Finite Group Databasebeta

← #22586841

Subgroups — #22586841

Order 9088.

334 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 65 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
1#17986509C211yesmaximal, characteristic
2#17985961C211yesmaximal, characteristic, complemented
3#95×D71C211yesmaximal, characteristic, complemented
4#17985737C211yesmaximal, characteristic, complemented
5#17985735C211yesmaximal, characteristic, complemented
6#17985733C211yesmaximal, characteristic, complemented
7#327×C71C211yesmaximal, characteristic, complemented
8C2×#11765864C411yescharacteristic, complemented
9C22×C4×D71—21no—
10C22×C4×D71C2211yescharacteristic
11C2×#11765848—21no—
12#14556110—21no—
13#14556108C411yescharacteristic, complemented
14#14556107C2211yescharacteristic
15#14556099C411yescharacteristic, complemented
16#14556096—21no—
17D4×Dic71—21no—
18(C22⋊D71)⋊C4—21no—
19#14556086C411yescharacteristic, complemented
20C22⋊(C71⋊C8)C2211yescharacteristic
21(C2×D71)⋊C8—21no—
22C71⋊#95—21no—
23C71⋊#95C2211yescharacteristic
24#121×C71C2211yescharacteristic
25C22⋊C8×C71C2211yescharacteristic
26#95×C71C2211yescharacteristic
27C23×D71D411yescharacteristic
28#11765864—42no—
29C2×Dic142C2×C411yescharacteristic, complemented
30C2×C22⋊D71—21no—
31C2×C22⋊D71—21no—
32C22×Dic71—21no—
33C22×Dic71—21no—
34C22×Dic71D411yescharacteristic
35C2×C4×D71—24no—
36C2×C4×D71—42no—
37C2×C4×D71—42no—
38C2×C4×D71—42no—
39C2×C4×D71C2×C411yescharacteristic
40C2×C4×D71C2×C411yescharacteristic
41C2×C4×D71D411yescharacteristic
42C2×C4×D71D411yescharacteristic
43#11765848—42no—
44C2×C71⋊C8—21no—
45C4⋊Dic71—21no—
46C4⋊Dic71C2×C411yescharacteristic, complemented
47Dic71⋊C4—21no—
48C22⋊Dic71—21no—
49C22⋊Dic71—21no—
50(C2×D71)⋊C4—21no—
51(C2×D71)⋊C4—21no—
52C4×Dic71—21no—
53C2×D4×C71—21no—
54C2×D4×C71C2×C411yescharacteristic, complemented
55C22×C4×C71—21no—
56C22×C4×C71C2311yescharacteristic
57C2×C8×C71—21no—
58C4⋊C4×C71C2×C411yescharacteristic, complemented
59C22⋊C4×C71—21no—
60C23×C71—21no—
61C23×C71C2×D411yescharacteristic
62C22×D71—21no—
63C22×D71—21no—
64C22×D71—22no—
65C22×D71—22no—
66C22×D71C22⋊C411yescharacteristic
67C22×D71C22⋊C411yescharacteristic
68D4×C71—41no—
69D4×C71—41no—
70D4×C71—41no—
71C2×C4×C71—22no—
72C2×C4×C71—21no—
73C2×C4×C71—41no—
74C2×C4×C71—41no—
75C2×C4×C71—41no—
76C2×C4×C71C22×C411yescharacteristic
77C2×C4×C71C2×D411yescharacteristic
78C22⋊D71—42no—
79C22⋊D71—42no—
80C2×Dic71—22no—
81C2×Dic71—21no—
82C2×Dic71—21no—
83C2×Dic71—41no—
84C2×Dic71—42no—
85C2×Dic71—41no—
86C2×Dic71—41no—
87C2×Dic71C22⋊C411yescharacteristic
88C2×Dic71C22⋊C411yescharacteristic
89C4×D71—22no—
90C4×D71—22no—
91C4×D71—41no—
92C4×D71—41no—
93C4×D71—48no—
94Dic142—41no—
95C8×C71—41no—
96C71⋊C8—41no—
97C22×C71—21no—
98C22×C71—21no—
99C22×C71—41no—
100C22×C71—41no—
101C22×C71—41no—
102C22×C71C2×C22⋊C411yescharacteristic
103C22×C71AGL(1,ℤ/8)11yescharacteristic
104C22×C71#13711yescharacteristic
105C2×D71—24no—
106C2×D71—24no—
107C2×D71—21no—
108C2×D71—21no—
109C2×D71—21no—
110C2×D71—21no—
111C2×D71#9912yes—
112C2×D71#10412yes—
113C4×C71—21no—
114C4×C71—21no—
115C4×C71—42no—
116C4×C71—41no—
117Dic71—21no—
118Dic71—21no—
119Dic71—42no—
120Dic71—41no—
121Dic71—41no—
122C2×C71—21no—
123C2×C71—21no—
124C2×C71—81no—
125C2×C71C2×#9911yescharacteristic
