Finite Group Databasebeta

← #22591597

Subgroups — #22591597

Order 9088.

560 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 560 conjugacy classes of subgroups — too many to draw. Only its 157 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#17985937C211yesmaximal, characteristic, complemented
3#17985934C211yesmaximal, characteristic, complemented
4#17985930C211yesmaximal, characteristic, complemented
5#17985927C212yesmaximal, complemented
6#17985925C211yesmaximal, characteristic, complemented
7#17985914C212yesmaximal
8C4×Dic71⋊C4C212yesmaximal, complemented
9C4×(C2×D71)⋊C4C212yesmaximal, complemented
10C4×C22⋊Dic71C211yesmaximal, characteristic, complemented
11C2×#117×C71C211yesmaximal, characteristic, complemented
12C22×C22⋊D71C2211yescharacteristic
13C22×C4×D71C2211yescharacteristic
14C4×C22⋊D71—24no—
15C2×Dic71⋊C4C2212yes—
16C2×Dic71⋊C4C2212yes—
17C2×Dic71⋊C4C2211yescharacteristic
18C2×C22⋊Dic71C2212yes—
19C2×C22⋊Dic71C2211yescharacteristic
20C2×(C2×D71)⋊C4C2212yes—
21C2×(C2×D71)⋊C4C2212yes—
22C2×(C2×D71)⋊C4C2211yescharacteristic
23C2×C4×Dic71C2212yes—
24C2×C4×Dic71C2211yescharacteristic
25C71⋊#95C2211yescharacteristic
26C71⋊#95C2211yescharacteristic
27C71⋊#95C2212yes—
28C71⋊#95C2212yes—
29C71⋊#95C2212yes—
30C71⋊#95C2212yes—
31C23×C4×C71C2211yescharacteristic
32#117×C71—24no—
33C2×C4⋊C4×C71C2211yescharacteristic
34C2×C4⋊C4×C71C2211yescharacteristic
35C2×C22⋊C4×C71C2212yes—
36C2×C42×C71C2212yes—
37C23×D71C2311yescharacteristic
38C2×C22⋊D71—22no—
39C2×C22⋊D71—22no—
40C2×C22⋊D71—22no—
41C22×Dic71C2311yescharacteristic
42C22×Dic71C2312yes—
43C22×Dic71C2311yescharacteristic
44C22×Dic71C2311yescharacteristic
45C22×Dic71C2312yes—
46C2×C4×D71—22no—
47C2×C4×D71—24no—
48Dic71⋊C4—22no—
49Dic71⋊C4—24no—
50Dic71⋊C4—24no—
51C22⋊Dic71—22no—
52C22⋊Dic71—24no—
53(C2×D71)⋊C4—24no—
54(C2×D71)⋊C4—22no—
55(C2×D71)⋊C4—24no—
56C4×Dic71—22no—
57C4×Dic71—24no—
58C24×C71C2311yescharacteristic
59C22×C4×C71—24no—
60C22×C4×C71D414yes—
61C22×C4×C71C2312yes—
62C22×C4×C71C2312yes—
63C22×C4×C71C2311yescharacteristic
64C22×C4×C71C2311yescharacteristic
65C4⋊C4×C71—22no—
66C4⋊C4×C71—22no—
67C22⋊C4×C71—24no—
68C42×C71—24no—
69C23×C71—22no—
70C23×C71—22no—
71C23×C71—21no—
72C23×C71—21no—
73C23×C71C2×D412yes—
74C23×C71C2411yescharacteristic
75C22×D71—21no—
76C22×D71—22no—
77C22×D71—22no—
78C22×D71—21no—
79C22×D71#4112yes—
80C2×C4×C71—22no—
81C2×C4×C71—28no—
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×C71C2×D412yes—
89C2×C4×C71C2×D412yes—
90C2×C4×C71#4114yes—
91C22⋊D71—28no—
92C2×Dic71—21no—
93C2×Dic71—22no—
94C2×Dic71—22no—
95C2×Dic71—21no—
96C2×Dic71—21no—
97C2×Dic71—21no—
98C2×Dic71—22no—
99C2×Dic71—22no—
100C2×Dic71—22no—
101C2×Dic71—22no—
102C2×Dic71—22no—
103C2×Dic71#4114yes—
104C2×Dic71#4112yes—
105C4×D71—44no—
106C22×C71—22no—
107C22×C71—22no—
108C22×C71—22no—
109C22×C71—42no—
110C22×C71—42no—
111C22×C71C22×D411yescharacteristic
112C22×C71C2×#4111yescharacteristic
113C22×C71C2×#4111yescharacteristic
114C22×C71C2×#4112yes—
115C22×C71C2×#4112yes—
116C2×D71—22no—
117C2×D71—24no—
118C2×D71—41no—
119C2×D71—42no—
120C2×D71—41no—
121C4×C71—22no—
122C4×C71—42no—
123C4×C71—42no—
124C4×C71#12314yes—
125Dic71—22no—
126Dic71—24no—
127Dic71—41no—
128Dic71—42no—
129Dic71—41no—
130C2×C71—42no—
