Finite Group Databasebeta

← #22597852

Subgroups — #22597852

Order 9088.

494 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 494 conjugacy classes of subgroups — too many to draw. Only its 149 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
2C22×#11765853C211yesmaximal, characteristic, complemented
3#17986575C211yesmaximal, characteristic, complemented
4#17986214C214yesmaximal, complemented
5#17986204C214yesmaximal, complemented
6#17986026C211yesmaximal, characteristic, complemented
7#17986007C211yesmaximal, characteristic, complemented
8#17985993C211yesmaximal, characteristic, complemented
9#518×C71C211yesmaximal, characteristic, complemented
10C2×#11765865C2211yescharacteristic
11C22×Dic142C2211yescharacteristic
12#14556154—24no—
13C2×#11765853—24no—
14C2×#11765853C2212yes—
15C2×#11765853C2214yes—
16C2×#11765850C2212yes—
17C2×#11765849C2211yescharacteristic
18C22×C71⋊C8C2211yescharacteristic
19#14556128C2212yescomplemented
20#14556116C2212yescomplemented
21#14556112—22no—
22#14556108—22no—
23C4⋊(C71⋊C8)C2214yescomplemented
24#14556051C2212yescomplemented
25#14556051C2214yescomplemented
26#14556048C2212yescomplemented
27C22×D4×C71C2211yescharacteristic
28C4×D4×C71—22no—
29C4×D4×C71C2214yes—
30#117×C71C2211yescharacteristic
31#117×C71C2211yescharacteristic
32C2×C22⋊C4×C71—22no—
33#11765865—22no—
34#11765865—22no—
35C2×Dic142—22no—
36C2×Dic142—22no—
37C2×Dic142—21no—
38C2×Dic142C2312yes—
39C2×Dic142C2311yescharacteristic
40C2×D284C2311yescharacteristic
41C2×C22⋊D71—21no—
42C22×Dic71—21no—
43C2×C4×D71—21no—
44#11765853—28no—
45#11765853—24no—
46#11765850—24no—
47#11765849—22no—
48C2×C71⋊C8—22no—
49C2×C71⋊C8C2312yes—
50C2×C71⋊C8C2312yes—
51C4⋊Dic71—24no—
52(C2×D71)⋊C4—24no—
53C24×C71—21no—
54C2×D4×C71—21no—
55C2×D4×C71—22no—
56C2×D4×C71D414yes—
57C2×D4×C71C2312yes—
58C22×C4×C71—24no—
59C22×C4×C71C2311yescharacteristic
60C4⋊C4×C71D412yes—
61C4⋊C4×C71C2312yes—
62C22⋊C4×C71—21no—
63C22⋊C4×C71—24no—
64C22⋊C4×C71D412yes—
65C42×C71—21no—
66C42×C71C2312yes—
67C23×C71—22no—
68C23×C71—24no—
69C23×C71—82no—
70C23×C71C2×D411yescharacteristic
71C22×D71—21no—
72D4×C71—24no—
73D4×C71—22no—
74D4×C71C2×D414yes—
75C2×C4×C71—22no—
76C2×C4×C71—84no—
77C2×C4×C71C2×D414yes—
78C2×C4×C71C2×D411yescharacteristic
79C2×C4×C71C2×D412yes—
80C2×C4×C71#4112yes—
81C2×C4×C71C2411yescharacteristic
82C22⋊D71—42no—
83C2×Dic71—22no—
84C2×Dic71—21no—
85C2×Dic71—42no—
86D284—21no—
87D284—21no—
88C4×D71—42no—
89Dic142—22no—
90Dic142—21no—
91Dic142—22no—
92Dic142—21no—
93Dic142—42no—
94C71⋊C8—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—24no—
105C4×C71—42no—
106C4×C71#12112yes—
107C4×C71C22×D411yescharacteristic
108C4×C71C2×#4111yescharacteristic
109Dic71—42no—
110Dic71—41no—
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×SD16—711no—
