Finite Group Databasebeta

← #26102898

Subgroups — #26102898

Order 12032.

240 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 53 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#20243454C211yesmaximal, characteristic
2#20241312C211yesmaximal, characteristic
3C47×#1314C211yesmaximal, characteristic
4C22×C4×Dic47C411yescharacteristic
5C2×C22⋊(C47⋊C8)C411yescharacteristic
6C22⋊(C47⋊C16)—21no—
7#15688877—21no—
8#15688877—21no—
9C2×C22⋊C8×C47C2211yescharacteristic
10C47×C22⋊C16—21no—
11C23×Dic47C811yescharacteristic
12C23×Dic47C811yescharacteristic
13C22×C47⋊C8—21no—
14C2×C4×Dic47—21no—
15C2×C4×Dic47—21no—
16C2×C4×Dic47—41no—
17C2×C4×Dic47—41no—
18C2×C47⋊C16—41no—
19C22⋊(C47⋊C8)—41no—
20C23×C4×C47C2×C411yescharacteristic
21C22×C8×C47D411yescharacteristic
22C22×C8×C47D411yescharacteristic
23C2×C16×C47—41no—
24C22⋊C8×C47—41no—
25C24×C47C2×C811yescharacteristic
26C22×C4×C47—21no—
27C22×C4×C47—21no—
28C22×C4×C47—41no—
29C22×C4×C47—41no—
30C22×C4×C47C22⋊C411yescharacteristic
31C22×C4×C47M1611yescharacteristic
32C22×Dic47—21no—
33C22×Dic47—21no—
34C22×Dic47—21no—
35C22×Dic47—42no—
36C22×Dic47—41no—
37C22×Dic47—41no—
38C22×Dic47—41no—
39C22×Dic47C1612yescomplemented
40C2×C8×C47—21no—
41C2×C8×C47—21no—
42C2×C8×C47—41no—
43C2×C8×C47—41no—
44C4×Dic47—42no—
45C4×Dic47—41no—
46C4×Dic47—41no—
47C2×C47⋊C8—41no—
48C2×C47⋊C8—41no—
49C2×C47⋊C8—41no—
50C16×C47—81no—
51C47⋊C16—81no—
52C23×C47—21no—
53C23×C47—21no—
54C23×C47—41no—
55C23×C47—41no—
56C23×C47C22⋊C811yescharacteristic
57C23×C47C2×C1611yescharacteristic
58C23×C47M3211yescharacteristic
59C2×C4×C47—21no—
60C2×C4×C47—21no—
61C2×C4×C47—21no—
62C2×C4×C47—42no—
63C2×C4×C47—41no—
64C2×C4×C47—41no—
65C2×C4×C47—41no—
66C2×C4×C47#9911yescharacteristic
67C2×C4×C47#10011yescharacteristic
68C2×Dic47—22no—
69C2×Dic47—22no—
70C2×Dic47—42no—
71C2×Dic47—41no—
72C2×Dic47—42no—
73C2×Dic47—41no—
74C2×Dic47—41no—
75C2×Dic47—41no—
76C2×Dic47—42no—
77C2×Dic47—42no—
78C8×C47—41no—
79C8×C47—41no—
80C47⋊C8—81no—
81#7055—471nomaximal, Sylow
82C22×C47—21no—
83C22×C47—21no—
84C22×C47—21no—
85C22×C47—21no—
86C22×C47—41no—
87C22×C47—41no—
88C22×C47—42no—
89C22×C47—41no—
90C22×C47—41no—
91C22×C47#32311yescharacteristic
92C22×C47C22⋊C1611yescharacteristic
93C22×C47#34911yescharacteristic
94C4×C47—21no—
95C4×C47—41no—
96C4×C47#35211yescharacteristic
97C4×C47#35211yescharacteristic
98Dic47—42no—
99Dic47—42no—
100#1817—471no—
101#1314—471no—
102#1314D4711yescharacteristic
103C2×C47—21no—
104C2×C47—21no—
105C2×C47—41no—
106C2×C47—41no—
107C2×C47#131411yescharacteristic
108C2×C47#131711yescharacteristic
109C2×C47#132011yescharacteristic
