Finite Group Databasebeta

← #22594258

Subgroups — #22594258

Order 9088.

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

normal subgroups only

This group has 520 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×C4×C22⋊D71C211yesmaximal, characteristic
2#17985930C211yesmaximal, characteristic, complemented
3#17985927C212yesmaximal, complemented
4#17985925C211yesmaximal, characteristic, complemented
5#17985918C211yesmaximal, characteristic, complemented
6#17985914C212yesmaximal
7#17985910C211yesmaximal, characteristic, complemented
8#17985880C212yesmaximal, complemented
9#17985870C212yesmaximal, complemented
10#17985862C211yesmaximal, characteristic, complemented
11C2×C22⋊Q8×C71C211yesmaximal, characteristic, complemented
12C22×C22⋊D71C2211yescharacteristic
13C22×C4×D71C2211yescharacteristic
14C4×C22⋊D71—24no—
15C2×Dic71⋊C4C2212yes—
16C2×Dic71⋊C4C2211yescharacteristic
17C2×Dic71⋊C4C2212yes—
18C2×C22⋊Dic71C2211yescharacteristic
19C2×C22⋊Dic71C2212yes—
20C2×(C2×D71)⋊C4C2212yes—
21C2×(C2×D71)⋊C4C2211yescharacteristic
22C2×(C2×D71)⋊C4C2212yes—
23C2×C4×Dic71C2211yescharacteristic
24C71⋊#95C2212yes—
25C71⋊#95C2211yescharacteristic
26C71⋊#95C2211yescharacteristic
27C71⋊#95C2212yes—
28C71⋊#95C2212yes—
29C71⋊#95C2212yes—
30C71⋊#95C2212yes—
31C22×Q8×C71C2211yescharacteristic
32C23×C4×C71C2211yescharacteristic
33C22⋊Q8×C71—24no—
34C2×C4⋊C4×C71C2212yes—
35C2×C4⋊C4×C71C2211yescharacteristic
36C2×C22⋊C4×C71C2212yes—
37C23×D71C2311yescharacteristic
38C2×C22⋊D71—22no—
39C2×C22⋊D71—22no—
40C2×C22⋊D71—22no—
41C22×Dic71C2311yescharacteristic
42C22×Dic71C2311yescharacteristic
43C22×Dic71C2312yes—
44C22×Dic71C2312yes—
45C22×Dic71C2311yescharacteristic
46C2×C4×D71—22no—
47C2×C4×D71—42no—
48Dic71⋊C4—22no—
49Dic71⋊C4—24no—
50Dic71⋊C4—42no—
51C22⋊Dic71—22no—
52C22⋊Dic71—42no—
53(C2×D71)⋊C4—24no—
54(C2×D71)⋊C4—22no—
55(C2×D71)⋊C4—42no—
56C4×Dic71—22no—
57C24×C71C2311yescharacteristic
58C2×Q8×C71—22no—
59C2×Q8×C71—42no—
60C22×C4×C71—42no—
61C22×C4×C71D414yes—
62C22×C4×C71C2312yes—
63C22×C4×C71C2312yes—
64C22×C4×C71C2311yescharacteristic
65C22×C4×C71C2311yescharacteristic
66C4⋊C4×C71—22no—
67C4⋊C4×C71—24no—
68C22⋊C4×C71—24no—
69C23×C71—21no—
70C23×C71—21no—
71C23×C71—22no—
72C23×C71—22no—
73C23×C71C2×D412yes—
74C23×C71C2411yescharacteristic
75C22×D71—21no—
76C22×D71—21no—
77C22×D71—22no—
78C22×D71—22no—
79C22×D71#4112yes—
80Q8×C71—44no—
81C2×C4×C71—22no—
82C2×C4×C71—22no—
83C2×C4×C71—22no—
84C2×C4×C71—22no—
85C2×C4×C71—22no—
86C2×C4×C71—22no—
87C2×C4×C71—22no—
88C2×C4×C71—22no—
89C2×C4×C71—44no—
90C2×C4×C71C2×D412yes—
91C2×C4×C71C2×D412yes—
92C22⋊D71—28no—
93C2×Dic71—21no—
94C2×Dic71—21no—
95C2×Dic71—21no—
96C2×Dic71—21no—
97C2×Dic71—22no—
98C2×Dic71—22no—
99C2×Dic71—22no—
100C2×Dic71—22no—
101C2×Dic71—22no—
102C2×Dic71—22no—
103C2×Dic71—22no—
104C2×Dic71—22no—
105C2×Dic71#4112yes—
106C4×D71—44no—
107C22×C71—22no—
108C22×C71—22no—
109C22×C71—22no—
110C22×C71—42no—
111C22×C71—42no—
112C22×C71C22×D411yescharacteristic
113C22×C71C2×#4111yescharacteristic
114C22×C71C2×#4111yescharacteristic
115C22×C71H(5,𝔽2)12yes—
116C22×C712−1+412yes—
117C2×D71—22no—
118C2×D71—24no—
119C2×D71—41no—
120C2×D71—41no—
121C2×D71—42no—
122C4×C71—22no—
123C4×C71—22no—
124C4×C71—42no—
125C4×C71—42no—
126Dic71—22no—
127Dic71—41no—
