Finite Group Databasebeta

← #26109732

Subgroups — #26109732

Order 12032.

404 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 404 conjugacy classes of subgroups — too many to draw. Only its 99 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#20245883C211yesmaximal, characteristic
2#20241700C212yesmaximal
3#20241220C212yesmaximal, complemented
4#20241217C211yesmaximal, characteristic, complemented
5C47×#1274C211yesmaximal, characteristic, complemented
6C2×#13093981C2211yescharacteristic
7#15689176C2212yes—
8#15689012—22no—
9#15689005—22no—
10C4×Dic47⋊C4C412yes—
11C4×(C2×D47)⋊C4C412yes—
12C42×Dic47C2211yescharacteristic
13#15688848—22no—
14#15688846—22no—
15C47×#431C2212yes—
16C47×#376C2211yescharacteristic
17C22×C4×D47C2×C411yescharacteristic
18#13093981—22no—
19#13093981—21no—
20#13093981—21no—
21C2×Dic47⋊C4—22no—
22C2×Dic47⋊C4D412yes—
23C2×(C2×D47)⋊C4—22no—
24C2×(C2×D47)⋊C4D412yes—
25C2×C4×Dic47—22no—
26C2×C4×Dic47C2×C412yes—
27C2×C4×Dic47C2×C411yescharacteristic
28C4⋊(C47⋊C8)—22no—
29C22⋊(C47⋊C8)—22no—
30C4×C47⋊C8—22no—
31C47⋊#95—22no—
32C47⋊#95—22no—
33C2×C42×C47C2×C412yes—
34C2×C42×C47C2311yescharacteristic
35C4⋊C8×C47—22no—
36C22⋊C8×C47—22no—
37C4×C8×C47—22no—
38#95×C47—22no—
39C23×D47C22⋊C411yescharacteristic
40C22×C4×C47—22no—
41C22×C4×C47C22×C411yescharacteristic
42C22×C4×C47C2×D412yes—
43C22×Dic47—22no—
44C22×Dic47C22⋊C411yescharacteristic
45C22×Dic47C22⋊C412yes—
46C2×C4×D47—21no—
47C2×C4×D47—21no—
48C2×C4×D47—22no—
49C2×C4×D47—22no—
50C2×C8×C47—42no—
51C42×C47—22no—
52C42×C47—22no—
53C42×C47#4112yes—
54C42×C47#4111yescharacteristic
55C42×C47#4111yescharacteristic
56(C2×D47)⋊C4—22no—
57(C2×D47)⋊C4—22no—
58(C2×D47)⋊C4—44no—
59Dic47⋊C4—22no—
60Dic47⋊C4—22no—
61Dic47⋊C4—44no—
62C4×Dic47—21no—
63C4×Dic47—21no—
64C4×Dic47—22no—
65C4×Dic47—22no—
66C4×Dic47—44no—
67C2×C47⋊C8—42no—
68C23×C47C2×C22⋊C411yescharacteristic
69C22×D47—21no—
70C22×D47—22no—
71C22×D47—42no—
72C2×C4×C47—42no—
73C2×C4×C47—42no—
74C2×C4×C47—42no—
75C2×C4×C47—42no—
76C2×C4×C47#11711yescharacteristic
77C2×C4×C47#11711yescharacteristic
78C2×C4×C47#12312yes—
79C2×C4×C47#12312yes—
80C2×C4×C47#13112yes—
81C2×C4×C47#13112yes—
82C2×Dic47—24no—
83C2×Dic47—22no—
84C2×Dic47—21no—
85C2×Dic47—22no—
86C2×Dic47—42no—
87C2×Dic47—42no—
88C2×Dic47—42no—
89C4×D47—41no—
90C4×D47—42no—
91C4×D47—41no—
92C8×C47—82no—
93C47⋊C8—82no—
94#8733—471nomaximal, Sylow
95C22×C47—22no—
96C22×C47#38111yescharacteristic
97C22×C47#40812yes—
98C2×D47—41no—
99C2×D47—42no—
100C2×D47—82no—
101C4×C47—22no—
102C4×C47—22no—
103C4×C47—82no—
104C4×C47#43311yescharacteristic
105C4×C47#43312yes—
106C4×C47#43311yescharacteristic
107Dic47—44no—
108Dic47—41no—
109Dic47—42no—
110#2298—471no—
111#1458—472no—
112#1276—472no—
113#1274—471no—
114#1274D4711yescharacteristic, complemented
