Finite Group Databasebeta

← #22594250

Subgroups — #22594250

Order 9088.

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

normal subgroups only

This group has 450 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×#14556122C211yesmaximal, characteristic
2C2×C4×Dic142C211yesmaximal, characteristic, complemented
3#17985931C211yesmaximal, characteristic, complemented
4#17985929C212yesmaximal, complemented
5#17985926C211yesmaximal, characteristic
6#17985921C212yesmaximal
7#17985920C211yesmaximal, characteristic, complemented
8#17985912C211yesmaximal, characteristic
9#17985891C212yesmaximal
10#17985884C212yesmaximal, complemented
11#382×C71C211yesmaximal, characteristic, complemented
12C22×Dic142C2211yescharacteristic
13#14556122—24no—
14C4×Dic142—24no—
15C2×C4⋊Dic71C2212yes—
16C2×C4⋊Dic71C2211yescharacteristic
17C2×C4⋊Dic71C2212yes—
18C2×Dic71⋊C4C2211yescharacteristic
19C2×Dic71⋊C4C2211yescharacteristic
20C2×Dic71⋊C4C2212yes—
21C2×Dic71⋊C4C2211yescharacteristic
22C2×(C2×D71)⋊C4C2211yescharacteristic
23C2×(C2×D71)⋊C4C2212yes—
24C2×(C2×D71)⋊C4C2211yescharacteristic
25C2×(C2×D71)⋊C4C2211yescharacteristic
26C2×(C2×D71)⋊C4C2212yes—
27C2×C4×Dic71C2212yes—
28C71⋊#95C2212yes—
29C71⋊#95C2212yes—
30C71⋊#95C2212yes—
31C71⋊#95C2212yes—
32C2×C4⋊C4×C71C2212yes—
33C2×C42×C71C2211yescharacteristic
34#95×C71C2212yes—
35#95×C71C2211yescharacteristic
36#95×C71C2211yescharacteristic
37C23×D71C2311yescharacteristic
38C2×Dic142—22no—
39C2×Dic142—24no—
40C22×Dic71C2311yescharacteristic
41C22×Dic71C2312yes—
42C22×Dic71C2311yescharacteristic
43C22×Dic71C2311yescharacteristic
44C22×Dic71C2312yes—
45C4⋊Dic71—22no—
46C4⋊Dic71—24no—
47C4⋊Dic71—42no—
48Dic71⋊C4—24no—
49Dic71⋊C4—22no—
50Dic71⋊C4—22no—
51Dic71⋊C4—22no—
52(C2×D71)⋊C4—22no—
53(C2×D71)⋊C4—22no—
54(C2×D71)⋊C4—22no—
55(C2×D71)⋊C4—24no—
56(C2×D71)⋊C4—42no—
57C4×Dic71—24no—
58C22×C4×C71C2312yes—
59C22×C4×C71C2312yes—
60C22×C4×C71C2311yescharacteristic
61C22×C4×C71C2311yescharacteristic
62C22×C4×C71C2311yescharacteristic
63C4⋊C4×C71—42no—
64C42×C71D414yes—
65C23×C71C2411yescharacteristic
66C22×D71—21no—
67C22×D71—22no—
68C22×D71—21no—
69C22×D71—22no—
70C22×D71—21no—
71C2×C4×C71—22no—
72C2×C4×C71—21no—
73C2×C4×C71—22no—
74C2×C4×C71—21no—
75C2×C4×C71—22no—
76C2×C4×C71—22no—
77C2×C4×C71—22no—
78C2×C4×C71—22no—
79C2×C4×C71—22no—
80C2×C4×C71—22no—
81C2×C4×C71C2×D412yes—
82C2×C4×C71C2×D412yes—
83C2×C4×C71C2×D412yes—
84C2×Dic71—22no—
85C2×Dic71—22no—
86C2×Dic71—22no—
87C2×Dic71—21no—
88C2×Dic71—21no—
89C2×Dic71—21no—
90C2×Dic71—22no—
91C2×Dic71—22no—
92C2×Dic71—21no—
93C2×Dic71—22no—
94C2×Dic71—22no—
95C2×Dic71—21no—
96C2×Dic71#4114yes—
97Dic142—28no—
98C22×C71C22×D411yescharacteristic
99C22×C71C2×#4112yes—
100C22×C71H(5,𝔽2)11yescharacteristic
101C22×C712−1+412yes—
102C22×C712−1+411yescharacteristic
103C2×D71—41no—
104C2×D71—41no—
105C2×D71—41no—
106C2×D71—42no—
107C2×D71—42no—
108C4×C71—22no—
109C4×C71—22no—
110C4×C71—22no—
111C4×C71—42no—
112C4×C71—42no—
113Dic71—24no—
114Dic71—41no—
115Dic71—41no—
