Finite Group Databasebeta

← #11739129

Subgroups — #11739129

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#90416C211yesmaximal, characteristic, complemented
2D7×#124C211yesmaximal, characteristic, complemented
3C7×#489C211yesmaximal, characteristic, complemented
4#89861C211yesmaximal, characteristic, complemented
5#89830C211yesmaximal, characteristic, complemented
6#89591C212yesmaximal, complemented
7#89578C212yesmaximal, complemented
8#89559C212yesmaximal, complemented
9#89545C212yesmaximal, complemented
10#89484C211yesmaximal, characteristic, complemented
11#89477C211yesmaximal, characteristic, complemented
12C2×#1161—21no—
13C2×Q8×D7C2211yescharacteristic
14C2×D4×D7C2211yescharacteristic
15C7×C4⋊Q8C2211yescharacteristic, complemented
16C7×#124C2211yescharacteristic
17#6510—21no—
18#6510C2211yescharacteristic
19C2×#1143C2212yescomplemented
20#6499C2211yescharacteristic
21C2×#1141C2212yes—
22C2×D56C2212yes—
23C2×#1132C2212yescomplemented
24#6458—21no—
25#6458C2212yes—
26#6456—21no—
27#6456C2212yescomplemented
28#6448—21no—
29#6448—21no—
30D7×C22⋊C4—21no—
31D7×C22⋊C4—21no—
32#6438C2211yescharacteristic
33#6437C2211yescharacteristic, complemented
34C42×D7C2211yescharacteristic
35C7×#103C2212yescomplemented
36C7×#102C2212yes—
37C7×#97C2211yescharacteristic, complemented
38(C2×D7)⋊C8C2212yes—
39Dic7⋊C8C2212yescomplemented
40#6384C2212yescomplemented
41#6383C2212yes—
42#6377C2211yescharacteristic, complemented
43#3213—71nomaximal, Sylow
44C23×D7—21no—
45C2×C7×Q8—21no—
46C2×C7×Q8C2311yescharacteristic
47C2×C7×D4C2311yescharacteristic
48C2×C22⋊D7—21no—
49#1161—44no—
50Q8×D7—42no—
51D4×D7—42no—
52C2×D28—22no—
53C2×D28—21no—
54C2×D28—21no—
55C2×D28C2312yes—
56C2×D28C2311yescharacteristic
57C2×C4×D7—21no—
58C2×C4×D7—22no—
59C2×C4×D7D411yescharacteristic
60C2×C4×D7D411yescharacteristic
61C2×C4×D7C2311yescharacteristic
62C2×Dic14—21no—
63C2×Dic14C2311yescharacteristic
64C2×C7×C8C2312yes—
65C7×C4⋊C4—21no—
66C7×C4⋊C4C2312yes—
67C7×C22⋊C4—21no—
68C7×C22⋊C4—21no—
69C42×C7C2311yescharacteristic
70C22⋊Dic7—21no—
71C22⋊Dic7—21no—
72#1143—42no—
73#1141—42no—
74(C2×D7)⋊C4—21no—
75(C2×D7)⋊C4—21no—
76(C2×D7)⋊C4—21no—
77(C2×D7)⋊C4—21no—
78(C2×D7)⋊C4—22no—
79(C2×D7)⋊C4—22no—
80C4×Dic7D411yescharacteristic
81C4×Dic7D411yescharacteristic
82C4×Dic7C2311yescharacteristic
83C2×C7⋊C8C2312yes—
84D56—42no—
85#1132—42no—
86#532—71no—
87C2×#124—71no—
88#496—71no—
89#496—71no—
90#489—71no—
91#489D711yescharacteristic, complemented
