Finite Group Databasebeta

← D7×#497 #11739313

Subgroups — D7×#497 #11739313

Order 896.

564 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 564 conjugacy classes of subgroups — too many to draw. Only its 145 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×D7×Q16C212yesmaximal, complemented
2C2×D7×SD16C212yesmaximal
3D7×C4⋊Q8C211yesmaximal, characteristic, complemented
4D7×#124C211yesmaximal, characteristic
5C7×#497C211yesmaximal, characteristic, complemented
6#89962C212yesmaximal, complemented
7#89939C212yesmaximal, complemented
8#89858C211yesmaximal, characteristic, complemented
9#89832C211yesmaximal, characteristic, complemented
10#89487C211yesmaximal, characteristic, complemented
11D7×#97C211yesmaximal, characteristic, complemented
12C2×Q8×D7C2212yes—
13C2×Q8×D7C2211yescharacteristic
14C2×D4×D7C2211yescharacteristic
15C2×C7×Q16C2212yescomplemented
16C2×C7×SD16C2212yes—
17C7×C4⋊Q8C2211yescharacteristic, complemented
18C7×#124C2211yescharacteristic
19#6510C2211yescharacteristic
20#6507C2212yescomplemented
21C2×#1144C2212yescomplemented
22C2×#1144C2212yescomplemented
23C2×#1143C2212yes—
24#6499C2211yescharacteristic
25C2×#1142C2212yes—
26D7×Q16—28no—
27D7×SD16—28no—
28C2×Dic28C2212yescomplemented
29C2×#1132C2212yes—
30C2×C8×D7C2212yes—
31D7×C4⋊C4—22no—
32#6451—22no—
33#6448—22no—
34D7×C22⋊C4—22no—
35#6438C2211yescharacteristic
36C42×D7C2211yescharacteristic
37#6432C2211yescharacteristic, complemented
38C7×#97C2211yescharacteristic, complemented
39#6389C2212yescomplemented
40#6377C2211yescharacteristic, complemented
41C2×#497—71nomaximal, Sylow
42C23×D7—21no—
43C2×C7×Q8C2312yes—
44C2×C7×Q8C2311yescharacteristic
45C2×C7×D4C2311yescharacteristic
46C2×C22⋊D7—21no—
47Q8×D7—24no—
48Q8×D7—24no—
49Q8×D7—24no—
50D4×D7—24no—
51C2×D28C2311yescharacteristic
52C2×C4×D7—21no—
53C2×C4×D7—22no—
54C2×C4×D7D412yes—
55C2×C4×D7C2311yescharacteristic
56C2×Dic14—21no—
57C2×Dic14—22no—
58C2×Dic14C2312yes—
59C2×Dic14C2311yescharacteristic
60C7×Q16—24no—
61C7×SD16—24no—
62C2×C7×C8C2312yes—
63C7×C4⋊C4—22no—
64C7×C22⋊C4—22no—
65C42×C7C2311yescharacteristic
66C22⋊Dic7—22no—
67#1144—24no—
68#1144—24no—
69#1143—24no—
70#1142—24no—
71(C2×D7)⋊C4—22no—
72(C2×D7)⋊C4—22no—
73C4⋊Dic7—22no—
74Dic7⋊C4—24no—
75C4×Dic7D412yes—
76C4×Dic7C2311yescharacteristic
77C2×C7⋊C8C2312yes—
78Dic28—24no—
79#1132—24no—
80C8×D7D418yes—
81C22×Q16—72no—
82C22×SD16—72no—
83C2×C4⋊Q8—71no—
84C2×#124—71no—
85#497—72no—
86#497—72no—
87#497—71no—
88#497—71no—
89#497—71no—
90#497D711yescharacteristic, direct product factor
91C2×#97—71no—
92C23×C7—21no—
93C22×D7—21no—
94C22×D7—42no—
95C22×D7—44no—
96C22×D7C2×D411yescharacteristic
97C7×Q8—22no—
98C7×Q8—22no—
99C7×Q8—22no—
100C7×D4—22no—
101C2×C4×C7—22no—
102C2×C4×C7—21no—
103C2×C4×C7C2×D412yes—
104C2×C4×C7C2411yescharacteristic
105C22⋊D7—42no—
106C2×Dic7—22no—
107C2×Dic7—21no—
108C2×Dic7C2×D412yes—
109C2×Dic7C2×D411yescharacteristic
110D28—22no—
111C4×D7—24no—
112C4×D7—42no—
113C4×D7—44no—
114C4×D7C2×D414yes—
115Dic14—22no—
116Dic14—22no—
117Dic14—22no—
118Dic14—42no—
119Dic14—44no—
120C7×C8C2×D414yes—
121C7⋊C8C2×D414yes—
122C22×Q8—71no—
123C22×Q8—72no—
124C22×D4—71no—
125C2×Q16—72no—
126C2×Q16—72no—
127C2×Q16—72no—
128C2×Q16—148no—
129C2×Q16C2×D712yescomplemented
130C2×SD16—72no—
131C2×SD16—72no—
132C2×SD16—72no—
133C2×SD16—148no—
