Finite Group Databasebeta

← #26192360

Subgroups — #26192360

Order 12032.

1068 conjugacy classes of subgroups; only the 179 normal subgroups are shown.

normal subgroups only

This group has 1068 conjugacy classes of subgroups — too many to draw. Only its 179 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#20253838C211yesmaximal, characteristic
2#20253510C212yesmaximal, complemented
3#20242509C214yesmaximal, complemented
4#20242447C214yesmaximal, complemented
5#20242102C211yesmaximal, characteristic, complemented
6#20241835C212yesmaximal, complemented
7C2×C47×#559C211yesmaximal, characteristic, complemented
8C2×#13094050—22no—
9#15689673C2211yescharacteristic
10#15689510—22no—
11#15689499C2212yes—
12#15689495C2212yes—
13#15689436—22no—
14#15689418C2214yescomplemented
15#15689323C2214yescomplemented
16#15689262—22no—
17#15689221—24no—
18#15689180C2212yes—
19C4×#10625329C2211yescharacteristic
20#15688867C2214yes—
21Q8⋊(C47⋊C8)C2218yescomplemented
22C47×#559—22no—
23C47×#559—24no—
24C47×#559C2214yes—
25C2×#127×C47C2211yescharacteristic
26C2×#124×C47—22no—
27C2×#123×C47—22no—
28C2×C4×Q8×C47C2212yes—
29C2×C4×D4×C47—22no—
30C2×#10625350—21no—
31#13094050—48no—
32C2×#10625335—24no—
33C2×#10625336—24no—
34C2×#10625334—24no—
35C2×#10625329—21no—
36#13094024C2311yescharacteristic
37#13094021C2311yescharacteristic
38#13094012—21no—
39#13093998—22no—
40C4×D188—22no—
41#13093957—24no—
42C4⋊(C47⋊C8)C2314yes—
43#13093948—24no—
44#13093944—24no—
45C22⋊(C47⋊C8)—22no—
46#13093935—22no—
47#13093935—21no—
48C4×C47⋊C8C2312yes—
49C22×Q8×C47D412yes—
50C22×D4×C47—22no—
51C22×D4×C47D412yes—
52C23×C4×C47—22no—
53#127×C47—22no—
54#127×C47D412yes—
55#127×C47C2312yes—
56#124×C47—28no—
57#124×C47—28no—
58#123×C47—28no—
59#123×C47—28no—
60C4×Q8×C47—22no—
61C4×Q8×C47—22no—
62C4×Q8×C47D414yes—
63C4×Q8×C47C2314yes—
64C4×D4×C47—28no—
65C4×D4×C47—28no—
66C2×C4⋊C4×C47—22no—
67C2×C4⋊C4×C47D412yes—
68C2×C22⋊C4×C47—24no—
69C2×C22⋊C4×C47—22no—
70C2×C42×C47—22no—
71C2×C42×C47C2311yescharacteristic
72C24×C47—22no—
73C2×Q8×C47—22no—
74C2×Q8×C47—44no—
75C2×Q8×C47C2×D414yes—
76C2×D4×C47—24no—
77C2×D4×C47—22no—
78C2×D4×C47—24no—
79C2×D4×C47—48no—
80C2×D4×C47—44no—
81C2×D4×C47C2×D414yes—
82C22×C4×C47—22no—
83C22×C4×C47—22no—
84C22×C4×C47—28no—
85C22×C4×C47—84no—
86C22×C4×C47C2×D412yes—
87C22×C4×C47C2×D411yescharacteristic
88C2×C22⋊D47—41no—
89#10625350—44no—
90C2×D188—22no—
91C2×D188—21no—
92C2×C4×D47—41no—
93C2×Dic94—21no—
94C4⋊C4×C47—24no—
95C4⋊C4×C47—22no—
96C4⋊C4×C47—24no—
97C4⋊C4×C47C2×D414yes—
98C22⋊C4×C47—216no—
