Finite Group Databasebeta

← #11739294

Subgroups — #11739294

Order 896.

444 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 111 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
1#90407C211yesmaximal, characteristic, complemented
2D7×#124C211yesmaximal, characteristic, complemented
3C7×#496C211yesmaximal, characteristic, complemented
4#89942C212yesmaximal, complemented
5#89937C212yesmaximal, complemented
6#89928C212yesmaximal, complemented
7#89925C212yesmaximal, complemented
8#89846C211yesmaximal, characteristic, complemented
9#89827C211yesmaximal, characteristic, complemented
10#89486C211yesmaximal, characteristic, complemented
11#89477C211yesmaximal, characteristic, complemented
12C2×Q8×D7C2211yescharacteristic
13C2×#1159—21no—
14C2×D4×D7C2211yescharacteristic
15C2×C7×SD16C2212yescomplemented
16C2×C7×D8C2212yes—
17C7×#127C2211yescharacteristic, complemented
18C7×#124C2211yescharacteristic
19#6510C2211yescharacteristic
20#6501—21no—
21#6501C2212yes—
22#6499—21no—
23#6499C2211yescharacteristic
24D4×Dic7—21no—
25D4×Dic7C2212yescomplemented
26C2×#1142C2212yescomplemented
27C2×#1141C2212yes—
28#6448—21no—
29#6448—21no—
30D7×C22⋊C4—21no—
31D7×C22⋊C4—21no—
32#6438C2211yescharacteristic
33C42×D7C2211yescharacteristic
34#6432C2211yescharacteristic, complemented
35C7×#97C2211yescharacteristic, complemented
36#6409C2212yescomplemented
37#6406C2212yes—
38#6395C2212yes—
39(C2×D7)⋊C8C2212yes—
40#6390C2212yescomplemented
41Dic7⋊C8C2212yescomplemented
42#6377C2211yescharacteristic, complemented
43#3213—71nomaximal, Sylow
44C23×D7—21no—
45C2×C7×Q8C2311yescharacteristic
46C2×C7×D4—21no—
47C2×C7×D4—21no—
48C2×C7×D4C2311yescharacteristic
49C2×C7×D4C2312yes—
50C2×C22⋊D7—22no—
51C2×C22⋊D7—21no—
52C22×Dic7—22no—
53Q8×D7—42no—
54#1159—44no—
55D4×D7—42no—
56C2×D28C2311yescharacteristic
57C2×C4×D7—21no—
58C2×C4×D7D411yescharacteristic
59C2×C4×D7D411yescharacteristic
60C2×C4×D7C2311yescharacteristic
61C2×Dic14—21no—
62C2×Dic14—21no—
63C2×Dic14C2311yescharacteristic
64C7×SD16—42no—
65C7×D8—42no—
66C2×C7×C8C2312yes—
67C7×C22⋊C4—21no—
68C7×C22⋊C4—21no—
69C42×C7C2311yescharacteristic
70C22⋊Dic7—21no—
71C22⋊Dic7—21no—
72C22⋊Dic7—22no—
73C22⋊Dic7—22no—
74#1142—42no—
75#1141—42no—
76(C2×D7)⋊C4—21no—
77(C2×D7)⋊C4—21no—
78(C2×D7)⋊C4—21no—
79(C2×D7)⋊C4—21no—
80C4⋊Dic7—21no—
81C4⋊Dic7C2312yes—
82C4×Dic7D411yescharacteristic
83C4×Dic7D411yescharacteristic
84C4×Dic7C2311yescharacteristic
85C2×C7⋊C8C2312yes—
86#532—71no—
87C2×#124—71no—
88#496—71no—
89#496D711yescharacteristic, complemented
90#489—71no—
91#489—71no—
92#463—72no—
93#463—72no—
94#448—72no—
95#447—72no—
96#432—71no—
97C23×C7—21no—
98C23×C7—22no—
99C22×D7—21no—
100C22×D7—42no—
101C22×D7—42no—
102C22×D7—42no—
103C22×D7C2×D411yescharacteristic
104C7×Q8—22no—
105C7×D4—22no—
106C7×D4—22no—
107C7×D4—22no—
108C7×D4—42no—
109C7×D4—42no—
110C2×C4×C7—21no—
111C2×C4×C7C2×D411yescharacteristic
112C2×C4×C7C2×D411yescharacteristic
113C2×C4×C7C2411yescharacteristic
114C22⋊D7—42no—
115C22⋊D7—44no—
116C2×Dic7—22no—
117C2×Dic7—21no—
118C2×Dic7—44no—
119C2×Dic7C2×D411yescharacteristic
120C2×Dic7C2×D411yescharacteristic
121C2×Dic7C2×D411yescharacteristic
122D28—22no—
123C4×D7—22no—
124C4×D7—22no—
125C4×D7—41no—
126C4×D7—42no—
127Dic14—22no—
128Dic14—42no—
129Dic14—42no—
130C7×C8—22no—
131C7⋊C8—22no—
132C2×#41—141no—
133C22×Q8—71no—
134C22×D4—71no—
135C2×SD16—72no—
136C2×SD16C2×D712yescomplemented
137C2×D8—72no—
138C2×D8C2×D712yes—
139C4⋊Q8—71no—
140#127C2×D711yescharacteristic, complemented
141#124—71no—
142#124—71no—
143#124—71no—
144#124—141no—
145#124—141no—
146#124—141no—
147#124C2×D711yescharacteristic
148#121—72no—