131C2×C71C2×#12311yescharacteristic
132C2×C71#52511yescharacteristic
133C2×C71#52912yes—
134C2×C71#52912yes—
135C2×C71#53211yescharacteristic
136D71—42no—
137#2395—711nomaximal, Sylow
138C71#239511yescharacteristic, complemented, Sylow
139C2×C4×D4—711no—
140C2×#117D7111yescharacteristic, complemented
141#400—711no—
142#399—711no—
143#398—711no—
144#397—711no—
145#397—712no—
146#394—712no—
147C4×C4⋊C4—712no—
148C4×C22⋊C4—711no—
149C4×C22⋊C4—712no—
150C22×D4—711no—
151C23×C4—711no—
152C23×C4C2×D7111yescharacteristic
153C4×D4—1424no—
154#117—24no—
155C2×C4⋊C4—711no—
156C2×C4⋊C4—712no—
157C2×C4⋊C4—712no—
158C2×C4⋊C4C2×D7111yescharacteristic
159C2×C4⋊C4C2×D7111yescharacteristic
160C2×C22⋊C4—711no—
161C2×C22⋊C4—712no—
162C2×C22⋊C4—711no—
163C2×C22⋊C4—712no—
164C2×C22⋊C4—712no—
165C2×C22⋊C4C2×D7112yes—
166C2×C42—711no—
167C2×C42—712no—
168C2×C42C2×D7112yes—
169#95—712no—
170#95—711no—
171#95—712no—
172#95—711no—
173#95—712no—
174#95—712no—
175C24—711no—
176C24C22×D7111yescharacteristic
177C2×D4—1422no—
178C2×D4—1422no—
179C2×D4—1422no—
180C22×C4—24no—
181C22×C4—712no—
182C22×C4—711no—
183C22×C4—712no—
184C22×C4—711no—
185C22×C4—711no—
186C22×C4—1422no—
187C22×C4—1424no—
188C22×C4C22⋊D7114yes—
189C22×C4C22×D7112yes—
190C22×C4C22×D7112yes—
191C22×C4C22×D7111yescharacteristic
192C22×C4C22×D7111yescharacteristic
193C4⋊C4—22no—
194C4⋊C4—22no—
195C4⋊C4—1424no—
196C4⋊C4—1424no—
197C4⋊C4—1422no—
198C22⋊C4—24no—
199C22⋊C4—1424no—
200C22⋊C4—1424no—
201C22⋊C4—1424no—
202C22⋊C4—1422no—
203C22⋊C4—1422no—
204C42—24no—
205C42—1422no—
206C42—1424no—
207C23—22no—
208C23—21no—
209C23—21no—
210C23—22no—
211C23—712no—
212C23—1421no—
213C23—1422no—
214C23—1421no—
215C23—1422no—
216C23C2×C22⋊D7112yes—
217C23C23×D7111yescharacteristic
218D4—1428no—
219C2×C4—22no—
220C2×C4—28no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—2844no—
228C2×C4—714no—
229C2×C4—712no—
230C2×C4—1421no—
231C2×C4—1421no—
232C2×C4—1422no—
233C2×C4—1422no—
234C2×C4—1422no—
235C2×C4—1422no—
236C2×C4—1422no—
237C2×C4—1421no—
238C2×C4—1422no—
239C2×C4—1421no—
240C2×C4—1422no—
241C2×C4C2×C22⋊D7112yes—
242C2×C4C2×C22⋊D7112yes—
243C2×C4#1176586514yes—
244C22—22no—
245C22—22no—
246C22—22no—
247C22—42no—
248C22—42no—
249C22—2841no—
250C22—2842no—
251C22—2841no—
252C22—1422no—
253C22—1424no—
254C22C22×C22⋊D7111yescharacteristic
255C22#41×D7111yescharacteristic
256C22#41×D7111yescharacteristic
257C22#41×D7112yes—
258C22C2×#1176586512yes—
259C4—22no—
260C4—42no—
261C4—42no—
262C4—2841no—
263C4—2842no—
264C4—2841no—
265C4—1422no—
266C4—1424no—
267C4#1455611114yes—
268C2—42no—
269C2—2842no—
270C2C2×#1455611111yescharacteristic
271C2#1798650611yescharacteristic
272C2#1798652712yes—
273C2#1798652712yes—
274C2#1798654511yescharacteristic
subgroupG/H
centerC22×C4C22×D71
commutator, socleC23×C71C24
FrattiniC23C23×D71
FittingC2×#117×C71C2
hypercenterC2×#117D71
hypocenterC71#2395
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C22×C4 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
2. #17985937C2semidirect
3. #17985934C2semidirect
4. #17985930C2semidirect
5. #17985927C2semidirect
6. #17985925C2semidirect
8. C4×Dic71⋊C4C2semidirect
9. C4×(C2×D71)⋊C4C2semidirect
10. C4×C22⋊Dic71C2semidirect
11. C2×#117×C71C2semidirect
138. C71#2395semidirect
140. C2×#117D71semidirect