Finite Group Databasebeta

← #11739162

Subgroups — #11739162

Order 896.

490 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 490 conjugacy classes of subgroups — too many to draw. Only its 113 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×#127C211yesmaximal, characteristic, complemented
2#90382C211yesmaximal, characteristic, complemented
3C7×#490C211yesmaximal, characteristic, complemented
4#89846C211yesmaximal, characteristic, complemented
5#89839C211yesmaximal, characteristic, complemented
6#89558C212yesmaximal, complemented
7#89548C212yesmaximal, complemented
8#89545C212yesmaximal, complemented
9#89534C212yesmaximal, complemented
10#89484C211yesmaximal, characteristic, complemented
11#89477C211yesmaximal, characteristic, complemented
12C2×D4×D7—21no—
13C2×D4×D7—21no—
14C2×D4×D7C2212yes—
15C7×#127C2211yescharacteristic
16C7×#125C2211yescharacteristic, complemented
17#6503—21no—
18#6503—21no—
19#6503C2212yes—
20C2×#1142C2212yescomplemented
21C2×#1141C2212yescomplemented
22C2×D56C2212yescomplemented
23C2×#1132C2212yescomplemented
24#6461—21no—
25#6460C2211yescharacteristic
26#6458C2211yescharacteristic
27#6457—21no—
28#6456—21no—
29#6456C2211yescharacteristic, complemented
30#6455—21no—
31#6450C2211yescharacteristic, complemented
32#6438C2211yescharacteristic, complemented
33#6437C2211yescharacteristic
34C42×D7C2211yescharacteristic
35C7×#102C2212yescomplemented
36C7×#102C2212yescomplemented
37C7×#97C2211yescharacteristic, complemented
38(C2×D7)⋊C8C2212yes—
39Dic7⋊C8C2212yescomplemented
40#6383C2212yescomplemented
41#6383C2212yescomplemented
42#6377C2211yescharacteristic, complemented
43#3217—71nomaximal, Sylow
44C23×D7—22no—
45C2×C7×D4—21no—
46C2×C7×D4—21no—
47C2×C7×D4C2312yes—
48C2×C22⋊D7—22no—
49C2×C22⋊D7—22no—
50C2×C22⋊D7—22no—
51D4×D7—42no—
52D4×D7—42no—
53D4×D7—44no—
54D4×D7—44no—
55C2×D28—21no—
56C2×D28—21no—
57C2×D28—21no—
58C2×D28C2312yes—
59C2×D28C2311yescharacteristic
60C2×C4×D7—21no—
61C2×C4×D7D411yescharacteristic
62C2×C4×D7D411yescharacteristic
63C2×C4×D7C2311yescharacteristic
64C2×Dic14C2311yescharacteristic
65C2×C7×C8C2312yes—
66C7×C4⋊C4—21no—
67C7×C4⋊C4—21no—
68C7×C4⋊C4C2311yescharacteristic
69C7×C4⋊C4C2311yescharacteristic
70C42×C7C2311yescharacteristic
71#1142—42no—
72#1141—42no—
73(C2×D7)⋊C4—21no—
74(C2×D7)⋊C4—21no—
75(C2×D7)⋊C4—21no—
76(C2×D7)⋊C4—21no—
77(C2×D7)⋊C4—21no—
78C4⋊Dic7—21no—
79Dic7⋊C4—21no—
80C4×Dic7—21no—
81C4×Dic7D411yescharacteristic
82C4×Dic7D411yescharacteristic
83C4×Dic7C2311yescharacteristic
84C2×C7⋊C8C2312yes—
85D56—42no—
86#1132—42no—
87C2×#127—71no—
88#529—71no—
89#496—71no—
90#496—71no—
91#490—71no—
92#490D711yescharacteristic, complemented
93#460—72no—
94#459—72no—
95#450—72no—
96#447—72no—
97#432—71no—
98C23×C7—22no—
99C22×D7—21no—
100C22×D7—22no—
101C22×D7—44no—
102C22×D7—44no—
103C22×D7—44no—
104C22×D7C2×D411yescharacteristic
105C7×D4—22no—
106C7×D4—22no—
107C7×D4—42no—
108C7×D4—42no—
109C2×C4×C7—21no—
110C2×C4×C7—21no—
111C2×C4×C7C2×D411yescharacteristic
112C2×C4×C7C2×D411yescharacteristic
113C2×C4×C7C2411yescharacteristic
114C22⋊D7—44no—
115C22⋊D7—44no—
116C22⋊D7—44no—
117C2×Dic7—21no—
118C2×Dic7C2×D411yescharacteristic
119C2×Dic7C2×D411yescharacteristic
120C2×Dic7C2×D411yescharacteristic
121D28—22no—
122D28—22no—
123D28—22no—
124D28—42no—
125D28—42no—
126D28—42no—
127C4×D7—22no—
128C4×D7—22no—
129C4×D7—22no—
130C4×D7—42no—
131Dic14—22no—
132C7×C8—22no—
133C7⋊C8—22no—
134C22×D4—72no—
135C22×D4—141no—
136C22×D4—141no—
137C2×SD16—72no—
138C2×SD16—72no—
139C2×D8—72no—
140C2×D8—72no—
141#127—72no—
142#127—71no—
143#127—141no—
144#127—141no—
145#127C2×D711yescharacteristic
146#126—141no—
147#125C2×D711yescharacteristic, complemented
148#124—71no—
149#123—141no—
150C22⋊Q8—71no—
151#121—71no—
152C4×Q8—71no—
153C4×D4—71no—
154C4×D4—141no—
