Finite Group Databasebeta

← #27360197

Subgroups — #27360197

Order 13568.

1630 conjugacy classes of subgroups; only the 195 normal subgroups are shown.

normal subgroups only

This group has 1630 conjugacy classes of subgroups — too many to draw. Only its 195 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285089C211yesmaximal, characteristic, complemented
2#20284814C212yesmaximal, complemented
3#20273894C218yesmaximal, complemented
4#20273429C211yesmaximal, characteristic, complemented
5#20273158C212yesmaximal, complemented
6C2×D42×C53C211yesmaximal, characteristic
7#16811180C2211yescharacteristic, complemented
8#16810987—22no—
9#16810969C2214yescomplemented
10#16810958—22no—
11#16810903C2214yescomplemented
12#16810828C2214yescomplemented
13#16810774—22no—
14#16810723—24no—
15#16810680C2212yescomplemented
16#16810628—22no—
17C4×#11719917C2211yescharacteristic, complemented
18#16810367C2214yescomplemented
19D4⋊#88322C2218yescomplemented
20C23×D4×C53—22no—
21D42×C53—22no—
22D42×C53—24no—
23D42×C53C2214yes—
24C2×#127×C53C2211yescharacteristic
25C2×#121×C53—22no—
26C2×#120×C53—22no—
27C2×C4×D4×C53C2212yes—
28C2×#11719923—24no—
29C2×#11719917—21no—
30#13133217C2312yescomplemented
31#13133209—21no—
32#13133196—22no—
33C4×Dic106—22no—
34#13133176—21no—
35#13133150—24no—
36C4⋊#88322C2314yescomplemented
37#13133140—24no—
38#13133138—24no—
39#13133138—28no—
40C22⋊#88322—22no—
41#13133128—21no—
42#13133128—22no—
43C4×#88322C2312yescomplemented
44C25×C53—22no—
45C22×D4×C53—24no—
46C22×D4×C53—28no—
47C22×D4×C53—28no—
48C22×D4×C53—22no—
49C22×D4×C53—416no—
50C22×D4×C53D412yes—
51C22×D4×C53D412yes—
52C23×C4×C53—22no—
53#127×C53—22no—
54#127×C53D412yes—
55#127×C53C2312yes—
56#121×C53—28no—
57#121×C53—28no—
58#120×C53—28no—
59#120×C53—28no—
60C4×D4×C53—22no—
61C4×D4×C53—22no—
62C4×D4×C53D414yes—
63C4×D4×C53C2314yes—
64C2×C4⋊C4×C53D412yes—
65C2×C22⋊C4×C53—22no—
66C2×C22⋊C4×C53—22no—
67C2×C42×C53C2311yescharacteristic
68C24×C53—22no—
69C24×C53—28no—
70C24×C53—22no—
71C24×C53—416no—
72C24×C53—84no—
73C2×D4×C53—28no—
74C2×D4×C53—216no—
75C2×D4×C53—22no—
76C2×D4×C53—24no—
77C2×D4×C53—22no—
78C2×D4×C53—24no—
79C2×D4×C53—48no—
80C2×D4×C53—44no—
81C2×D4×C53—432no—
82C2×D4×C53—416no—
83C2×D4×C53—416no—
84C2×D4×C53—416no—
85C2×D4×C53C2×D414yes—
86C2×D4×C53C2×D414yes—
87C22×C4×C53—22no—
88C22×C4×C53—24no—
89C22×C4×C53—24no—
90C22×C4×C53—48no—
91C22×C4×C53C2×D411yescharacteristic
92C22×C4×C53C2×D412yes—
93C2×Dic106—22no—
94C4⋊C4×C53—22no—
95C4⋊C4×C53C2×D414yes—
96C22⋊C4×C53—28no—
97C22⋊C4×C53—24no—
98C22⋊C4×C53—24no—
99C42×C53C2×D411yescharacteristic
