Finite Group Databasebeta

← #11739229

Subgroups — #11739229

Order 896.

596 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 596 conjugacy classes of subgroups — too many to draw. Only its 137 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×#6475C211yesmaximal, characteristic, complemented
2C2×#6474C212yesmaximal, complemented
3C2×D7×D8C211yesmaximal, characteristic, complemented
4#90406C212yesmaximal, complemented
5C7×#493C211yesmaximal, characteristic, complemented
6#89927C211yesmaximal, characteristic, complemented
7#89926C212yesmaximal, complemented
8#89923C211yesmaximal, characteristic, complemented
9#89846C212yesmaximal, complemented
10#89468C211yesmaximal, characteristic, complemented
11#89461C211yesmaximal, characteristic, complemented
12C2×#1159C2212yes—
13C2×D4×D7C2212yes—
14C2×C7×D8C2211yescharacteristic
15C2×C7×D8C2211yescharacteristic, complemented
16C2×C7×D8C2212yescomplemented
17C7×#127C2212yescomplemented
18#6503C2212yescomplemented
19#6502—24no—
20#6500—24no—
21#6499C2212yescomplemented
22C2×#1142C2212yescomplemented
23C2×#1142C2212yescomplemented
24C2×#1141C2212yes—
25C2×#1141C2212yescomplemented
26#6475—24no—
27#6474—28no—
28D7×D8—24no—
29C2×Dic28C2211yescharacteristic, complemented
30C2×D56C2211yescharacteristic
31C2×#1132C2212yescomplemented
32C2×#1131C2211yescharacteristic
33C2×C8×D7C2211yescharacteristic
34#6438C2212yescomplemented
35#6435C2211yescharacteristic
36C4×C7×C8C2211yescharacteristic, complemented
37#6389C2211yescharacteristic, complemented
38C8×Dic7C2211yescharacteristic, complemented
39#6377C2211yescharacteristic, complemented
40#3155—71nomaximal, Sylow
41C23×D7—22no—
42C2×C7×D4—24no—
43C2×C7×D4C2312yes—
44C2×C7×D4C2312yes—
45C2×C22⋊D7—22no—
46C2×C22⋊D7—22no—
47C2×C22⋊D7—24no—
48C22×Dic7—22no—
49#1159—24no—
50#1159—24no—
51D4×D7—24no—
52D4×D7—24no—
53C2×D28C2312yes—
54C2×C4×D7C2311yescharacteristic
55C2×Dic14C2312yes—
56C7×D8—22no—
57C7×D8—24no—
58C7×D8—22no—
59C2×C7×C8C2311yescharacteristic
60C2×C7×C8C2311yescharacteristic
61C42×C7C2311yescharacteristic
62C22⋊Dic7—24no—
63#1142—24no—
64#1142—24no—
65#1141—24no—
66#1141—24no—
67(C2×D7)⋊C4—24no—
68(C2×D7)⋊C4D412yes—
69Dic7⋊C4D412yes—
70C4×Dic7C2311yescharacteristic
71C2×C7⋊C8C2311yescharacteristic
72C2×C7⋊C8C2311yescharacteristic
73Dic28—22no—
74D56—22no—
75#1132—24no—
76#1131D414yescomplemented
77C8×D7D414yescomplemented
78C2×AGL(1,ℤ/8)—72no—
79C2×#135—71no—
80C22×D8—71no—
81#535—72no—
82#496—72no—
83#496—72no—
84#495—71no—
85#495—71no—
86#493—71no—
87#493D711yescharacteristic, complemented
88#405—71no—
89C23×C7—22no—
90C23×C7—22no—
91C22×D7—22no—
92C22×D7—44no—
93C22×D7—84no—
94C22×D7C2×D411yescharacteristic
95C7×D4—22no—
96C7×D4—22no—
97C7×D4—22no—
98C7×D4—22no—
99C7×D4—84no—
100C2×C4×C7C2×D412yes—
101C2×C4×C7C2411yescharacteristic
102C22⋊D7—44no—
103C22⋊D7—44no—
104C22⋊D7—84no—
105C2×Dic7—22no—
106C2×Dic7—44no—
107C2×Dic7C2×D412yes—
108C2×Dic7C2×D411yescharacteristic
109D28—22no—
110D28—22no—
111C4×D7C2×D412yes—
112C4×D7C2×D412yes—
113Dic14—22no—
114Dic14—22no—
115C7×C8C2×D412yes—
116C7×C8C2×D412yes—
117C7⋊C8C2×D412yes—
118C7⋊C8C2×D412yes—
119C2×#41—72no—
120C22×D4—72no—
121AGL(1,ℤ/8)—148no—
122#135—144no—
123C2×Q16—71no—
124C2×SD16—72no—
125C2×SD16—72no—
126C2×SD16—72no—
127C2×D8—72no—
128C2×D8—72no—
129C2×D8—71no—
130C2×D8—144no—
131C2×D8C2×D711yescharacteristic
132C2×D8C2×D711yescharacteristic, complemented
133C2×D8C2×D712yescomplemented
134C2×M16—71no—
135C22×C8—71no—
136#127—72no—
137#127C2×D712yescomplemented
138#124—72no—
139#124—72no—
140#121—144no—
141#120—144no—
142#117—71no—
143#97—71no—
144#97—71no—
145C4×C8—71no—
146C4×C8C2×D711yescharacteristic, complemented
147C22×C7—42no—
148C22×C7—42no—
149C22×C7—44no—
150C22×C7—44no—
151C22×C7C22×D411yescharacteristic
152C2×D7—22no—
153C2×D7—42no—
154C2×D7—44no—
155C2×D7—88no—
156C2×D7#12712yes—
157C4×C7—22no—
158C4×C7C22×D411yescharacteristic
159C4×C7C22×D411yescharacteristic
160Dic7—22no—
161Dic7—42no—
162Dic7#12712yes—
163C24—142no—
164#41—144no—
165#41—144no—
166C2×Q8—72no—
167C2×D4—24no—
168C2×D4—72no—
169C2×D4—144no—
170C2×D4—144no—
171C2×D4—142no—
172C2×D4—144no—
173C2×D4—142no—
174C2×D4C22×D712yes—
175C2×D4C22×D712yes—
176C22×C4—71no—
177C22×C4—142no—
178Q16—142no—
179SD16—144no—
180SD16—144no—
181SD16—144no—
182D8—22no—
183D8—22no—
184D8—24no—
185D8—144no—
186D8—144no—
187D8—142no—
188M16—74no—
189C2×C8—71no—
190C2×C8—71no—
191C2×C8—74no—
192C2×C8C22×D711yescharacteristic
193C2×C8C22×D711yescharacteristic
194C4⋊C4—72no—
195C22⋊C4—72no—
196C22⋊C4—144no—
197C22⋊C4—144no—
198C42—71no—
199C42C22×D711yescharacteristic
200C2×C7—82no—
201C2×C7—82no—
202C2×C7C2×#12711yescharacteristic
203C2×C7#57612yes—
204D7—22no—
205D7—82no—
206C23—22no—
207C23—22no—
208C23—71no—
209C23—142no—
210C23—284no—
211C23—564no—
212Q8—142no—
213Q8—142no—
214D4—22no—
215D4—22no—
216D4—22no—
217D4—22no—
218D4—84no—
219D4—142no—
220D4—142no—
221D4—284no—
222D4—284no—
223D4—564no—
224C2×C4—71no—
225C2×C4—72no—
226C2×C4—72no—
227C2×C4—72no—
228C2×C4—142no—
229C2×C4—284no—
230C2×C4D4×D712yes—
231C2×C4C23×D711yescharacteristic
232C8—72no—
233C8—72no—
234C8D4×D712yes—
235C8D4×D712yes—
236C7#315511yescharacteristic, complemented, Sylow
237C22—42no—
238C22—42no—
239C22—44no—
240C22—44no—
241C22—72no—
242C22—142no—
243C22—282no—
244C22—284no—
245C22—568no—
246C22C2×D4×D711yescharacteristic
247C4—22no—
248C4—72no—
249C4—142no—
250C4—282no—
251C4C2×D4×D711yescharacteristic
252C4C2×D4×D711yescharacteristic
253C2—82no—
254C2—82no—
255C2—142no—
256C2—562no—
257C2D7×#12711yescharacteristic
258C2#9044812yes—
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#493C2
socleC22×C7C22×D4
hypercenter#493D7
hypocenterC7#3155
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ #493