99C22⋊C4×C47—28no—
100C42×C47—24no—
101C42×C47—24no—
102C42×C47C2×D411yescharacteristic
103C42×C47C2×D412yes—
104C42×C47C2411yescharacteristic
105#10625336—48no—
106#10625335—48no—
107#10625334—48no—
108(C2×D47)⋊C4—42no—
109C4⋊Dic47—22no—
110#10625329—24no—
111C2×C47⋊C8—22no—
112C2×C47⋊C8—24no—
113C23×C47—24no—
114C23×C47—48no—
115C23×C47—44no—
116C23×C47#12011yescharacteristic
117C22×D47—41no—
118Q8×C47—24no—
119Q8×C47—24no—
120Q8×C47—44no—
121D4×C47—44no—
122D4×C47—48no—
123D4×C47—48no—
124D4×C47—82no—
125C2×C4×C47—22no—
126C2×C4×C47—24no—
127C2×C4×C47—24no—
128C2×C4×C47—24no—
129C2×C4×C47—44no—
130C2×C4×C47—44no—
131C2×C4×C47—88no—
132C2×C4×C47#12012yes—
133C2×C4×C47#12011yescharacteristic
134C2×C4×C47C22×D411yescharacteristic
135C2×C4×C47C22×D412yes—
136C22⋊D47—82no—
137C2×Dic47—41no—
138D188—42no—
139D188—44no—
140C4×D47—82no—
141Dic94—42no—
142C47⋊C8—24no—
143C47⋊C8—44no—
144#18526—471nomaximal, Sylow
145C22×C47—22no—
146C22×C47—48no—
147C22×C47—44no—
148C22×C47—88no—
149C22×C47#45312yes—
150C22×C47C2×#12011yescharacteristic
151C2×D47—82no—
152C2×D47—81no—
153C4×C47—22no—
154C4×C47—21no—
155C4×C47—21no—
156C4×C47—44no—
157C4×C47—44no—
158C4×C47#57514yes—
159Dic47—81no—
160C2×#559D4711yescharacteristic, complemented
161#3151—471no—
162#3112—472no—
163#1630—474no—
164#1620—474no—
165#1555—471no—
166#1491—472no—
167C2×C47—22no—
168C2×C47—84no—
169C2×C47#300512yes—
170C2×C47C2×#45311yescharacteristic
171D47—161no—
172C2×AGL(1,ℤ/8)—942no—
173#559—22no—
174#559—24no—
175#559C2×D4714yes—
176#532—471no—
177C2×#127C2×D4711yescharacteristic
178C2×#124—22no—
179C2×#123—22no—
180C2×C4×Q8C2×D4712yes—
181C2×C4×D4—22no—
182#496—942no—
183#493—472no—
184#492—472no—
185#480—942no—
186#477—474no—
187#460—474no—
188#448—942no—
189#440—944no—
190#432—472no—
191C4×M16—471no—
192#331—474no—
193Q8⋊C8—478no—
194C47#1852611yescharacteristic, complemented, Sylow
195C2×#41—941no—
196C22×Q8C22⋊D4712yes—
197C22×D4—22no—
198C22×D4C22⋊D4712yes—
199C23×C4—22no—
200AGL(1,ℤ/8)—1888no—
201C2×SD16—944no—
202C2×SD16—944no—
203C2×D8—944no—
204C2×M16—941no—
205C4⋊Q8—471no—
206#127—22no—
207#127—471no—
208#127C22⋊D4712yes—
209#127C22×D4712yes—
210#124—28no—
211#124—28no—
212#124—941no—
213#123—28no—
214#123—28no—
215#121—942no—
216C4×Q8—22no—
217C4×Q8—22no—
218C4×Q8C22⋊D4714yes—
219C4×Q8C22×D4714yes—
220C4×D4—28no—
221C4×D4—28no—
222C4×D4—942no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4C22⋊D4712yes—
225C2×C22⋊C4—24no—
226C2×C22⋊C4—22no—
227C2×C42—22no—
228C2×C42C22×D4711yescharacteristic
