Finite Group Databasebeta

← #26129271

Subgroups — #26129271

Order 12032.

558 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 558 conjugacy classes of subgroups — too many to draw. Only its 63 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#20244733C211yesmaximal, characteristic, complemented
2#20242327C212yesmaximal, complemented
3#20241391C212yesmaximal, complemented
4#20241334C211yesmaximal, characteristic, complemented
5C47×#2537C211yesmaximal, characteristic
6#15689754C2211yescharacteristic
7#15689274—22no—
8#15689163—21no—
9#15688892C2211yescharacteristic
10#15688883—21no—
11#15688859—22no—
12#15688849C2212yes—
13C2×#121×C47—22no—
14C2×C4×D4×C47—21no—
15C47×#394—21no—
16C47×#394C2212yes—
17C47×#384C2211yescharacteristic
18C47×#380—22no—
19C2×#10625335—42no—
20C2×#10625329—21no—
21#13094023—41no—
22#13094015—41no—
23#13094003—22no—
24#13093983—21no—
25#13093957—41no—
26#13093955—41no—
27#13093945—42no—
28C22⋊(C47⋊C8)—22no—
29C22×D4×C47—21no—
30C22×D4×C47—21no—
31C22×D4×C47D412yes—
32C23×C4×C47—22no—
33#121×C47—48no—
34C4×D4×C47—42no—
35C4×D4×C47—42no—
36C2×C4⋊C4×C47—21no—
37C2×C4⋊C4×C47—22no—
38C2×C4⋊C4×C47D411yescharacteristic
39C2×C4⋊C4×C47C2311yescharacteristic
40C2×C22⋊C4×C47—22no—
41C2×C22⋊C4×C47—22no—
42C2×C22⋊C4×C47—22no—
43C2×C42×C47D411yescharacteristic
44#95×C47—21no—
45#95×C47D412yes—
46C24×C47—22no—
47C2×D4×C47—24no—
48C2×D4×C47—42no—
49C2×D4×C47—42no—
50C2×D4×C47—42no—
51C2×D4×C47—44no—
52C2×D4×C47—44no—
53C2×D4×C47—81no—
54C22×C4×C47—21no—
55C22×C4×C47—22no—
56C22×C4×C47—21no—
57C22×C4×C47—44no—
58C22×C4×C47—44no—
59C22×C4×C47—44no—
60C22×C4×C47C2×D411yescharacteristic
61C22×C4×C47C2×D412yes—
62C2×Dic94—22no—
63C4⋊C4×C47—41no—
64C4⋊C4×C47—42no—
65C4⋊C4×C47—44no—
66C4⋊C4×C47—41no—
67C22⋊C4×C47—44no—
68C22⋊C4×C47—44no—
69C22⋊C4×C47—42no—
70C22⋊C4×C47—42no—
71C42×C47—21no—
72C42×C47SD1612yescomplemented
73C22⋊Dic47—41no—
74#10625335—84no—
75C4⋊Dic47—22no—
76C4⋊Dic47—21no—
77C4⋊Dic47—21no—
78Dic47⋊C4—42no—
79C4×Dic47—41no—
80#10625329—22no—
81#10625329—41no—
82C2×C47⋊C8—41no—
83C2×C47⋊C8—42no—
84C23×C47—24no—
85C23×C47—42no—
86C23×C47—42no—
87C23×C47—42no—
88C23×C47—42no—
89C23×C47—42no—
90C23×C47—42no—
91C23×C47#12011yescharacteristic
92D4×C47—48no—
93D4×C47—44no—
94D4×C47—82no—
95D4×C47—84no—
96C2×C4×C47—22no—
97C2×C4×C47—22no—
98C2×C4×C47—22no—
99C2×C4×C47—21no—
100C2×C4×C47—21no—
101C2×C4×C47—22no—
102C2×C4×C47—42no—
103C2×C4×C47—42no—
104C2×C4×C47—42no—
105C2×C4×C47—42no—
106C2×C4×C47—42no—
107C2×C4×C47—41no—
108C2×C4×C47—88no—
109C2×C4×C47C2×SD1611yescharacteristic
110C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
111C2×Dic47—41no—
112C2×Dic47—41no—
113Dic94—82no—
114C47⋊C8—42no—
115C47⋊C8—82no—
116#12246—471nomaximal, Sylow
117C22×C47—21no—
118C22×C47—21no—
119C22×C47—44no—
120C22×C47—44no—
121C22×C47—44no—
122C22×C47—82no—
123C22×C47—82no—
124C22×C47—82no—
125C22×C47—82no—
126C22×C47#45011yescharacteristic
127C22×C47#45611yescharacteristic
128C22×C47B(4,𝔽2)11yescharacteristic
129C4×C47—24no—
130C4×C47—21no—
131C4×C47—21no—
132C4×C47—42no—
133C4×C47—82no—
134C4×C47—81no—
135Dic47—81no—
136Dic47—81no—
137#2537D4711yescharacteristic
138#2060—471no—
139#1599—472no—
140#1345—472no—
141#1325—471no—
142C2×C47—21no—
