Finite Group Databasebeta

← D7×#489 #11739123

Subgroups — D7×#489 #11739123

Order 896.

436 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 436 conjugacy classes of subgroups — too many to draw. Only its 125 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
1D7×C4⋊Q8C211yesmaximal, characteristic, complemented
2D7×#124C211yesmaximal, characteristic
3C7×#489C211yesmaximal, characteristic, complemented
4#89861C211yesmaximal, characteristic, complemented
5#89827C211yesmaximal, characteristic, complemented
6D7×#103C212yesmaximal, complemented
7#89567C212yesmaximal, complemented
8D7×#102C212yesmaximal
9#89537C212yesmaximal, complemented
10#89486C211yesmaximal, characteristic, complemented
11D7×#97C211yesmaximal, characteristic, complemented
12C2×Q8×D7—21no—
13C2×Q8×D7C2211yescharacteristic
14C2×D4×D7C2211yescharacteristic
15C7×C4⋊Q8C2211yescharacteristic, complemented
16C7×#124C2211yescharacteristic
17#6510C2211yescharacteristic
18#6507—21no—
19#6499C2211yescharacteristic
20C2×C8×D7C2212yes—
21D7×C4⋊C4—21no—
22D7×C4⋊C4C2212yes—
23#6451—21no—
24#6451C2212yescomplemented
25#6448—21no—
26#6448—21no—
27D7×C22⋊C4—21no—
28D7×C22⋊C4—21no—
29#6438C2211yescharacteristic
30C42×D7C2211yescharacteristic
31#6432C2211yescharacteristic, complemented
32C7×#103C2212yescomplemented
33C7×#102C2212yes—
34C7×#97C2211yescharacteristic, complemented
35#6409C2212yescomplemented
36#6406C2212yes—
37#6395C2212yes—
38#6390C2212yescomplemented
39#6389C2212yescomplemented
40#6384C2212yescomplemented
41#6383C2212yes—
42#6377C2211yescharacteristic, complemented
43C2×#489—71nomaximal, Sylow
44C23×D7—21no—
45C2×C7×Q8—21no—
46C2×C7×Q8C2311yescharacteristic
47C2×C7×D4C2311yescharacteristic
48C2×C22⋊D7—21no—
49Q8×D7—24no—
50Q8×D7—44no—
51D4×D7—24no—
52C2×D28C2311yescharacteristic
53C2×C4×D7—21no—
54C2×C4×D7—22no—
55C2×C4×D7D411yescharacteristic
56C2×C4×D7D411yescharacteristic
57C2×C4×D7C2311yescharacteristic
58C2×Dic14—21no—
59C2×Dic14—21no—
60C2×Dic14—22no—
61C2×Dic14C2311yescharacteristic
62C2×C7×C8C2312yes—
63C7×C4⋊C4—21no—
64C7×C4⋊C4C2312yes—
65C7×C22⋊C4—21no—
66C7×C22⋊C4—21no—
67C42×C7C2311yescharacteristic
68C22⋊Dic7—21no—
69C22⋊Dic7—21no—
70(C2×D7)⋊C4—21no—
71(C2×D7)⋊C4—21no—
72(C2×D7)⋊C4—21no—
73(C2×D7)⋊C4—21no—
74C4⋊Dic7—21no—
75C4⋊Dic7C2312yes—
76Dic7⋊C4—22no—
77Dic7⋊C4—22no—
78C4×Dic7D411yescharacteristic
79C4×Dic7D411yescharacteristic
80C4×Dic7C2311yescharacteristic
81C2×C7⋊C8C2312yes—
82C8×D7—24no—
83C2×C4⋊Q8—71no—
84C2×#124—71no—
85#489—71no—
86#489—71no—
87#489—72no—
88#489—72no—
89#489—71no—
90#489D711yescharacteristic, direct product factor
91C2×#103—72no—
92C2×#102—72no—
93C2×#97—71no—
94C23×C7—21no—
95C22×D7—21no—
96C22×D7—42no—
97C22×D7—42no—
98C22×D7—42no—
99C22×D7C2×D411yescharacteristic
100C7×Q8—22no—
101C7×Q8—42no—
102C7×D4—22no—
103C2×C4×C7—22no—
104C2×C4×C7—21no—
105C2×C4×C7C2×D411yescharacteristic
106C2×C4×C7C2×D411yescharacteristic
107C2×C4×C7C2411yescharacteristic
108C22⋊D7—42no—
109C2×Dic7—22no—
110C2×Dic7—21no—
111C2×Dic7C2×D411yescharacteristic
112C2×Dic7C2×D411yescharacteristic
113C2×Dic7C2×D411yescharacteristic
114D28—22no—
115C4×D7—22no—
116C4×D7—22no—
117C4×D7—42no—
118C4×D7—44no—
119C4×D7#4114yes—
120Dic14—22no—
121Dic14—44no—
122Dic14—42no—
123Dic14—42no—
124C7×C8—22no—
125C7⋊C8—22no—
126C22×Q8—71no—
127C22×Q8—141no—
128C22×D4—71no—
129C22×C8—72no—
130C4⋊Q8—72no—
131C4⋊Q8—71no—
132C4⋊Q8—141no—
133C4⋊Q8—141no—
134C4⋊Q8C2×D711yescharacteristic, complemented
135#124—71no—
136#124—71no—
137#124—71no—
138#124—141no—
139#124—141no—
140#124C2×D711yescharacteristic
141C2×C4⋊C4—72no—
