Finite Group Databasebeta

← #26125577

Subgroups — #26125577

Order 12032.

382 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 382 conjugacy classes of subgroups — too many to draw. Only its 83 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#20252078C211yesmaximal, characteristic
2#20245007C212yesmaximal, complemented
3#20241340C212yesmaximal, complemented
4#20241324C211yesmaximal, characteristic, complemented
5C47×#1320C211yesmaximal, characteristic, complemented
6C2×#13094032C2211yescharacteristic
7C23×C47⋊C8C411yescharacteristic
8#15689634C411yescharacteristic
9M32×D47—22no—
10#15689191—22no—
11(C22⋊D47)⋊C8—22no—
12#15689151C2211yescharacteristic
13#15689091C411yescharacteristic
14#15689081C411yescharacteristic
15#15688905C2212yes—
16#15688904—22no—
17C47×#407C2211yescharacteristic
18C47×#349C2212yes—
19C2×#10625350C2×C411yescharacteristic
20#13094032—41no—
21#13094032—41no—
22C2×#10625324D412yes—
23C22×C47⋊C8—21no—
24C22×C47⋊C8—21no—
25C22×C47⋊C8—41no—
26C22×C47⋊C8—41no—
27C22×C47⋊C8C2×C411yescharacteristic
28C22×C47⋊C8C2×C411yescharacteristic
29C2×C8×D47D412yes—
30#13094007C812yes—
31#13094003—21no—
32#13093998—21no—
33#13093993C812yes—
34C4×C22⋊D47—22no—
35#13093983C2×C411yescharacteristic
36#13093964—22no—
37#13093963—42no—
38C16×D47—42no—
39Dic47⋊C8—22no—
40(C2×D47)⋊C8—22no—
41#13093934—21no—
42C8×Dic47—21no—
43C23×C4×C47C2×C411yescharacteristic
44C2×M16×C47C2×C411yescharacteristic
45C2×M16×C47C2311yescharacteristic
46M32×C47—22no—
47C22⋊C8×C47—22no—
48C24×C47C2×C811yescharacteristic
49C22×C4×C47—21no—
50C22×C4×C47—21no—
51C22×C4×C47—41no—
52C22×C4×C47—41no—
53C22×C4×C47C2×C811yescharacteristic
54C22×C4×C47C22×C411yescharacteristic
55C2×C22⋊D47—22no—
56C2×C22⋊D47C2×C812yes—
57#10625350—41no—
58#10625350—41no—
59C2×D188M1611yescharacteristic
60C2×C4×D47C22⋊C412yes—
61C2×Dic94M1611yescharacteristic
62M16×C47—41no—
63M16×C47—41no—
64C2×C8×C47—21no—
65C2×C8×C47C2×D412yes—
66C22⋊Dic47—22no—
67C22⋊Dic47C2×C812yes—
68(C2×D47)⋊C4—22no—
69C4⋊Dic47—21no—
70Dic47⋊C4—22no—
71C4×Dic47—21no—
72C2×C47⋊C8—21no—
73C2×C47⋊C8—21no—
74C2×C47⋊C8—21no—
75C2×C47⋊C8—42no—
76C2×C47⋊C8—41no—
77C2×C47⋊C8—41no—
78C2×C47⋊C8—41no—
79C2×C47⋊C8C22⋊C411yescharacteristic
80C2×C47⋊C8C22⋊C411yescharacteristic
81#10625324—42no—
82C8×D47—22no—
83C8×D47—22no—
84C16×C47—42no—
85C47⋊C16—42no—
86C23×C47—21no—
87C23×C47—41no—
88C23×C47—42no—
89C23×C47C22×C811yescharacteristic
90C22×D47C22⋊C812yes—
91C2×C4×C47—21no—
92C2×C4×C47—21no—
93C2×C4×C47—21no—
94C2×C4×C47—42no—
95C2×C4×C47—41no—
96C2×C4×C47—41no—
97C2×C4×C47—41no—
98C2×C4×C47C2×C22⋊C411yescharacteristic
99C2×C4×C47C2×M1611yescharacteristic
100C22⋊D47—42no—
101C22⋊D47—44no—
102C2×Dic47—22no—
103C2×Dic47C22⋊C812yes—
104D188—41no—
105C4×D47—22no—
106C4×D47—22no—
107Dic94—41no—
108C8×C47—22no—
109C8×C47—41no—
110C47⋊C8—21no—
111C47⋊C8—41no—
112C47⋊C8#9911yescharacteristic
113C47⋊C8#9911yescharacteristic
114#11483—471nomaximal, Sylow
115C22×C47—22no—
116C22×C47—21no—
117C22×C47—41no—
118C22×C47—41no—
119C22×C47—42no—
120C22×C47—42no—
121C22×C47—41no—
122C22×C47C2×C22⋊C811yescharacteristic
123C2×D47—22no—
124C2×D47—42no—
125C4×C47—21no—
126C4×C47—41no—
127C4×C47C2×#9911yescharacteristic
128C4×C47#41311yescharacteristic
