Finite Group Databasebeta

← #20246101

Subgroups — #20246101

Order 6016.

612 conjugacy classes of subgroups; only the 237 normal subgroups are shown.

normal subgroups only

This group has 612 conjugacy classes of subgroups — too many to draw. Only its 237 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×Q8×Dic47C211yesmaximal, characteristic, complemented
2C2×D4×Dic47C211yesmaximal, characteristic
3#15689030C211yesmaximal, characteristic, complemented
4#15689026C211yesmaximal, characteristic
5#15689016C214yesmaximal, complemented
6#15688998C214yesmaximal, complemented
7#15688993C211yesmaximal, characteristic, complemented
8#15688969C211yesmaximal, characteristic, complemented
9C2×#124×C47C211yesmaximal, characteristic
10C23×Dic47C2212yes—
11Q8×Dic47—24no—
12D4×Dic47—24no—
13C2×C4⋊Dic47C2211yescharacteristic
14C2×C4⋊Dic47C2211yescharacteristic
15C2×C4⋊Dic47C2212yes—
16C2×C4⋊Dic47C2214yes—
17C2×C22⋊Dic47C2212yes—
18C2×C22⋊Dic47C2214yes—
19C2×C4×Dic47C2211yescharacteristic
20C2×C4×Dic47C2212yes—
21C2×C4×Dic47C2211yescharacteristic
22C47⋊#95C2212yes—
23C47⋊#95C2212yes—
24C47⋊#95C2214yes—
25C22×Q8×C47C2211yescharacteristic
26C22×D4×C47C2211yescharacteristic
27#124×C47C418yescomplemented
28C2×C22⋊C4×C47C2214yes—
29C2×C42×C47C2211yescharacteristic
30C24×C47C2312yes—
31C2×Q8×C47—22no—
32C2×Q8×C47—22no—
33C2×Q8×C47C2×C414yescomplemented
34C2×D4×C47—22no—
35C2×D4×C47—22no—
36C2×D4×C47C2×C414yes—
37C22×C4×C47C2312yes—
38C22×C4×C47C2312yes—
39C22×C4×C47C2311yescharacteristic
40C22×Dic47—24no—
41C22×Dic47—44no—
42C22×Dic47C2312yes—
43C22×Dic47C2312yes—
44C22×Dic47C2312yes—
45C22×Dic47C2311yescharacteristic
46C22×Dic47C2311yescharacteristic
47C22⋊C4×C47C2×C4116yescomplemented
48C42×C47C2×C414yescomplemented
49C22⋊Dic47—24no—
50C22⋊Dic47—44no—
51C4⋊Dic47—22no—
52C4⋊Dic47—22no—
53C4⋊Dic47—24no—
54C4⋊Dic47—44no—
55C4×Dic47—22no—
56C4×Dic47—24no—
57C4×Dic47D414yes—
58C23×C47—22no—
59C23×C47—22no—
60C23×C47—24no—
61C23×C47—24no—
62C23×C47C22×C414yes—
63C23×C47C2411yescharacteristic
64Q8×C47—28no—
65D4×C47—28no—
66C2×C4×C47—22no—
67C2×C4×C47—22no—
68C2×C4×C47—22no—
69C2×C4×C47—22no—
70C2×C4×C47C22×C414yes—
71C2×C4×C47C22×C414yes—
72C2×C4×C47C22×C412yes—
73C2×C4×C47C2×D412yes—
74C2×C4×C47#4112yes—
75C2×Dic47—22no—
76C2×Dic47—22no—
77C2×Dic47—22no—
78C2×Dic47—22no—
79C2×Dic47—22no—
80C2×Dic47—24no—
81C2×Dic47—24no—
82C2×Dic47—22no—
83C2×Dic47—22no—
84C2×Dic47—48no—
85C2×Dic47C2×D412yes—
86C2×Dic47C2×D412yes—
87C22×C47—24no—
88C22×C47—28no—
89C22×C47—42no—
90C22×C47—42no—
91C22×C47—44no—
92C22×C47C23×C411yescharacteristic
93C22×C47C22×D411yescharacteristic
94C22×C47C2×#4111yescharacteristic
95C22×C47H(5,𝔽2)12yes—
96C22×C472−1+412yes—
97C4×C47—24no—
98C4×C47—24no—
99C4×C47C4×D414yes—
100Dic47—22no—
101Dic47—22no—
102Dic47—42no—
103Dic47—42no—
104Dic47—42no—
105#2326—471nomaximal, Sylow
106C2×C47—44no—
107C2×C47C2×C4×D411yescharacteristic
