Finite Group Databasebeta

← #20250406

Subgroups — #20250406

Order 6016.

528 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 149 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×#13094000C212yesmaximal, complemented
2#15689060C212yesmaximal, complemented
3#15689055C212yesmaximal, complemented
4#15689045C212yesmaximal, complemented
5#15689025C211yesmaximal, characteristic
6#15689021C211yesmaximal, characteristic, complemented
7#15689001C212yesmaximal
8#15688997C212yesmaximal, complemented
9C47×#391C211yesmaximal, characteristic, complemented
10C22×D188C2211yescharacteristic
11C22×C4×D47C2212yes—
12#13094000—28no—
13C2×C4⋊Dic47C2211yescharacteristic
14C2×C4⋊Dic47C2212yes—
15C2×C4⋊Dic47C2211yescharacteristic
16C2×C4⋊Dic47C2211yescharacteristic
17C2×C4⋊Dic47C2212yes—
18C2×Dic47⋊C4C2212yes—
19C2×Dic47⋊C4C2212yes—
20C2×(C2×D47)⋊C4C2212yes—
21C2×(C2×D47)⋊C4C2212yes—
22C2×(C2×D47)⋊C4C2212yes—
23C2×(C2×D47)⋊C4C2212yes—
24C2×(C2×D47)⋊C4C2212yes—
25C47⋊#95C2212yes—
26C47⋊#95C2212yes—
27C22×Q8×C47C2211yescharacteristic
28C2×C4⋊C4×C47C2211yescharacteristic
29C2×C42×C47C2211yescharacteristic
30#95×C47C2212yes—
31#95×C47C2212yes—
32C23×D47C2312yes—
33C2×Q8×C47—41no—
34C2×Q8×C47—41no—
35C2×Q8×C47D414yescomplemented
36C22×C4×C47C2312yes—
37C22×C4×C47C2311yescharacteristic
38C22×C4×C47C2312yes—
39C22×C4×C47C2311yescharacteristic
40C22×C4×C47C2311yescharacteristic
41C22×Dic47C2312yes—
42C22×Dic47C2312yes—
43C22×Dic47C2312yes—
44C2×D188—41no—
45C2×D188—41no—
46C2×D188Q814yescomplemented
47C2×C4×D47—24no—
48C2×C4×D47—42no—
49C2×C4×D47—42no—
50C4⋊C4×C47—41no—
51C42×C47—41no—
52(C2×D47)⋊C4—24no—
53(C2×D47)⋊C4—24no—
54(C2×D47)⋊C4—24no—
55(C2×D47)⋊C4—42no—
56(C2×D47)⋊C4—42no—
57C4⋊Dic47—24no—
58C4⋊Dic47—22no—
59C4⋊Dic47—41no—
60C4⋊Dic47—42no—
61C4⋊Dic47—41no—
62Dic47⋊C4—24no—
63Dic47⋊C4—24no—
64C23×C47C2411yescharacteristic
65C22×D47—22no—
66C22×D47—22no—
67C22×D47—22no—
68C22×D47—22no—
69C22×D47—22no—
70C22×D47—22no—
71C22×D47C2×Q814yes—
72Q8×C47—44no—
73C2×C4×C47—21no—
74C2×C4×C47—21no—
75C2×C4×C47—21no—
76C2×C4×C47—21no—
77C2×C4×C47—21no—
78C2×C4×C47—22no—
79C2×C4×C47—22no—
80C2×C4×C47—21no—
81C2×C4×C47—22no—
82C2×C4×C47—22no—
83C2×C4×C47—22no—
84C2×C4×C47C2×D412yes—
85C2×C4×C47C2×D414yes—
86C2×C4×C47C2×Q812yes—
87C2×C4×C47#4112yes—
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—
97D188—44no—
98C4×D47—48no—
99C22×C47C22×D411yescharacteristic
100C22×C47C22×Q811yescharacteristic
101C22×C47C2×#4111yescharacteristic
102C22×C47H(5,𝔽2)11yescharacteristic
103C22×C47H(5,𝔽2)11yescharacteristic
104C22×C47H(5,𝔽2)11yescharacteristic
105C22×C472−1+411yescharacteristic
106C2×D47—24no—
107C2×D47—24no—
108C2×D47—24no—
109C2×D47—42no—
110C2×D47—42no—
111C2×D47—42no—
112C2×D47—42no—
113C4×C47—24no—
114C4×C47—22no—
115C4×C47—22no—
116C4×C47—22no—
117C4×C47—42no—
118Dic47—42no—
119Dic47—42no—
120Dic47—42no—
121#2732—471nomaximal, Sylow
122C2×C47#53411yescharacteristic
123C2×C47#53911yescharacteristic
124C2×C47#54311yescharacteristic
