Finite Group Databasebeta

← #26103098

Subgroups — #26103098

Order 12032.

368 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 368 conjugacy classes of subgroups — too many to draw. Only its 87 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#20241231C212yesmaximal
2C47×#1445C211yesmaximal, characteristic
3C2×C22⋊(C47⋊C8)—22no—
4C22×C4⋊C4×C47C411yescharacteristic
5C2×#95×C47—22no—
6C2×#95×C47C411yescharacteristic
7C2×#95×C47C2211yescharacteristic
8C22×C47⋊C8—42no—
9C22⋊(C47⋊C8)—24no—
10C22⋊(C47⋊C8)—42no—
11C23×C4×C47—21no—
12C23×C4×C47—21no—
13C23×C4×C47C2×C411yescharacteristic
14C23×C4×C47D412yes—
15C2×C4⋊C4×C47—21no—
16C2×C4⋊C4×C47—41no—
17C2×C4⋊C4×C47—41no—
18C2×C4⋊C4×C47C812yescomplemented
19#95×C47—41no—
20#95×C47—44no—
21#95×C47—41no—
22C24×C47C22⋊C411yescharacteristic
23C22×C4×C47—21no—
24C22×C4×C47—22no—
25C22×C4×C47—21no—
26C22×C4×C47—22no—
27C22×C4×C47—22no—
28C22×C4×C47—22no—
29C22×C4×C47—42no—
30C22×C4×C47—42no—
31C22×C4×C47—42no—
32C22×C4×C47—42no—
33C22×C4×C47—41no—
34C22×C4×C47—42no—
35C22×C4×C47—42no—
36C22×C4×C47—41no—
37C22×C4×C47—41no—
38C22×C4×C47—41no—
39C22×C4×C47—42no—
40C22×C4×C47C2×C811yescharacteristic
41C22×C4×C47M1611yescharacteristic
42C22×C4×C47SD1612yes—
43C22×C4×C47Q1612yes—
44C4⋊C4×C47—42no—
45C4⋊C4×C47—42no—
46C2×C47⋊C8—44no—
47C2×C47⋊C8—82no—
48C2×C47⋊C8—82no—
49C23×C47—22no—
50C23×C47—21no—
51C23×C47—21no—
52C23×C47C22⋊C811yescharacteristic
53C23×C47#9911yescharacteristic
54C23×C47#10011yescharacteristic
55C23×C47#10312yes—
56C23×C47#10412yes—
57C2×C4×C47—21no—
58C2×C4×C47—21no—
59C2×C4×C47—22no—
60C2×C4×C47—22no—
61C2×C4×C47—42no—
62C2×C4×C47—42no—
63C2×C4×C47—42no—
64C2×C4×C47—42no—
65C2×C4×C47—41no—
66C2×C4×C47—41no—
67C2×C4×C47—42no—
68C2×C4×C47—42no—
69C2×C4×C47—42no—
70C2×C4×C47—44no—
71C2×C4×C47—44no—
72C2×C4×C47—42no—
73C2×C4×C47—82no—
74C2×C4×C47—81no—
75C2×C4×C47—82no—
76C2×C4×C47—82no—
77C2×C4×C47—81no—
78C2×C4×C47—81no—
79C2×C4×C47—81no—
80C47⋊C8—84no—
81#7132—471nomaximal, Sylow
82C22×C47—21no—
83C22×C47—21no—
84C22×C47—21no—
85C22×C47—21no—
86C22×C47—22no—
87C22×C47—22no—
88C22×C47—22no—
89C22×C47—22no—
90C22×C47#32311yescharacteristic
91C22×C47Q8⋊C812yes—
92C22×C47#32812yes—
93C22×C47#32911yescharacteristic
94C22×C47#33311yescharacteristic
95C22×C47#35212yes—
96C22×C47#35311yescharacteristic
97C22×C47#35511yescharacteristic
98C4×C47—41no—
99C4×C47—41no—
100C4×C47—42no—
101C4×C47—42no—
102C4×C47—82no—
103C4×C47—82no—
104#1445D4711yescharacteristic
105#1280—472no—
106C2×C47—21no—
107C2×C47—22no—
108C2×C47—21no—
109C2×C47#127211yescharacteristic
110C2×C47#131711yescharacteristic
111C2×C47#131811yescharacteristic
112C2×C47#135212yes—
113C2×C47#135312yes—
114C22×C4⋊C4Dic4711yescharacteristic
