Finite Group Databasebeta

← #26184032

Subgroups — #26184032

Order 12032.

938 conjugacy classes of subgroups; only the 355 normal subgroups are shown.

normal subgroups only

This group has 938 conjugacy classes of subgroups — too many to draw. Only its 355 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#20257198C211yesmaximal, characteristic
2#20246309C211yesmaximal, characteristic
3#20241953C216yesmaximal
4#20241888C216yesmaximal
5C47×#2982C211yesmaximal, characteristic
6Q8×Dic94C414yescomplemented
7#15689852C414yescomplemented
8#15689849—24no—
9#15689754—23no—
10#15689680C2216yes—
11#15689601—21no—
12C2×Q8×Dic47—21no—
13#15689022—23no—
14#15688989C2213yes—
15C4×C4⋊Dic47C2213yes—
16#15688861C22112yes—
17#15688861C2214yes—
18C2×C4×Q8×C47C2211yescharacteristic
19C8×Q8×C47—24no—
20C47×#431C2216yes—
21#13094024C2×C416yes—
22#13094024C2×C416yes—
23#13094023—212no—
24#13094016—26no—
25#13094015—212no—
26#13094003—26no—
27#13094003C2314yes—
28Q8×Dic47—24no—
29Q8×Dic47—24no—
30Q8×Dic47—44no—
31C4×Dic94—26no—
32C4×Dic94C2×C4112yes—
33#13093983—26no—
34C2×C4⋊Dic47—23no—
35C2×C4⋊Dic47—23no—
36C2×C4×Dic47—23no—
37C47⋊#95—23no—
38C47⋊#95C2311yescharacteristic
39C47⋊#95C2313yes—
40C22×Q8×C47D411yescharacteristic
41C4×Q8×C47C2×C414yes—
42C4×Q8×C47D414yes—
43C2×C4⋊C4×C47D413yes—
44C2×C42×C47C2313yes—
45C4⋊C8×C47—26no—
46C22⋊C8×C47C2314yes—
47C4×C8×C47—26no—
48C2×Q8×C47C22⋊C414yes—
49C2×Q8×C47SD1614yes—
50C2×Q8×C47Q1614yes—
51C22×C4×C47C2×D413yes—
52C22×C4×C47C2×D413yes—
53C22×C4×C47C2411yescharacteristic
54C22×Dic47—21no—
55C22×Dic47—23no—
56C2×Dic94—26no—
57C2×Dic94C22×C414yes—
58C2×C8×C47—24no—
59C4⋊C4×C47C22⋊C4112yes—
60C42×C47C22×C416yes—
61C42×C47C2×D416yes—
62C22⋊Dic47—24no—
63C4⋊Dic47—26no—
64C4⋊Dic47—26no—
65C4⋊Dic47—26no—
66C4⋊Dic47—46no—
67C4⋊Dic47—46no—
68C4⋊Dic47C22×C414yes—
69Dic47⋊C4—212no—
70Dic47⋊C4—24no—
71C4×Dic47—212no—
72C4×Dic47—46no—
73C23×C47C22×D411yescharacteristic
74Q8×C47#103116yes—
75C2×C4×C47—43no—
76C2×C4×C47—41no—
77C2×C4×C47C2×C22⋊C416yes—
78C2×C4×C47C2×C22⋊C416yes—
79C2×C4×C47C2×SD1616yes—
80C2×C4×C47C2×Q1616yes—
81C2×C4×C47C23×C411yescharacteristic
82C2×C4×C47C22×D411yescharacteristic
83C2×Dic47—22no—
84C2×Dic47—24no—
85C2×Dic47—24no—
86C2×Dic47—41no—
87C2×Dic47—41no—
88C2×Dic47—43no—
89C2×Dic47—43no—
90C2×Dic47—43no—
91Dic94—44no—
92Dic94—86no—
93C8×C47—44no—
94#17619—471nomaximal, Sylow
95C22×C47—21no—
96C22×C47—21no—
97C22×C47C22×C22⋊C411yescharacteristic
98C22×C47C22×SD1611yescharacteristic
99C22×C47C22×Q1611yescharacteristic
100C4×C47—21no—
101C4×C47—21no—
102C4×C47—26no—
103C4×C47C2×#103112yes—
104Dic47—42no—
105Dic47—44no—
106Dic47—44no—
107Dic47—83no—
108#3502—471no—
109#2982D4711yescharacteristic
110#2351—471no—
111#1518—476no—
112#1503—476no—
113C2×C47—21no—
114C2×C47—21no—
115C2×C47#288411yescharacteristic
