Finite Group Databasebeta

← #26125593

Subgroups — #26125593

Order 12032.

430 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 430 conjugacy classes of subgroups — too many to draw. Only its 109 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
1C2×(C22⋊D47)⋊C8C211yesmaximal, characteristic
2#20245184C212yesmaximal, complemented
3#20241312C211yesmaximal, characteristic, complemented
4#20241311C212yesmaximal, complemented
5C47×#1314C211yesmaximal, characteristic, complemented
6C23×C47⋊C8C411yescharacteristic
7C22×C8×D47C411yescharacteristic
8C2×C4×C22⋊D47C411yescharacteristic
9(C22⋊D47)⋊C8—22no—
10(C22⋊D47)⋊C8—22no—
11C2×Dic47⋊C8C411yescharacteristic
12C2×(C2×D47)⋊C8C2211yescharacteristic
13C2×C8×Dic47C2211yescharacteristic
14Dic47⋊C16—22no—
15C22⋊(C47⋊C16)C2212yes—
16C16×Dic47—22no—
17C2×C22⋊C8×C47C2211yescharacteristic
18C47×C22⋊C16C2212yes—
19C22×C22⋊D47C811yescharacteristic
20C22×C4×D47C2×C411yescharacteristic
21C22×C47⋊C8—21no—
22C22×C47⋊C8—21no—
23C22×C47⋊C8—42no—
24C22×C47⋊C8C2×C411yescharacteristic
25C22×C47⋊C8C2×C411yescharacteristic
26C2×C8×D47—21no—
27C2×C8×D47—21no—
28C2×C8×D47—42no—
29C4×C22⋊D47—22no—
30C4×C22⋊D47—22no—
31C2×Dic47⋊C4C811yescharacteristic
32C2×C22⋊Dic47C811yescharacteristic
33C2×(C2×D47)⋊C4C811yescharacteristic
34C2×C4×Dic47C2×C411yescharacteristic
35C2×C47⋊C16—22no—
36Dic47⋊C8—22no—
37(C2×D47)⋊C8—22no—
38C8×Dic47D412yes—
39C8×Dic47D412yes—
40C23×C4×C47C2×C411yescharacteristic
41C22×C8×C47C2×C411yescharacteristic
42C22×C8×C47C2311yescharacteristic
43C2×C16×C47—22no—
44C22⋊C8×C47—22no—
45C24×C47C2×C811yescharacteristic
46C23×D47C2×C811yescharacteristic
47C22×C4×C47—21no—
48C22×C4×C47—21no—
49C22×C4×C47—41no—
50C22×C4×C47—41no—
51C22×C4×C47C2×C811yescharacteristic
52C22×C4×C47C22×C411yescharacteristic
53C2×C22⋊D47—22no—
54C2×C22⋊D47—22no—
55C2×C22⋊D47C1614yescomplemented
56C22×Dic47C2×C811yescharacteristic
57C22×Dic47C2×C811yescharacteristic
58C22×Dic47C2×C811yescharacteristic
59C2×C4×D47—21no—
60C2×C4×D47—21no—
61C2×C4×D47—41no—
62C2×C4×D47—41no—
63C2×C8×C47—21no—
64C2×C8×C47—41no—
65C2×C8×C47—41no—
66C2×C8×C47C2×D412yes—
67C22⋊Dic47—22no—
68(C2×D47)⋊C4—22no—
69Dic47⋊C4—22no—
70C4×Dic47C22⋊C412yes—
71C4×Dic47M1612yes—
72C2×C47⋊C8—22no—
73C2×C47⋊C8—21no—
74C2×C47⋊C8—42no—
75C2×C47⋊C8—42no—
76C2×C47⋊C8—41no—
77C2×C47⋊C8C22⋊C411yescharacteristic
78C2×C47⋊C8C22⋊C411yescharacteristic
79C8×D47—42no—
80C8×D47—42no—
81C16×C47—42no—
82C47⋊C16—42no—
83C23×C47—21no—
84C23×C47—21no—
85C23×C47—41no—
86C23×C47—41no—
87C23×C47C2×C1612yes—
88C23×C47C22×C811yescharacteristic
89C22×D47—21no—
90C22×D47—21no—
91C22×D47—41no—
92C22×D47—41no—
93C22×D47C2×C1612yes—
94C2×C4×C47—21no—
95C2×C4×C47—21no—
96C2×C4×C47—21no—
97C2×C4×C47—42no—
98C2×C4×C47—41no—
99C2×C4×C47—41no—
100C2×C4×C47—41no—
101C2×C4×C47C2×C22⋊C411yescharacteristic
102C2×C4×C47C2×M1611yescharacteristic
103C22⋊D47—28no—
104C2×Dic47—21no—
105C2×Dic47—41no—
106C2×Dic47—41no—
107C2×Dic47C22⋊C812yes—
108C2×Dic47C22⋊C812yes—
109C2×Dic47C2×C1612yes—
110C2×Dic47M3212yes—
111C4×D47—42no—
112C4×D47—41no—
113C4×D47—41no—
114C8×C47—22no—
115C8×C47—41no—
116C47⋊C8—21no—
117C47⋊C8—41no—
118C47⋊C8#10012yes—
119#11487—471nomaximal, Sylow
120C22×C47—21no—
121C22×C47—21no—
122C22×C47—22no—
123C22×C47—42no—
124C22×C47—42no—
125C22×C47—41no—
126C22×C47—41no—
127C22×C47C2×C22⋊C811yescharacteristic
128C22×C47C22×C1611yescharacteristic
129C22×C47C2×M3211yescharacteristic
130C2×D47—22no—
131C2×D47—42no—
132C2×D47—41no—
133C2×D47—41no—
134C2×D47—42no—
135C4×C47—21no—
136C4×C47—41no—
137C4×C47#41011yescharacteristic
138C4×C47C2×#10011yescharacteristic
139Dic47—22no—
140Dic47—41no—
141Dic47C22⋊C1614yes—
