Finite Group Databasebeta

← #26103599

Subgroups — #26103599

Order 12032.

374 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 374 conjugacy classes of subgroups — too many to draw. Only its 71 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#20241718C212yesmaximal
2C47×#1445C211yesmaximal, characteristic
3#15689087—22no—
4#15688849—22no—
5#15688849C414yescomplemented
6C22×C4⋊C4×C47C2211yescharacteristic
7C2×#95×C47—22no—
8C2×#95×C47C411yescharacteristic
9C2×#95×C47C411yescharacteristic
10C2×#10625329—42no—
11C22⋊(C47⋊C8)—24no—
12C22⋊(C47⋊C8)—42no—
13C23×C4×C47—21no—
14C23×C4×C47—21no—
15C23×C4×C47C2×C411yescharacteristic
16C23×C4×C47D412yes—
17C2×C4⋊C4×C47—41no—
18C2×C4⋊C4×C47—41no—
19C2×C4⋊C4×C47C2×C412yes—
20C2×C4⋊C4×C47D411yescharacteristic
21C2×C4⋊C4×C47Q811yescharacteristic
22#95×C47—44no—
23#95×C47—41no—
24#95×C47—41no—
25C24×C47C22⋊C411yescharacteristic
26C22×C4×C47—21no—
27C22×C4×C47—21no—
28C22×C4×C47—22no—
29C22×C4×C47—22no—
30C22×C4×C47—22no—
31C22×C4×C47—22no—
32C22×C4×C47—42no—
33C22×C4×C47—42no—
34C22×C4×C47—42no—
35C22×C4×C47—42no—
36C22×C4×C47—42no—
37C22×C4×C47—41no—
38C22×C4×C47—41no—
39C22×C4×C47—42no—
40C22×C4×C47—42no—
41C22×C4×C47—41no—
42C22×C4×C47—41no—
43C22×C4×C47C4211yescharacteristic
44C22×C4×C47C22⋊C412yes—
45C22×C4×C47C4⋊C411yescharacteristic
46C22×C4×C47C4⋊C412yes—
47C4⋊C4×C47—42no—
48C4⋊C4×C47—41no—
49C4⋊C4×C47—41no—
50#10625329—82no—
51#10625329—82no—
52C2×C47⋊C8—44no—
53C23×C47—22no—
54C23×C47—21no—
55C23×C47—21no—
56C23×C47—21no—
57C23×C47—21no—
58C23×C47#9511yescharacteristic
59C23×C47#9911yescharacteristic
60C23×C47#9911yescharacteristic
61C2×C4×C47—22no—
62C2×C4×C47—21no—
63C2×C4×C47—22no—
64C2×C4×C47—21no—
65C2×C4×C47—22no—
66C2×C4×C47—22no—
67C2×C4×C47—42no—
68C2×C4×C47—42no—
69C2×C4×C47—42no—
70C2×C4×C47—42no—
71C2×C4×C47—42no—
72C2×C4×C47—42no—
73C2×C4×C47—44no—
74C2×C4×C47—44no—
75C2×C4×C47—42no—
76C2×C4×C47—42no—
77C2×C4×C47—82no—
78C2×C4×C47—82no—
79C2×C4×C47—82no—
80C2×C4×C47—81no—
81C2×C4×C47—81no—
82C2×C4×C47—81no—
83C2×C4×C47—81no—
84C47⋊C8—84no—
85#7326—471nomaximal, Sylow
86C22×C47—22no—
87C22×C47—22no—
88C22×C47—22no—
89C22×C47—22no—
90C22×C47—21no—
91C22×C47—21no—
92C22×C47—21no—
93C22×C47—21no—
94C22×C47—21no—
95C22×C47—21no—
96C22×C47#34211yescharacteristic
97C22×C47#34412yes—
98C22×C47#35311yescharacteristic
99C22×C47#35411yescharacteristic
100C22×C47#35511yescharacteristic
101C22×C47#35611yescharacteristic
102C4×C47—41no—
103C4×C47—41no—
104C4×C47—42no—
105C4×C47—42no—
106C4×C47—82no—
107C4×C47—82no—
108#1463—472no—
109#1445D4711yescharacteristic
110C2×C47—21no—
111C2×C47—21no—
112C2×C47—22no—
113C2×C47—21no—
114C2×C47—21no—
115C2×C47#130611yescharacteristic
116C2×C47#139311yescharacteristic
