Finite Group Databasebeta

← #26104233

Subgroups — #26104233

Order 12032.

550 conjugacy classes of subgroups; only the 163 normal subgroups are shown.

normal subgroups only

This group has 550 conjugacy classes of subgroups — too many to draw. Only its 163 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
1C4×C22⋊(C47⋊C8)C212yesmaximal
2#20241211C214yesmaximal, complemented
3C47×#2296C211yesmaximal, characteristic
4C2×C22⋊(C47⋊C8)—22no—
5C2×C4×C47⋊C8—22no—
6#15688878C2214yes—
7C2×#117×C47C411yescharacteristic
8C2×#117×C47C411yescharacteristic
9C22×C42×C47C2211yescharacteristic
10C47×#389C412yes—
11C47×#389C2212yes—
12C47×#385—22no—
13C4×C4⋊C4×C47—22no—
14C22×C47⋊C8—24no—
15C22⋊(C47⋊C8)—28no—
16C4×C47⋊C8—44no—
17C23×C4×C47C2×C411yescharacteristic
18C23×C4×C47D412yes—
19#117×C47—22no—
20#117×C47—42no—
21#117×C47C814yescomplemented
22C2×C4⋊C4×C47—22no—
23C2×C4⋊C4×C47—22no—
24C2×C4⋊C4×C47C2×C412yes—
25C2×C4⋊C4×C47C2×C412yes—
26C2×C22⋊C4×C47—21no—
27C2×C22⋊C4×C47—21no—
28C2×C42×C47—21no—
29C2×C42×C47—21no—
30C2×C42×C47—42no—
31C2×C42×C47C2×C411yescharacteristic
32C2×C42×C47D412yes—
33C2×C42×C47C2311yescharacteristic
34#95×C47—22no—
35#95×C47—22no—
36C24×C47C22⋊C411yescharacteristic
37C22×C4×C47—22no—
38C22×C4×C47—22no—
39C22×C4×C47—22no—
40C22×C4×C47—24no—
41C22×C4×C47—24no—
42C22×C4×C47—22no—
43C22×C4×C47—42no—
44C22×C4×C47C22⋊C412yes—
45C22×C4×C47C22⋊C411yescharacteristic
46C22×C4×C47C2×C812yes—
47C22×C4×C47M1612yes—
48C22×C4×C47C22×C411yescharacteristic
49C22×C4×C47C2×D412yes—
50C4⋊C4×C47—22no—
51C4⋊C4×C47—42no—
52C4⋊C4×C47—44no—
53C4⋊C4×C47—44no—
54C4⋊C4×C47C2×C814yes—
55C22⋊C4×C47—24no—
56C22⋊C4×C47—42no—
57C42×C47—24no—
58C42×C47—24no—
59C42×C47—41no—
60C42×C47—41no—
61C42×C47—42no—
62C2×C47⋊C8—28no—
63C2×C47⋊C8—44no—
64C2×C47⋊C8—44no—
65C23×C47—21no—
66C23×C47—21no—
67C23×C47—22no—
68C23×C47—22no—
69C23×C47C22⋊C812yes—
70C23×C47C2×C22⋊C411yescharacteristic
71C2×C4×C47—22no—
72C2×C4×C47—22no—
73C2×C4×C47—21no—
74C2×C4×C47—21no—
75C2×C4×C47—22no—
76C2×C4×C47—22no—
77C2×C4×C47—24no—
78C2×C4×C47—24no—
79C2×C4×C47—24no—
80C2×C4×C47—48no—
81C2×C4×C47—42no—
82C2×C4×C47—42no—
83C2×C4×C47—42no—
84C2×C4×C47—42no—
85C2×C4×C47—42no—
86C2×C4×C47C22⋊C812yes—
87C2×C4×C47#9912yes—
88C2×C4×C47#10112yes—
89C2×C4×C47#10414yes—
90C2×C4×C47C22×C811yescharacteristic
91C2×C4×C47C2×M1611yescharacteristic
92C2×C4×C47#13514yes—
93C47⋊C8—48no—
94#7564—471nomaximal, Sylow
95C22×C47—22no—
96C22×C47—24no—
97C22×C47—41no—
98C22×C47—42no—
99C22×C47—41no—
100C22×C47C2×C22⋊C811yescharacteristic
101C22×C47C2×#9911yescharacteristic
102C22×C47C2×#10111yescharacteristic
103C22×C47#41612yes—
104C22×C47C2×#10412yes—
105C4×C47—21no—
106C4×C47—21no—
107C4×C47—24no—
108C4×C47—24no—
109C4×C47—44no—
110C4×C47—82no—
111C4×C47#32414yes—
112#2296D4711yescharacteristic
113C4×C22⋊C8—472no—
114#1271—474no—
115C2×C47—42no—
116C2×C47C2×#32411yescharacteristic
117C2×C47#147612yes—
118C2×C47#150112yes—
119C2×C47#152611yescharacteristic
