Finite Group Databasebeta

← #20249211

Subgroups — #20249211

Order 6016.

488 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 488 conjugacy classes of subgroups — too many to draw. Only its 143 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×#13094022C211yesmaximal, characteristic, complemented
2C2×#13093982C211yesmaximal, characteristic
3#15689065C212yesmaximal, complemented
4#15689058C212yesmaximal
5#15689043C211yesmaximal, characteristic, complemented
6#15689036C211yesmaximal, characteristic
7#15689034C211yesmaximal, characteristic
8#15689018C212yesmaximal, complemented
9#15688987C211yesmaximal, characteristic, complemented
10#15688982C212yesmaximal
11C47×#395C211yesmaximal, characteristic, complemented
12C22×Dic94C2211yescharacteristic
13C22×C4×D47C2211yescharacteristic
14#13094022—24no—
15#13093982—24no—
16C2×C4⋊Dic47C2211yescharacteristic
17C2×Dic47⋊C4C2212yes—
18C2×Dic47⋊C4C2212yes—
19C2×Dic47⋊C4C2212yes—
20C2×Dic47⋊C4C2212yes—
21C2×(C2×D47)⋊C4C2211yescharacteristic
22C2×(C2×D47)⋊C4C2211yescharacteristic
23C2×(C2×D47)⋊C4C2212yes—
24C2×(C2×D47)⋊C4C2212yes—
25C2×C4×Dic47C2211yescharacteristic
26C2×C4×Dic47C2211yescharacteristic
27C47⋊#95C2212yes—
28C47⋊#95C2211yescharacteristic
29C47⋊#95C2212yes—
30C47⋊#95C2212yes—
31C47⋊#95C2212yes—
32C22×Q8×C47C2211yescharacteristic
33C2×C4⋊C4×C47C2211yescharacteristic
34C2×C4⋊C4×C47C2212yes—
35#95×C47C2212yes—
36#95×C47C2211yescharacteristic
37C23×D47C2311yescharacteristic
38C2×Q8×C47—22no—
39C2×Q8×C47—42no—
40C22×C4×C47C2311yescharacteristic
41C22×C4×C47C2312yes—
42C22×C4×C47C2312yes—
43C22×C4×C47C2311yescharacteristic
44C22×C4×C47C2311yescharacteristic
45C22×Dic47C2311yescharacteristic
46C22×Dic47C2312yes—
47C22×Dic47C2312yes—
48C22×Dic47C2311yescharacteristic
49C22×Dic47C2311yescharacteristic
50C2×C4×D47—22no—
51C2×C4×D47—22no—
52C2×C4×D47Q814yescomplemented
53C2×Dic94—22no—
54C2×Dic94—42no—
55C4⋊C4×C47—22no—
56C4⋊C4×C47—42no—
57(C2×D47)⋊C4—22no—
58(C2×D47)⋊C4—22no—
59(C2×D47)⋊C4—42no—
60(C2×D47)⋊C4—42no—
61C4⋊Dic47—22no—
62Dic47⋊C4—24no—
63Dic47⋊C4—24no—
64Dic47⋊C4—24no—
65Dic47⋊C4—42no—
66C4×Dic47—22no—
67C4×Dic47D414yes—
68C23×C47C2411yescharacteristic
69C22×D47—21no—
70C22×D47—21no—
71C22×D47—22no—
72C22×D47—22no—
73C22×D47C2×Q812yes—
74Q8×C47—44no—
75C2×C4×C47—21no—
76C2×C4×C47—21no—
77C2×C4×C47—22no—
78C2×C4×C47—21no—
79C2×C4×C47—22no—
80C2×C4×C47—22no—
81C2×C4×C47—22no—
82C2×C4×C47—22no—
83C2×C4×C47—22no—
84C2×C4×C47—22no—
85C2×C4×C47—22no—
86C2×C4×C47C2×D412yes—
87C2×C4×C47C2×Q812yes—
88C2×Dic47—22no—
89C2×Dic47—21no—
90C2×Dic47—22no—
91C2×Dic47—22no—
92C2×Dic47—22no—
93C2×Dic47—22no—
94C2×Dic47—22no—
95C2×Dic47—22no—
96C2×Dic47—22no—
97C2×Dic47C2×D412yes—
98C2×Dic47C2×D412yes—
99C2×Dic47C2×Q812yes—
100C2×Dic47#4112yes—
101C4×D47—28no—
102Dic94—44no—
103C22×C47C22×D411yescharacteristic
104C22×C47C22×Q811yescharacteristic
105C22×C47C2×#4111yescharacteristic
106C22×C47H(5,𝔽2)12yes—
107C22×C472−1+412yes—
108C2×D47—22no—
109C2×D47—24no—
110C2×D47—41no—
111C2×D47—41no—
112C2×D47—42no—
113C4×C47—22no—
114C4×C47—22no—
115C4×C47—41no—
116C4×C47—42no—
117C4×C47—42no—
118Dic47—22no—