108C2×C47#52012yes—
109C2×C47#53511yescharacteristic
110C2×C47#53611yescharacteristic
111C2×C47#54212yes—
112C2×#124D4711yescharacteristic
113C2×C4×Q8—471no—
114C2×C4×D4—471no—
115#391—471no—
116#390—471no—
117#388—474no—
118#385—474no—
119#384—471no—
120#376—471no—
121C47#232611yescharacteristic, complemented, Sylow
122C22×Q8C2×D4711yescharacteristic
123C22×D4C2×D4711yescharacteristic
124C23×C4—472no—
125#124Dic4718yescomplemented
126C4×Q8—944no—
127C4×D4—944no—
128C2×C4⋊C4—471no—
129C2×C4⋊C4—471no—
130C2×C4⋊C4—472no—
131C2×C4⋊C4—474no—
132C2×C22⋊C4—472no—
133C2×C22⋊C4—474no—
134C2×C22⋊C4C2×D4714yes—
135C2×C42—471no—
136C2×C42—472no—
137C2×C42—471no—
138C2×C42C2×D4711yescharacteristic
139#95—472no—
140#95—472no—
141#95—474no—
142C24C22×D4712yes—
143C2×Q8—22no—
144C2×Q8—22no—
145C2×Q8C2×Dic4714yescomplemented
146C2×D4—22no—
147C2×D4—22no—
148C2×D4C2×Dic4714yes—
149C22×C4—471no—
150C22×C4—471no—
151C22×C4—472no—
152C22×C4—472no—
153C22×C4—472no—
154C22×C4—944no—
155C22×C4—1884no—
156C22×C4C22×D4712yes—
157C22×C4C22×D4712yes—
158C22×C4C22×D4711yescharacteristic
159C4⋊C4—944no—
160C4⋊C4—942no—
161C4⋊C4—942no—
162C4⋊C4—1884no—
163C22⋊C4—944no—
164C22⋊C4—1884no—
165C22⋊C4C2×Dic47116yescomplemented
166C42—474no—
167C42—944no—
168C42—942no—
169C42C2×Dic4714yescomplemented
170C23—22no—
171C23—22no—
172C23—24no—
173C23—24no—
174C23C22×Dic4714yes—
175C23C23×D4711yescharacteristic
176Q8—28no—
177D4—28no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—472no—
183C2×C4—472no—
184C2×C4—942no—
185C2×C4—942no—
186C2×C4—942no—
187C2×C4—942no—
188C2×C4—942no—
189C2×C4—942no—
190C2×C4—944no—
191C2×C4—944no—
192C2×C4—942no—
193C2×C4—1888no—
194C2×C4D4×D4712yes—
195C2×C4#1062535212yes—
196C2×C4C22×Dic4714yes—
197C2×C4C22×Dic4714yes—
198C2×C4C22×Dic4712yes—
199C22—24no—
200C22—28no—
201C22—44no—
202C22—42no—
203C22—42no—
204C22C23×Dic4711yescharacteristic
205C22C2×D4×D4711yescharacteristic
206C22C2×#1062535211yescharacteristic
207C22#1309407012yes—
208C22#1309407112yes—
209C4—24no—
210C4—24no—
211C4—942no—
212C4—942no—
213C4—1882no—
214C4—1882no—
215C4—1882no—
216C4D4×Dic4714yes—
217C2—44no—
218C2C2×D4×Dic4711yescharacteristic
219C2#1568960612yes—
220C2#1568969111yescharacteristic
221C2#1568969311yescharacteristic
222C2#1568974212yes—
subgroupG/H
center, FrattiniC23C23×D47
commutatorC22×C47C23×C4
FittingC2×#124×C47C2
socleC23×C47C24
hypercenterC2×#124D47
hypocenterC47#2326
LCSG ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
1. C2×Q8×Dic47C2semidirect
3. #15689030C2semidirect
5. #15689016C2semidirect
6. #15688998C2semidirect
7. #15688993C2semidirect
8. #15688969C2semidirect
27. #124×C47C4semidirect
33. C2×Q8×C47C2×C4semidirect
47. C22⋊C4×C47C2×C4semidirect
48. C42×C47C2×C4semidirect
121. C47#2326semidirect
125. #124Dic47semidirect
145. C2×Q8C2×Dic47semidirect
165. C22⋊C4C2×Dic47semidirect
169. C42C2×Dic47semidirect