115C2×C22⋊C8—942no—
116C2×#95—22no—
117C2×#95Dic4711yescharacteristic
118C2×#95C2×D4711yescharacteristic
119C47#713211yescharacteristic, complemented, Sylow
120C23×C4—21no—
121C23×C4—21no—
122C23×C4C2×Dic4711yescharacteristic
123C23×C4C22⋊D4712yes—
124C22×C8—1882no—
125C2×C4⋊C4—21no—
126C2×C4⋊C4—41no—
127C2×C4⋊C4—41no—
128C2×C4⋊C4C47⋊C812yescomplemented
129C22⋊C8—944no—
130C22⋊C8—1882no—
131#95—41no—
132#95—41no—
133#95—44no—
134C24C22⋊Dic4711yescharacteristic
135C22×C4—21no—
136C22×C4—22no—
137C22×C4—21no—
138C22×C4—22no—
139C22×C4—22no—
140C22×C4—22no—
141C22×C4—42no—
142C22×C4—42no—
143C22×C4—42no—
144C22×C4—42no—
145C22×C4—42no—
146C22×C4—41no—
147C22×C4—42no—
148C22×C4—41no—
149C22×C4—41no—
150C22×C4—41no—
151C22×C4—42no—
152C22×C4C2×C47⋊C811yescharacteristic
153C22×C4#1062532911yescharacteristic
154C22×C4#1062533612yes—
155C22×C4#1062533712yes—
156C2×C8—3762no—
157C2×C8—3762no—
158C2×C8—1884no—
159C4⋊C4—42no—
160C4⋊C4—42no—
161C23—22no—
162C23—21no—
163C23—21no—
164C23C22⋊(C47⋊C8)11yescharacteristic
165C23#1309393911yescharacteristic
166C23#1309394111yescharacteristic
167C23#1309394812yes—
168C23#1309395112yes—
169C2×C4—21no—
170C2×C4—21no—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—42no—
174C2×C4—42no—
175C2×C4—42no—
176C2×C4—42no—
177C2×C4—41no—
178C2×C4—41no—
179C2×C4—42no—
180C2×C4—42no—
181C2×C4—44no—
182C2×C4—42no—
183C2×C4—44no—
184C2×C4—42no—
185C2×C4—82no—
186C2×C4—82no—
187C2×C4—81no—
188C2×C4—82no—
189C2×C4—81no—
190C2×C4—81no—
191C2×C4—81no—
192C8—3764no—
193C22—22no—
194C22—22no—
195C22—22no—
196C22—22no—
197C22—21no—
198C22—21no—
199C22—21no—
200C22—21no—
201C22#1568884711yescharacteristic
202C22Q8⋊(C47⋊C8)12yes—
203C22#1568886012yes—
204C22#1568886311yescharacteristic
205C22#1568887211yescharacteristic
206C22#1568891312yes—
207C22#1568891611yescharacteristic
208C22#1568892011yescharacteristic
209C4—42no—
210C4—42no—
211C4—41no—
212C4—41no—
213C4—82no—
214C4—82no—
215C2—21no—
216C2—22no—
217C2—21no—
218C2#2024121411yescharacteristic
219C2#2024131811yescharacteristic
220C2#2024132011yescharacteristic
221C2#2024141212yes—
222C2#2024141512yes—
subgroupG/H
centerC23#13093939
commutatorC22×C4×C47C2×C8
FrattiniC2×#95C2×D47
FittingC47×#1445C2
socleC23×C47#99
prime radicalC24×C47C22⋊C4
cosocleC2×#95×C47C22
hypercenter#1445D47
hypocenterC47#7132
LCSG ▷ C22×C4×C47 ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C24 ◁ C2×#95 ◁ #1445

Complements

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

NH ≅ G/Ntype
18. C2×C4⋊C4×C47C8semidirect
119. C47#7132semidirect
128. C2×C4⋊C4C47⋊C8semidirect

Maximal subgroups

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

subgroupindex
#202412312
C47×#14452
#713247