Finite Group Databasebeta

← #11739268

Subgroups — #11739268

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×#6482C211yesmaximal, characteristic, complemented
2C2×#6477C212yesmaximal, complemented
3C2×D7×D8C211yesmaximal, characteristic, complemented
4#90365C212yesmaximal, complemented
5C7×#495C211yesmaximal, characteristic, complemented
6#89963C211yesmaximal, characteristic, complemented
7#89948C212yesmaximal, complemented
8#89923C211yesmaximal, characteristic, complemented
9#89830C212yesmaximal, complemented
10#89467C211yesmaximal, characteristic, complemented
11#89461C211yesmaximal, characteristic, complemented
12C2×#1161C2212yes—
13C2×D4×D7C2212yes—
14C2×C7×Q16C2211yescharacteristic, complemented
15C2×C7×SD16C2212yescomplemented
16C2×C7×D8C2211yescharacteristic
17C7×#124C2212yescomplemented
18#6510C2212yescomplemented
19C2×#1143C2212yescomplemented
20C2×#1143C2212yescomplemented
21#6503C2212yescomplemented
22C2×#1141C2212yes—
23C2×#1141C2212yescomplemented
24#6482—24no—
25#6477—28no—
26D7×D8—24no—
27C2×D56C2211yescharacteristic
28C2×D56C2211yescharacteristic, complemented
29C2×D56C2212yescomplemented
30C2×#1131C2211yescharacteristic
31C2×C8×D7C2211yescharacteristic
32#6447—24no—
33#6445—24no—
34#6437C2212yescomplemented
35#6435C2211yescharacteristic
36C4×C7×C8C2211yescharacteristic, complemented
37#6389C2211yescharacteristic, complemented
38C8×Dic7C2211yescharacteristic, complemented
39#6377C2211yescharacteristic, complemented
40#3155—71nomaximal, Sylow
41C23×D7—22no—
42C2×C7×Q8C2312yes—
43C2×C7×D4C2312yes—
44C2×C22⋊D7—22no—
45C2×C22⋊D7—24no—
46#1161—24no—
47#1161—24no—
48D4×D7—24no—
49D4×D7—24no—
50C2×D28—22no—
51C2×D28—24no—
52C2×D28C2312yes—
53C2×D28C2312yes—
54C2×C4×D7—22no—
55C2×C4×D7C2311yescharacteristic
56C7×Q16—22no—
57C7×SD16—24no—
58C7×D8—22no—
59C2×C7×C8C2311yescharacteristic
60C2×C7×C8C2311yescharacteristic
61C7×C22⋊C4—24no—
62C42×C7C2311yescharacteristic
63#1143—24no—
64#1143—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
73D56—22no—
74D56—24no—
75D56—22no—
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#495D711yescharacteristic, complemented
86#493—71no—
87#493—71no—
88#405—71no—
89C23×C7—22no—
90C22×D7—22no—
91C22×D7—22no—
92C22×D7—44no—
93C22×D7—84no—
94C22×D7C2×D411yescharacteristic
95C7×Q8—22no—
96C7×Q8—22no—
97C7×D4—22no—
98C7×D4—22no—
99C2×C4×C7—22no—
100C2×C4×C7C2×D412yes—
101C2×C4×C7C2411yescharacteristic
102C22⋊D7—44no—
103C22⋊D7—84no—
104C2×Dic7C2×D412yes—
105C2×Dic7C2×D411yescharacteristic
106D28—22no—
107D28—22no—
108D28—22no—
109D28—22no—
110D28—44no—
111D28—84no—
112C4×D7—44no—
113C4×D7C2×D412yes—
114C4×D7C2×D412yes—
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×Q16C2×D711yescharacteristic, complemented
124C2×SD16—72no—
125C2×SD16—72no—
126C2×SD16C2×D712yescomplemented
127C2×D8—72no—
128C2×D8—72no—
129C2×D8—71no—
130C2×D8—71no—
131C2×D8—72no—
132C2×D8—144no—
133C2×D8C2×D711yescharacteristic
134C2×M16—71no—
135C22×C8—71no—
136#127—72no—
137#127—72no—
138#124—72no—
139#124C2×D712yescomplemented
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—44no—
149C22×C7C22×D411yescharacteristic
150C2×D7—22no—
151C2×D7—42no—
152C2×D7—42no—
153C2×D7—44no—
154C2×D7—44no—
155C2×D7—88no—
156C2×D7#12712yes—
157C4×C7—22no—
158C4×C7—42no—
159C4×C7C22×D411yescharacteristic
160C4×C7C22×D411yescharacteristic
161Dic7—22no—
162Dic7#12712yes—
163C24—142no—
164#41—144no—
165#41—144no—
166C2×Q8C22×D712yes—
167C2×D4—72no—
168C2×D4—72no—
169C2×D4—142no—
170C2×D4—144no—
171C2×D4—144no—
172C2×D4—142no—
173C2×D4—144no—
174C2×D4—144no—
175C2×D4C22×D712yes—
176C22×C4—71no—
177C22×C4—142no—
178Q16—22no—
179SD16—24no—
180SD16—144no—
181SD16—144no—
182D8—22no—
183D8—144no—
184D8—144no—
185D8—142no—
186D8—142no—
187D8—144no—
188M16—74no—
189C2×C8—71no—
190C2×C8—71no—
191C2×C8—74no—
192C2×C8C22×D711yescharacteristic
193C2×C8C22×D711yescharacteristic
194C4⋊C4—72no—
195C22⋊C4—24no—
196C22⋊C4—72no—
197C22⋊C4—144no—
198C42—71no—
199C42C22×D711yescharacteristic
200C2×C7—82no—
201C2×C7C2×#12711yescharacteristic
202C2×C7#57612yes—
203D7—22no—
204D7—82no—
205D7—82no—
206C23—22no—
207C23—71no—
208C23—142no—
209C23—142no—
210C23—284no—
211C23—564no—
212Q8—22no—
213Q8—22no—
214D4—22no—
215D4—22no—
216D4—142no—
217D4—142no—
218D4—142no—
219D4—142no—
220D4—284no—
221D4—284no—
222D4—564no—
223D4—564no—
224C2×C4—22no—
225C2×C4—71no—
226C2×C4—72no—
227C2×C4—72no—
228C2×C4—72no—
229C2×C4—284no—
230C2×C4D4×D712yes—
231C2×C4C23×D711yescharacteristic
232C8—72no—
233C8—72no—
234C8D4×D712yes—
235C8D4×D712yes—
236C7#315511yescharacteristic, complemented, Sylow
237C22—42no—
238C22—44no—
239C22—72no—
240C22—142no—
241C22—282no—
242C22—282no—
243C22—284no—
244C22—284no—
245C22—568no—
246C22C2×D4×D711yescharacteristic
247C4—22no—
248C4—42no—
249C4—72no—
250C4—142no—
251C4C2×D4×D711yescharacteristic
252C4C2×D4×D711yescharacteristic
253C2—82no—
254C2—142no—
255C2—562no—
256C2—562no—
257C2D7×#12711yescharacteristic
258C2#9046012yes—
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#495C2
socleC22×C7C22×D4
hypercenter#495D7
hypocenterC7#3155
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ #495