116Dic71—42no—
117Dic71—41no—
118C2×C71#52511yescharacteristic
119C2×C71#53611yescharacteristic
120C2×C71#53711yescharacteristic
121C2×C71#54112yes—
122C2×C71#54212yes—
123D71—81no—
124#2637—711nomaximal, Sylow
125C71#263711yescharacteristic, complemented, Sylow
126C2×#126—711no—
127C2×C4×Q8—711no—
128#398—712no—
129#398—711no—
130#397—711no—
131#396—712no—
132#395—711no—
133#393—711no—
134#388—712no—
135#387—712no—
136#382D7111yescharacteristic, complemented
137C22×Q8—711no—
138#126—1424no—
139C4×Q8—1424no—
140C2×C4⋊C4—712no—
141C2×C4⋊C4—712no—
142C2×C4⋊C4—711no—
143C2×C4⋊C4—711no—
144C2×C4⋊C4—711no—
145C2×C4⋊C4—711no—
146C2×C4⋊C4—712no—
147C2×C4⋊C4C2×D7112yes—
148C2×C22⋊C4—712no—
149C2×C22⋊C4—711no—
150C2×C22⋊C4—711no—
151C2×C22⋊C4—711no—
152C2×C22⋊C4—712no—
153C2×C42—712no—
154C2×C42C2×D7111yescharacteristic
155#95—712no—
156#95—712no—
157#95—712no—
158#95—712no—
159#95C2×D7112yes—
160#95C2×D7111yescharacteristic
161#95C2×D7111yescharacteristic
162C24—711no—
163C2×Q8—1424no—
164C2×Q8—1422no—
165C22×C4—712no—
166C22×C4—711no—
167C22×C4—712no—
168C22×C4—711no—
169C22×C4—711no—
170C22×C4C22×D7112yes—
171C22×C4C22×D7112yes—
172C22×C4C22×D7111yescharacteristic
173C22×C4C22×D7111yescharacteristic
174C22×C4C22×D7111yescharacteristic
175C4⋊C4—42no—
176C4⋊C4—2842no—
177C4⋊C4—1424no—
178C4⋊C4—1424no—
179C4⋊C4—1422no—
180C4⋊C4—1422no—
181C4⋊C4—1422no—
182C4⋊C4—1422no—
183C22⋊C4—2842no—
184C22⋊C4—1424no—
185C22⋊C4—1422no—
186C22⋊C4—1422no—
187C22⋊C4—1422no—
188C42—1424no—
189C42D28414yes—
190C23—1421no—
191C23—1422no—
192C23—1421no—
193C23—1422no—
194C23—1421no—
195C23C23×D7111yescharacteristic
196Q8—1428no—
197C2×C4—22no—
198C2×C4—21no—
199C2×C4—22no—
200C2×C4—21no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—714no—
208C2×C4—1422no—
209C2×C4—1421no—
210C2×C4—1421no—
211C2×C4—1421no—
212C2×C4—1422no—
213C2×C4—1422no—
214C2×C4—1421no—
215C2×C4—1422no—
216C2×C4—1422no—
217C2×C4—1421no—
218C2×C4—1422no—
219C2×C4—1422no—
220C2×C4C2×D28412yes—
221C2×C4C2×D28412yes—
222C2×C4C2×D28412yes—
223C22—2841no—
224C22—2841no—
225C22—2842no—
226C22—2842no—
227C22—2841no—
228C22C22×D28411yescharacteristic
229C22#41×D7112yes—
230C22#1455617311yescharacteristic
231C22#1455617512yes—
232C22#1455617611yescharacteristic
233C4—22no—
234C4—22no—
235C4—22no—
236C4—42no—
237C4—42no—
238C4—2841no—
239C4—2842no—
240C4—2841no—
241C4—2841no—
242C4—1424no—
243C2—5681no—
244C2#1798651111yescharacteristic
245C2#1798657511yescharacteristic
246C2#1798658011yescharacteristic
247C2#1798660812yes—
248C2#1798661312yes—
subgroupG/H
center, FrattiniC23C23×D71
commutator, socleC23×C71C24
Fitting#382×C71C2
hypercenter#382D71
hypocenterC71#2637
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C23 ◁ #382

Complements

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

NH ≅ G/Ntype
2. C2×C4×Dic142C2semidirect
3. #17985931C2semidirect
4. #17985929C2semidirect
7. #17985920C2semidirect
10. #17985884C2semidirect
11. #382×C71C2semidirect
125. C71#2637semidirect
136. #382D71semidirect