Finite Group Databasebeta

← #26243042

Subgroups — #26243042

Order 12032.

626 conjugacy classes of subgroups; only the 201 normal subgroups are shown.

normal subgroups only

This group has 626 conjugacy classes of subgroups — too many to draw. Only its 201 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#20246686C211yesmaximal, characteristic
2#20244103C213yesmaximal, complemented
3#20243993C213yesmaximal
4#20243942C213yesmaximal, complemented
5#20243569C213yesmaximal
6#20243533C211yesmaximal, characteristic, complemented
7C47×#1843C211yesmaximal, characteristic, complemented
8C22×#10625324C2211yescharacteristic
9C2×Dic47⋊C8C2213yes—
10C2×(C2×D47)⋊C8C2213yes—
11C2×C8×Dic47C2213yes—
12C47⋊#401C411yescharacteristic
13#15689064C413yes—
14#15689052C413yes—
15#15689047C411yescharacteristic
16C4×Dic47⋊C4C2213yes—
17C4×(C2×D47)⋊C4C2213yes—
18#15688878C2213yes—
19#15688878C2213yes—
20#15688877C2213yes—
21#15688877C2213yes—
22C2×C4⋊C8×C47C2213yes—
23C2×C4⋊C8×C47C2213yes—
24C47×#376C2211yescharacteristic
25C22×C4×D47C2311yescharacteristic
26C2×#10625324—43no—
27C22×C47⋊C8C2313yes—
28C22×C47⋊C8C2311yescharacteristic
29C2×Dic47⋊C4C2×C413yes—
30C2×Dic47⋊C4C2×C413yes—
31C2×(C2×D47)⋊C4C2×C413yes—
32C2×(C2×D47)⋊C4C2×C413yes—
33C2×C4×Dic47C2313yes—
34Dic47⋊C8—43no—
35(C2×D47)⋊C8—43no—
36C8×Dic47—43no—
37C47⋊#95C2×C413yes—
38C47⋊#95C2×C413yes—
39C47⋊#95C2×C413yes—
40C47⋊#95C2×C413yes—
41C22×C8×C47C2313yes—
42C22×C8×C47C2311yescharacteristic
43C2×C42×C47C2313yes—
44C4⋊C8×C47—43no—
45C4⋊C8×C47—43no—
46#95×C47C2×C411yescharacteristic
47#95×C47C2×C413yes—
48C23×D47C22×C411yescharacteristic
49C22×C4×C47C22×C413yes—
50C22×C4×C47C22×C413yes—
51C22×C4×C47C2411yescharacteristic
52C22×Dic47C22×C411yescharacteristic
53C22×Dic47C22×C413yes—
54C22×Dic47C22×C413yes—
55C2×C4×D47—23no—
56C2×C4×D47—23no—
57C2×C8×C47—23no—
58C2×C8×C47—23no—
59C2×C8×C47—23no—
60C2×C8×C47—23no—
61C42×C47—23no—
62C42×C47—23no—
63(C2×D47)⋊C4—26no—
64(C2×D47)⋊C4—43no—
65Dic47⋊C4—26no—
66Dic47⋊C4—26no—
67C4×Dic47—23no—
68C4×Dic47—23no—
69C2×C47⋊C8—23no—
70C2×C47⋊C8—23no—
71C2×C47⋊C8—26no—
72#10625324—44no—
73C23×C47C23×C411yescharacteristic
74C22×D47—23no—
75C22×D47—23no—
76C22×D47—21no—
77C2×C4×C47—23no—
78C2×C4×C47—23no—
79C2×C4×C47—23no—
80C2×C4×C47—23no—
81C2×C4×C47—23no—
82C2×C4×C47—23no—
83C2×C4×C47H(5,𝔽2)13yes—
84C2×C4×C472−1+413yes—
85C2×Dic47—23no—
86C2×Dic47—23no—
87C2×Dic47—23no—
88C2×Dic47—23no—
89C2×Dic47—23no—
90C2×Dic47—23no—
91C2×Dic47#13116yes—
92C4×D47—41no—
93C4×D47—43no—
94C8×C47—41no—
95C8×C47—43no—
96C47⋊C8—41no—
97C47⋊C8—43no—
98#23381—471nomaximal, Sylow
99C22×C47#52013yes—
100C22×C47C2×#13113yes—
101C22×C47#56811yescharacteristic
102C2×D47—41no—
103C2×D47—43no—
104C2×D47—43no—
105C4×C47—43no—
106C4×C47—43no—
107C4×C47#56011yescharacteristic
108C4×C47#56313yes—
109Dic47—26no—
110Dic47—41no—
111Dic47—43no—
112#2401—471no—