100C42×C53C2×D412yes—
101C42×C53C2411yescharacteristic
102C22⋊Dic53—41no—
103#11719923—48no—
104C4⋊Dic53—22no—
105C4⋊Dic53—22no—
106#11719919—42no—
107C4×Dic53—41no—
108#11719917—42no—
109C2×#88322—22no—
110C2×#88322—24no—
111C23×C53—28no—
112C23×C53—24no—
113C23×C53—24no—
114C23×C53—44no—
115C23×C53—48no—
116C23×C53—432no—
117C23×C53—44no—
118C23×C53—44no—
119C23×C53—88no—
120C23×C53—832no—
121C23×C53#12011yescharacteristic
122D4×C53—28no—
123D4×C53—24no—
124D4×C53—44no—
125D4×C53—48no—
126D4×C53—432no—
127D4×C53—48no—
128D4×C53—82no—
129D4×C53AGL(1,ℤ/8)18yes—
130C2×C4×C53—22no—
131C2×C4×C53—24no—
132C2×C4×C53—24no—
133C2×C4×C53—48no—
134C2×C4×C53—44no—
135C2×C4×C53—48no—
136C2×C4×C53#12012yes—
137C2×C4×C53#12011yescharacteristic
138C2×C4×C53C22×D411yescharacteristic
139C2×C4×C53C22×D412yes—
140C2×Dic53—42no—
141Dic106—44no—
142#88322—42no—
143#88322—44no—
144#18529—531nomaximal, Sylow
145C22×C53—22no—
146C22×C53—28no—
147C22×C53—416no—
148C22×C53—44no—
149C22×C53—44no—
150C22×C53—48no—
151C22×C53—44no—
152C22×C53—88no—
153C22×C53—832no—
154C22×C53#45512yes—
155C22×C53C2×#12011yescharacteristic
156C4×C53—22no—
157C4×C53—21no—
158C4×C53—21no—
159C4×C53—44no—
160C4×C53C2×AGL(1,ℤ/8)14yes—
161Dic53—82no—
162C2×D42D5311yescharacteristic
163#3138—531no—
164#3110—532no—
165#1639—538no—
166#1555—531no—
167#1490—532no—
168C2×C53—22no—
169C2×C53—48no—
170C2×C53—84no—
171C2×C53#300312yes—
172C2×C53C2×#45511yescharacteristic
173C23×D4—22no—
174D42—22no—
175D42—24no—
176D42C2×D5314yes—
177#533—531no—
178C2×#127C2×D5311yescharacteristic
179C2×#121—22no—
180C2×#120—22no—
181C2×C4×D4C2×D5312yes—
182#490—1062no—
183#486—534no—
184#484—1062no—
185#474—534no—
186#461—534no—
187#450—1062no—
188#440—1064no—
189#432—532no—
190#418—1062no—
191C4×M16—531no—
192#331—534no—
193D4⋊C8—538no—
194C53#1852911yescharacteristic, complemented, Sylow
195C25—22no—
196C22×D4—24no—
197C22×D4—28no—
198C22×D4—28no—
199C22×D4—22no—
200C22×D4—416no—
201C22×D4C22⋊D5312yes—
202C22×D4C22⋊D5312yes—
203C23×C4—22no—
204C2×SD16—1064no—
205C2×M16—1061no—
206C4⋊Q8—532no—
207#127—22no—
208#127C22⋊D5312yes—
209#127C22×D5312yes—
210#125—1061no—
211C22⋊Q8—1062no—
212#121—28no—
213#121—28no—
214#120—28no—
215#120—28no—
216C4×Q8—1062no—
217C4×D4—22no—
218C4×D4—22no—
219C4×D4C22⋊D5314yes—
220C4×D4C22×D5314yes—
221#117—1061no—
222C2×C4⋊C4C22⋊D5312yes—
223C2×C22⋊C4—22no—
224C2×C22⋊C4—22no—
225C2×C42C22×D5311yescharacteristic
226#107—1064no—
227C4⋊C8—534no—
228#103—1064no—
229#102—1068no—
230#102—1064no—
231C22⋊C8—1062no—
232#97—1061no—
233#97—1062no—
234C4×C8—532no—
235C24—22no—
236C24—28no—
237C24—22no—