142C2×C8×D4—471no—
143#2162—472no—
144#1314—471no—
145#1314—472no—
146#1314D4711yescharacteristic, complemented
147C2×C47—21no—
148C2×C47—21no—
149C2×C47—42no—
150C2×C47#146011yescharacteristic
151C2×C47C2×C22⋊C1611yescharacteristic
152C2×C47#211511yescharacteristic
153D47—42no—
154C23×C8—471no—
155C23×C8—471no—
156C2×C4×D4—471no—
157C8×D4—942no—
158C8×D4—942no—
159C2×C4⋊C8—471no—
160C2×C22⋊C8—471no—
161C2×C22⋊C8C2×D4711yescharacteristic
162C2×C4×C8—471no—
163C4⋊C16—942no—
164C22⋊C16—472no—
165C22⋊C16C2×D4712yes—
166C4×C16—942no—
167C47#1148711yescharacteristic, complemented, Sylow
168C22×D4—471no—
169C23×C4—471no—
170C23×C4C4×D4711yescharacteristic
171C22×C8—471no—
172C22×C8—471no—
173C22×C8—941no—
174C22×C8—941no—
175C22×C8—941no—
176C22×C8—941no—
177C22×C8—1882no—
178C22×C8—1882no—
179C22×C8C4×D4711yescharacteristic
180C22×C8C22×D4711yescharacteristic
181C4×D4—942no—
182C4×D4—942no—
183C2×C4⋊C4—471no—
184C2×C22⋊C4—471no—
185C2×C22⋊C4—471no—
186C2×C42—471no—
187C2×C16—22no—
188C2×C16—942no—
189C4⋊C8—942no—
190C22⋊C8—22no—
191C22⋊C8—942no—
192C4×C8—472no—
193C4×C8—472no—
194C24—471no—
195C24C8×D4711yescharacteristic
196C2×D4—474no—
197C2×D4—942no—
198C2×D4—942no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—41no—
202C22×C4—41no—
203C22×C4—471no—
204C22×C4—471no—
205C22×C4—471no—
206C22×C4—941no—
207C22×C4—941no—
208C22×C4—1881no—
209C22×C4—1881no—
210C22×C4C8×D4711yescharacteristic
211C22×C4C2×C4×D4711yescharacteristic
212C2×C8—21no—
213C2×C8—41no—
214C2×C8—41no—
215C2×C8—471no—
216C2×C8—471no—
217C2×C8—942no—
218C2×C8—941no—
219C2×C8—1882no—
220C2×C8—1882no—
221C2×C8—1882no—
222C2×C8—1882no—
223C2×C8—1881no—
224C2×C8D4×D4712yes—
225C4⋊C4—942no—
226C22⋊C4—942no—
227C22⋊C4—942no—
228C42—472no—
229C42—472no—
230C16—42no—
231C16—1882no—
232C23—21no—
233C23—21no—
234C23—41no—
235C23—41no—
236C23—472no—
237C23—941no—
238C23—941no—
239C23—1881no—
240C23—1881no—
241C23C16×D4712yes—
242C23C2×C8×D4711yescharacteristic
243D4—948no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—42no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—472no—
252C2×C4—472no—
253C2×C4—472no—
254C2×C4—472no—
255C2×C4—941no—
256C2×C4—1882no—
257C2×C4—1881no—
258C2×C4—1881no—
259C2×C4—1881no—
260C2×C4—1881no—
261C2×C4C22⋊C4×D4711yescharacteristic
262C2×C4M16×D4711yescharacteristic
263C8—22no—
264C8—41no—
265C8—472no—
266C8—941no—
267C8—1881no—
268C22—21no—
269C22—21no—
270C22—22no—
271C22—42no—
272C22—42no—
273C22—41no—
274C22—41no—
275C22—942no—
276C22—1882no—
277C22—1882no—
278C22—1881no—
279C22—1881no—
280C22C22⋊C8×D4711yescharacteristic
281C22C2×C16×D4711yescharacteristic
282C22M32×D4711yescharacteristic
283C4—21no—
284C4—41no—
285C4—474no—
286C4—942no—
287C4—1881no—
288C4#1568909511yescharacteristic
289C4#100×D4711yescharacteristic
290C2—21no—
291C2—21no—
292C2—42no—
293C2—1882no—
294C2#2024170711yescharacteristic
295C2C22⋊C16×D4711yescharacteristic
296C2#2024500711yescharacteristic
subgroupG/H
centerC2×C4C22⋊C4×D47
commutatorC22×C47C22×C16
FrattiniC22×C8C22×D47
FittingC47×#1314C2
socleC22×C47C2×C22⋊C8
prime radicalC22×C22⋊D47C8
cosocleC22×C8×C47C23
hypercenter#1314D47
hypocenterC47#11487
LCSG ▷ C22×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ #1314

Complements

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

NH ≅ G/Ntype
2. #20245184C2semidirect
3. #20241312C2semidirect
4. #20241311C2semidirect
5. C47×#1314C2semidirect
55. C2×C22⋊D47C16semidirect
146. #1314D47semidirect
167. C47#11487semidirect

Maximal subgroups

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

subgroupindex
C2×(C22⋊D47)⋊C82
#202451842
#202413122
#202413112
C47×#13142
#1148747