142C2×C4⋊C4—141no—
143C2×C22⋊C4—141no—
144C2×C22⋊C4—141no—
145C2×C42—71no—
146#103—72no—
147#103—72no—
148#103—72no—
149#103C2×D712yescomplemented
150#102—72no—
151#102—72no—
152#102—72no—
153#102C2×D712yes—
154#97—72no—
155#97—71no—
156#97C2×D711yescharacteristic, complemented
157C22×C7—41no—
158C22×C7—41no—
159C22×C7—41no—
160C22×C7C22×D411yescharacteristic
161C2×D7—41no—
162C2×D7—41no—
163C2×D7—41no—
164C2×D7—84no—
165C2×D7#12412yes—
166C2×D7AGL(1,ℤ/8)12yes—
167C2×D7#13712yes—
168C4×C7—21no—
169C4×C7—21no—
170C4×C7—41no—
171C4×C7—42no—
172C4×C7C2×#4112yes—
173Dic7—21no—
174Dic7—21no—
175Dic7—41no—
176Dic7—42no—
177Dic7#12412yes—
178C24—141no—
179C2×Q8—21no—
180C2×Q8—71no—
181C2×Q8—141no—
182C2×Q8—142no—
183C2×Q8—144no—
184C2×Q8—141no—
185C2×Q8—284no—
186C2×Q8C22×D711yescharacteristic
187C2×D4—71no—
188C2×D4—144no—
189C2×D4—141no—
190C2×D4C22×D711yescharacteristic
191C22×C4—71no—
192C22×C4—71no—
193C22×C4—71no—
194C22×C4—141no—
195C22×C4—142no—
196C2×C8—72no—
197C2×C8—144no—
198C2×C8C22×D712yes—
199C4⋊C4—21no—
200C4⋊C4—72no—
201C4⋊C4—142no—
202C4⋊C4—142no—
203C4⋊C4—141no—
204C4⋊C4C22×D712yes—
205C22⋊C4—21no—
206C22⋊C4—21no—
207C22⋊C4—141no—
208C22⋊C4—141no—
209C22⋊C4—141no—
210C22⋊C4—141no—
211C22⋊C4—141no—
212C22⋊C4—141no—
213C42—71no—
214C42—71no—
215C42—71no—
216C42C22×D711yescharacteristic
217C2×C7—81no—
218C2×C7C2×#12411yescharacteristic
219C2×C7C2×AGL(1,ℤ/8)11yescharacteristic
220C2×C7C2×#13711yescharacteristic
221D7—81no—
222D7#48914yesdirect product factor
223C23—21no—
224C23—71no—
225C23—141no—
226C23—282no—
227C23—282no—
228C23—282no—
229Q8—22no—
230Q8—42no—
231Q8—142no—
232Q8—282no—
233Q8—282no—
234Q8—284no—
235D4—22no—
236D4—142no—
237D4—282no—
238C2×C4—21no—
239C2×C4—22no—
240C2×C4—71no—
241C2×C4—74no—
242C2×C4—71no—
243C2×C4—71no—
244C2×C4—142no—
245C2×C4—142no—
246C2×C4—141no—
247C2×C4—142no—
248C2×C4—282no—
249C2×C4—284no—
250C2×C4D4×D711yescharacteristic
251C2×C4D4×D711yescharacteristic
252C2×C4C23×D711yescharacteristic
253C8—22no—
254C8—142no—
255C7C2×#48911yescharacteristic, complemented, Sylow
256C22—41no—
257C22—41no—
258C22—41no—
259C22—72no—
260C22—72no—
261C22—72no—
262C22—281no—
263C22—281no—
264C22—281no—
265C22—564no—
266C22C2×D4×D711yescharacteristic
267C4—21no—
268C4—21no—
269C4—42no—
270C4—41no—
271C4—72no—
272C4—141no—
273C4—141no—
274C4—282no—
275C4—281no—
276C4D7×#4112yes—
277C2—74no—
278C2—81no—
279C2—561no—
280C2D7×#12411yescharacteristic
281C2D7×AGL(1,ℤ/8)11yescharacteristic
282C2D7×#13711yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#489C2
socleC22×C7C22×D4
prime radicalC2×D4×D7C22
hypercenter#489D7
hypocenterC7C2×#489
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ #489

Complements

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

NH ≅ G/Ntype
1. D7×C4⋊Q8C2semidirect
3. C7×#489C2semidirect
4. #89861C2semidirect
5. #89827C2semidirect
6. D7×#103C2semidirect
7. #89567C2semidirect
9. #89537C2semidirect
10. #89486C2semidirect
11. D7×#97C2semidirect
15. C7×C4⋊Q8C22semidirect
24. #6451C22semidirect
31. #6432C22semidirect
32. C7×#103C22semidirect
34. C7×#97C22semidirect
35. #6409C22semidirect
38. #6390C22semidirect
39. #6389C22semidirect
40. #6384C22semidirect
42. #6377C22semidirect
90. #489D7direct (N × H)
134. C4⋊Q8C2×D7semidirect
149. #103C2×D7semidirect
156. #97C2×D7semidirect
222. D7#489direct (N × H)
255. C7C2×#489semidirect

Maximal subgroups

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

subgroupindex
D7×C4⋊Q82
D7×#1242
C7×#4892
#898612
#898272
D7×#1032
#895672
D7×#1022
#895372
#894862
D7×#972
C2×#4897