Finite Group Databasebeta

← #26103627

Subgroups — #26103627

Order 12032.

356 conjugacy classes of subgroups; only the 43 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 43 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#20241318C212yesmaximal, complemented
2C47×#1782C211yesmaximal, characteristic
3#15688847—22no—
4C22×#41×C47C411yescharacteristic
5C2×#99×C47—22no—
6C47×#381C411yescharacteristic
7C47×#381C2211yescharacteristic
8C22⋊(C47⋊C8)—42no—
9C2×#41×C47—41no—
10C2×#41×C47—41no—
11C2×#41×C47—44no—
12C22×Q8×C47C811yescharacteristic, complemented
13C22×D4×C47—21no—
14C22×D4×C47C811yescharacteristic, complemented
15C23×C4×C47—21no—
16C23×C4×C47C2×C411yescharacteristic
17C2×C22⋊C4×C47—21no—
18C2×C22⋊C4×C47D412yes—
19#99×C47—44no—
20#95×C47—41no—
21#95×C47—41no—
22C24×C47—21no—
23C24×C47C22⋊C411yescharacteristic
24#41×C47—44no—
25#41×C47—44no—
26#41×C47—44no—
27#41×C47—44no—
28C2×Q8×C47—42no—
29C2×Q8×C47—41no—
30C2×D4×C47—24no—
31C2×D4×C47—41no—
32C2×D4×C47—42no—
33C2×D4×C47—41no—
34C2×D4×C47—41no—
35C2×D4×C47—42no—
36C22×C4×C47—21no—
37C22×C4×C47—21no—
38C22×C4×C47—21no—
39C22×C4×C47—22no—
40C22×C4×C47—42no—
41C22×C4×C47—42no—
42C22×C4×C47—42no—
43C22×C4×C47—42no—
44C22×C4×C47—41no—
45C22×C4×C47C2×C811yescharacteristic
46C22×C4×C47M1611yescharacteristic
47C22⋊C4×C47—42no—
48C22⋊C4×C47—42no—
49C2×C47⋊C8—82no—
50C23×C47—22no—
51C23×C47—22no—
52C23×C47—41no—
53C23×C47—41no—
54C23×C47—81no—
55C23×C47—82no—
56C23×C47C22⋊C811yescharacteristic
57C23×C47#9911yescharacteristic
58C23×C47#10011yescharacteristic
59Q8×C47—42no—
60Q8×C47—42no—
61D4×C47—44no—
62D4×C47—42no—
63D4×C47—42no—
64D4×C47—44no—
65C2×C4×C47—22no—
66C2×C4×C47—21no—
67C2×C4×C47—21no—
68C2×C4×C47—41no—
69C2×C4×C47—42no—
70C2×C4×C47—48no—
71C2×C4×C47—41no—
72C2×C4×C47—42no—
73C2×C4×C47—41no—
74C2×C4×C47—41no—
75C2×C4×C47—42no—
76C2×C4×C47—82no—
77C2×C4×C47—81no—
78C2×C4×C47—82no—
79C47⋊C8—162no—
80#7339—471nomaximal, Sylow
81C22×C47—21no—
82C22×C47—21no—
83C22×C47—41no—
84C22×C47—41no—
85C22×C47—41no—
86C22×C47—42no—
87C22×C47—82no—
88C22×C47—82no—
89C22×C47—82no—
90C22×C47—82no—
91C22×C47#32311yescharacteristic
92C22×C47AΣL(1,𝔽16)12yes—
93C4×C47—41no—
94C4×C47—42no—
95C4×C47—41no—
96C4×C47—82no—
97C4×C47—161no—
98#1782D4711yescharacteristic
99#1317—472no—
100C2×C47—22no—
101C2×C47—21no—
102C2×C47—41no—
103C2×C47—41no—
104C2×C47—82no—
105C2×C47#131611yescharacteristic
106C22×#41Dic4711yescharacteristic
107C2×#99—22no—
108#381Dic4711yescharacteristic
109#381C2×D4711yescharacteristic
110#323—942no—
111C47#733911yescharacteristic, complemented, Sylow
112C2×#41—41no—
113C2×#41—41no—
114C2×#41—44no—
115C22×Q8C47⋊C811yescharacteristic, complemented
116C22×D4—21no—
117C22×D4C47⋊C811yescharacteristic, complemented
118C23×C4—21no—
119C23×C4C2×Dic4711yescharacteristic
120C2×C22⋊C4—21no—
121C2×C22⋊C4C22⋊D4712yes—
122#99—44no—
123C22⋊C8—1882no—
124#95—41no—
125#95—41no—
126C24—21no—
127C24C22⋊Dic4711yescharacteristic
128#41—44no—
129#41—44no—
130#41—44no—
131#41—44no—
132C2×Q8—42no—
133C2×Q8—41no—
134C2×D4—24no—
135C2×D4—41no—
136C2×D4—42no—
137C2×D4—41no—
138C2×D4—41no—
139C2×D4—42no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—22no—
144C22×C4—42no—
145C22×C4—42no—
146C22×C4—42no—
147C22×C4—42no—
148C22×C4—41no—
149C22×C4C2×C47⋊C811yescharacteristic
150C22×C4#1062532911yescharacteristic
151C2×C8—3762no—
152C22⋊C4—42no—
153C22⋊C4—42no—
154C23—22no—
155C23—22no—
156C23—41no—
157C23—41no—
158C23—81no—
159C23—82no—
160C23C22⋊(C47⋊C8)11yescharacteristic
161C23#1309393911yescharacteristic
162C23#1309394111yescharacteristic
163Q8—42no—
164Q8—42no—
165D4—44no—
166D4—42no—
167D4—42no—
168D4—44no—
169C2×C4—22no—
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—41no—
173C2×C4—42no—
174C2×C4—48no—
175C2×C4—41no—
176C2×C4—42no—
177C2×C4—41no—
178C2×C4—41no—
179C2×C4—42no—
180C2×C4—82no—
181C2×C4—81no—
182C2×C4—82no—
183C8—7522no—
184C22—21no—
185C22—21no—
186C22—41no—
187C22—41no—
188C22—41no—
189C22—42no—
190C22—82no—
191C22—82no—
192C22—82no—
193C22—82no—
194C22#1568884711yescharacteristic
195C22#1568891012yes—
196C4—41no—
197C4—42no—
198C4—41no—
199C4—82no—
200C4—161no—
201C2—22no—
202C2—21no—
203C2—41no—
204C2—41no—
205C2—82no—
206C2#2024131611yescharacteristic
subgroupG/H
centerC2#20241316
commutatorC22×C4×C47C2×C8
Frattini#381C2×D47
FittingC47×#1782C2
socleC2×C47#1316
prime radicalC22×#41×C47C4
cosocleC47×#381C22
hypercenter#1782D47
hypocenterC47#7339
LCSG ▷ C22×C4×C47 ▷ C23×C47 ▷ C22×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2 ◁ C23 ◁ C24 ◁ #381 ◁ #1782

Complements

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

NH ≅ G/Ntype
1. #20241318C2semidirect
12. C22×Q8×C47C8semidirect
14. C22×D4×C47C8semidirect
111. C47#7339semidirect
115. C22×Q8C47⋊C8semidirect
117. C22×D4C47⋊C8semidirect

Maximal subgroups

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

subgroupindex
#202413182
C47×#17822
#733947