Finite Group Databasebeta

← #22597849

Subgroups — #22597849

Order 9088.

494 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 494 conjugacy classes of subgroups — too many to draw. Only its 141 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×#14556154C211yesmaximal, characteristic
2#17986573C211yesmaximal, characteristic, complemented
3#17986474C211yesmaximal, characteristic, complemented
4#17986216C214yesmaximal, complemented
5#17986202C214yesmaximal, complemented
6#17986025C211yesmaximal, characteristic, complemented
7#17986008C211yesmaximal, characteristic, complemented
8#17985996C211yesmaximal, characteristic, complemented
9C22×SD16×C71C211yesmaximal, characteristic, complemented
10C2×#11765865C2211yescharacteristic
11#14556154—24no—
12C2×#11765853C2212yes—
13C2×#11765850C2212yes—
14C2×#11765849C2211yescharacteristic
15#14556126C2212yescomplemented
16#14556117C2212yescomplemented
17#14556111—22no—
18#14556104—22no—
19D4×Dic71—22no—
20D4×Dic71C2214yes—
21#14556087C2211yescharacteristic
22#14556087C2211yescharacteristic
23C2×C22⋊Dic71—22no—
24Dic71⋊C8C2214yescomplemented
25#14556053C2212yescomplemented
26#14556052C2214yescomplemented
27#14556050C2212yescomplemented
28C22×Q8×C71C2211yescharacteristic
29C22×D4×C71C2211yescharacteristic
30C2×SD16×C71—24no—
31C2×SD16×C71C2214yes—
32C22×C8×C71C2211yescharacteristic
33#11765865—22no—
34#11765865—22no—
35C2×Dic142C2311yescharacteristic
36C2×D284C2311yescharacteristic
37C2×C22⋊D71—21no—
38C22×Dic71—24no—
39C2×C4×D71—21no—
40#11765853—24no—
41#11765850—24no—
42#11765849—22no—
43C2×C71⋊C8C2312yes—
44C4⋊Dic71D412yes—
45C4⋊Dic71C2312yes—
46Dic71⋊C4—24no—
47C22⋊Dic71—21no—
48C22⋊Dic71—24no—
49C22⋊Dic71D412yes—
50(C2×D71)⋊C4—24no—
51C4×Dic71—21no—
52C4×Dic71C2312yes—
53C24×C71—21no—
54C2×Q8×C71—21no—
55C2×Q8×C71—22no—
56C2×Q8×C71—22no—
57C2×Q8×C71C2312yes—
58C2×D4×C71—21no—
59C2×D4×C71—22no—
60C2×D4×C71D414yes—
61C2×D4×C71C2312yes—
62C22×C4×C71—21no—
63C22×C4×C71C2311yescharacteristic
64SD16×C71—28no—
65C2×C8×C71—22no—
66C2×C8×C71C2312yes—
67C23×C71—22no—
68C23×C71—24no—
69C23×C71—82no—
70C23×C71C2×D411yescharacteristic
71C22×D71—21no—
72Q8×C71—22no—
73Q8×C71—22no—
74Q8×C71—42no—
75D4×C71—24no—
76D4×C71—22no—
77D4×C71C2×D414yes—
78C2×C4×C71—22no—
79C2×C4×C71—42no—
80C2×C4×C71C2×D411yescharacteristic
81C2×C4×C71C2×D412yes—
82C2×C4×C71#4112yes—
83C2×C4×C71C2411yescharacteristic
84C22⋊D71—42no—
85C2×Dic71—22no—
86C2×Dic71—21no—
87C2×Dic71—84no—
88C2×Dic71C2×D414yes—
89D284—21no—
90D284—21no—
91C4×D71—42no—
92Dic142—21no—
93Dic142—21no—
94C8×C71—22no—
95C71⋊C8—22no—
96C22×C71—22no—
97C22×C71—24no—
98C22×C71—44no—
99C22×C71—84no—
100C22×C71#12112yes—
101C22×C71C22×D411yescharacteristic
102C2×D71—41no—
103C2×D71—42no—
104C4×C71—42no—
105C4×C71#12112yes—
106C4×C71C22×D411yescharacteristic
107C4×C71C2×#4111yescharacteristic
108Dic71—24no—
109Dic71—41no—
110Dic71—42no—
111C2×C71—22no—
112C2×C71—44no—
