Finite Group Databasebeta

← #22599549

Subgroups — #22599549

Order 9088.

322 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 111 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#17986555C211yesmaximal, characteristic
2#17986554C211yesmaximal, characteristic
3#17986409C211yesmaximal, characteristic, complemented
4#17986408C211yesmaximal, characteristic, complemented
5#17986367C211yesmaximal, characteristic, complemented
6#17986339C212yesmaximal
7#17986338C212yesmaximal
8#17986292C212yesmaximal, complemented
9#17986282C212yesmaximal, complemented
10#17986054C211yesmaximal, characteristic
11#491×C71C211yesmaximal, characteristic, complemented
12#14556128C2211yescharacteristic
13#14556127C2211yescharacteristic
14#14556127C2211yescharacteristic
15#14556120C2211yescharacteristic
16#14556119—21no—
17#14556108—21no—
18#14556106—21no—
19#14556106—21no—
20#14556104C2212yes—
21Q8×Dic71C2212yes—
22#14556093—21no—
23#14556093—21no—
24#14556093—21no—
25#14556086—21no—
26#14556086C2211yescharacteristic
27#14556086C2211yescharacteristic
28C42×D71C2211yescharacteristic
29#14556062C2212yes—
30#14556061C2212yes—
31#14556060C2212yes—
32#14556059C2212yes—
33Dic71⋊C8C2212yes—
34#14556052C2212yes—
35#14556052C2212yes—
36(C2×D71)⋊C8C2212yes—
37#14556039C2211yescharacteristic
38C4⋊Q8×C71C2211yescharacteristic
39#125×C71C2211yescharacteristic
40#103×C71C2212yes—
41#103×C71C2212yes—
42#97×C71C2211yescharacteristic
43C2×Dic142—21no—
44C2×Dic142—21no—
45C2×Dic142—21no—
46C2×C4×D71D411yescharacteristic
47C2×C4×D71D411yescharacteristic
48C2×C4×D71C2311yescharacteristic
49C2×C71⋊C8C2312yes—
50C4⋊Dic71—21no—
51C4⋊Dic71—22no—
52C4⋊Dic71C2311yescharacteristic
53C4⋊Dic71C2311yescharacteristic
54C4⋊Dic71C2312yes—
55Dic71⋊C4—21no—
56Dic71⋊C4—21no—
57Dic71⋊C4—22no—
58Dic71⋊C4—21no—
59Dic71⋊C4—21no—
60(C2×D71)⋊C4—22no—
61(C2×D71)⋊C4—21no—
62(C2×D71)⋊C4—21no—
63C4×Dic71—22no—
64C4×Dic71—21no—
65C4×Dic71—21no—
66C4×Dic71D411yescharacteristic
67C4×Dic71D411yescharacteristic
68C4×Dic71C2311yescharacteristic
69C2×Q8×C71C2312yes—
70C2×C8×C71C2312yes—
71C4⋊C4×C71—21no—
72C4⋊C4×C71—21no—
73C4⋊C4×C71—21no—
74C4⋊C4×C71—21no—
75C4⋊C4×C71C2311yescharacteristic
76C4⋊C4×C71C2311yescharacteristic
77C42×C71C2311yescharacteristic
78C22×D71C2×D411yescharacteristic
79Q8×C71—24no—
80C2×C4×C71—21no—
81C2×C4×C71—21no—
82C2×C4×C71—22no—
83C2×C4×C71C2×D411yescharacteristic
84C2×C4×C71C2×D411yescharacteristic
85C2×C4×C71C2411yescharacteristic
86C2×Dic71—21no—
87C2×Dic71—21no—
88C2×Dic71—22no—
89C2×Dic71C2×D411yescharacteristic
90C2×Dic71C2×D411yescharacteristic
91C2×Dic71C2×D411yescharacteristic
92C4×D71—22no—
93C4×D71—22no—
94C4×D71—41no—
95Dic142—42no—
96Dic142—42no—
97Dic142—42no—
98C8×C71—22no—
99C71⋊C8—22no—
100C22×C71C22×D411yescharacteristic
101C2×D71—21no—
102C2×D71—21no—
103C2×D71—21no—
104C4×C71—21no—
105C4×C71—21no—
106C4×C71—42no—
107C4×C71—41no—
108C4×C71—41no—
109C4×C712−1+412yes—