110C22×C42—471no—
111C2×C22⋊C8—471no—
112C2×C22⋊C8C2×D4711yescharacteristic
113C22⋊C16—21no—
114C22⋊C16—941no—
115#336—941no—
116#336—941no—
117C47#705511yescharacteristic, complemented, Sylow
118C23×C4—471no—
119C23×C4—471no—
120C23×C4C4×D4711yescharacteristic
121C22×C8—941no—
122C22×C8D18811yescharacteristic
123C22×C8C22⋊D4711yescharacteristic
124C2×C42—941no—
125C2×C42—941no—
126C2×C42—1881no—
127C2×C42—1881no—
128C2×C16—41no—
129C2×C16—1881no—
130C22⋊C8—41no—
131C22⋊C8—1881no—
132C24C8×D4711yescharacteristic
133C22×C4—21no—
134C22×C4—21no—
135C22×C4—41no—
136C22×C4—41no—
137C22×C4—472no—
138C22×C4—941no—
139C22×C4—941no—
140C22×C4—941no—
141C22×C4—1882no—
142C22×C4—1881no—
143C22×C4—1881no—
144C22×C4—1881no—
145C22×C4#1062532411yescharacteristic
146C22×C4(C2×D47)⋊C411yescharacteristic
147C2×C8—21no—
148C2×C8—21no—
149C2×C8—41no—
150C2×C8—41no—
151C2×C8—1881no—
152C2×C8—1881no—
153C2×C8—1881no—
154C42—1882no—
155C42—1881no—
156C42—1881no—
157C16—81no—
158C16—3761no—
159C23—21no—
160C23—21no—
161C23—41no—
162C23—41no—
163C23(C2×D47)⋊C811yescharacteristic
164C23C16×D4711yescharacteristic
165C23#1309396311yescharacteristic
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—42no—
170C2×C4—41no—
171C2×C4—41no—
172C2×C4—41no—
173C2×C4—942no—
174C2×C4—942no—
175C2×C4—1882no—
176C2×C4—1881no—
177C2×C4—1882no—
178C2×C4—1881no—
179C2×C4—1881no—
180C2×C4—1881no—
181C2×C4—1882no—
182C2×C4—1882no—
183C2×C4#1309393811yescharacteristic
184C2×C4#1309394011yescharacteristic
185C8—41no—
186C8—41no—
187C8—3761no—
188C22—21no—
189C22—21no—
190C22—21no—
191C22—21no—
192C22—41no—
193C22—41no—
194C22—42no—
195C22—41no—
196C22—41no—
197C22#1568884611yescharacteristic
198C22(C2×D47)⋊C1611yescharacteristic
199C22#1568890411yescharacteristic
200C4—21no—
201C4—41no—
202C4—1882no—
203C4—1882no—
204C4#1568891211yescharacteristic
205C4#1568891411yescharacteristic
206C2—21no—
207C2—21no—
208C2—41no—
209C2—41no—
210C2#2024131111yescharacteristic
211C2#2024131711yescharacteristic
212C2#2024132311yescharacteristic
subgroupG/H
centerC2×C4#13093938
commutatorC23×C47C2×C16
FrattiniC2×C22⋊C8C2×D47
FittingC47×#1314C2
socleC22×C47#323
prime radicalC24×C47C2×C8
cosocleC2×C22⋊C8×C47C22
hypercenter#1314D47
hypocenterC47#7055
LCSG ▷ C23×C47 ▷ C22×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×C22⋊C8 ◁ #1314

Complements

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

NH ≅ G/Ntype
39. C22×Dic47C16semidirect
117. C47#7055semidirect

Maximal subgroups

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

subgroupindex
#202434542
#202413122
C47×#13142
#705547