120C2×C47#154411yescharacteristic
121C2×#117Dic4711yescharacteristic
122C2×#117Dic4711yescharacteristic
123C22×C42C2×D4711yescharacteristic
124C2×C22⋊C8—942no—
125C2×C4×C8—942no—
126#389Dic4712yes—
127#389C2×D4712yes—
128#385—22no—
129C4×C4⋊C4—22no—
130#336—474no—
131C47#756411yescharacteristic, complemented, Sylow
132C23×C4C2×Dic4711yescharacteristic
133C23×C4C22⋊D4712yes—
134C22×C8—944no—
135#117—22no—
136#117—42no—
137#117C47⋊C814yescomplemented
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4C2×Dic4712yes—
141C2×C4⋊C4C2×Dic4712yes—
142C2×C22⋊C4—21no—
143C2×C22⋊C4—21no—
144C2×C42—21no—
145C2×C42—21no—
146C2×C42—42no—
147C2×C42C2×Dic4711yescharacteristic
148C2×C42C22⋊D4712yes—
149C2×C42C22×D4711yescharacteristic
150C22⋊C8—948no—
151C4×C8—1884no—
152#95—22no—
153#95—22no—
154C24C22⋊Dic4711yescharacteristic
155C22×C4—22no—
156C22×C4—22no—
157C22×C4—22no—
158C22×C4—24no—
159C22×C4—24no—
160C22×C4—22no—
161C22×C4—42no—
162C22×C4C2×C47⋊C812yes—
163C22×C4#1062532912yes—
164C22×C4C22⋊Dic4712yes—
165C22×C4C22⋊Dic4711yescharacteristic
166C22×C4C22×Dic4711yescharacteristic
167C22×C4C2×C22⋊D4712yes—
168C2×C8—948no—
169C2×C8—1884no—
170C2×C8—1884no—
171C4⋊C4—22no—
172C4⋊C4—42no—
173C4⋊C4—44no—
174C4⋊C4—44no—
175C4⋊C4C2×C47⋊C814yes—
176C22⋊C4—24no—
177C22⋊C4—42no—
178C42—24no—
179C42—24no—
180C42—41no—
181C42—41no—
182C42—42no—
183C23—21no—
184C23—21no—
185C23—22no—
186C23—22no—
187C23C22⋊(C47⋊C8)12yes—
188C23C2×C22⋊Dic4711yescharacteristic
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—24no—
196C2×C4—24no—
197C2×C4—24no—
198C2×C4—48no—
199C2×C4—42no—
200C2×C4—42no—
201C2×C4—42no—
202C2×C4—42no—
203C2×C4—42no—
204C2×C4C22⋊(C47⋊C8)12yes—
205C2×C4#1309393912yes—
206C2×C4#1309394312yes—
207C2×C4#1309395114yes—
208C2×C4C22×C47⋊C811yescharacteristic
209C2×C4C2×#1062532911yescharacteristic
210C2×C4#1309404414yes—
211C8—1888no—
212C22—22no—
213C22—24no—
214C22—41no—
215C22—41no—
216C22—42no—
217C22C2×C22⋊(C47⋊C8)11yescharacteristic
218C22C2×#1309393911yescharacteristic
219C22C2×#1309394311yescharacteristic
220C22#1568911912yes—
221C22C2×#1309395112yes—
222C4—21no—
223C4—21no—
224C4—24no—
225C4—24no—
226C4—44no—
227C4—82no—
228C4#1568884914yes—
229C2—42no—
230C2C2×#1568884911yescharacteristic
231C2#2024177012yes—
232C2#2024187912yes—
233C2#2024198311yescharacteristic
234C2#2024205311yescharacteristic
subgroupG/H
centerC22×C4C22⋊Dic47
commutatorC2×C4×C47C22×C8
FrattiniC2×C42C22×D47
FittingC47×#2296C2
socleC23×C47C2×C22⋊C4
prime radicalC24×C47C22⋊C4
cosocleC2×C42×C47C23
hypercenter#2296D47
hypocenterC47#7564
LCSG ▷ C2×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22×C4 ◁ C22×C42 ◁ #2296

Complements

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

NH ≅ G/Ntype
2. #20241211C2semidirect
21. #117×C47C8semidirect
131. C47#7564semidirect
137. #117C47⋊C8semidirect

Maximal subgroups

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

subgroupindex
C4×C22⋊(C47⋊C8)2
#202412112
C47×#22962
#756447