110Dic71—21no—
111Dic71—21no—
112Dic71—41no—
113Dic71—41no—
114Dic71—42no—
115Dic71#13712yes—
116C2×C71#53611yescharacteristic
117C2×C71C2×#13711yescharacteristic
118C2×C71#57511yescharacteristic
119D71—41no—
120#3237—711nomaximal, Sylow
121C71#323711yescharacteristic, complemented, Sylow
122#533—711no—
123#533—711no—
124#501—711no—
125#501—711no—
126#491—711no—
127#491D7111yescharacteristic, complemented
128#485—712no—
129#485—712no—
130#477—712no—
131#475—712no—
132#432—711no—
133C4⋊Q8—711no—
134C4⋊Q8—711no—
135C4⋊Q8—711no—
136C4⋊Q8C2×D7111yescharacteristic
137#125—711no—
138#125—1421no—
139#125C2×D7111yescharacteristic
140C22⋊Q8—712no—
141C22⋊Q8—1421no—
142C22⋊Q8—1421no—
143C22⋊Q8—1421no—
144C4×Q8—712no—
145C4×Q8—1421no—
146C4×Q8—1421no—
147C4×Q8—1421no—
148#117—711no—
149#117—711no—
150#117—1421no—
151C2×C42—711no—
152#107—712no—
153#107—712no—
154#106—712no—
155#106—712no—
156C4⋊C8—712no—
157#103—712no—
158#103—712no—
159#103C2×D7112yes—
160#103C2×D7112yes—
161C22⋊C8—712no—
162#97—711no—
163#97C2×D7111yescharacteristic
164C2×Q8—1421no—
165C2×Q8—1421no—
166C2×Q8—1421no—
167C2×Q8C22×D7112yes—
168C22×C4—711no—
169C22×C4—711no—
170C22×C4—711no—
171C2×C8—712no—
172C2×C8C22×D7112yes—
173C4⋊C4—21no—
174C4⋊C4—21no—
175C4⋊C4—21no—
176C4⋊C4—21no—
177C4⋊C4—711no—
178C4⋊C4—711no—
179C4⋊C4—712no—
180C4⋊C4—1422no—
181C4⋊C4—1421no—
182C4⋊C4—1421no—
183C4⋊C4—1422no—
184C4⋊C4—1421no—
185C4⋊C4—1421no—
186C4⋊C4—1421no—
187C4⋊C4C22×D7111yescharacteristic
188C4⋊C4C22×D7111yescharacteristic
189C22⋊C4—1421no—
190C22⋊C4—1421no—
191C22⋊C4—1422no—
192C42—711no—
193C42—711no—
194C42—711no—
195C42—1421no—
196C42—1421no—
197C42—1422no—
198C42C22×D7111yescharacteristic
199C23—711no—
200Q8—24no—
201Q8—2842no—
202Q8—2842no—
203Q8—2842no—
204C2×C4—22no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—2841no—
208C2×C4—711no—
209C2×C4—711no—
210C2×C4—711no—
211C2×C4—1422no—
212C2×C4—1421no—
213C2×C4—1421no—
214C2×C4—1422no—
215C2×C4—1422no—
216C2×C4D4×D7111yescharacteristic
217C2×C4D4×D7111yescharacteristic
218C2×C4C23×D7111yescharacteristic
219C8—22no—
220C8—1422no—
221C22—1421no—
222C22—1421no—
223C22—1421no—
224C22C2×D4×D7111yescharacteristic
225C4—21no—
226C4—21no—
227C4—42no—
228C4—41no—
229C4—41no—
230C4—2841no—
231C4—2842no—
232C4—2841no—
233C4—712no—
234C4—1421no—
235C4—1421no—
236C4#1455617512yes—
237C2—2841no—
238C2#1798656911yescharacteristic
239C2#137×D7111yescharacteristic
240C2#1798679811yescharacteristic
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#491×C71C2
socleC22×C71C22×D4
hypercenter#491D71
hypocenterC71#3237
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C42 ◁ #491

Complements

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

NH ≅ G/Ntype
3. #17986409C2semidirect
4. #17986408C2semidirect
5. #17986367C2semidirect
8. #17986292C2semidirect
9. #17986282C2semidirect
11. #491×C71C2semidirect
121. C71#3237semidirect
127. #491D71semidirect