113#1939—473no—
114#1923—473no—
115#1916—473no—
116#1851—473no—
117#1843D4711yescharacteristic, complemented
118#1840—471no—
119C2×C47#238311yescharacteristic
120C2×C47#299313yes—
121C2×C47#299413yes—
122D47—81no—
123C22×M16—471no—
124C2×C4⋊C8—473no—
125C2×C4⋊C8C2×D4713yes—
126C2×C4⋊C8C2×D4713yes—
127C2×C22⋊C8—473no—
128C2×C4×C8—473no—
129#401—471no—
130#400—473no—
131#397—473no—
132#396—471no—
133C4×C4⋊C4—473no—
134C4×C22⋊C4—473no—
135#376C2×D4711yescharacteristic
136#336—473no—
137#336—473no—
138#336—473no—
139#336—473no—
140C47#2338111yescharacteristic, complemented, Sylow
141C23×C4—471no—
142C2×M16—1883no—
143C22×C8—471no—
144C22×C8—473no—
145C22×C8C22×D4711yescharacteristic
146C22×C8C22×D4713yes—
147C2×C4⋊C4—473no—
148C2×C4⋊C4—473no—
149C2×C22⋊C4—473no—
150C2×C22⋊C4—473no—
151C2×C42—473no—
152C2×C42C22×D4713yes—
153C4⋊C8—43no—
154C4⋊C8—43no—
155C4⋊C8—1883no—
156C22⋊C8—1883no—
157C4×C8—1883no—
158#95—473no—
159#95—473no—
160#95—473no—
161#95—473no—
162#95C4×D4711yescharacteristic
163#95C4×D4713yes—
164C24—471no—
165C22×C4—473no—
166C22×C4—471no—
167C22×C4—473no—
168C22×C4—943no—
169C22×C4—943no—
170C22×C4C2×C4×D4713yes—
171C22×C4C2×C4×D4713yes—
172C22×C4C23×D4711yescharacteristic
173M16—1884no—
174C2×C8—23no—
175C2×C8—23no—
176C2×C8—23no—
177C2×C8—23no—
178C2×C8—943no—
179C2×C8—943no—
180C2×C8—946no—
181C4⋊C4—946no—
182C4⋊C4—946no—
183C22⋊C4—946no—
184C22⋊C4—1883no—
185C42—23no—
186C42—23no—
187C42—943no—
188C42—943no—
189C23—943no—
190C23—943no—
191C23—941no—
192C23C22×C4×D4711yescharacteristic
193C2×C4—23no—
194C2×C4—23no—
195C2×C4—23no—
196C2×C4—23no—
197C2×C4—23no—
198C2×C4—23no—
199C2×C4—476no—
200C2×C4—943no—
201C2×C4—943no—
202C2×C4—943no—
203C2×C4—943no—
204C2×C4—943no—
205C2×C4—943no—
206C2×C4—1881no—
207C2×C4—1883no—
208C2×C4#1309406913yes—
209C2×C4#1309407213yes—
210C8—43no—
211C8—41no—
212C8—1881no—
213C8—1883no—
214C22—1881no—
215C22—1883no—
216C22—1883no—
217C22#1568960413yes—
218C22#131×D4713yes—
219C22#1568988811yescharacteristic
220C4—43no—
221C4—43no—
222C4—946no—
223C4—1881no—
224C4—1883no—
225C4#1568986011yescharacteristic
226C4#1568987213yes—
227C2—3761no—
228C2#2024658111yescharacteristic
229C2#2025253513yes—
230C2#2025254613yes—
subgroupG/H
center, FrattiniC22×C4C23×D47
commutator, socleC23×C47C23×C4
FittingC47×#1843C2
prime radicalC23×D47C22×C4
cosocleC22×C4×C47C24
hypercenter#1843D47
hypocenterC47#23381
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C22×C4 ◁ #1843

Complements

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

NH ≅ G/Ntype
2. #20244103C2semidirect
4. #20243942C2semidirect
6. #20243533C2semidirect
7. C47×#1843C2semidirect
117. #1843D47semidirect
140. C47#23381semidirect

Maximal subgroups

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

subgroupindex
#202466862
#202441032
#202439932
#202439422
#202435692
#202435332
C47×#18432
#2338147