238C24—416no—
239C24—84no—
240C2×Q8—1062no—
241C2×D4—28no—
242C2×D4—216no—
243C2×D4—22no—
244C2×D4—24no—
245C2×D4—22no—
246C2×D4—24no—
247C2×D4—48no—
248C2×D4—44no—
249C2×D4—432no—
250C2×D4—416no—
251C2×D4—416no—
252C2×D4—416no—
253C2×D4C2×C22⋊D5314yes—
254C2×D4C2×C22⋊D5314yes—
255C22×C4—22no—
256C22×C4—24no—
257C22×C4—24no—
258C22×C4—48no—
259C22×C4C2×C22⋊D5312yes—
260C22×C4C2×C22⋊D5311yescharacteristic
261SD16—2128no—
262M16—2122no—
263C2×C8—1062no—
264C2×C8—1064no—
265C4⋊C4—22no—
266C4⋊C4—1062no—
267C4⋊C4—1062no—
268C4⋊C4—2122no—
269C4⋊C4C2×C22⋊D5314yes—
270C22⋊C4—28no—
271C22⋊C4—24no—
272C22⋊C4—24no—
273C22⋊C4—2121no—
274C42—2121no—
275C42C2×C22⋊D5311yescharacteristic
276C42C2×C22⋊D5312yes—
277C42C23×D5311yescharacteristic
278C23—28no—
279C23—24no—
280C23—24no—
281C23—44no—
282C23—48no—
283C23—432no—
284C23—44no—
285C23—44no—
286C23—88no—
287C23—832no—
288C23#1313318611yescharacteristic
289Q8—2124no—
290D4—28no—
291D4—24no—
292D4—44no—
293D4—48no—
294D4—432no—
295D4—48no—
296D4—82no—
297D4#1313324318yes—
298C2×C4—22no—
299C2×C4—24no—
300C2×C4—24no—
301C2×C4—48no—
302C2×C4—44no—
303C2×C4—48no—
304C2×C4—2122no—
305C2×C4#1313318611yescharacteristic
306C2×C4#1313318612yes—
307C2×C4C22×C22⋊D5311yescharacteristic
308C2×C4C22×C22⋊D5312yes—
309C8—2122no—
310C8—2124no—
311C22—22no—
312C22—28no—
313C22—416no—
314C22—44no—
315C22—44no—
316C22—48no—
317C22—44no—
318C22—88no—
319C22—832no—
320C22#1681080112yes—
321C22C2×#1313318611yescharacteristic
322C4—22no—
323C4—21no—
324C4—21no—
325C4—44no—
326C4—4242no—
327C4C2×#1313324314yes—
328C2—22no—
329C2—48no—
330C2—84no—
331C2#2028394212yes—
332C2C2×#1681080111yescharacteristic
subgroupG/H
centerC22C2×#13133186
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
Fitting, prime radicalC2×D42×C53C2
socleC22×C53C2×#120
hypercenterC2×D42D53
hypocenterC53#18529
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×D42

Complements

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

NH ≅ G/Ntype
1. #20285089C2semidirect
2. #20284814C2semidirect
3. #20273894C2semidirect
4. #20273429C2semidirect
5. #20273158C2semidirect
7. #16811180C22semidirect
9. #16810969C22semidirect
11. #16810903C22semidirect
12. #16810828C22semidirect
15. #16810680C22semidirect
17. C4×#11719917C22semidirect
18. #16810367C22semidirect
19. D4⋊#88322C22semidirect
30. #13133217C23semidirect
36. C4⋊#88322C23semidirect
43. C4×#88322C23semidirect
194. C53#18529semidirect

Maximal subgroups

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

subgroupindex
#202850892
#202848142
#202738942
#202734292
#202731582
C2×D42×C532
#1852953