Finite Group Databasebeta

← #20250225

Subgroups — #20250225

Order 6016.

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

normal subgroups only

This group has 512 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×#13094011C211yesmaximal, characteristic
2#15689051C212yesmaximal, complemented
3#15689051C212yesmaximal, complemented
4#15689043C212yesmaximal, complemented
5#15689004C212yesmaximal, complemented
6#15689003C211yesmaximal, characteristic
7#15688996C212yesmaximal, complemented
8#15688981C212yesmaximal, complemented
9C2×#123×C47C211yesmaximal, characteristic
10C22×Dic94C2211yescharacteristic
11C23×Dic47C2211yescharacteristic
12#13094011—24no—
13C2×Dic47⋊C4C2212yes—
14C2×Dic47⋊C4C2212yes—
15C2×Dic47⋊C4C2212yes—
16C2×C22⋊Dic47C2212yes—
17C2×C22⋊Dic47C2212yes—
18C2×C22⋊Dic47C2212yes—
19C2×C22⋊Dic47C2211yescharacteristic
20C2×C4×Dic47C2211yescharacteristic
21C47⋊#95C2212yes—
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—24no—
45C22×Dic47—41no—
46C22×Dic47C2312yes—
47C22×Dic47C2312yes—
48C22×Dic47C2312yes—
49C22×Dic47C2312yes—
50C2×Dic94—22no—
51C2×Dic94—41no—
52C2×Dic94—41no—
53C4⋊C4×C47—24no—
54C22⋊C4×C47—22no—
55C22⋊C4×C47—24no—
56C22⋊Dic47—24no—
57C22⋊Dic47—24no—
58C22⋊Dic47—24no—
59C22⋊Dic47—41no—
60Dic47⋊C4—24no—
61Dic47⋊C4—42no—
62Dic47⋊C4—42no—
63C4×Dic47—22no—
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—
99Dic94—44no—
100C22×C47—22no—
101C22×C47—22no—
102C22×C47—24no—
103C22×C47—24no—
104C22×C47—42no—
105C22×C47—42no—
106C22×C47—41no—
107C22×C47—41no—
108C22×C47—41no—
109C22×C47—41no—
110C22×C47C22×D411yescharacteristic
111C22×C47C2×#4112yes—
112C22×C47H(5,𝔽2)11yescharacteristic
113C22×C47H(5,𝔽2)11yescharacteristic
114C22×C472−1+412yes—
115C4×C47—22no—
116C4×C47—42no—
117C4×C47—42no—
118Dic47—24no—
119Dic47—42no—
120Dic47—42no—
121Dic47—42no—
122#2717—471nomaximal, Sylow
123C2×C47—42no—
124C2×C47—42no—
125C2×C47#53411yescharacteristic
126C2×C47#54212yes—
127C2×C47#54712yes—
128C2×C47#55111yescharacteristic
129C2×C47#56311yescharacteristic
130C2×#124—471no—
131C2×#123D4711yescharacteristic
132#397—472no—
133#397—472no—
134#395—472no—
135#387—472no—
136#386—471no—
137#385—472no—
138#380—472no—
139C47#271711yescharacteristic, complemented, Sylow
140C22×Q8—471no—
141C22×D4C2×D4711yescharacteristic
142C23×C4—471no—
143C23×C4C2×D4711yescharacteristic
144#124—944no—
145#123—24no—
146C2×C4⋊C4—472no—
147C2×C4⋊C4—472no—
148C2×C4⋊C4—472no—
149C2×C4⋊C4C2×D4712yes—
150C2×C22⋊C4—472no—
151C2×C22⋊C4—472no—
152C2×C22⋊C4—472no—
153C2×C22⋊C4—471no—
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—
162#95—472no—
163C24C22×D4711yescharacteristic
164C24C22×D4711yescharacteristic
165C2×Q8—942no—
166C2×Q8—1881no—
167C2×Q8—1881no—
168C2×D4—41no—
169C2×D4—41no—
170C2×D4C22⋊D4714yes—
171C22×C4—22no—
172C22×C4—41no—
173C22×C4—41no—
174C22×C4—472no—
175C22×C4—472no—
176C22×C4—472no—
177C22×C4—472no—
178C22×C4—944no—
179C22×C4—1881no—
180C22×C4C22×D4712yes—
181C22×C4C22×D4712yes—
182C22×C4C22×D4711yescharacteristic
183C4⋊C4—24no—
184C4⋊C4—944no—
185C4⋊C4—1882no—
186C4⋊C4—1882no—
187C22⋊C4—22no—
188C22⋊C4—24no—
189C22⋊C4—944no—
190C22⋊C4—944no—
191C22⋊C4—944no—
192C22⋊C4—1881no—
193C42—942no—
194C23—22no—
195C23—22no—
196C23—21no—
197C23—21no—
198C23—21no—
199C23—21no—
200C23—22no—
201C23—22no—
202C23C2×C22⋊D4712yes—
203C23C2×C22⋊D4712yes—
204C23C23×D4711yescharacteristic
205Q8—1884no—
206D4—44no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—44no—
216C2×C4—474no—
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—942no—
227C2×C4—942no—
228C2×C4—1884no—
229C2×C4C2×C22⋊D4712yes—
230C22—24no—
231C22—24no—
232C22—22no—
233C22—22no—
234C22—42no—
235C22—42no—
236C22—41no—
237C22—41no—
238C22—41no—
239C22—41no—
240C22C22×C22⋊D4711yescharacteristic
241C22#41×D4712yes—
242C22#1309406911yescharacteristic
243C22#1309406911yescharacteristic
244C22#1309407112yes—
245C4—22no—
246C4—42no—
247C4—42no—
248C4—944no—
249C4—1882no—
250C4—1882no—
251C4—1882no—
252C2—42no—
253C2—42no—
254C2#1568968211yescharacteristic
255C2#1568974112yes—
256C2#1568977212yes—
257C2#1568980111yescharacteristic
258C2#1568987611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D47
commutator, socleC23×C47C24
FittingC2×#123×C47C2
hypercenterC2×#123D47
hypocenterC47#2717
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
2. #15689051C2semidirect
3. #15689051C2semidirect
4. #15689043C2semidirect
5. #15689004C2semidirect
7. #15688996C2semidirect
8. #15688981C2semidirect
139. C47#2717semidirect