155#117—141no—
156C2×C42—71no—
157C4⋊C8—72no—
158#102—72no—
159#102—72no—
160#102C2×D712yescomplemented
161#102C2×D712yescomplemented
162C22⋊C8—72no—
163#97—71no—
164#97C2×D711yescharacteristic, complemented
165C22×C7—42no—
166C22×C7—42no—
167C22×C7—42no—
168C22×C7C22×D411yescharacteristic
169C2×D7—21no—
170C2×D7—21no—
171C2×D7—42no—
172C2×D7—41no—
173C2×D7—41no—
174C2×D7—42no—
175C2×D7—42no—
176C2×D7—41no—
177C2×D7—88no—
178C2×D7AGL(1,ℤ/8)12yes—
179C4×C7—21no—
180C4×C7—21no—
181C4×C7—41no—
182C4×C7—41no—
183C4×C7H(5,𝔽2)12yes—
184Dic7—21no—
185Dic7—21no—
186Dic7—41no—
187Dic7AGL(1,ℤ/8)12yes—
188C24—142no—
189C2×Q8—71no—
190C2×D4—21no—
191C2×D4—21no—
192C2×D4—72no—
193C2×D4—71no—
194C2×D4—141no—
195C2×D4—142no—
196C2×D4—142no—
197C2×D4—142no—
198C2×D4—141no—
199C2×D4—141no—
200C2×D4—282no—
201C2×D4—282no—
202C2×D4—284no—
203C2×D4—284no—
204C2×D4C22×D712yes—
205C22×C4—71no—
206C22×C4—71no—
207C22×C4—71no—
208C22×C4—141no—
209SD16—282no—
210SD16—282no—
211D8—282no—
212D8—282no—
213C2×C8—72no—
214C2×C8C22×D712yes—
215C4⋊C4—21no—
216C4⋊C4—21no—
217C4⋊C4—141no—
218C4⋊C4—141no—
219C4⋊C4C22×D711yescharacteristic
220C4⋊C4C22×D711yescharacteristic
221C22⋊C4—141no—
222C22⋊C4—141no—
223C22⋊C4—141no—
224C22⋊C4—141no—
225C22⋊C4—141no—
226C42—71no—
227C42—71no—
228C42—71no—
229C42—141no—
230C42C22×D711yescharacteristic
231C2×C7—82no—
232C2×C7#53511yescharacteristic
233C2×C7C2×AGL(1,ℤ/8)11yescharacteristic
234C2×C7C2×AGL(1,ℤ/8)11yescharacteristic
235D7—22no—
236D7—81no—
237D7—82no—
238C23—22no—
239C23—71no—
240C23—141no—
241C23—142no—
242C23—284no—
243C23—284no—
244C23—284no—
245Q8—142no—
246D4—22no—
247D4—22no—
248D4—42no—
249D4—42no—
250D4—142no—
251D4—142no—
252D4—142no—
253D4—282no—
254D4—282no—
255D4—282no—
256D4—284no—
257D4—284no—
258D4—284no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—71no—
262C2×C4—71no—
263C2×C4—71no—
264C2×C4—142no—
265C2×C4—142no—
266C2×C4—142no—
267C2×C4—141no—
268C2×C4—282no—
269C2×C4D4×D711yescharacteristic
270C2×C4D4×D711yescharacteristic
271C2×C4C23×D711yescharacteristic
272C8—22no—
273C8—142no—
274C7#321711yescharacteristic, complemented, Sylow
275C22—42no—
276C22—42no—
277C22—42no—
278C22—72no—
279C22—141no—
280C22—141no—
281C22—281no—
282C22—282no—
283C22—282no—
284C22—281no—
285C22—282no—
286C22—281no—
287C22—568no—
288C22C2×D4×D711yescharacteristic
289C4—21no—
290C4—21no—
291C4—41no—
292C4—41no—
293C4—72no—
294C4—141no—
295C4—141no—
296C4—281no—
297C4#655312yes—
298C2—82no—
299C2—142no—
300C2—562no—
301C2—561no—
302C2#9036511yescharacteristic
303C2D7×AGL(1,ℤ/8)11yescharacteristic
304C2D7×AGL(1,ℤ/8)11yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#490C2
socleC22×C7C22×D4
hypercenter#490D7
hypocenterC7#3217
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ #490

Complements

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

NH ≅ G/Ntype
1. D7×#127C2semidirect
2. #90382C2semidirect
3. C7×#490C2semidirect
4. #89846C2semidirect
5. #89839C2semidirect
6. #89558C2semidirect
7. #89548C2semidirect
8. #89545C2semidirect
9. #89534C2semidirect
10. #89484C2semidirect
11. #89477C2semidirect
16. C7×#125C22semidirect
20. C2×#1142C22semidirect
21. C2×#1141C22semidirect
22. C2×D56C22semidirect
23. C2×#1132C22semidirect
29. #6456C22semidirect
31. #6450C22semidirect
32. #6438C22semidirect
35. C7×#102C22semidirect
36. C7×#102C22semidirect
37. C7×#97C22semidirect
39. Dic7⋊C8C22semidirect
40. #6383C22semidirect
41. #6383C22semidirect
42. #6377C22semidirect
92. #490D7semidirect
147. #125C2×D7semidirect
160. #102C2×D7semidirect
161. #102C2×D7semidirect
164. #97C2×D7semidirect
274. C7#3217semidirect

Maximal subgroups

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

subgroupindex
D7×#1272
#903822
C7×#4902
#898462
#898392
#895582
#895482
#895452
#895342
#894842
#894772
#32177