115C2×C47—22no—
116C2×C47#181811yescharacteristic
117C2×C47#182712yes—
118D47—81no—
119C2×#117—471no—
120#431—472no—
121#431C2×D4712yes—
122#388—942no—
123#387—942no—
124C4×C4⋊C4—472no—
125C4×C22⋊C4—472no—
126#376C2×D4711yescharacteristic
127C43—471no—
128#324—942no—
129#323—942no—
130C47#873311yescharacteristic, complemented, Sylow
131C23×C4—471no—
132#117—942no—
133#117—941no—
134#117—941no—
135C2×C4⋊C4—472no—
136C2×C4⋊C4—942no—
137C2×C22⋊C4—472no—
138C2×C22⋊C4—942no—
139C2×C42—471no—
140C2×C42—472no—
141C2×C42—942no—
142C2×C42C4×D4712yes—
143C2×C42C22×D4711yescharacteristic
144C4⋊C8—22no—
145C4⋊C8—942no—
146C22⋊C8—22no—
147C22⋊C8—942no—
148C4×C8—22no—
149C4×C8—942no—
150#95—22no—
151#95—942no—
152#95—942no—
153C24—471no—
154C22×C4—22no—
155C22×C4—471no—
156C22×C4—472no—
157C22×C4—942no—
158C22×C4—941no—
159C22×C4—941no—
160C22×C4—942no—
161C22×C4—942no—
162C22×C4C2×C4×D4711yescharacteristic
163C22×C4D4×D4712yes—
164C2×C8—42no—
165C2×C8—1882no—
166C4⋊C4—942no—
167C4⋊C4—942no—
168C4⋊C4—1884no—
169C22⋊C4—942no—
170C22⋊C4—942no—
171C22⋊C4—1884no—
172C42—22no—
173C42—22no—
174C42—941no—
175C42—941no—
176C42—942no—
177C42—942no—
178C42—1884no—
179C42#1062535012yes—
180C42#1062535211yescharacteristic
181C42#1062535411yescharacteristic
182C23—942no—
183C23—941no—
184C23—1882no—
185C23C22⋊C4×D4711yescharacteristic
186C2×C4—42no—
187C2×C4—42no—
188C2×C4—42no—
189C2×C4—42no—
190C2×C4—944no—
191C2×C4—942no—
192C2×C4—941no—
193C2×C4—942no—
194C2×C4—1882no—
195C2×C4—1882no—
196C2×C4—1881no—
197C2×C4—1882no—
198C2×C4—1881no—
199C2×C4—1882no—
200C2×C4#1309398111yescharacteristic
201C2×C4#1309398211yescharacteristic
202C2×C4#1309400612yes—
203C2×C4#1309400912yes—
204C2×C4#1309403012yes—
205C2×C4#1309403212yes—
206C8—82no—
207C8—3762no—
208C22—22no—
209C22—3762no—
210C22—1881no—
211C22—1882no—
212C22#1568898511yescharacteristic
213C22#1568908812yes—
214C4—22no—
215C4—22no—
216C4—82no—
217C4—1884no—
218C4—1881no—
219C4—1882no—
220C4#1568918211yescharacteristic
221C4#1568918312yes—
222C4#1568918511yescharacteristic
223C2—22no—
224C2—3761no—
225C2#2024345811yescharacteristic
226C2#2024349012yes—
subgroupG/H
centerC2×C4#13093981
commutatorC22×C4×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#1274C2
socleC22×C47#381
prime radicalC23×D47C22⋊C4
cosocleC2×C42×C47C23
hypercenter#1274D47
hypocenterC47#8733
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ #376 ◁ #1274

Complements

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

NH ≅ G/Ntype
3. #20241220C2semidirect
4. #20241217C2semidirect
5. C47×#1274C2semidirect
114. #1274D47semidirect
130. C47#8733semidirect

Maximal subgroups

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

subgroupindex
#202458832
#202417002
#202412202
#202412172
C47×#12742
#873347