117C2×C47#139411yescharacteristic
118C22×C4⋊C4C2×D4711yescharacteristic
119#407—942no—
120C2×#95—22no—
121C2×#95Dic4711yescharacteristic
122C2×#95Dic4711yescharacteristic
123#324—474no—
124#324—942no—
125C47#732611yescharacteristic, complemented, Sylow
126C23×C4—21no—
127C23×C4—21no—
128C23×C4C2×Dic4711yescharacteristic
129C23×C4C22⋊D4712yes—
130C2×M16—1882no—
131C2×C4⋊C4—41no—
132C2×C4⋊C4—41no—
133C2×C4⋊C4Dic9411yescharacteristic
134C2×C4⋊C4C4×D4712yes—
135C2×C4⋊C4D18811yescharacteristic
136C22⋊C8—944no—
137C22⋊C8—1882no—
138#95—41no—
139#95—41no—
140#95—44no—
141C24C22⋊Dic4711yescharacteristic
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—22no—
145C22×C4—22no—
146C22×C4—22no—
147C22×C4—22no—
148C22×C4—42no—
149C22×C4—42no—
150C22×C4—42no—
151C22×C4—42no—
152C22×C4—42no—
153C22×C4—42no—
154C22×C4—41no—
155C22×C4—41no—
156C22×C4—42no—
157C22×C4—41no—
158C22×C4—41no—
159C22×C4C4×Dic4711yescharacteristic
160C22×C4Dic47⋊C412yes—
161C22×C4C4⋊Dic4711yescharacteristic
162C22×C4(C2×D47)⋊C412yes—
163M16—3762no—
164M16—3762no—
165C2×C8—1884no—
166C4⋊C4—41no—
167C4⋊C4—42no—
168C4⋊C4—41no—
169C23—22no—
170C23—21no—
171C23—21no—
172C23—21no—
173C23—21no—
174C23C47⋊#9511yescharacteristic
175C23#1309393911yescharacteristic
176C23#1309393911yescharacteristic
177C2×C4—22no—
178C2×C4—21no—
179C2×C4—22no—
180C2×C4—21no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—42no—
184C2×C4—42no—
185C2×C4—42no—
186C2×C4—42no—
187C2×C4—42no—
188C2×C4—42no—
189C2×C4—44no—
190C2×C4—44no—
191C2×C4—42no—
192C2×C4—42no—
193C2×C4—82no—
194C2×C4—82no—
195C2×C4—82no—
196C2×C4—81no—
197C2×C4—81no—
198C2×C4—81no—
199C2×C4—81no—
200C8—3764no—
201C22—22no—
202C22—22no—
203C22—22no—
204C22—22no—
205C22—21no—
206C22—21no—
207C22—21no—
208C22—21no—
209C22—21no—
210C22—21no—
211C22#1568889011yescharacteristic
212C22#1568889312yes—
213C22#1568891611yescharacteristic
214C22#1568891811yescharacteristic
215C22#1568892011yescharacteristic
216C22#1568892211yescharacteristic
217C4—42no—
218C4—42no—
219C4—41no—
220C4—41no—
221C4—82no—
222C4—82no—
223C2—21no—
224C2—21no—
225C2—22no—
226C2—21no—
227C2—21no—
228C2#2024129611yescharacteristic
229C2#2024151311yescharacteristic
230C2#2024151511yescharacteristic
subgroupG/H
centerC22#15688890
commutatorC22×C4×C47C42
FrattiniC22×C4⋊C4C2×D47
FittingC47×#1445C2
socleC22×C47#342
prime radicalC24×C47C22⋊C4
cosocleC22×C4⋊C4×C47C22
hypercenter#1445D47
hypocenterC47#7326
LCSG ▷ C22×C4×C47 ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ #1445

Complements

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

NH ≅ G/Ntype
5. #15688849C4semidirect
125. C47#7326semidirect

Maximal subgroups

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

subgroupindex
#202417182
C47×#14452
#732647