroupindex
#202852742
#202756592
#202756342
C4×#131331492
#202732152
#202732112
C8×M16×C532
#971753