Finite Group Databasebeta

← #26125921

Subgroups — #26125921

Order 12032.

394 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 95 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#20241732C211yesmaximal, characteristic
2#20241712C211yesmaximal, characteristic
3#20241322C214yesmaximal
4C47×#2546C211yesmaximal, characteristic
5#15689180—21no—
6#15689176—21no—
7#15688849—21no—
8#15688849—21no—
9#15688849C2212yes—
10#15688849C2212yes—
11C2×C4⋊Q8×C47C411yescharacteristic
12C2×C4⋊Q8×C47C412yes—
13C2×C4×Q8×C47C2211yescharacteristic
14C47×#398—21no—
15C47×#398—21no—
16C47×#391C2212yes—
17C47×#389—21no—
18C47×#389—21no—
19C4×C4⋊C4×C47C411yescharacteristic
20C4⋊(C47⋊C8)—41no—
21C4⋊(C47⋊C8)—41no—
22C22⋊(C47⋊C8)—22no—
23C22⋊(C47⋊C8)—22no—
24#13093935—41no—
25C4×C47⋊C8—41no—
26C22×Q8×C47C2×C412yes—
27C22×Q8×C47C2×C411yescharacteristic
28C4⋊Q8×C47—41no—
29C4⋊Q8×C47—41no—
30C4⋊Q8×C47—44no—
31C4×Q8×C47—41no—
32C4×Q8×C47—41no—
33C2×C4⋊C4×C47—21no—
34C2×C4⋊C4×C47—21no—
35C2×C4⋊C4×C47—22no—
36C2×C4⋊C4×C47—22no—
37C2×C4⋊C4×C47—21no—
38C2×C4⋊C4×C47D411yescharacteristic
39C2×C4⋊C4×C47D411yescharacteristic
40C2×C4⋊C4×C47C2311yescharacteristic
41C2×C42×C47C2×C411yescharacteristic
42C2×C42×C47C2×C412yes—
43C2×C42×C47D411yescharacteristic
44C2×C42×C47D411yescharacteristic
45#95×C47—21no—
46#95×C47—22no—
47#95×C47—22no—
48C2×Q8×C47—22no—
49C2×Q8×C47—22no—
50C2×Q8×C47—24no—
51C2×Q8×C47—41no—
52C2×Q8×C47—42no—
53C2×Q8×C47—42no—
54C22×C4×C47—22no—
55C22×C4×C47—22no—
56C22×C4×C47C22⋊C411yescharacteristic
57C22×C4×C47C22⋊C411yescharacteristic
58C22×C4×C47C22⋊C411yescharacteristic
59C22×C4×C47C22⋊C411yescharacteristic
60C22×C4×C47C22×C411yescharacteristic
61C22×C4×C47C2×D411yescharacteristic
62C22×C4×C47C2×D411yescharacteristic
63C4⋊C4×C47—24no—
64C4⋊C4×C47—41no—
65C4⋊C4×C47—41no—
66C4⋊C4×C47—41no—
67C4⋊C4×C47—42no—
68C4⋊C4×C47—44no—
69C4⋊C4×C47—41no—
70C4⋊C4×C47—41no—
71C4⋊C4×C47—44no—
72C42×C47—21no—
73C42×C47—21no—
74C42×C47—21no—
75C42×C47—21no—
76C42×C47—21no—
77C42×C47—21no—
78C42×C47—42no—
79C2×C47⋊C8—42no—
80C2×C47⋊C8—42no—
81C23×C47C2×C22⋊C411yescharacteristic
82Q8×C47—44no—
83Q8×C47—84no—
84C2×C4×C47—21no—
85C2×C4×C47—21no—
86C2×C4×C47—21no—
87C2×C4×C47—21no—
88C2×C4×C47—22no—
89C2×C4×C47—24no—
90C2×C4×C47—21no—
91C2×C4×C47—21no—
92C2×C4×C47—21no—
93C2×C4×C47—21no—
94C2×C4×C47—21no—
95C2×C4×C47—21no—
96C2×C4×C47—42no—
97C2×C4×C47—42no—
98C2×C4×C47—41no—
99C2×C4×C47—41no—
100C2×C4×C47—41no—
101C2×C4×C47—42no—
102C2×C4×C47—42no—
103C2×C4×C47—42no—
104C2×C4×C47#9911yescharacteristic
105C2×C4×C47#9911yescharacteristic
106C2×C4×C47#10012yes—
107C2×C4×C47#13112yes—
108C47⋊C8—82no—
109C47⋊C8—82no—
110#11568—471nomaximal, Sylow
111C22×C47—21no—
112C22×C47—21no—
113C22×C47#40811yescharacteristic
114C22×C47C2×#9911yescharacteristic
115C22×C47C2×#10011yescharacteristic
116C4×C47—21no—
117C4×C47—22no—
118C4×C47—22no—
119C4×C47—44no—
120C4×C47—41no—
121C4×C47—41no—
122C4×C47—41no—
123C4×C47—41no—
124C4×C47—82no—
125C4×C47#35612yes—
126#2546D4711yescharacteristic
127#1467—471no—
128#1461—471no—
129#1319—474no—
130C2×C47—21no—
131C2×C47—21no—
