Finite Group Databasebeta

← #20241958

Subgroups — #20241958

Order 6016.

288 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 288 conjugacy classes of subgroups — too many to draw. Only its 71 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
1C22⋊Q8×D47C211yesmaximal, characteristic
2C22⋊C8×D47C211yesmaximal, characteristic
3#15688982C211yesmaximal, characteristic
4#15688861C211yesmaximal, characteristic, complemented
5#15688860C211yesmaximal, characteristic, complemented
6#15688859C211yesmaximal, characteristic, complemented
7C47×#328C211yesmaximal, characteristic, complemented
8C2×Q8×D47C411yescharacteristic, complemented
9C22×C4×D47C2211yescharacteristic
10C2×C8×D47—21no—
11#13094004C411yescharacteristic, complemented
12#13094003C2211yescharacteristic
13#13094002—21no—
14#13094001—21no—
15#13094000C411yescharacteristic, complemented
16C2×Dic47⋊C4—21no—
17C4⋊C4×D47—21no—
18C4⋊C4×D47C411yescharacteristic, complemented
19C2×(C2×D47)⋊C4—21no—
20C22⋊C4×D47—21no—
21C22⋊(C47⋊C8)C2211yescharacteristic
22(C2×D47)⋊C8—21no—
23C47⋊#95C2211yescharacteristic
24C22⋊Q8×C47C2211yescharacteristic
25C22⋊C8×C47C2211yescharacteristic
26#95×C47C2211yescharacteristic
27C23×D47D411yescharacteristic
28C2×Q8×C47C2×C411yescharacteristic
29C22×C4×C47—21no—
30C22×C4×C47C2311yescharacteristic
31C22×Dic47—21no—
32C22×Dic47D411yescharacteristic
33Q8×D47—42no—
34C2×C4×D47—21no—
35C2×C4×D47—21no—
36C2×C4×D47—42no—
37C2×C4×D47C2×C411yescharacteristic
38C2×C4×D47C2×C411yescharacteristic
39C2×C4×D47D411yescharacteristic
40C2×C4×D47D411yescharacteristic
41C2×Dic94—21no—
42C2×Dic94C2×C411yescharacteristic
43C2×C8×C47—21no—
44C4⋊C4×C47—21no—
45C4⋊C4×C47C2×C411yescharacteristic
46C22⋊C4×C47—21no—
47C22⋊Dic47—21no—
48(C2×D47)⋊C4—21no—
49(C2×D47)⋊C4—21no—
50(C2×D47)⋊C4—42no—
51C4⋊Dic47—21no—
52C4⋊Dic47C2×C411yescharacteristic
53Dic47⋊C4—21no—
54Dic47⋊C4—21no—
55Dic47⋊C4—21no—
56Dic47⋊C4—42no—
57C2×C47⋊C8—21no—
58C8×D47—42no—
59C23×C47C2×D411yescharacteristic
60C22×D47—41no—
61C22×D47—41no—
62C22×D47C22⋊C411yescharacteristic
63C22×D47C22⋊C411yescharacteristic
64C22×D47SD1612yes—
65C22×D47Q1612yes—
66Q8×C47—41no—
67C2×C4×C47—21no—
68C2×C4×C47—21no—
69C2×C4×C47—41no—
70C2×C4×C47—41no—
71C2×C4×C47—41no—
72C2×C4×C47—41no—
73C2×C4×C47C22×C411yescharacteristic
74C2×C4×C47C2×D411yescharacteristic
75C2×Dic47—22no—
76C2×Dic47—21no—
77C2×Dic47—21no—
78C2×Dic47—41no—
79C2×Dic47—41no—
80C2×Dic47—41no—
81C2×Dic47C22⋊C411yescharacteristic
82C2×Dic47C22⋊C411yescharacteristic
83C4×D47—22no—
84C4×D47—22no—
85C4×D47—41no—
86C4×D47—41no—
87C4×D47—42no—
88C4×D47—42no—
89Dic94—42no—
90Dic94—41no—
91C8×C47—41no—
92C47⋊C8—41no—
93C22×C47—21no—
94C22×C47—21no—
95C22×C47C2×C22⋊C411yescharacteristic
96C22×C47C2×SD1611yescharacteristic
97C22×C47C2×Q1611yescharacteristic
98C2×D47—21no—
99C2×D47—21no—
100C2×D47—21no—
101C2×D47—21no—
102C2×D47—42no—
103C2×D47—42no—
104C2×D47#10312yes—
105C2×D47#10312yes—
106C4×C47—21no—
107C4×C47—21no—
108C4×C47—41no—
109C4×C47—41no—
110C4×C47—81no—
111Dic47—21no—
112Dic47—41no—
113Dic47—41no—
