Finite Group Databasebeta

← #22597853

Subgroups — #22597853

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×#14556156C211yesmaximal, characteristic
2C22×#11765854C211yesmaximal, characteristic, complemented
3#17986575C211yesmaximal, characteristic, complemented
4#17986473C211yesmaximal, characteristic, complemented
5#17986215C214yesmaximal, complemented
6#17986205C214yesmaximal, complemented
7#17986009C211yesmaximal, characteristic, complemented
8#17985996C211yesmaximal, characteristic, complemented
9#424×C71C211yesmaximal, characteristic, complemented
10C2×#11765865C2211yescharacteristic
11C22×Dic142C2211yescharacteristic
12C22×D284C2211yescharacteristic
13#14556156—24no—
14C2×#11765854—24no—
15C2×#11765854C2212yes—
16C2×#11765854C2214yes—
17C2×D568C2212yes—
18#14556128C2212yescomplemented
19#14556116C2212yescomplemented
20#14556112—22no—
21#14556108—22no—
22C4×D284C2214yes—
23#14556090—22no—
24#14556087C2211yescharacteristic
25C22⋊C4×D71—22no—
26#14556053C2212yescomplemented
27#14556053C2214yescomplemented
28#14556050C2212yescomplemented
29C2×M16×C71C2211yescharacteristic
30C22×C8×C71C2211yescharacteristic
31#117×C71C2211yescharacteristic
32C4⋊C8×C71C2214yescomplemented
33C23×D71—21no—
34#11765865—22no—
35#11765865—22no—
36C2×Dic142—22no—
37C2×Dic142—22no—
38C2×Dic142—21no—
39C2×Dic142C2312yes—
40C2×Dic142C2311yescharacteristic
41C2×D284—21no—
42C2×D284—22no—
43C2×D284D414yes—
44C2×D284C2311yescharacteristic
45C2×D284C2312yes—
46C2×C22⋊D71—21no—
47C22×Dic71—21no—
48C2×C4×D71—21no—
49C2×C4×D71—24no—
50#11765854—28no—
51#11765854—24no—
52D568—24no—
53C4⋊Dic71—24no—
54C4⋊Dic71C2312yes—
55C22⋊Dic71—21no—
56(C2×D71)⋊C4—24no—
57(C2×D71)⋊C4—24no—
58C4×Dic71—21no—
59C22×C4×C71C2311yescharacteristic
60M16×C71—22no—
61C2×C8×C71—22no—
62C2×C8×C71C2312yes—
63C2×C8×C71C2312yes—
64C4⋊C4×C71D412yes—
65C22⋊C4×C71D412yes—
66C42×C71C2312yes—
67C23×C71C2×D411yescharacteristic
68C22×D71—22no—
69C22×D71—24no—
70C22×D71—21no—
71C22×D71—82no—
72C2×C4×C71C2×D412yes—
73C2×C4×C71C2×D414yes—
74C2×C4×C71C2×D411yescharacteristic
75C2×C4×C71#4112yes—
76C2×C4×C71C2411yescharacteristic
77C22⋊D71—42no—
78C2×Dic71—22no—
79C2×Dic71—22no—
80C2×Dic71—21no—
81C2×Dic71—42no—
82D284—24no—
83D284—21no—
84D284—22no—
85D284—21no—
86D284C2×D414yes—
87C4×D71—42no—
88C4×D71—84no—
89Dic142—22no—
90Dic142—21no—
91Dic142—22no—
92Dic142—21no—
93Dic142—42no—
94C8×C71—22no—
95C8×C71—22no—
96C22×C71—22no—
97C22×C71#12112yes—
98C22×C71C22×D411yescharacteristic
99C2×D71—24no—
100C2×D71—44no—
101C2×D71—41no—
102C2×D71—42no—
103C2×D71—84no—
104C4×C71—24no—
105C4×C71#12112yes—
106C4×C71C22×D411yescharacteristic
107C4×C71C2×#4111yescharacteristic
108Dic71—42no—
109Dic71—41no—
110Dic71—42no—
111C2×C71—22no—
112C2×C71C2×#12111yescharacteristic
113C2×C71#57712yes—
114D71—44no—
115D71—81no—
116#3042—711nomaximal, Sylow