116C2×C47C22×#10311yescharacteristic
117C2×C47#290111yescharacteristic
118Q82—474no—
119#557—474no—
120#557—944no—
121#544—943no—
122#533—476no—
123#519—941no—
124C2×C4×Q8—941no—
125C2×C4×Q8C2×D4711yescharacteristic
126C8×Q8—24no—
127#431C2×D4716yes—
128#389—943no—
129#382—473no—
130C4×C4⋊C4—473no—
131#328—474no—
132#328—4712no—
133C47#1761911yescharacteristic, complemented, Sylow
134C22×Q8D18811yescharacteristic
135C4⋊Q8—476no—
136C4⋊Q8—476no—
137C4⋊Q8—9412no—
138#125—9412no—
139#125—946no—
140C22⋊Q8—474no—
141C22⋊Q8—946no—
142C4×Q8—4712no—
143C4×Q8—944no—
144C4×Q8—944no—
145C4×Q8—946no—
146C4×Q8—1884no—
147C4×Q8C4×D4714yes—
148C4×Q8C22⋊D4714yes—
149#117—946no—
150C2×C4⋊C4—943no—
151C2×C4⋊C4—943no—
152C2×C4⋊C4D18813yes—
153C2×C42—943no—
154C2×C42C22×D4713yes—
155C4⋊C8—26no—
156C22⋊C8C22×D4714yes—
157C4×C8—26no—
158#95—471no—
159#95—473no—
160#95—943no—
161C2×Q8—474no—
162C2×Q8—946no—
163C2×Q8#1062532514yes—
164C2×Q8Dic18814yes—
165C2×Q8(C2×D47)⋊C414yes—
166C22×C4—941no—
167C22×C4—943no—
168C22×C4C2×D18813yes—
169C22×C4C2×D18813yes—
170C22×C4C23×D4711yescharacteristic
171C2×C8—24no—
172C4⋊C4—474no—
173C4⋊C4—9412no—
174C4⋊C4—946no—
175C4⋊C4—946no—
176C4⋊C4—946no—
177C4⋊C4—944no—
178C4⋊C4—1886no—
179C4⋊C4—1886no—
180C4⋊C4(C2×D47)⋊C4112yes—
181C22⋊C4—944no—
182C42—9412no—
183C42—1886no—
184C42C2×C4×D4716yes—
185C42C2×C22⋊D4716yes—
186C23C22×D18811yescharacteristic
187Q8—3766no—
188Q8—1884no—
189Q8#13093949116yes—
190C2×C4—43no—
191C2×C4—41no—
192C2×C4—942no—
193C2×C4—944no—
194C2×C4—944no—
195C2×C4—1883no—
196C2×C4—1883no—
197C2×C4—1883no—
198C2×C4—1881no—
199C2×C4—1881no—
200C2×C4C2×(C2×D47)⋊C416yes—
201C2×C4C2×(C2×D47)⋊C416yes—
202C2×C4C2×#1062532516yes—
203C2×C4C2×Dic18816yes—
204C2×C4C22×C4×D4711yescharacteristic
205C2×C4C22×C22⋊D4711yescharacteristic
206C8—44no—
207C22—21no—
208C22—21no—
209C22C22×(C2×D47)⋊C411yescharacteristic
210C22C22×#1062532511yescharacteristic
211C22C22×Dic18811yescharacteristic
212C4—26no—
213C4—21no—
214C4—21no—
215C4—3763no—
216C4—1882no—
217C4—1884no—
218C4—1884no—
219C4C2×#13093949112yes—
220C2—21no—
221C2—21no—
222C2#2025185111yescharacteristic
223C2C22×#1309394911yescharacteristic
224C2#2025193311yescharacteristic
subgroupG/H
centerC22C22×(C2×D47)⋊C4
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#2982C2
socleC22×C47C22×C22⋊C4
prime radicalC23×C47C22×D4
cosocleC22×C4×C47C24
hypercenter#2982D47
hypocenterC47#17619
LCSG ▷ C2×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C4×Q8 ◁ #2982

Complements

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

NH ≅ G/Ntype
6. Q8×Dic94C4semidirect
7. #15689852C4semidirect
133. C47#17619semidirect

Maximal subgroups

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

subgroupindex
#202571982
#202463092
#202419532
#202418882
C47×#29822
#1761947