125C2×C47D4×Q811yescharacteristic
126C2×C47#55011yescharacteristic
127C2×C47#55411yescharacteristic
128C2×C47#56211yescharacteristic
129D47—44no—
130C2×C22⋊Q8—472no—
131#399—472no—
132#398—472no—
133#396—472no—
134#391D4711yescharacteristic, complemented
135#390—471no—
136#389—471no—
137#386—472no—
138#385—472no—
139C47#273211yescharacteristic, complemented, Sylow
140C22×Q8C2×D4711yescharacteristic
141C22×D4—471no—
142C23×C4—472no—
143C22⋊Q8—948no—
144C2×C4⋊C4—472no—
145C2×C4⋊C4—472no—
146C2×C4⋊C4—472no—
147C2×C4⋊C4—471no—
148C2×C4⋊C4—471no—
149C2×C4⋊C4—471no—
150C2×C4⋊C4—472no—
151C2×C4⋊C4C2×D4711yescharacteristic
152C2×C22⋊C4—472no—
153C2×C22⋊C4—472no—
154C2×C22⋊C4—472no—
155C2×C22⋊C4—472no—
156C2×C22⋊C4—472no—
157C2×C42C2×D4711yescharacteristic
158#95—472no—
159#95—472no—
160#95C2×D4712yes—
161#95C2×D4712yes—
162C24—472no—
163C2×Q8—41no—
164C2×Q8—41no—
165C2×Q8D18814yescomplemented
166C2×D4—474no—
167C2×D4—1881no—
168C2×D4—1881no—
169C22×C4—472no—
170C22×C4—472no—
171C22×C4—472no—
172C22×C4—944no—
173C22×C4—1882no—
174C22×C4—1882no—
175C22×C4C22×D4712yes—
176C22×C4C22×D4712yes—
177C22×C4C22×D4711yescharacteristic
178C22×C4C22×D4711yescharacteristic
179C22×C4C22×D4711yescharacteristic
180C4⋊C4—41no—
181C4⋊C4—944no—
182C4⋊C4—944no—
183C4⋊C4—944no—
184C4⋊C4—942no—
185C4⋊C4—1881no—
186C4⋊C4—1881no—
187C4⋊C4—1882no—
188C22⋊C4—944no—
189C22⋊C4—944no—
190C22⋊C4—944no—
191C22⋊C4—1882no—
192C22⋊C4—1882no—
193C42—41no—
194C23—474no—
195C23—942no—
196C23—942no—
197C23—942no—
198C23—942no—
199C23—942no—
200C23—942no—
201C23C23×D4711yescharacteristic
202Q8—44no—
203D4—1884no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
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—1888no—
225C2×C4C2×D18812yes—
226C2×C4C2×D18814yes—
227C2×C4Q8×D4712yes—
228C2×C4#1062535412yes—
229C22—944no—
230C22—944no—
231C22—944no—
232C22—1882no—
233C22—1882no—
234C22—1882no—
235C22—1882no—
236C22C22×D18811yescharacteristic
237C22C2×Q8×D4711yescharacteristic
238C22C2×#1062535411yescharacteristic
239C22#1309406911yescharacteristic
240C22#1309406911yescharacteristic
241C22#1309407011yescharacteristic
242C22#1309407111yescharacteristic
243C4—22no—
244C4—24no—
245C4—22no—
246C4—22no—
247C4—42no—
248C4—1882no—
249C4—1882no—
250C4—1882no—
251C2—1884no—
252C2#1568968511yescharacteristic
253C2#1568972211yescharacteristic
254C2#1568974811yescharacteristic
255C2Q8×D18811yescharacteristic
256C2#1568979511yescharacteristic
257C2#1568983411yescharacteristic
258C2#1568986711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D47
commutator, socleC23×C47C24
FittingC47×#391C2
hypercenter#391D47
hypocenterC47#2732
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
1. C2×#13094000C2semidirect
2. #15689060C2semidirect
3. #15689055C2semidirect
4. #15689045C2semidirect
6. #15689021C2semidirect
8. #15688997C2semidirect
9. C47×#391C2semidirect
35. C2×Q8×C47D4semidirect
46. C2×D188Q8semidirect
134. #391D47semidirect
139. C47#2732semidirect
165. C2×Q8D188semidirect