120#536—711no—
121#518D7111yescharacteristic, complemented
122#463—714no—
123#461—714no—
124#424—711no—
125#419—711no—
126#416—711no—
127C2×#41—711no—
128C22×Q8—711no—
129C22×D4C2×D7111yescharacteristic
130AGL(1,ℤ/8)—1424no—
131C2×SD16—714no—
132C2×SD16—712no—
133C2×SD16—1424no—
134C2×D8—712no—
135C2×M16—711no—
136C22×C8—711no—
137C4⋊Q8—712no—
138#124—712no—
139#123—1422no—
140C22⋊Q8—1422no—
141C4×D4—22no—
142C4×D4C2×D7114yes—
143#117C2×D7111yescharacteristic
144#117C2×D7111yescharacteristic
145C2×C22⋊C4—22no—
146C4⋊C8—714no—
147#103—712no—
148#103—714no—
149#102—712no—
150C24—21no—
151#41—1422no—
152#41—1422no—
153C2×Q8—712no—
154C2×Q8—711no—
155C2×Q8—1422no—
156C2×Q8—1422no—
157C2×Q8—1421no—
158C2×D4—21no—
159C2×D4—22no—
160C2×D4—711no—
161C2×D4—1421no—
162C2×D4D28414yes—
163C2×D4C22×D7112yes—
164C22×C4—24no—
165C22×C4—1421no—
166C22×C4—1421no—
167C22×C4C22×D7111yescharacteristic
168SD16—1428no—
169SD16—1424no—
170D8—1424no—
171M16—1422no—
172C2×C8—712no—
173C2×C8—712no—
174C2×C8—1422no—
175C4⋊C4—1424no—
176C4⋊C4C22⋊D7112yes—
177C4⋊C4C22×D7112yes—
178C22⋊C4—21no—
179C22⋊C4—24no—
180C22⋊C4—1424no—
181C22⋊C4C22⋊D7112yes—
182C42—21no—
183C42C22×D7112yes—
184C23—22no—
185C23—24no—
186C23—82no—
187C23—1421no—
188C23C2×C22⋊D7111yescharacteristic
189Q8—2842no—
190Q8—1422no—
191Q8—1421no—
192Q8—1422no—
193Q8—1421no—
194D4—22no—
195D4—24no—
196D4—2842no—
197D4—1421no—
198D4—1421no—
199D4C2×D28414yes—
200C2×C4—22no—
201C2×C4—84no—
202C2×C4—2842no—
203C2×C4—2842no—
204C2×C4—1422no—
205C2×C4—1421no—
206C2×C4C2×C22⋊D7114yes—
207C2×C4C2×C22⋊D7111yescharacteristic
208C2×C4C2×D28412yes—
209C2×C4#1176586512yes—
210C2×C4C23×D7111yescharacteristic
211C8—1422no—
212C8—1422no—
213C22—22no—
214C22—24no—
215C22—44no—
216C22—84no—
217C22—2841no—
218C22—2842no—
219C22#1455610212yes—
220C22C22×C22⋊D7111yescharacteristic
221C4—24no—
222C4—42no—
223C4—2841no—
224C4—2842no—
225C4#1455610212yes—
226C4C22×D28411yescharacteristic
227C4C2×#1176586511yescharacteristic
228C2—22no—
229C2—44no—
230C2—5681no—
231C2C2×#1455610211yescharacteristic
232C2#1798681012yes—
subgroupG/H
centerC22C22×C22⋊D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#518×C71C2
socleC22×C71C22×D4
hypercenter#518D71
hypocenterC71#3042
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ #117 ◁ #518

Complements

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

NH ≅ G/Ntype
2. C22×#11765853C2semidirect
3. #17986575C2semidirect
4. #17986214C2semidirect
5. #17986204C2semidirect
6. #17986026C2semidirect
7. #17986007C2semidirect
8. #17985993C2semidirect
9. #518×C71C2semidirect
19. #14556128C22semidirect
20. #14556116C22semidirect
23. C4⋊(C71⋊C8)C22semidirect
24. #14556051C22semidirect
25. #14556051C22semidirect
26. #14556048C22semidirect
117. C71#3042semidirect
121. #518D71semidirect