Finite Group Databasebeta

← #20250316

Subgroups — #20250316

Order 6016.

504 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 137 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
1C2×#13094005C211yesmaximal, characteristic
2#15689056C212yesmaximal, complemented
3#15689051C212yesmaximal, complemented
4#15689051C212yesmaximal, complemented
5#15689026C211yesmaximal, characteristic
6#15689018C212yesmaximal, complemented
7#15688996C212yesmaximal, complemented
8#15688995C212yesmaximal, complemented
9C2×#123×C47C211yesmaximal, characteristic
10C23×Dic47C2211yescharacteristic
11#13094005—24no—
12C2×C4⋊Dic47C2211yescharacteristic
13C2×Dic47⋊C4C2212yes—
14C2×Dic47⋊C4C2212yes—
15C2×Dic47⋊C4C2212yes—
16C2×Dic47⋊C4C2212yes—
17C2×C22⋊Dic47C2212yes—
18C2×C22⋊Dic47C2212yes—
19C2×C22⋊Dic47C2212yes—
20C2×C22⋊Dic47C2211yescharacteristic
21C2×C4×Dic47C2211yescharacteristic
22C47⋊#95C2212yes—
23C47⋊#95C2212yes—
24C47⋊#95C2212yes—
25C47⋊#95C2212yes—
26C47⋊#95C2212yes—
27C22×D4×C47C2211yescharacteristic
28C23×C4×C47C2211yescharacteristic
29#123×C47—24no—
30C2×C4⋊C4×C47C2212yes—
31C2×C22⋊C4×C47C2212yes—
32C2×C22⋊C4×C47C2211yescharacteristic
33C24×C47C2311yescharacteristic
34C24×C47C2311yescharacteristic
35C2×D4×C47—41no—
36C2×D4×C47—41no—
37C2×D4×C47D414yes—
38C22×C4×C47—22no—
39C22×C4×C47—41no—
40C22×C4×C47—41no—
41C22×C4×C47C2312yes—
42C22×C4×C47C2312yes—
43C22×C4×C47C2311yescharacteristic
44C22×Dic47—22no—
45C22×Dic47—41no—
46C22×Dic47—41no—
47C22×Dic47C2312yes—
48C22×Dic47C2312yes—
49C22×Dic47C2312yes—
50C22×Dic47C2312yes—
51C4⋊C4×C47—24no—
52C22⋊C4×C47—22no—
53C22⋊C4×C47—24no—
54C22⋊Dic47—22no—
55C22⋊Dic47—24no—
56C22⋊Dic47—24no—
57C22⋊Dic47—24no—
58C4⋊Dic47—41no—
59Dic47⋊C4—24no—
60Dic47⋊C4—24no—
61Dic47⋊C4—42no—
62Dic47⋊C4—42no—
63C4×Dic47—41no—
64C23×C47—22no—
65C23×C47—22no—
66C23×C47—21no—
67C23×C47—21no—
68C23×C47—21no—
69C23×C47—21no—
70C23×C47—22no—
71C23×C47—22no—
72C23×C47C2×D412yes—
73C23×C47C2×D412yes—
74C23×C47C2411yescharacteristic
75D4×C47—44no—
76C2×C4×C47—21no—
77C2×C4×C47—21no—
78C2×C4×C47—22no—
79C2×C4×C47—22no—
80C2×C4×C47—22no—
81C2×C4×C47—22no—
82C2×C4×C47—22no—
83C2×C4×C47—22no—
84C2×C4×C47—44no—
85C2×C4×C47C2×D412yes—
86C2×Dic47—22no—
87C2×Dic47—22no—
88C2×Dic47—22no—
89C2×Dic47—22no—
90C2×Dic47—22no—
91C2×Dic47—22no—
92C2×Dic47—22no—
93C2×Dic47—22no—
94C2×Dic47—22no—
95C2×Dic47—22no—
96C2×Dic47—22no—
97C2×Dic47—44no—
98C2×Dic47#4114yes—
99C22×C47—22no—
100C22×C47—22no—
101C22×C47—24no—
102C22×C47—24no—
103C22×C47—42no—
104C22×C47—42no—
105C22×C47—41no—
106C22×C47—41no—
107C22×C47—41no—
108C22×C47—41no—
109C22×C47C22×D411yescharacteristic
110C22×C47C2×#4112yes—
111C22×C47H(5,𝔽2)11yescharacteristic
112C22×C47H(5,𝔽2)11yescharacteristic
113C22×C472−1+412yes—
114C4×C47—22no—
115C4×C47—42no—
116C4×C47—42no—
117Dic47—24no—
118Dic47—42no—
119Dic47—42no—
120Dic47—42no—