114Dic47—81no—
115Dic47#9912yes—
116#1520—471nomaximal, Sylow
117C2×C47—21no—
118C2×C47—21no—
119C2×C47C2×#9911yescharacteristic
120C2×C47C2×#10311yescharacteristic
121C2×C47#42111yescharacteristic
122D47—22no—
123D47—22no—
124C2×C22⋊Q8—471no—
125C2×C22⋊C8—471no—
126#380—471no—
127#328—471no—
128#328—471no—
129#328—471no—
130#328D4711yescharacteristic, complemented
131C47#152011yescharacteristic, complemented, Sylow
132C22×Q8—471no—
133C23×C4—471no—
134C22×C8—941no—
135C22⋊Q8—471no—
136C22⋊Q8—471no—
137C22⋊Q8—471no—
138C22⋊Q8—941no—
139C22⋊Q8—941no—
140C22⋊Q8C2×D4711yescharacteristic
141C2×C4⋊C4—471no—
142C2×C4⋊C4—941no—
143C2×C4⋊C4—941no—
144C2×C22⋊C4—941no—
145C2×C22⋊C4—941no—
146C22⋊C8—471no—
147C22⋊C8—941no—
148C22⋊C8C2×D4711yescharacteristic
149#95—471no—
150#95C2×D4711yescharacteristic
151C24—471no—
152C2×Q8—471no—
153C2×Q8—941no—
154C2×Q8—1882no—
155C2×Q8C4×D4711yescharacteristic
156C22×C4—21no—
157C22×C4—471no—
158C22×C4—471no—
159C22×C4—471no—
160C22×C4—471no—
161C22×C4—471no—
162C22×C4—941no—
163C22×C4—941no—
164C22×C4—941no—
165C22×C4—1882no—
166C22×C4C22×D4711yescharacteristic
167C2×C8—21no—
168C2×C8—941no—
169C2×C8—1882no—
170C4⋊C4—21no—
171C4⋊C4—471no—
172C4⋊C4—941no—
173C4⋊C4—941no—
174C4⋊C4—941no—
175C4⋊C4—941no—
176C4⋊C4—1882no—
177C4⋊C4C4×D4711yescharacteristic
178C22⋊C4—21no—
179C22⋊C4—941no—
180C22⋊C4—941no—
181C22⋊C4—941no—
182C22⋊C4—1882no—
183C23—471no—
184C23—471no—
185C23—472no—
186C23—472no—
187C23—1881no—
188C23—1881no—
189C23D4×D4711yescharacteristic
190Q8—41no—
191Q8—1881no—
192Q8—1882no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4—41no—
198C2×C4—41no—
199C2×C4—471no—
200C2×C4—471no—
201C2×C4—942no—
202C2×C4—942no—
203C2×C4—942no—
204C2×C4—941no—
205C2×C4—941no—
206C2×C4—1881no—
207C2×C4—1881no—
208C2×C4—1882no—
209C2×C4—1882no—
210C2×C4—1881no—
211C2×C4—1881no—
212C2×C4—1881no—
213C2×C4C2×C4×D4711yescharacteristic
214C2×C4D4×D4711yescharacteristic
215C8—41no—
216C8—1881no—
217C22—21no—
218C22—21no—
219C22—472no—
220C22—472no—
221C22—941no—
222C22—941no—
223C22—941no—
224C22—941no—
225C22—1882no—
226C22—1882no—
227C22C22⋊C4×D4711yescharacteristic
228C22SD16×D4711yescharacteristic
229C22Q16×D4711yescharacteristic
230C4—21no—
231C4—21no—
232C4—41no—
233C4—41no—
234C4—81no—
235C4—472no—
236C4—941no—
237C4—3761no—
238C4—1881no—
239C4—1881no—
240C2—21no—
241C2—21no—
242C2—942no—
243C2—942no—
244C2#99×D4711yescharacteristic
245C2#103×D4711yescharacteristic
246C2#1568913811yescharacteristic
subgroupG/H
centerC22C22⋊C4×D47
commutatorC2×C4×C47C22×C4
FrattiniC22×C4C22×D47
FittingC47×#328C2
socleC22×C47C2×C22⋊C4
hypercenter#328D47
hypocenterC47#1520
LCSG ▷ C2×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C22×C4 ◁ #328

Complements

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

NH ≅ G/Ntype
4. #15688861C2semidirect
5. #15688860C2semidirect
6. #15688859C2semidirect
7. C47×#328C2semidirect
8. C2×Q8×D47C4semidirect
11. #13094004C4semidirect
15. #13094000C4semidirect
18. C4⋊C4×D47C4semidirect
130. #328D47semidirect
131. C47#1520semidirect