129Dic47—22no—
130Dic47—42no—
131#2929—471no—
132#2115—472no—
133#1328—472no—
134#1320—471no—
135#1320D4711yescharacteristic, complemented
136C2×C47—21no—
137C2×C47—42no—
138C2×C47—41no—
139C2×C47#145911yescharacteristic
140D47—42no—
141C2×#131—471no—
142C23×C8—471no—
143#525—471no—
144C2×M32—942no—
145#435—942no—
146C8×D4—942no—
147#424—471no—
148#408—471no—
149#407C2×D4711yescharacteristic
150#405—471no—
151#349—472no—
152#349—942no—
153#349C2×D4712yes—
154C47#1148311yescharacteristic, complemented, Sylow
155C2×#41—471no—
156C23×C4C4×D4711yescharacteristic
157#131—1881no—
158#131—1881no—
159C2×M16—472no—
160C2×M16C4×D4711yescharacteristic
161C2×M16C22×D4711yescharacteristic
162C22×C8—471no—
163C22×C8—472no—
164C22×C8—471no—
165C22×C8—941no—
166C22×C8—941no—
167C22×C8—1881no—
168C22×C8—1881no—
169#123—472no—
170C22⋊Q8—941no—
171#121—941no—
172#120—472no—
173C4×D4—942no—
174#117—471no—
175M32—22no—
176M32—942no—
177M32—1882no—
178C2×C16—1882no—
179C4⋊C8—942no—
180C22⋊C8—22no—
181C22⋊C8—942no—
182#97—941no—
183C4×C8—941no—
184C24C8×D4711yescharacteristic
185#41—1881no—
186#41—1881no—
187C2×Q8—471no—
188C2×D4—472no—
189C2×D4—471no—
190C2×D4—942no—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—41no—
194C22×C4—41no—
195C22×C4—472no—
196C22×C4C8×D4711yescharacteristic
197C22×C4C2×C4×D4711yescharacteristic
198M16—41no—
199M16—41no—
200M16—1882no—
201C2×C8—21no—
202C2×C8—471no—
203C2×C8—471no—
204C2×C8—942no—
205C2×C8—942no—
206C2×C8—941no—
207C2×C8—941no—
208C2×C8—941no—
209C2×C8—1881no—
210C2×C8—1881no—
211C2×C8—1882no—
212C2×C8—1881no—
213C2×C8D4×D4712yes—
214C4⋊C4—942no—
215C4⋊C4—941no—
216C22⋊C4—472no—
217C22⋊C4—942no—
218C22⋊C4—942no—
219C42—941no—
220C16—42no—
221C16—1882no—
222C23—21no—
223C23—41no—
224C23—42no—
225C23—472no—
226C23C2×C8×D4711yescharacteristic
227Q8—1881no—
228D4—1881no—
229D4—1882no—
230D4—1884no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—42no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—472no—
239C2×C4—942no—
240C2×C4—942no—
241C2×C4—942no—
242C2×C4C22⋊C4×D4711yescharacteristic
243C2×C4M16×D4711yescharacteristic
244C8—22no—
245C8—41no—
246C8—471no—
247C8—471no—
248C8—941no—
249C8—1881no—
250C22—22no—
251C22—21no—
252C22—41no—
253C22—41no—
254C22—42no—
255C22—42no—
256C22—41no—
257C22—942no—
258C22—1882no—
259C22C22⋊C8×D4711yescharacteristic
260C4—21no—
261C4—41no—
262C4—942no—
263C4—1882no—
264C4#99×D4711yescharacteristic
265C4#1568910411yescharacteristic
266C2—21no—
267C2—42no—
268C2—41no—
269C2—1882no—
270C2#2024170311yescharacteristic
subgroupG/H
centerC4#99×D47
commutatorC23×C47C22×C8
FrattiniC2×M16C22×D47
FittingC47×#1320C2
socleC2×C47#1459
prime radical#15689634C4
cosocleC2×M16×C47C23
hypercenter#1320D47
hypocenterC47#11483
LCSG ▷ C23×C47 ▷ C22×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×M16 ◁ #1320

Complements

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

NH ≅ G/Ntype
2. #20245007C2semidirect
3. #20241340C2semidirect
4. #20241324C2semidirect
5. C47×#1320C2semidirect
135. #1320D47semidirect
154. C47#11483semidirect

Maximal subgroups

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

subgroupindex
#202520782
#202450072
#202413402
#202413242
C47×#13202
#1148347