143C2×C47—21no—
144C2×C47—84no—
145C2×C47#160411yescharacteristic
146C2×C47#219611yescharacteristic
147C2×C47#220411yescharacteristic
148#544—471no—
149C2×#121—22no—
150C2×C4×D4—21no—
151#450—942no—
152#428—941no—
153#394—21no—
154#394C2×D4712yes—
155#384C2×D4711yescharacteristic
156#380—22no—
157#343—471no—
158#339—941no—
159#328—942no—
160#324—472no—
161C47#1224611yescharacteristic, complemented, Sylow
162C22×D4—21no—
163C22×D4—21no—
164C22×D4C22⋊D4712yes—
165C23×C4—22no—
166C2×SD16—1882no—
167C2×M16—941no—
168C4⋊Q8—1881no—
169#125—1881no—
170C22⋊Q8—942no—
171#121—48no—
172C4×D4—42no—
173C4×D4—42no—
174#117—941no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4C22⋊D4711yescharacteristic
178C2×C4⋊C4C22×D4711yescharacteristic
179C2×C22⋊C4—22no—
180C2×C22⋊C4—22no—
181C2×C22⋊C4—22no—
182C2×C42C22⋊D4711yescharacteristic
183#107—1881no—
184#106—1881no—
185#102—1882no—
186C22⋊C8—942no—
187#95—21no—
188#95C22⋊D4712yes—
189C24—22no—
190C2×Q8—942no—
191C2×D4—24no—
192C2×D4—42no—
193C2×D4—42no—
194C2×D4—42no—
195C2×D4—44no—
196C2×D4—44no—
197C2×D4—81no—
198C22×C4—21no—
199C22×C4—22no—
200C22×C4—21no—
201C22×C4—44no—
202C22×C4—44no—
203C22×C4—44no—
204C22×C4C2×C22⋊D4711yescharacteristic
205C22×C4C2×C22⋊D4712yes—
206SD16—3764no—
207M16—942no—
208M16—1881no—
209C2×C8—1882no—
210C2×C8—1881no—
211C4⋊C4—41no—
212C4⋊C4—41no—
213C4⋊C4—42no—
214C4⋊C4—44no—
215C4⋊C4—942no—
216C4⋊C4—941no—
217C4⋊C4—941no—
218C4⋊C4—1882no—
219C22⋊C4—44no—
220C22⋊C4—44no—
221C22⋊C4—42no—
222C22⋊C4—42no—
223C22⋊C4—1881no—
224C42—21no—
225C42—1881no—
226C42#1062533512yescomplemented
227C23—24no—
228C23—42no—
229C23—42no—
230C23—42no—
231C23—42no—
232C23—42no—
233C23—42no—
234C23#1309399311yescharacteristic
235Q8—3762no—
236D4—48no—
237D4—44no—
238D4—82no—
239D4—84no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—22no—
246C2×C4—42no—
247C2×C4—42no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—41no—
252C2×C4—88no—
253C2×C4—1881no—
254C2×C4—1881no—
255C2×C4C2×#1062533511yescharacteristic
256C2×C4#1309405011yescharacteristic
257C8—3762no—
258C8—1882no—
259C22—21no—
260C22—21no—
261C22—44no—
262C22—44no—
263C22—44no—
264C22—82no—
265C22—82no—
266C22—82no—
267C22—82no—
268C22#1568927411yescharacteristic
269C22#1568930611yescharacteristic
270C22#1568931011yescharacteristic
271C4—24no—
272C4—21no—
273C4—21no—
274C4—42no—
275C4—82no—
276C4—81no—
277C4—3761no—
278C4—3761no—
279C2—21no—
280C2—21no—
281C2—84no—
282C2#2024235711yescharacteristic
283C2#2024537211yescharacteristic
284C2#2024541411yescharacteristic
subgroupG/H
centerC22#15689310
commutator, cosocleC2×C4⋊C4×C47C23
FrattiniC2×C4⋊C4C22×D47
Fitting, prime radicalC47×#2537C2
socleC22×C47B(4,𝔽2)
hypercenter#2537D47
hypocenterC47#12246
LCSG ▷ C2×C4⋊C4×C47 ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2537

Complements

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

NH ≅ G/Ntype
1. #20244733C2semidirect
2. #20242327C2semidirect
3. #20241391C2semidirect
4. #20241334C2semidirect
72. C42×C47SD16semidirect
161. C47#12246semidirect
226. C42#10625335semidirect

Maximal subgroups

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

subgroupindex
#202447332
#202423272
#202413912
#202413342
C47×#25372
#1224647