Finite Group Databasebeta

← #26147334

Subgroups — #26147334

Order 12032.

1530 conjugacy classes of subgroups; only the 411 normal subgroups are shown.

normal subgroups only

This group has 1530 conjugacy classes of subgroups — too many to draw. Only its 411 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×#15688993C211yesmaximal, characteristic, complemented
2C2×#15688991C211yesmaximal, characteristic, complemented
3#20241641C216yesmaximal, complemented
4#20241633C216yesmaximal
5C22×#126×C47C211yesmaximal, characteristic
6C22×C4⋊Dic47C2216yes—
7C22×C22⋊Dic47C2214yes—
8#15688993—24no—
9#15688991—24no—
10C2×C47⋊#95C2216yes—
11C2×C47⋊#95C22112yes—
12C2×#126×C47—24no—
13C2×#126×C47—24no—
14C2×#126×C47C418yescomplemented
15C22×C4⋊C4×C47C2213yes—
16C22×C22⋊C4×C47C2213yes—
17C22×C42×C47C2211yescharacteristic
18C23×Dic47C2314yes—
19C23×Dic47C2314yes—
20C2×C4⋊Dic47—212no—
21C2×C4⋊Dic47—412no—
22C2×C22⋊Dic47—412no—
23C47⋊#95—212no—
24C47⋊#95—412no—
25C25×C47C2311yescharacteristic
26C23×C4×C47C2313yes—
27C23×C4×C47C2313yes—
28#126×C47—232no—
29C2×C4⋊C4×C47—26no—
30C2×C4⋊C4×C47—26no—
31C2×C4⋊C4×C47C2×C4112yes—
32C2×C22⋊C4×C47—26no—
33C2×C22⋊C4×C47—26no—
34C2×C22⋊C4×C47C2×C4112yes—
35C2×C42×C47C2×C414yes—
36C2×C42×C47D414yes—
37C2×C42×C47D414yes—
38C24×C47—21no—
39C24×C47—21no—
40C24×C47—26no—
41C24×C47—26no—
42C24×C47C22×C412yes—
43C24×C47C2411yescharacteristic
44C22×C4×C47—26no—
45C22×C4×C47—26no—
46C22×C4×C47—23no—
47C22×C4×C47—23no—
48C22×C4×C47—26no—
49C22×C4×C47—26no—
50C22×C4×C47C22×C416yes—
51C22×C4×C47C22×C416yes—
52C22×C4×C47C2×D416yes—
53C22×C4×C47C2×D416yes—
54C22×Dic47—24no—
55C22×Dic47—24no—
56C22×Dic47—212no—
57C22×Dic47—212no—
58C22×Dic47—212no—
59C22×Dic47—212no—
60C4⋊C4×C47—224no—
61C22⋊C4×C47—224no—
62C42×C47C22⋊C4116yescomplemented
63C22⋊Dic47—416no—
64C4⋊Dic47—424no—
65C23×C47—22no—
66C23×C47—212no—
67C23×C47—43no—
68C23×C47—43no—
69C23×C47—46no—
70C23×C47—44no—
71C23×C47—412no—
72C23×C47C23×C411yescharacteristic
73C23×C47C22×D411yescharacteristic
74C23×C47C22×D411yescharacteristic
75C23×C47H(5,𝔽2)16yes—
76C23×C472−1+416yes—
77C2×C4×C47—212no—
78C2×C4×C47—26no—
79C2×C4×C47—26no—
80C2×C4×C47—212no—
81C2×C4×C47—26no—
82C2×C4×C47—212no—
83C2×C4×C47—43no—
84C2×C4×C47—43no—
85C2×C4×C47—46no—
86C2×C4×C47C2×C22⋊C4112yes—
87C2×Dic47—412no—
88C2×Dic47—412no—
89C2×Dic47—44no—
90C2×Dic47—44no—
91C2×Dic47—412no—
92C2×Dic47—412no—
93#14665—471nomaximal, Sylow
94C22×C47—46no—
95C22×C47—48no—
96C22×C47—81no—
97C22×C47—83no—
98C22×C47—84no—
99C22×C47C22×C22⋊C411yescharacteristic
100C22×C47#52016yes—
101C22×C47#53513yes—
102C22×C47#53613yes—
103C22×C47#53716yes—
104C22×C47#56014yes—
105C22×C47#563112yes—
106C4×C47—212no—
107C4×C47—46no—
108Dic47—84no—
109Dic47—84no—
110C22×#126D4711yescharacteristic
111C2×#384—471no—
112C2×#383—471no—
113#1441—476no—
114#1439—476no—
115C2×C47—82no—
116C2×C47#231113yes—
117C2×C47#238314yes—
118C2×C47#281811yescharacteristic
119C2×C47#282113yes—