132C2×C47#147111yescharacteristic
133C2×C47#212911yescharacteristic
134C2×C47C2×#35611yescharacteristic
135C2×C4⋊Q8Dic4711yescharacteristic
136C2×C4⋊Q8Dic4712yes—
137C2×C4×Q8C2×D4711yescharacteristic
138#432—941no—
139#431—941no—
140#398—21no—
141#398—21no—
142#391C2×D4712yes—
143#389—21no—
144#389—21no—
145C4×C4⋊C4Dic4711yescharacteristic
146#324—472no—
147#324—472no—
148#324—941no—
149#324—941no—
150C47#1156811yescharacteristic, complemented, Sylow
151C22×Q8C2×Dic4712yes—
152C22×Q8C2×Dic4711yescharacteristic
153C4⋊Q8—41no—
154C4⋊Q8—41no—
155C4⋊Q8—44no—
156C4×Q8—41no—
157C4×Q8—41no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4C22⋊D4711yescharacteristic
164C2×C4⋊C4C22⋊D4711yescharacteristic
165C2×C4⋊C4C22×D4711yescharacteristic
166C2×C42C2×Dic4711yescharacteristic
167C2×C42C2×Dic4712yes—
168C2×C42C22⋊D4711yescharacteristic
169C2×C42C22⋊D4711yescharacteristic
170C4⋊C8—1881no—
171C4⋊C8—1881no—
172C22⋊C8—942no—
173C22⋊C8—942no—
174#97—1881no—
175C4×C8—1881no—
176#95—21no—
177#95—22no—
178#95—22no—
179C2×Q8—22no—
180C2×Q8—22no—
181C2×Q8—24no—
182C2×Q8—41no—
183C2×Q8—42no—
184C2×Q8—42no—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4C22⋊Dic4711yescharacteristic
188C22×C4C22⋊Dic4711yescharacteristic
189C22×C4C22⋊Dic4711yescharacteristic
190C22×C4C22⋊Dic4711yescharacteristic
191C22×C4C22×Dic4711yescharacteristic
192C22×C4C2×C22⋊D4711yescharacteristic
193C22×C4C2×C22⋊D4711yescharacteristic
194C2×C8—1882no—
195C2×C8—1882no—
196C4⋊C4—24no—
197C4⋊C4—41no—
198C4⋊C4—41no—
199C4⋊C4—41no—
200C4⋊C4—42no—
201C4⋊C4—44no—
202C4⋊C4—41no—
203C4⋊C4—41no—
204C4⋊C4—44no—
205C42—21no—
206C42—21no—
207C42—21no—
208C42—21no—
209C42—21no—
210C42—21no—
211C42—42no—
212C23C2×C22⋊Dic4711yescharacteristic
213Q8—44no—
214Q8—84no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—22no—
220C2×C4—24no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—42no—
228C2×C4—42no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4—41no—
232C2×C4—42no—
233C2×C4—42no—
234C2×C4—42no—
235C2×C4#1309393911yescharacteristic
236C2×C4#1309393911yescharacteristic
237C2×C4#1309394112yes—
238C2×C4#1309403112yes—
239C8—3762no—
240C8—3762no—
241C22—21no—
242C22—21no—
243C22#1568909011yescharacteristic
244C22C2×#1309393911yescharacteristic
245C22C2×#1309394111yescharacteristic
246C4—21no—
247C4—22no—
248C4—22no—
249C4—44no—
250C4—41no—
251C4—41no—
252C4—41no—
253C4—41no—
254C4—82no—
255C4#1568892212yes—
256C2—21no—
257C2—21no—
258C2#2024174811yescharacteristic
259C2#2024506311yescharacteristic
260C2C2×#1568892211yescharacteristic
subgroupG/H
centerC22C2×#13093939
commutatorC22×C4×C47C22×C4
FrattiniC2×C4⋊C4C22×D47
FittingC47×#2546C2
socleC22×C47C2×#99
prime radicalC23×C47C2×C22⋊C4
cosocleC2×C4⋊C4×C47C23
hypercenter#2546D47
hypocenterC47#11568
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #2546

Complements

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

NH ≅ G/Ntype
150. C47#11568semidirect

Maximal subgroups

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

subgroupindex
#202417322
#202417122
#202413222
C47×#25462
#1156847