149#121—141no—
150C4×D4—72no—
151C4×D4—141no—
152C2×C22⋊C4—141no—
153C2×C22⋊C4—141no—
154C2×C42—71no—
155C4⋊C8—72no—
156#103—72no—
157#103—72no—
158#102—72no—
159#102—72no—
160C22⋊C8—72no—
161#97—71no—
162#97C2×D711yescharacteristic, complemented
163C22×C7—41no—
164C22×C7—42no—
165C22×C7—42no—
166C22×C7—41no—
167C22×C7—42no—
168C22×C7—41no—
169C22×C7C22×D411yescharacteristic
170C2×D7—21no—
171C2×D7—21no—
172C2×D7—41no—
173C2×D7—41no—
174C2×D7—41no—
175C2×D7—84no—
176C2×D7AGL(1,ℤ/8)12yes—
177C4×C7—21no—
178C4×C7—21no—
179C4×C7—41no—
180C4×C7H(5,𝔽2)12yes—
181Dic7—21no—
182Dic7—21no—
183Dic7—21no—
184Dic7—41no—
185Dic7—42no—
186C24—141no—
187#41—284no—
188C2×Q8—71no—
189C2×Q8—141no—
190C2×Q8—141no—
191C2×Q8—282no—
192C2×Q8C22×D711yescharacteristic
193C2×D4—21no—
194C2×D4—21no—
195C2×D4—71no—
196C2×D4—141no—
197C2×D4—142no—
198C2×D4—282no—
199C2×D4C22×D711yescharacteristic
200C2×D4C22×D712yes—
201C22×C4—71no—
202C22×C4—71no—
203C22×C4—71no—
204C22×C4—141no—
205C22×C4—142no—
206SD16—42no—
207SD16—282no—
208D8—42no—
209D8—282no—
210C2×C8—72no—
211C2×C8C22×D712yes—
212C4⋊C4—72no—
213C4⋊C4—141no—
214C22⋊C4—21no—
215C22⋊C4—21no—
216C22⋊C4—141no—
217C22⋊C4—141no—
218C22⋊C4—141no—
219C22⋊C4—141no—
220C22⋊C4—142no—
221C22⋊C4—142no—
222C22⋊C4—141no—
223C22⋊C4—141no—
224C42—71no—
225C42—71no—
226C42—71no—
227C42C22×D711yescharacteristic
228C2×C7—81no—
229C2×C7—82no—
230C2×C7#53511yescharacteristic
231C2×C7C2×AGL(1,ℤ/8)11yescharacteristic
232C2×C7#57511yescharacteristic
233D7—22no—
234D7—81no—
235C23—21no—
236C23—22no—
237C23—71no—
238C23—141no—
239C23—282no—
240C23—282no—
241C23—282no—
242Q8—22no—
243Q8—142no—
244Q8—282no—
245Q8—282no—
246D4—22no—
247D4—22no—
248D4—22no—
249D4—42no—
250D4—42no—
251D4—142no—
252D4—282no—
253D4—284no—
254C2×C4—21no—
255C2×C4—71no—
256C2×C4—71no—
257C2×C4—71no—
258C2×C4—142no—
259C2×C4—142no—
260C2×C4—141no—
261C2×C4—142no—
262C2×C4—281no—
263C2×C4—282no—
264C2×C4—284no—
265C2×C4D4×D711yescharacteristic
266C2×C4D4×D711yescharacteristic
267C2×C4C23×D711yescharacteristic
268C8—22no—
269C8—142no—
270C7#321311yescharacteristic, complemented, Sylow
271C22—41no—
272C22—42no—
273C22—42no—
274C22—41no—
275C22—42no—
276C22—41no—
277C22—72no—
278C22—141no—
279C22—141no—
280C22—281no—
281C22—281no—
282C22—281no—
283C22—564no—
284C22C2×D4×D711yescharacteristic
285C4—21no—
286C4—21no—
287C4—41no—
288C4—141no—
289C4—141no—
290C4—141no—
291C4—281no—
292C4—282no—
293C4#654812yes—
294C2—81no—
295C2—82no—
296C2—142no—
297C2—561no—
298C2#9040611yescharacteristic
299C2D7×AGL(1,ℤ/8)11yescharacteristic
300C2#9046411yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#496C2
socleC22×C7C22×D4
hypercenter#496D7
hypocenterC7#3213
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ #496

Complements

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

NH ≅ G/Ntype
1. #90407C2semidirect
2. D7×#124C2semidirect
3. C7×#496C2semidirect
4. #89942C2semidirect
5. #89937C2semidirect
6. #89928C2semidirect
7. #89925C2semidirect
8. #89846C2semidirect
9. #89827C2semidirect
10. #89486C2semidirect
11. #89477C2semidirect
15. C2×C7×SD16C22semidirect
17. C7×#127C22semidirect
25. D4×Dic7C22semidirect
26. C2×#1142C22semidirect
34. #6432C22semidirect
35. C7×#97C22semidirect
36. #6409C22semidirect
40. #6390C22semidirect
41. Dic7⋊C8C22semidirect
42. #6377C22semidirect
89. #496D7semidirect
136. C2×SD16C2×D7semidirect
140. #127C2×D7semidirect
162. #97C2×D7semidirect
270. C7#3213semidirect

Maximal subgroups

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

subgroupindex
#904072
D7×#1242
C7×#4962
#899422
#899372
#899282
#899252
#898462
#898272
#894862
#894772
#32137