126C2×C71#41711yescharacteristic
127C2×C71C2×#10411yescharacteristic
128D71—22no—
129D71—22no—
130#1504—711nomaximal, Sylow
131C71#150411yescharacteristic, complemented, Sylow
132#525—711no—
133#407—711no—
134C2×#95—711no—
135#328—711no—
136#328—711no—
137#327—711no—
138#327D7111yescharacteristic, complemented
139C2×#41—711no—
140C23×C4—711no—
141C23×C4—1421no—
142C2×M16—1421no—
143#123—1421no—
144C22⋊Q8—711no—
145C22⋊Q8—711no—
146#121—711no—
147#121C2×D7111yescharacteristic
148#120—1421no—
149C4×D4—1421no—
150C4×D4—1421no—
151#117—711no—
152C22⋊C8—711no—
153C22⋊C8—1421no—
154C22⋊C8C2×D7111yescharacteristic
155#95—711no—
156#95—1421no—
157#95C2×D7111yescharacteristic
158C24—711no—
159#41—2842no—
160C2×Q8—711no—
161C2×D4—21no—
162C2×D4—1421no—
163C2×D4—1421no—
164C2×D4C4×D7111yescharacteristic, complemented
165C22×C4—21no—
166C22×C4—2842no—
167C22×C4—2842no—
168C22×C4—2842no—
169C22×C4—711no—
170C22×C4—711no—
171C22×C4—711no—
172C22×C4—711no—
173C22×C4—711no—
174C22×C4—1424no—
175C22×C4—1421no—
176C22×C4—1421no—
177C22×C4C22×D7111yescharacteristic
178M16—2842no—
179C2×C8—21no—
180C2×C8—1421no—
181C4⋊C4—711no—
182C4⋊C4—1421no—
183C4⋊C4—1421no—
184C4⋊C4C4×D7111yescharacteristic, complemented
185C22⋊C4—21no—
186C22⋊C4—1421no—
187C22⋊C4—1421no—
188C22⋊C4—1421no—
189C22⋊C4—1421no—
190C42—1421no—
191C23—21no—
192C23—711no—
193C23—711no—
194C23—1422no—
195C23—1422no—
196C23—1421no—
197C23—1421no—
198C23D4×D7111yescharacteristic
199Q8—2841no—
200D4—41no—
201D4—41no—
202D4—41no—
203D4—2842no—
204D4—2842no—
205C2×C4—22no—
206C2×C4—21no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—2841no—
211C2×C4—2841no—
212C2×C4—2848no—
213C2×C4—2841no—
214C2×C4—2841no—
215C2×C4—2842no—
216C2×C4—2841no—
217C2×C4—711no—
218C2×C4—711no—
219C2×C4—1422no—
220C2×C4—1422no—
221C2×C4—1422no—
222C2×C4—1421no—
223C2×C4—1421no—
224C2×C4C2×C4×D7111yescharacteristic
225C2×C4D4×D7111yescharacteristic
226C8—41no—
227C8—2841no—
228C22—21no—
229C22—21no—
230C22—41no—
231C22—41no—
232C22—41no—
233C22—712no—
234C22—712no—
235C22—1424no—
236C22—1424no—
237C22—1421no—
238C22—1421no—
239C22—1421no—
240C22—1421no—
241C22C22⋊C4×D7111yescharacteristic
242C22#1455615211yescharacteristic
243C22#1455615711yescharacteristic
244C4—21no—
245C4—21no—
246C4—42no—
247C4—41no—
248C4—2842no—
249C4—2841no—
250C4—2841no—
251C4—1421no—
252C4—1421no—
253C2—21no—
254C2—21no—
255C2—81no—
256C2—1422no—
257C2—1422no—
258C2#99×D7111yescharacteristic
259C2#1798599711yescharacteristic
260C2#104×D7111yescharacteristic
subgroupG/H
centerC22C22⋊C4×D71
commutatorC2×C4×C71C22×C4
FrattiniC22×C4C22×D71
Fitting#327×C71C2
socleC22×C71C2×C22⋊C4
hypercenter#327D71
hypocenterC71#1504
LCSG ▷ C2×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ #327

Complements

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

NH ≅ G/Ntype
2. #17985961C2semidirect
3. #95×D71C2semidirect
4. #17985737C2semidirect
5. #17985735C2semidirect
6. #17985733C2semidirect
7. #327×C71C2semidirect
8. C2×#11765864C4semidirect
13. #14556108C4semidirect
15. #14556099C4semidirect
19. #14556086C4semidirect
29. C2×Dic142C2×C4semidirect
46. C4⋊Dic71C2×C4semidirect
54. C2×D4×C71C2×C4semidirect
58. C4⋊C4×C71C2×C4semidirect
131. C71#1504semidirect
138. #327D71semidirect
164. C2×D4C4×D71semidirect
184. C4⋊C4C4×D71semidirect