Finite Group Databasebeta

← #26147332

Subgroups — #26147332

Order 12032.

1530 conjugacy classes of subgroups; only the 371 normal subgroups are shown.

normal subgroups only

This group has 1530 conjugacy classes of subgroups — too many to draw. Only its 371 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
1C22×#13094018C211yesmaximal, characteristic
2C2×#15688991C211yesmaximal, characteristic, complemented
3#20241638C216yesmaximal, complemented
4#20241630C216yesmaximal
5C2×C47×#384C211yesmaximal, characteristic, complemented
6C2×#13094018—24no—
7C2×#13094018—24no—
8C2×#13094018C418yescomplemented
9C22×Dic47⋊C4C2213yes—
10C22×(C2×D47)⋊C4C2213yes—
11C22×(C2×D47)⋊C4C2214yes—
12#15688991—24no—
13C2×C47⋊#95C2216yes—
14C2×C47⋊#95C22112yes—
15C22×C4⋊C4×C47C2216yes—
16C22×C42×C47C2211yescharacteristic
17C47×#384—24no—
18C24×D47C2311yescharacteristic
19C23×Dic47C2314yes—
20C23×Dic47C2313yes—
21#13094018—232no—
22C2×Dic47⋊C4—26no—
23C2×Dic47⋊C4—26no—
24C2×Dic47⋊C4C2×C4112yes—
25C2×(C2×D47)⋊C4—26no—
26C2×(C2×D47)⋊C4—26no—
27C2×(C2×D47)⋊C4—412no—
28C2×(C2×D47)⋊C4C2×C4112yes—
29C47⋊#95—212no—
30C47⋊#95—412no—
31C23×C4×C47C2314yes—
32C23×C4×C47C2313yes—
33C2×C4⋊C4×C47—212no—
34C2×C4⋊C4×C47—412no—
35C2×C42×C47C2×C414yes—
36C2×C42×C47D414yes—
37C2×C42×C47D414yes—
38C24×C47C2411yescharacteristic
39C23×D47—21no—
40C23×D47—21no—
41C23×D47—26no—
42C23×D47—26no—
43C23×D47C22×C412yes—
44C22×C4×C47—26no—
45C22×C4×C47—26no—
46C22×C4×C47—24no—
47C22×C4×C47—212no—
48C22×C4×C47—212no—
49C22×C4×C47C22×C416yes—
50C22×C4×C47C2×D416yes—
51C22×C4×C47C2×D416yes—
52C22×Dic47—23no—
53C22×Dic47—23no—
54C22×Dic47—26no—
55C22×Dic47—26no—
56C22×Dic47—24no—
57C22×Dic47—212no—
58C22×Dic47—212no—
59C22×Dic47C22×C416yes—
60C4⋊C4×C47—424no—
61C42×C47C22⋊C4116yescomplemented
62(C2×D47)⋊C4—224no—
63(C2×D47)⋊C4—416no—
64Dic47⋊C4—224no—
65C23×C47C23×C411yescharacteristic
66C23×C47C22×D411yescharacteristic
67C23×C47C22×D411yescharacteristic
68C23×C47H(5,𝔽2)16yes—
69C23×C472−1+416yes—
70C22×D47—22no—
71C22×D47—212no—
72C22×D47—43no—
73C22×D47—43no—
74C22×D47—46no—
75C22×D47—44no—
76C22×D47—412no—
77C2×C4×C47—212no—
78C2×C4×C47—26no—
79C2×C4×C47—26no—
80C2×C4×C47—212no—
81C2×C4×C47—412no—
82C2×C4×C47—44no—
83C2×C4×C47—412no—
84C2×C4×C47C2×C22⋊C4112yes—
85C2×Dic47—26no—
86C2×Dic47—212no—
87C2×Dic47—43no—
88C2×Dic47—43no—
89C2×Dic47—46no—
90C2×Dic47—412no—
91C2×Dic47—44no—
92C2×Dic47—412no—
93#14665—471nomaximal, Sylow
94C22×C47C22×C22⋊C411yescharacteristic
95C22×C47#52016yes—
96C22×C47#53513yes—
97C22×C47#53613yes—
98C22×C47#53716yes—
99C22×C47#56014yes—
100C22×C47#563112yes—
101C2×D47—46no—
102C2×D47—48no—
103C2×D47—81no—
104C2×D47—83no—
105C2×D47—84no—
106C4×C47—212no—
107C4×C47—84no—
108Dic47—46no—
109Dic47—84no—
110C22×#126—471no—
111C2×#384D4711yescharacteristic, complemented
112C2×#383—471no—
113#1441—476no—
114#1439—476no—
115C2×C47#231113yes—
116C2×C47#238314yes—
117C2×C47#281811yescharacteristic