120C2×C47#282314yes—
121C2×#126—24no—
122C2×#126—24no—
123C2×#126Dic4718yescomplemented
124C22×C4⋊C4—476no—
125C22×C4⋊C4C2×D4713yes—
126C22×C22⋊C4—474no—
127C22×C22⋊C4C2×D4713yes—
128C22×C42C2×D4711yescharacteristic
129#384—944no—
130#383—944no—
131C2×#95—476no—
132C2×#95—4712no—
133C47#1466511yescharacteristic, complemented, Sylow
134C25C22×D4711yescharacteristic
135C23×C4—474no—
136C23×C4—474no—
137C23×C4C22×D4713yes—
138C23×C4C22×D4713yes—
139#126—232no—
140C2×C4⋊C4—26no—
141C2×C4⋊C4—26no—
142C2×C4⋊C4—9412no—
143C2×C4⋊C4—18812no—
144C2×C4⋊C4C2×Dic47112yes—
145C2×C22⋊C4—26no—
146C2×C22⋊C4—26no—
147C2×C22⋊C4—18812no—
148C2×C22⋊C4C2×Dic47112yes—
149C2×C42C2×Dic4714yes—
150C2×C42C22⋊D4714yes—
151C2×C42C22⋊D4714yes—
152#95—9412no—
153#95—18812no—
154C24—21no—
155C24—21no—
156C24—26no—
157C24—26no—
158C24C22×Dic4712yes—
159C24C23×D4711yescharacteristic
160C22×C4—26no—
161C22×C4—26no—
162C22×C4—23no—
163C22×C4—23no—
164C22×C4—26no—
165C22×C4—26no—
166C22×C4—944no—
167C22×C4—944no—
168C22×C4—9412no—
169C22×C4—9412no—
170C22×C4—9412no—
171C22×C4—9412no—
172C22×C4C22×Dic4716yes—
173C22×C4C22×Dic4716yes—
174C22×C4C2×C22⋊D4716yes—
175C22×C4C2×C22⋊D4716yes—
176C4⋊C4—224no—
177C4⋊C4—18824no—
178C22⋊C4—224no—
179C22⋊C4—18816no—
180C42C22⋊Dic47116yescomplemented
181C23—22no—
182C23—212no—
183C23—44no—
184C23—412no—
185C23—43no—
186C23—43no—
187C23—46no—
188C23C23×Dic4711yescharacteristic
189C23C22×C22⋊D4711yescharacteristic
190C23C22×C22⋊D4711yescharacteristic
191C23#1309407016yes—
192C23#1309407116yes—
193C2×C4—212no—
194C2×C4—26no—
195C2×C4—26no—
196C2×C4—212no—
197C2×C4—26no—
198C2×C4—212no—
199C2×C4—43no—
200C2×C4—43no—
201C2×C4—46no—
202C2×C4—1884no—
203C2×C4—1884no—
204C2×C4—18812no—
205C2×C4—18812no—
206C2×C4—18812no—
207C2×C4—18812no—
208C2×C4C2×C22⋊Dic47112yes—
209C22—48no—
210C22—46no—
211C22—81no—
212C22—83no—
213C22—84no—
214C22C22×C22⋊Dic4711yescharacteristic
215C22#1568960616yes—
216C22#1568968713yes—
217C22#1568969413yes—
218C22#1568970116yes—
219C22#1568986414yes—
220C22#15689875112yes—
221C4—212no—
222C4—46no—
223C4—3764no—
224C4—3764no—
225C2—82no—
226C2#2024598013yes—
227C2#2024658014yes—
228C2#2025142811yescharacteristic
229C2#2025144613yes—
230C2#2025146014yes—
subgroupG/H
center, FrattiniC24C23×D47
commutatorC23×C47C23×C4
FittingC22×#126×C47C2
socle, cosocleC24×C47C24
prime radicalC25×C47C23
hypercenterC22×#126D47
hypocenterC47#14665
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C24 ◁ C22×#126

Complements

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

NH ≅ G/Ntype
1. C2×#15688993C2semidirect
2. C2×#15688991C2semidirect
3. #20241641C2semidirect
14. C2×#126×C47C4semidirect
62. C42×C47C22⋊C4semidirect
123. C2×#126Dic47semidirect
133. C47#14665semidirect
180. C42C22⋊Dic47semidirect

Maximal subgroups

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

subgroupindex
C2×#156889932
C2×#156889912
#202416412
#202416332
C22×#126×C472
#1466547