229#107—944no—
230C4⋊C8—474no—
231#103—944no—
232#102—944no—
233C22⋊C8—942no—
234#97—941no—
235#97—942no—
236C4×C8—472no—
237C24—22no—
238#41—1884no—
239C2×Q8—22no—
240C2×Q8—44no—
241C2×Q8—941no—
242C2×Q8C2×C22⋊D4714yes—
243C2×D4—24no—
244C2×D4—22no—
245C2×D4—24no—
246C2×D4—48no—
247C2×D4—44no—
248C2×D4—942no—
249C2×D4—941no—
250C2×D4—1881no—
251C2×D4C2×C22⋊D4714yes—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—28no—
255C22×C4—84no—
256C22×C4—1881no—
257C22×C4C2×C22⋊D4712yes—
258C22×C4C2×C22⋊D4711yescharacteristic
259SD16—1888no—
260SD16—1888no—
261D8—1888no—
262M16—944no—
263C2×C8—942no—
264C2×C8—944no—
265C4⋊C4—24no—
266C4⋊C4—22no—
267C4⋊C4—24no—
268C4⋊C4—942no—
269C4⋊C4C2×C22⋊D4714yes—
270C22⋊C4—216no—
271C22⋊C4—28no—
272C22⋊C4—1882no—
273C42—24no—
274C42—24no—
275C42C2×C22⋊D4712yes—
276C42C2×C22⋊D4711yescharacteristic
277C42C23×D4711yescharacteristic
278C23—24no—
279C23—48no—
280C23—44no—
281C23—1881no—
282C23#1309399311yescharacteristic
283Q8—24no—
284Q8—24no—
285Q8—44no—
286Q8—1882no—
287D4—44no—
288D4—48no—
289D4—48no—
290D4—82no—
291D4—3762no—
292D4—1882no—
293D4—1884no—
294C2×C4—22no—
295C2×C4—24no—
296C2×C4—24no—
297C2×C4—24no—
298C2×C4—44no—
299C2×C4—44no—
300C2×C4—88no—
301C2×C4—3762no—
302C2×C4—1881no—
303C2×C4#1309399312yes—
304C2×C4#1309399311yescharacteristic
305C2×C4C22×C22⋊D4711yescharacteristic
306C2×C4C22×C22⋊D4712yes—
307C8—944no—
308C8—1884no—
309C22—22no—
310C22—44no—
311C22—48no—
312C22—88no—
313C22—3761no—
314C22—3762no—
315C22#1568929112yes—
316C22C2×#1309399311yescharacteristic
317C4—21no—
318C4—21no—
319C4—22no—
320C4—44no—
321C4—44no—
322C4—3761no—
323C4#1568992114yes—
324C2—22no—
325C2—84no—
326C2—7521no—
327C2#2025263012yes—
328C2C2×#1568929111yescharacteristic
subgroupG/H
centerC22C2×#13093993
commutator, cosocleC42×C47C24
FrattiniC42C23×D47
FittingC2×C47×#559C2
socleC22×C47C2×#120
prime radical#20253838C2
hypercenterC2×#559D47
hypocenterC47#18526
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#559

Complements

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

NH ≅ G/Ntype
2. #20253510C2semidirect
3. #20242509C2semidirect
4. #20242447C2semidirect
5. #20242102C2semidirect
6. #20241835C2semidirect
7. C2×C47×#559C2semidirect
14. #15689418C22semidirect
15. #15689323C22semidirect
21. Q8⋊(C47⋊C8)C22semidirect
160. C2×#559D47semidirect
194. C47#18526semidirect

Maximal subgroups

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

subgroupindex
#202538382
#202535102
#202425092
#202424472
#202421022
#202418352
C2×C47×#5592
#1852647