128Dic71—41no—
129Dic71—42no—
130Dic71—42no—
131C2×C71—42no—
132C2×C71#52511yescharacteristic
133C2×C71#53411yescharacteristic
134C2×C71#53611yescharacteristic
135C2×C71#53912yes—
136C2×C71#54212yes—
137D71—42no—
138#2638—711nomaximal, Sylow
139C71#263811yescharacteristic, complemented, Sylow
140C2×C22⋊Q8D7111yescharacteristic, complemented
141C2×C4×D4—711no—
142#398—711no—
143#397—711no—
144#397—712no—
145#395—711no—
146#394—712no—
147#393—711no—
148#387—712no—
149#385—712no—
150#381—711no—
151C22×Q8C2×D7111yescharacteristic
152C22×D4—711no—
153C23×C4—711no—
154C23×C4C2×D7111yescharacteristic
155C22⋊Q8—24no—
156C4×D4—1424no—
157C2×C4⋊C4—712no—
158C2×C4⋊C4—711no—
159C2×C4⋊C4—712no—
160C2×C4⋊C4C2×D7112yes—
161C2×C4⋊C4C2×D7111yescharacteristic
162C2×C22⋊C4—712no—
163C2×C22⋊C4—712no—
164C2×C22⋊C4—711no—
165C2×C22⋊C4—711no—
166C2×C22⋊C4—712no—
167C2×C22⋊C4C2×D7112yes—
168C2×C42—711no—
169#95—712no—
170#95—711no—
171#95—711no—
172#95—712no—
173#95—712no—
174#95—712no—
175#95—712no—
176C24—711no—
177C24C22×D7111yescharacteristic
178C2×Q8—22no—
179C2×Q8—42no—
180C2×D4—1422no—
181C2×D4—1422no—
182C2×D4—1422no—
183C22×C4—42no—
184C22×C4—2842no—
185C22×C4—711no—
186C22×C4—711no—
187C22×C4—712no—
188C22×C4—712no—
189C22×C4—711no—
190C22×C4—1422no—
191C22×C4C22⋊D7114yes—
192C22×C4C22×D7112yes—
193C22×C4C22×D7112yes—
194C22×C4C22×D7111yescharacteristic
195C22×C4C22×D7111yescharacteristic
196C4⋊C4—22no—
197C4⋊C4—24no—
198C4⋊C4—2842no—
199C4⋊C4—1424no—
200C4⋊C4—1422no—
201C22⋊C4—24no—
202C22⋊C4—2842no—
203C22⋊C4—2842no—
204C22⋊C4—1424no—
205C22⋊C4—1422no—
206C22⋊C4—1422no—
207C42—1422no—
208C23—21no—
209C23—21no—
210C23—22no—
211C23—22no—
212C23—712no—
213C23—1421no—
214C23—1421no—
215C23—1422no—
216C23—1422no—
217C23C2×C22⋊D7112yes—
218C23C23×D7111yescharacteristic
219Q8—44no—
220D4—1428no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—44no—
230C2×C4—2844no—
231C2×C4—712no—
232C2×C4—1421no—
233C2×C4—1421no—
234C2×C4—1422no—
235C2×C4—1422no—
236C2×C4—1422no—
237C2×C4—1422no—
238C2×C4—1422no—
239C2×C4—1422no—
240C2×C4—1422no—
241C2×C4—1422no—
242C2×C4—1421no—
243C2×C4—1421no—
244C2×C4C2×C22⋊D7112yes—
245C2×C4C2×C22⋊D7112yes—
246C22—22no—
247C22—22no—
248C22—22no—
249C22—42no—
250C22—42no—
251C22—2841no—
252C22—2842no—
253C22—2841no—
254C22—1424no—
255C22—1422no—
256C22C22×C22⋊D7111yescharacteristic
257C22#41×D7111yescharacteristic
258C22#41×D7111yescharacteristic
259C22#1455617312yes—
260C22#1455617612yes—
261C4—22no—
262C4—22no—
263C4—42no—
264C4—42no—
265C4—2841no—
266C4—2842no—
267C4—2842no—
268C4—2841no—
269C4—1422no—
270C2—42no—
271C2—2842no—
272C2#1798650611yescharacteristic
273C2#1798655811yescharacteristic
274C2#1798657311yescharacteristic
275C2#1798659512yes—
276C2#1798661912yes—
subgroupG/H
center, FrattiniC23C23×D71
commutator, socleC23×C71C24
FittingC2×C22⋊Q8×C71C2
hypercenterC2×C22⋊Q8D71
hypocenterC71#2638
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C23 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. #17985930C2semidirect
3. #17985927C2semidirect
4. #17985925C2semidirect
5. #17985918C2semidirect
7. #17985910C2semidirect
8. #17985880C2semidirect
9. #17985870C2semidirect
10. #17985862C2semidirect
11. C2×C22⋊Q8×C71C2semidirect
139. C71#2638semidirect
140. C2×C22⋊Q8D71semidirect