121#2724—471nomaximal, Sylow
122C2×C47—42no—
123C2×C47—42no—
124C2×C47#53411yescharacteristic
125C2×C47#53912yes—
126C2×C47#54712yes—
127C2×C47#55911yescharacteristic
128C2×C47#56311yescharacteristic
129C2×#123—471no—
130C2×#123D4711yescharacteristic
131#398—472no—
132#397—472no—
133#397—472no—
134#390—471no—
135#389—472no—
136#385—472no—
137#385—472no—
138C47#272411yescharacteristic, complemented, Sylow
139C22×D4C2×D4711yescharacteristic
140C23×C4—471no—
141C23×C4C2×D4711yescharacteristic
142#123—24no—
143#123—944no—
144C2×C4⋊C4—472no—
145C2×C4⋊C4—472no—
146C2×C4⋊C4—472no—
147C2×C4⋊C4—472no—
148C2×C4⋊C4—471no—
149C2×C4⋊C4C2×D4712yes—
150C2×C22⋊C4—472no—
151C2×C22⋊C4—471no—
152C2×C22⋊C4—472no—
153C2×C22⋊C4—472no—
154C2×C22⋊C4C2×D4712yes—
155C2×C22⋊C4C2×D4711yescharacteristic
156C2×C42—471no—
157#95—472no—
158#95—472no—
159#95—472no—
160#95—472no—
161#95—472no—
162C24C22×D4711yescharacteristic
163C24C22×D4711yescharacteristic
164C2×D4—41no—
165C2×D4—41no—
166C2×D4C22⋊D4714yes—
167C22×C4—22no—
168C22×C4—41no—
169C22×C4—41no—
170C22×C4—472no—
171C22×C4—472no—
172C22×C4—472no—
173C22×C4—472no—
174C22×C4—942no—
175C22×C4—1881no—
176C22×C4—1881no—
177C22×C4C22×D4712yes—
178C22×C4C22×D4712yes—
179C22×C4C22×D4711yescharacteristic
180C4⋊C4—24no—
181C4⋊C4—944no—
182C4⋊C4—944no—
183C4⋊C4—1882no—
184C4⋊C4—1882no—
185C4⋊C4—1881no—
186C22⋊C4—22no—
187C22⋊C4—24no—
188C22⋊C4—944no—
189C22⋊C4—944no—
190C22⋊C4—942no—
191C22⋊C4—944no—
192C42—1881no—
193C23—22no—
194C23—22no—
195C23—21no—
196C23—21no—
197C23—21no—
198C23—21no—
199C23—22no—
200C23—22no—
201C23C2×C22⋊D4712yes—
202C23C2×C22⋊D4712yes—
203C23C23×D4711yescharacteristic
204D4—44no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—44no—
214C2×C4—474no—
215C2×C4—942no—
216C2×C4—942no—
217C2×C4—942no—
218C2×C4—942no—
219C2×C4—942no—
220C2×C4—942no—
221C2×C4—942no—
222C2×C4—942no—
223C2×C4—942no—
224C2×C4—942no—
225C2×C4—942no—
226C2×C4—1884no—
227C2×C4C2×C22⋊D4712yes—
228C22—24no—
229C22—24no—
230C22—22no—
231C22—22no—
232C22—41no—
233C22—41no—
234C22—41no—
235C22—41no—
236C22—42no—
237C22—42no—
238C22C22×C22⋊D4711yescharacteristic
239C22#41×D4712yes—
240C22#1309406911yescharacteristic
241C22#1309406911yescharacteristic
242C22#1309407112yes—
243C4—22no—
244C4—42no—
245C4—42no—
246C4—944no—
247C4—1882no—
248C4—1882no—
249C4—1882no—
250C2—42no—
251C2—42no—
252C2#1568968211yescharacteristic
253C2#1568971612yes—
254C2#1568977212yes—
255C2#1568985711yescharacteristic
256C2#1568987611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D47
commutator, socleC23×C47C24
FittingC2×#123×C47C2
hypercenterC2×#123D47
hypocenterC47#2724
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
2. #15689056C2semidirect
3. #15689051C2semidirect
4. #15689051C2semidirect
6. #15689018C2semidirect
7. #15688996C2semidirect
8. #15688995C2semidirect
138. C47#2724semidirect