117C71#304211yescharacteristic, complemented, Sylow
118C2×AGL(1,ℤ/8)—711no—
119C22×SD16—711no—
120#536—711no—
121#518—711no—
122#463—714no—
123#461—714no—
124#424D7111yescharacteristic, complemented
125#419—711no—
126#416—711no—
127C2×#41—711no—
128C22×Q8—711no—
129C22×D4—711no—
130AGL(1,ℤ/8)—1424no—
131C2×SD16—714no—
132C2×SD16—712no—
133C2×SD16—1424no—
134C2×D8—712no—
135C2×M16C2×D7111yescharacteristic
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#117C2×D7111yescharacteristic
145C2×C22⋊C4—1422no—
146C4⋊C8C2×D7114yescomplemented
147#103—712no—
148#103—714no—
149#102—712no—
150C24—1421no—
151#41—1422no—
152#41—1422no—
153C2×Q8—712no—
154C2×Q8—711no—
155C2×Q8—1422no—
156C2×Q8—1422no—
157C2×Q8—1421no—
158C2×D4—714no—
159C2×D4—712no—
160C2×D4—711no—
161C2×D4—1421no—
162C2×D4—1421no—
163C2×D4—1422no—
164C22×C4—1421no—
165C22×C4—1421no—
166C22×C4—1424no—
167C22×C4C22×D7111yescharacteristic
168SD16—1428no—
169SD16—1424no—
170D8—1424no—
171M16—22no—
172C2×C8—22no—
173C2×C8C22×D7112yes—
174C2×C8C22×D7112yes—
175C4⋊C4—712no—
176C4⋊C4—1424no—
177C4⋊C4D28412yes—
178C22⋊C4—1421no—
179C22⋊C4—1424no—
180C22⋊C4—1424no—
181C22⋊C4D28412yes—
182C42—1421no—
183C42C22×D7112yes—
184C23—5682no—
185C23—1421no—
186C23—1422no—
187C23—1424no—
188C23C2×D28411yescharacteristic
189Q8—2842no—
190Q8—1422no—
191Q8—1421no—
192Q8—1422no—
193Q8—1421no—
194D4—2842no—
195D4—714no—
196D4—1421no—
197D4—1421no—
198D4—1422no—
199D4—1424no—
200C2×C4—2842no—
201C2×C4—2842no—
202C2×C4—5684no—
203C2×C4—1422no—
204C2×C4—1422no—
205C2×C4—1421no—
206C2×C4D4×D7112yes—
207C2×C4C2×D28414yes—
208C2×C4C2×D28411yescharacteristic
209C2×C4#1176586612yes—
210C2×C4C23×D7111yescharacteristic
211C8—22no—
212C8—22no—
213C22—22no—
214C22—2844no—
215C22—2841no—
216C22—2842no—
217C22—5684no—
218C22—1424no—
219C22#1455610312yes—
220C22C22×D28411yescharacteristic
221C4—24no—
222C4—2841no—
223C4—2842no—
224C4—2842no—
225C4#1455610312yes—
226C4C2×D4×D7111yescharacteristic
227C4C2×#1176586611yescharacteristic
228C2—22no—
229C2—2844no—
230C2—5681no—
231C2C2×#1455610311yescharacteristic
232C2#1798681112yes—
subgroupG/H
centerC22C22×D284
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#424×C71C2
socleC22×C71C22×D4
hypercenter#424D71
hypocenterC71#3042
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ #117 ◁ #424

Complements

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

NH ≅ G/Ntype
2. C22×#11765854C2semidirect
3. #17986575C2semidirect
4. #17986473C2semidirect
5. #17986215C2semidirect
6. #17986205C2semidirect
7. #17986009C2semidirect
8. #17985996C2semidirect
9. #424×C71C2semidirect
18. #14556128C22semidirect
19. #14556116C22semidirect
26. #14556053C22semidirect
27. #14556053C22semidirect
28. #14556050C22semidirect
32. C4⋊C8×C71C22semidirect
117. C71#3042semidirect
124. #424D71semidirect
146. C4⋊C8C2×D71semidirect