Finite Group Databasebeta

← #22600979

Subgroups — #22600979

Order 9088.

370 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 123 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#17986724C211yesmaximal, characteristic, complemented
2#17986664C211yesmaximal, characteristic
3#17986348C212yesmaximal, complemented
4#17986282C212yesmaximal, complemented
5#17986281C211yesmaximal, characteristic, complemented
6#17986203C212yesmaximal
7#17986201C211yesmaximal, characteristic
8C8×Dic142C211yesmaximal, characteristic, complemented
9C4×#11765854C211yesmaximal, characteristic
10#17986084C212yesmaximal
11#487×C71C211yesmaximal, characteristic, complemented
12C2×Q8×D71—21no—
13C2×#11765854C2211yescharacteristic
14C2×#11765852C2212yes—
15#14556127C2212yes—
16#14556126C2212yes—
17#14556119—21no—
18#14556118—22no—
19#14556106—22no—
20#14556104—21no—
21Q8×Dic71C2212yes—
22C4×Dic142C2211yescharacteristic
23C4×Dic142C2211yescharacteristic
24#14556093—21no—
25C4×D284C2211yescharacteristic
26#14556090C2212yes—
27#14556060C2211yescharacteristic
28#14556059C2212yes—
29C4⋊(C71⋊C8)C2211yescharacteristic
30Dic71⋊C8C2212yes—
31#14556053C2211yescharacteristic
32#14556052C2212yes—
33#14556051C2212yes—
34#14556050C2211yescharacteristic
35C8×Dic71C2212yes—
36C4⋊Q8×C71C2211yescharacteristic
37C4⋊Q8×C71C2211yescharacteristic
38#103×C71C2212yes—
39#103×C71C2212yes—
40C4×C8×C71C2211yescharacteristic
41C2×Dic142—22no—
42C2×Dic142D411yescharacteristic
43C2×Dic142C2311yescharacteristic
44Q8×D71—44no—
45C2×D284C2311yescharacteristic
46C2×C4×D71—21no—
47C2×C4×D71—22no—
48#11765854—22no—
49#11765852—24no—
50C2×C71⋊C8C2312yes—
51C4⋊Dic71—22no—
52C4⋊Dic71D411yescharacteristic
53C4⋊Dic71C2311yescharacteristic
54C4⋊Dic71C2311yescharacteristic
55Dic71⋊C4—21no—
56Dic71⋊C4—22no—
57Dic71⋊C4—22no—
58Dic71⋊C4—22no—
59Dic71⋊C4D412yes—
60(C2×D71)⋊C4—21no—
61(C2×D71)⋊C4—22no—
62C4×Dic71—21no—
63C4×Dic71—22no—
64C4×Dic71C2312yes—
65C2×Q8×C71—21no—
66C2×Q8×C71C2312yes—
67C2×C8×C71C2311yescharacteristic
68C2×C8×C71C2311yescharacteristic
69C4⋊C4×C71—21no—
70C4⋊C4×C71—21no—
71C4⋊C4×C71—21no—
72C4⋊C4×C71C2312yes—
73C42×C71C2311yescharacteristic
74C22×D71—21no—
75Q8×C71—22no—
76Q8×C71—22no—
77Q8×C71—42no—
78C2×C4×C71—22no—
79C2×C4×C71—22no—
80C2×C4×C71C2×D411yescharacteristic
81C2×C4×C71C2×D411yescharacteristic
82C2×C4×C71C2411yescharacteristic
83C2×Dic71—21no—
84C2×Dic71—21no—
85C2×Dic71—21no—
86C2×Dic71C2×D412yes—
87C2×Dic71C2×D412yes—
88D284—21no—
89D284#4112yes—
90C4×D71—42no—
91C4×D71—44no—
92Dic142—21no—
93Dic142—21no—
94Dic142—44no—
95Dic142SD1614yescomplemented
96C8×C71—21no—
97C8×C71—21no—
98C71⋊C8—22no—
99C22×C71C22×D411yescharacteristic
100C2×D71—22no—
101C2×D71—41no—
102C2×D71—41no—
103C4×C71—21no—
104C4×C71—42no—
105C4×C71—42no—
106C4×C71C2×SD1612yes—
107C4×C71C2×#4111yescharacteristic
108C4×C712−1+411yescharacteristic
109Dic71—22no—
110Dic71—41no—
111Dic71—41no—