Complements

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

NH ≅ G/Ntype
1. C2×#6482C2semidirect
2. C2×#6477C2semidirect
3. C2×D7×D8C2semidirect
4. #90365C2semidirect
5. C7×#495C2semidirect
6. #89963C2semidirect
7. #89948C2semidirect
8. #89923C2semidirect
9. #89830C2semidirect
10. #89467C2semidirect
11. #89461C2semidirect
14. C2×C7×Q16C22semidirect
15. C2×C7×SD16C22semidirect
17. C7×#124C22semidirect
18. #6510C22semidirect
19. C2×#1143C22semidirect
20. C2×#1143C22semidirect
21. #6503C22semidirect
23. C2×#1141C22semidirect
28. C2×D56C22semidirect
29. C2×D56C22semidirect
34. #6437C22semidirect
36. C4×C7×C8C22semidirect
37. #6389C22semidirect
38. C8×Dic7C22semidirect
39. #6377C22semidirect
76. #1131D4semidirect
77. C8×D7D4semidirect
85. #495D7semidirect
123. C2×Q16C2×D7semidirect
126. C2×SD16C2×D7semidirect
139. #124C2×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×#64822
C2×#64772
C2×D7×D82
#903652
C7×#4952
#899632
#899482
#899232
#898302
#894672
#894612
#31557