134C2×SD16C2×D712yes—
135C22×C8—72no—
136C4⋊Q8—72no—
137C4⋊Q8—71no—
138C4⋊Q8—142no—
139C4⋊Q8C2×D711yescharacteristic, complemented
140#124—71no—
141#124—71no—
142#124—71no—
143#124—142no—
144#124C2×D711yescharacteristic
145C2×C4⋊C4—142no—
146C2×C22⋊C4—142no—
147C2×C42—71no—
148#97—72no—
149#97—71no—
150#97C2×D711yescharacteristic, complemented
151C22×C7—41no—
152C22×C7—42no—
153C22×C7C22×D411yescharacteristic
154C2×D7—41no—
155C2×D7—42no—
156C2×D7—84no—
157C2×D7#12712yes—
158C2×D7#13714yes—
159C4×C7—22no—
160C4×C7—42no—
161C4×C7—41no—
162C4×C7C22×D412yes—
163Dic7—22no—
164Dic7—42no—
165Dic7—41no—
166Dic7#12712yes—
167C24—141no—
168C2×Q8—71no—
169C2×Q8—72no—
170C2×Q8—141no—
171C2×Q8—142no—
172C2×Q8—144no—
173C2×Q8—144no—
174C2×Q8—144no—
175C2×Q8C22×D711yescharacteristic
176C2×Q8C22×D712yes—
177C2×D4—71no—
178C2×D4—144no—
179C2×D4—141no—
180C2×D4C22×D711yescharacteristic
181C22×C4—71no—
182C22×C4—72no—
183C22×C4—141no—
184C22×C4—142no—
185Q16—24no—
186Q16—144no—
187Q16—144no—
188Q16—144no—
189SD16—24no—
190SD16—144no—
191SD16—144no—
192SD16—144no—
193C2×C8—72no—
194C2×C8—78no—
195C2×C8C22×D712yes—
196C4⋊C4—22no—
197C4⋊C4—144no—
198C4⋊C4—142no—
199C22⋊C4—22no—
200C22⋊C4—142no—
201C22⋊C4—142no—
202C22⋊C4—142no—
203C42—72no—
204C42—71no—
205C42C22×D711yescharacteristic
206C2×C7—81no—
207C2×C7C2×#12711yescharacteristic
208C2×C7C2×#13712yes—
209D7—81no—
210D7#49714yesdirect product factor
211C23—21no—
212C23—71no—
213C23—141no—
214C23—282no—
215C23—284no—
216Q8—22no—
217Q8—22no—
218Q8—22no—
219Q8—142no—
220Q8—142no—
221Q8—142no—
222Q8—282no—
223Q8—284no—
224D4—22no—
225D4—142no—
226D4—282no—
227C2×C4—21no—
228C2×C4—22no—
229C2×C4—71no—
230C2×C4—74no—
231C2×C4—72no—
232C2×C4—144no—
233C2×C4—141no—
234C2×C4—142no—
235C2×C4—282no—
236C2×C4—284no—
237C2×C4D4×D712yes—
238C2×C4C23×D711yescharacteristic
239C8—74no—
240C8D4×D714yes—
241C7C2×#49711yescharacteristic, complemented, Sylow
242C22—41no—
243C22—42no—
244C22—72no—
245C22—74no—
246C22—281no—
247C22—282no—
248C22—564no—
249C22C2×D4×D711yescharacteristic
250C4—22no—
251C4—41no—
252C4—42no—
253C4—72no—
254C4—142no—
255C4—281no—
256C4—282no—
257C4C2×D4×D712yes—
258C2—74no—
259C2—81no—
260C2—561no—
261C2D7×#12711yescharacteristic
262C2D7×#13712yes—
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#497C2
socleC22×C7C22×D4
prime radicalC2×D4×D7C22
hypercenter#497D7
hypocenterC7C2×#497
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ #497

Complements

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

NH ≅ G/Ntype
1. C2×D7×Q16C2semidirect
3. D7×C4⋊Q8C2semidirect
5. C7×#497C2semidirect
6. #89962C2semidirect
7. #89939C2semidirect
8. #89858C2semidirect
9. #89832C2semidirect
10. #89487C2semidirect
11. D7×#97C2semidirect
15. C2×C7×Q16C22semidirect
17. C7×C4⋊Q8C22semidirect
20. #6507C22semidirect
21. C2×#1144C22semidirect
22. C2×#1144C22semidirect
28. C2×Dic28C22semidirect
37. #6432C22semidirect
38. C7×#97C22semidirect
39. #6389C22semidirect
40. #6377C22semidirect
90. #497D7direct (N × H)
129. C2×Q16C2×D7semidirect
139. C4⋊Q8C2×D7semidirect
150. #97C2×D7semidirect
210. D7#497direct (N × H)
241. C7C2×#497semidirect

Maximal subgroups

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

subgroupindex
C2×D7×Q162
C2×D7×SD162
D7×C4⋊Q82
D7×#1242
C7×#4972
#899622
#899392
#898582
#898322
#894872
D7×#972
C2×#4977