112Dic71—41no—
113Dic71C2×SD1614yes—
114C2×C71#54811yescharacteristic
115C2×C71C22×SD1611yescharacteristic
116C2×C71#57811yescharacteristic
117D71—42no—
118#3359—711nomaximal, Sylow
119C71#335911yescharacteristic, complemented, Sylow
120#557—711no—
121D4×Q8—711no—
122#487—712no—
123#487D7111yescharacteristic, complemented
124#475—711no—
125#475—712no—
126#461—711no—
127#461—712no—
128C8×Q8—711no—
129C4×SD16—712no—
130C4×SD16—711no—
131C22×Q8—1421no—
132C2×SD16—712no—
133C2×SD16—711no—
134C4⋊Q8—712no—
135C4⋊Q8—712no—
136C4⋊Q8C2×D7111yescharacteristic
137C4⋊Q8C2×D7111yescharacteristic
138#125—1422no—
139#125—1421no—
140C22⋊Q8—1422no—
141C22⋊Q8—1421no—
142C4×Q8—711no—
143C4×Q8—712no—
144C4×Q8—711no—
145C4×Q8—1421no—
146C4×D4—712no—
147C4×D4—711no—
148#106—711no—
149#106—712no—
150C4⋊C8—712no—
151C4⋊C8—711no—
152#103—712no—
153#103—711no—
154#103—712no—
155#103C2×D7112yes—
156#103C2×D7112yes—
157#102—711no—
158C4×C8—712no—
159C4×C8C2×D7111yescharacteristic
160C2×Q8—21no—
161C2×Q8—2844no—
162C2×Q8—711no—
163C2×Q8—711no—
164C2×Q8—1422no—
165C2×Q8C22×D7112yes—
166C2×D4—711no—
167C22×C4—1422no—
168C22×C4—1421no—
169SD16—1422no—
170SD16—1424no—
171C2×C8—712no—
172C2×C8C22×D7111yescharacteristic
173C2×C8C22×D7111yescharacteristic
174C4⋊C4—21no—
175C4⋊C4—21no—
176C4⋊C4—21no—
177C4⋊C4—712no—
178C4⋊C4—711no—
179C4⋊C4—711no—
180C4⋊C4—711no—
181C4⋊C4—1422no—
182C4⋊C4—1422no—
183C4⋊C4—1422no—
184C4⋊C4—1422no—
185C4⋊C4—1421no—
186C4⋊C4C22×D7112yes—
187C22⋊C4—1422no—
188C22⋊C4—1421no—
189C42—712no—
190C42—1422no—
191C42—1421no—
192C42C22×D7111yescharacteristic
193C23—1421no—
194Q8—22no—
195Q8—22no—
196Q8—42no—
197Q8—2844no—
198Q8—714no—
199Q8—1421no—
200Q8—1421no—
201D4—712no—
202D4—1421no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—2844no—
206C2×C4—2842no—
207C2×C4—712no—
208C2×C4—712no—
209C2×C4—1421no—
210C2×C4—1421no—
211C2×C4—1421no—
212C2×C4D4×D7111yescharacteristic
213C2×C4D4×D7111yescharacteristic
214C2×C4C23×D7111yescharacteristic
215C8—21no—
216C8—21no—
217C8—1422no—
218C22—2841no—
219C22—2841no—
220C22—1422no—
221C22C2×D4×D7111yescharacteristic
222C4—21no—
223C4—42no—
224C4—42no—
225C4—2841no—
226C4—2841no—
227C4—2841no—
228C4—714no—
229C4—1422no—
230C4SD16×D7112yes—
231C4#41×D7111yescharacteristic
232C4#1455617511yescharacteristic
233C2—2842no—
234C2#1798665711yescharacteristic
235C2C2×SD16×D7111yescharacteristic
236C2#1798681211yescharacteristic
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#487×C71C2
socleC22×C71C22×D4
hypercenter#487D71
hypocenterC71#3359
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C42 ◁ #487

Complements

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

NH ≅ G/Ntype
1. #17986724C2semidirect
3. #17986348C2semidirect
4. #17986282C2semidirect
5. #17986281C2semidirect
8. C8×Dic142C2semidirect
11. #487×C71C2semidirect
95. Dic142SD16semidirect
119. C71#3359semidirect
123. #487D71semidirect