Finite Group Databasebeta

← #22588327

Subgroups — #22588327

Order 9088.

380 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 87 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×(C22⋊D71)⋊C4C211yesmaximal, characteristic
2C2×#14556042C212yesmaximal
3#17985857C212yesmaximal, complemented
4#17985766C211yesmaximal, characteristic, complemented
5#342×C71C211yesmaximal, characteristic, complemented
6C22×C22⋊D71C2211yescharacteristic
7C23×Dic71C2211yescharacteristic
8C22×C4×D71C411yescharacteristic, complemented
9(C22⋊D71)⋊C4—22no—
10(C22⋊D71)⋊C4—22no—
11C2×Dic71⋊C4—22no—
12C2×Dic71⋊C4C411yescharacteristic, complemented
13C2×C22⋊Dic71C2212yes—
14C2×(C2×D71)⋊C4C411yescharacteristic, complemented
15C2×C4×Dic71C411yescharacteristic, complemented
16#14556042—24no—
17C71⋊#95—22no—
18C2×C22⋊C4×C71C2212yes—
19C2×C22⋊C4×C71C2211yescharacteristic
20C23×D71C2×C411yescharacteristic
21C2×C22⋊D71—22no—
22C2×C22⋊D71—22no—
23C2×C22⋊D71D412yes—
24C2×C22⋊D71Q812yes—
25C22×Dic71—22no—
26C22×Dic71—42no—
27C22×Dic71C2×C411yescharacteristic
28C22×Dic71C2×C411yescharacteristic
29C22×Dic71C2×C411yescharacteristic
30C22×Dic71D412yes—
31C22×Dic71D412yes—
32C2×C4×D71—21no—
33C2×C4×D71—21no—
34C2×C4×D71—42no—
35Dic71⋊C4—22no—
36Dic71⋊C4—44no—
37C22⋊Dic71—22no—
38C22⋊Dic71—22no—
39(C2×D71)⋊C4—22no—
40C4×Dic71—21no—
41C4×Dic71—21no—
42C24×C71C2311yescharacteristic
43C22×C4×C71—22no—
44C22×C4×C71C2×C411yescharacteristic
45C22×C4×C71C2×C411yescharacteristic
46C22⋊C4×C71—22no—
47C22⋊C4×C71—22no—
48C22⋊C4×C71—22no—
49C23×C71—41no—
50C23×C71—41no—
51C23×C71C22×C411yescharacteristic
52C23×C71C2×D412yes—
53C23×C71C2×D411yescharacteristic
54C23×C71C2×Q811yescharacteristic
55C23×C71#4112yes—
56C22×D71—21no—
57C22×D71—21no—
58C22×D71—41no—
59C22×D71—41no—
60C22×D71C4⋊C412yescomplemented
61C2×C4×C71—21no—
62C2×C4×C71—21no—
63C2×C4×C71—41no—
64C2×C4×C71—41no—
65C2×C4×C71—42no—
66C2×C4×C71—42no—
67C2×C4×C71—42no—
68C22⋊D71—24no—
69C22⋊D71—24no—
70C2×Dic71—21no—
71C2×Dic71—21no—
72C2×Dic71—24no—
73C2×Dic71—22no—
74C2×Dic71—41no—
75C2×Dic71—42no—
76C2×Dic71—42no—
77C2×Dic71—42no—
78C2×Dic71—42no—
79C2×Dic71C22⋊C411yescharacteristic
80C2×Dic71C22⋊C412yes—
81C2×Dic71C22⋊C411yescharacteristic
82C2×Dic71C4⋊C412yescomplemented
83C4×D71—42no—
84C4×D71—42no—
85C22×C71—22no—
86C22×C71—21no—
87C22×C71—22no—
88C22×C71—21no—
89C22×C71—41no—
90C22×C71—41no—
91C22×C71—42no—
92C22×C71C2×C22⋊C411yescharacteristic
93C22×C71C2×C4⋊C411yescharacteristic
94C22×C71#11711yescharacteristic
95C22×C71#12112yes—
96C22×C71C22⋊Q812yes—
97C2×D71—22no—
98C2×D71—42no—
99C2×D71—41no—
100C2×D71—41no—
101C2×D71—42no—
102C4×C71—41no—
103C4×C71—41no—
104C4×C71—82no—
105Dic71—22no—
106Dic71—21no—
107Dic71—22no—
108Dic71—82no—
109Dic71#9912yescomplemented
110C2×C71—22no—
111C2×C71—21no—
112C2×C71—21no—
113C2×C71—22no—
114C2×C71#38011yescharacteristic
115C2×C71C2×#9911yescharacteristic