118C2×C47#282113yes—
119C2×C47#282314yes—
120D47—82no—
121C2×#126—478no—
122C2×#126—944no—
123C2×#126—944no—
124C22×C4⋊C4—473no—
125C22×C4⋊C4C2×D4716yes—
126C22×C22⋊C4—473no—
127C22×C22⋊C4—474no—
128C22×C42C2×D4711yescharacteristic
129#384—24no—
130#383—944no—
131C2×#95—476no—
132C2×#95—4712no—
133C47#1466511yescharacteristic, complemented, Sylow
134C25—471no—
135C23×C4—473no—
136C23×C4—474no—
137C23×C4C22×D4714yes—
138C23×C4C22×D4713yes—
139#126—9432no—
140C2×C4⋊C4—212no—
141C2×C4⋊C4—412no—
142C2×C4⋊C4—4712no—
143C2×C4⋊C4—946no—
144C2×C4⋊C4—946no—
145C2×C22⋊C4—4712no—
146C2×C22⋊C4—946no—
147C2×C22⋊C4—946no—
148C2×C22⋊C4—18812no—
149C2×C42C4×D4714yes—
150C2×C42D18814yes—
151C2×C42C22⋊D4714yes—
152#95—9412no—
153#95—18812no—
154C24—472no—
155C24—941no—
156C24—941no—
157C24—946no—
158C24—946no—
159C24C23×D4711yescharacteristic
160C22×C4—26no—
161C22×C4—26no—
162C22×C4—24no—
163C22×C4—212no—
164C22×C4—212no—
165C22×C4—476no—
166C22×C4—943no—
167C22×C4—943no—
168C22×C4—946no—
169C22×C4—946no—
170C22×C4—944no—
171C22×C4—9412no—
172C22×C4—9412no—
173C22×C4C2×C4×D4716yes—
174C22×C4C2×D18816yes—
175C22×C4C2×C22⋊D4716yes—
176C4⋊C4—424no—
177C4⋊C4—9424no—
178C22⋊C4—9424no—
179C22⋊C4—18816no—
180C42(C2×D47)⋊C4116yescomplemented
181C23—942no—
182C23—9412no—
183C23—1884no—
184C23—18812no—
185C23—1883no—
186C23—1883no—
187C23—1886no—
188C23C22×C4×D4711yescharacteristic
189C23C22×C22⋊D4711yescharacteristic
190C23C22×D18811yescharacteristic
191C23#1309406916yes—
192C23#1309407216yes—
193C2×C4—212no—
194C2×C4—26no—
195C2×C4—26no—
196C2×C4—212no—
197C2×C4—44no—
198C2×C4—412no—
199C2×C4—412no—
200C2×C4—946no—
201C2×C4—9412no—
202C2×C4—1884no—
203C2×C4—18812no—
204C2×C4—18812no—
205C2×C4—1883no—
206C2×C4—1883no—
207C2×C4—1886no—
208C2×C4C2×(C2×D47)⋊C4112yes—
209C22—3761no—
210C22—3763no—
211C22—3764no—
212C22—1888no—
213C22—1886no—
214C22C22×(C2×D47)⋊C411yescharacteristic
215C22#1568960416yes—
216C22#1568969013yes—
217C22#1568969713yes—
218C22#1568970416yes—
219C22#1568986014yes—
220C22#15689872112yes—
221C4—212no—
222C4—84no—
223C4—3764no—
224C4—1886no—
225C2—3762no—
226C2#2024598113yes—
227C2#2024658114yes—
228C2#2025142911yescharacteristic
229C2#2025144713yes—
230C2#2025146214yes—
subgroupG/H
center, FrattiniC24C23×D47
commutatorC23×C47C23×C4
FittingC2×C47×#384C2
socle, cosocleC24×C47C24
prime radicalC24×D47C23
hypercenterC2×#384D47
hypocenterC47#14665
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C24 ◁ C2×#384

Complements

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

NH ≅ G/Ntype
2. C2×#15688991C2semidirect
3. #20241638C2semidirect
5. C2×C47×#384C2semidirect
8. C2×#13094018C4semidirect
61. C42×C47C22⋊C4semidirect
111. C2×#384D47semidirect
133. C47#14665semidirect
180. C42(C2×D47)⋊C4semidirect

Maximal subgroups

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

subgroupindex
C22×#130940182
C2×#156889912
#202416382
#202416302
C2×C47×#3842
#1466547