Complements

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

NH ≅ G/Ntype
1. C2×#6475C2semidirect
2. C2×#6474C2semidirect
3. C2×D7×D8C2semidirect
4. #90406C2semidirect
5. C7×#493C2semidirect
6. #89927C2semidirect
7. #89926C2semidirect
8. #89923C2semidirect
9. #89846C2semidirect
10. #89468C2semidirect
11. #89461C2semidirect
15. C2×C7×D8C22semidirect
16. C2×C7×D8C22semidirect
17. C7×#127C22semidirect
18. #6503C22semidirect
21. #6499C22semidirect
22. C2×#1142C22semidirect
23. C2×#1142C22semidirect
25. C2×#1141C22semidirect
29. C2×Dic28C22semidirect
31. C2×#1132C22semidirect
34. #6438C22semidirect
36. C4×C7×C8C22semidirect
37. #6389C22semidirect
38. C8×Dic7C22semidirect
39. #6377C22semidirect
76. #1131D4semidirect
77. C8×D7D4semidirect
87. #493D7semidirect
132. C2×D8C2×D7semidirect
133. C2×D8C2×D7semidirect
137. #127C2×D7semidirect
146. C4×C8C2×D7semidirect
236. C7#3155semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#64752
C2×#64742
C2×D7×D82
#904062
C7×#4932
#899272
#899262
#899232
#898462
#894682
#894612
#31557