116C2×C71#41011yescharacteristic
117D71—42no—
118#1799—711nomaximal, Sylow
119C71#179911yescharacteristic, complemented, Sylow
120C2×C4×D4—711no—
121C2×#99—712no—
122#380—712no—
123#342—711no—
124#342D7111yescharacteristic, complemented
125C22×D4—711no—
126C23×C4—711no—
127C23×C4—711no—
128C4×D4—1422no—
129C4×D4—1422no—
130C2×C4⋊C4—711no—
131C2×C4⋊C4—1422no—
132C2×C22⋊C4—711no—
133C2×C22⋊C4—712no—
134C2×C22⋊C4C2×D7112yes—
135C2×C22⋊C4C2×D7111yescharacteristic
136C2×C42—711no—
137#99—1424no—
138#95—1422no—
139C24—711no—
140C24C22×D7111yescharacteristic
141C2×D4—712no—
142C2×D4—712no—
143C2×D4—1422no—
144C2×D4—1422no—
145C22×C4—22no—
146C22×C4—2842no—
147C22×C4—2842no—
148C22×C4—711no—
149C22×C4—712no—
150C22×C4—711no—
151C22×C4—711no—
152C22×C4—712no—
153C22×C4—1421no—
154C22×C4—1421no—
155C22×C4—1422no—
156C22×C4C4×D7111yescharacteristic
157C22×C4C4×D7111yescharacteristic
158C4⋊C4—2844no—
159C4⋊C4—1422no—
160C22⋊C4—22no—
161C22⋊C4—22no—
162C22⋊C4—22no—
163C22⋊C4—1422no—
164C22⋊C4—1422no—
165C22⋊C4—1422no—
166C42—1421no—
167C42—1421no—
168C23—41no—
169C23—41no—
170C23—2841no—
171C23—2841no—
172C23—712no—
173C23—1421no—
174C23—1421no—
175C23C2×C4×D7111yescharacteristic
176C23D4×D7112yes—
177C23D4×D7111yescharacteristic
178C23Q8×D7111yescharacteristic
179C23#1176586512yes—
180D4—1424no—
181D4—1424no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—42no—
187C2×C4—42no—
188C2×C4—42no—
189C2×C4—2842no—
190C2×C4—2842no—
191C2×C4—2842no—
192C2×C4—2842no—
193C2×C4—2842no—
194C2×C4—2842no—
195C2×C4—2841no—
196C2×C4—711no—
197C2×C4—711no—
198C2×C4—712no—
199C2×C4—712no—
200C2×C4—1424no—
201C2×C4—1422no—
202C2×C4—1421no—
203C2×C4—1421no—
204C22—22no—
205C22—21no—
206C22—22no—
207C22—21no—
208C22—41no—
209C22—41no—
210C22—42no—
211C22—2842no—
212C22—2842no—
213C22—2841no—
214C22—2841no—
215C22—1422no—
216C22C22⋊C4×D7111yescharacteristic
217C22C4⋊C4×D7111yescharacteristic
218C22#1455608511yescharacteristic
219C22#1455610112yes—
220C22#1455610612yes—
221C4—41no—
222C4—41no—
223C4—82no—
224C4—5682no—
225C4—712no—
226C4—1422no—
227C4—1421no—
228C4—1422no—
229C2—22no—
230C2—21no—
231C2—21no—
232C2—22no—
233C2—2842no—
234C2#1798585811yescharacteristic
235C2#99×D7111yescharacteristic
236C2#1798597111yescharacteristic
subgroupG/H
centerC22C22⋊C4×D71
commutatorC23×C71C22×C4
FrattiniC24C22×D71
Fitting#342×C71C2
socleC22×C71C2×C22⋊C4
hypercenter#342D71
hypocenterC71#1799
LCSG ▷ C23×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C24 ◁ #342

Complements

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

NH ≅ G/Ntype
3. #17985857C2semidirect
4. #17985766C2semidirect
5. #342×C71C2semidirect
8. C22×C4×D71C4semidirect
12. C2×Dic71⋊C4C4semidirect
14. C2×(C2×D71)⋊C4C4semidirect
15. C2×C4×Dic71C4semidirect
60. C22×D71C4⋊C4semidirect
82. C2×Dic71C4⋊C4semidirect
109. Dic71#99semidirect
119. C71#1799semidirect
124. #342D71semidirect