119Dic47—42no—
120Dic47—41no—
121Dic47—42no—
122Dic47C22⋊Q814yes—
123#2628—471nomaximal, Sylow
124C2×C47C2×C22⋊Q811yescharacteristic
125C2×C47#53511yescharacteristic
126C2×C47#53611yescharacteristic
127C2×C47#54212yes—
128C2×C47#54412yes—
129D47—42no—
130C2×C4⋊Q8—471no—
131C2×#117—471no—
132#400—472no—
133#399—472no—
134#395—471no—
135#395D4711yescharacteristic, complemented
136#393—471no—
137#393—471no—
138#389—472no—
139#382—471no—
140#380—472no—
141C47#262811yescharacteristic, complemented, Sylow
142C22×Q8—471no—
143C22×Q8C2×D4711yescharacteristic
144C23×C4—471no—
145C4⋊Q8—944no—
146#117—944no—
147C2×C4⋊C4—471no—
148C2×C4⋊C4—472no—
149C2×C4⋊C4—472no—
150C2×C4⋊C4—472no—
151C2×C4⋊C4—472no—
152C2×C4⋊C4C2×D4711yescharacteristic
153C2×C4⋊C4C2×D4712yes—
154C2×C22⋊C4—471no—
155C2×C22⋊C4—471no—
156C2×C22⋊C4—472no—
157C2×C22⋊C4—472no—
158C2×C42—471no—
159C2×C42—471no—
160#95—472no—
161#95—471no—
162#95—472no—
163#95—472no—
164#95—472no—
165#95C2×D4712yes—
166#95C2×D4711yescharacteristic
167C24—471no—
168C2×Q8—22no—
169C2×Q8—42no—
170C2×Q8—942no—
171C2×Q8—1882no—
172C22×C4—471no—
173C22×C4—471no—
174C22×C4—474no—
175C22×C4—471no—
176C22×C4—472no—
177C22×C4—472no—
178C22×C4—942no—
179C22×C4—942no—
180C22×C4C22×D4712yes—
181C22×C4C22×D4712yes—
182C22×C4C22×D4711yescharacteristic
183C22×C4C22×D4711yescharacteristic
184C22×C4C22×D4711yescharacteristic
185C4⋊C4—22no—
186C4⋊C4—42no—
187C4⋊C4—944no—
188C4⋊C4—944no—
189C4⋊C4—944no—
190C4⋊C4—942no—
191C4⋊C4—1882no—
192C22⋊C4—942no—
193C22⋊C4—942no—
194C22⋊C4—1882no—
195C22⋊C4—1882no—
196C42—474no—
197C42—942no—
198C23—472no—
199C23—941no—
200C23—941no—
201C23—942no—
202C23—942no—
203C23C23×D4711yescharacteristic
204Q8—44no—
205Q8—1884no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—22no—
209C2×C4—21no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—472no—
218C2×C4—472no—
219C2×C4—472no—
220C2×C4—472no—
221C2×C4—948no—
222C2×C4—942no—
223C2×C4—941no—
224C2×C4—942no—
225C2×C4—942no—
226C2×C4—942no—
227C2×C4—942no—
228C2×C4—942no—
229C2×C4—942no—
230C2×C4—942no—
231C2×C4D4×D4712yes—
232C2×C4Q8×D4712yes—
233C22—944no—
234C22—942no—
235C22—1882no—
236C22—1881no—
237C22—1881no—
238C22C2×D4×D4711yescharacteristic
239C22C2×Q8×D4711yescharacteristic
240C22#41×D4711yescharacteristic
241C22#1309406912yes—
242C22#1309407212yes—
243C4—22no—
244C4—22no—
245C4—42no—
246C4—42no—
247C4—41no—
248C4—474no—
249C4—942no—
250C4—1882no—
251C4—1882no—
252C4—1881no—
253C2—1882no—
254C2C22⋊Q8×D4711yescharacteristic
255C2#1568968811yescharacteristic
256C2#1568969811yescharacteristic
257C2#1568974312yes—
258C2#1568975012yes—
subgroupG/H
center, FrattiniC23C23×D47
commutator, socleC23×C47C24
FittingC47×#395C2
hypercenter#395D47
hypocenterC47#2628
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C23 ◁ #395

Complements

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

NH ≅ G/Ntype
1. C2×#13094022C2semidirect
3. #15689065C2semidirect
5. #15689043C2semidirect
8. #15689018C2semidirect
9. #15688987C2semidirect
11. C47×#395C2semidirect
52. C2×C4×D47Q8semidirect
135. #395D47semidirect
141. C47#2628semidirect