113C2×C71C2×#12111yescharacteristic
114C2×C71#57712yes—
115D71—81no—
116#3042—711nomaximal, Sylow
117C71#304211yescharacteristic, complemented, Sylow
118C2×AGL(1,ℤ/8)—711no—
119C22×SD16D7111yescharacteristic, complemented
120#536—711no—
121#518—711no—
122#463—714no—
123#461—714no—
124#424—711no—
125#419—711no—
126#416—711no—
127C2×#41—711no—
128C22×Q8C2×D7111yescharacteristic
129C22×D4C2×D7111yescharacteristic
130AGL(1,ℤ/8)—1424no—
131C2×SD16—24no—
132C2×SD16—712no—
133C2×SD16C2×D7114yes—
134C2×D8—712no—
135C2×M16—711no—
136C22×C8C2×D7111yescharacteristic
137C4⋊Q8—712no—
138#124—712no—
139#123—1422no—
140C22⋊Q8—1422no—
141C4×D4—714no—
142C4×D4—1422no—
143#117—711no—
144#117—711no—
145C2×C22⋊C4—1422no—
146C4⋊C8—714no—
147#103—714no—
148#103—712no—
149#102—712no—
150C24—21no—
151#41—1422no—
152#41—1422no—
153C2×Q8—21no—
154C2×Q8—22no—
155C2×Q8—22no—
156C2×Q8—711no—
157C2×Q8C22×D7112yes—
158C2×D4—21no—
159C2×D4—22no—
160C2×D4—711no—
161C2×D4—1421no—
162C2×D4C22⋊D7114yes—
163C2×D4C22×D7112yes—
164C22×C4—21no—
165C22×C4—1421no—
166C22×C4—1424no—
167C22×C4C22×D7111yescharacteristic
168SD16—28no—
169SD16—1424no—
170D8—1424no—
171M16—1422no—
172C2×C8—22no—
173C2×C8—712no—
174C2×C8C22×D7112yes—
175C4⋊C4—712no—
176C4⋊C4—712no—
177C4⋊C4—1424no—
178C22⋊C4—712no—
179C22⋊C4—1424no—
180C22⋊C4—1424no—
181C22⋊C4—1421no—
182C42—712no—
183C42—1421no—
184C23—22no—
185C23—24no—
186C23—82no—
187C23—1421no—
188C23D4×D7111yescharacteristic
189Q8—22no—
190Q8—22no—
191Q8—42no—
192Q8—1421no—
193Q8—1421no—
194D4—22no—
195D4—24no—
196D4—2842no—
197D4—1421no—
198D4—1421no—
199D4C2×C22⋊D7114yes—
200C2×C4—22no—
201C2×C4—42no—
202C2×C4—2842no—
203C2×C4—5684no—
204C2×C4—714no—
205C2×C4—1422no—
206C2×C4—1421no—
207C2×C4D4×D7111yescharacteristic
208C2×C4C2×C22⋊D7112yes—
209C2×C4#1176586412yes—
210C2×C4C23×D7111yescharacteristic
211C8—22no—
212C8—1422no—
213C22—22no—
214C22—24no—
215C22—44no—
216C22—84no—
217C22—2841no—
218C22—2842no—
219C22#1455610012yes—
220C22C2×D4×D7111yescharacteristic
221C4—42no—
222C4—2841no—
223C4—2842no—
224C4—1424no—
225C4#1455610012yes—
226C4C22×C22⋊D7111yescharacteristic
227C4C2×#1176586411yescharacteristic
228C2—22no—
229C2—44no—
230C2—5681no—
231C2C2×#1455610011yescharacteristic
232C2#1798680512yes—
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
FittingC22×SD16×C71C2
socleC22×C71C22×D4
hypercenterC22×SD16D71
hypocenterC71#3042
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×SD16

Complements

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

NH ≅ G/Ntype
2. #17986573C2semidirect
3. #17986474C2semidirect
4. #17986216C2semidirect
5. #17986202C2semidirect
6. #17986025C2semidirect
7. #17986008C2semidirect
8. #17985996C2semidirect
9. C22×SD16×C71C2semidirect
15. #14556126C22semidirect
16. #14556117C22semidirect
24. Dic71⋊C8C22semidirect
25. #14556053C22semidirect
26. #14556052C22semidirect
27. #14556050C22semidirect
117. C71#3042semidirect
119. C22×SD16D71semidirect