92#463—72no—
93#463—72no—
94#448—72no—
95#447—72no—
96#432—71no—
97C23×C7—21no—
98C22×D7—21no—
99C22×D7—22no—
100C22×D7—42no—
101C22×D7—42no—
102C22×D7—42no—
103C22×D7C2×D411yescharacteristic
104C7×Q8—22no—
105C7×Q8—42no—
106C7×D4—22no—
107C2×C4×C7—21no—
108C2×C4×C7—22no—
109C2×C4×C7C2×D411yescharacteristic
110C2×C4×C7C2×D411yescharacteristic
111C2×C4×C7C2411yescharacteristic
112C22⋊D7—42no—
113C2×Dic7—21no—
114C2×Dic7C2×D411yescharacteristic
115C2×Dic7C2×D411yescharacteristic
116C2×Dic7C2×D411yescharacteristic
117D28—22no—
118D28—22no—
119D28—22no—
120D28—42no—
121D28—42no—
122D28—44no—
123C4×D7—22no—
124C4×D7—22no—
125C4×D7—41no—
126C4×D7—44no—
127C4×D7—42no—
128Dic14—22no—
129Dic14—42no—
130C7×C8—22no—
131C7⋊C8—22no—
132C2×#41—141no—
133C22×Q8—71no—
134C22×D4—71no—
135C2×SD16—72no—
136C2×SD16—72no—
137C2×D8—72no—
138C2×D8—72no—
139C4⋊Q8C2×D711yescharacteristic, complemented
140#127—71no—
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#103C2×D712yescomplemented
158#102—72no—
159#102C2×D712yes—
160C22⋊C8—72no—
161#97—71no—
162#97C2×D711yescharacteristic, complemented
163C22×C7—41no—
164C22×C7—41no—
165C22×C7—41no—
166C22×C7C22×D411yescharacteristic
167C2×D7—21no—
168C2×D7—21no—
169C2×D7—41no—
170C2×D7—42no—
171C2×D7—42no—
172C2×D7—41no—
173C2×D7—42no—
174C2×D7—41no—
175C2×D7—84no—
176C2×D7AGL(1,ℤ/8)12yes—
177C4×C7—21no—
178C4×C7—21no—
179C4×C7—41no—
180C4×C7—42no—
181C4×C7H(5,𝔽2)12yes—
182Dic7—21no—
183Dic7—21no—
184Dic7—21no—
185Dic7—41no—
186C24—141no—
187#41—284no—
188C2×Q8—21no—
189C2×Q8—71no—
190C2×Q8—141no—
191C2×Q8—282no—
192C2×Q8C22×D711yescharacteristic
193C2×D4—71no—
194C2×D4—72no—
195C2×D4—142no—
196C2×D4—141no—
197C2×D4—141no—
198C2×D4—141no—
199C2×D4—282no—
200C2×D4C22×D711yescharacteristic
201C22×C4—71no—
202C22×C4—71no—
203C22×C4—71no—
204C22×C4—142no—
205C22×C4—141no—
206SD16—282no—
207SD16—282no—
208D8—282no—
209D8—282no—
210C2×C8—72no—
211C2×C8C22×D712yes—
212C4⋊C4—21no—
213C4⋊C4C22×D712yes—
214C22⋊C4—21no—
215C22⋊C4—21no—
216C22⋊C4—141no—
217C22⋊C4—141no—
218C22⋊C4—142no—
219C22⋊C4—142no—
220C22⋊C4—141no—
221C22⋊C4—141no—
222C22⋊C4—141no—
223C22⋊C4—141no—
224C42—71no—
225C42—71no—
226C42—71no—
227C42C22×D711yescharacteristic
228C2×C7—81no—
229C2×C7#53511yescharacteristic
230C2×C7C2×AGL(1,ℤ/8)11yescharacteristic
231C2×C7#57511yescharacteristic
232D7—22no—
233D7—82no—
234D7—81no—
235C23—21no—
236C23—71no—
237C23—141no—
238C23—142no—
239C23—282no—
240C23—282no—
241C23—282no—
242Q8—22no—
243Q8—42no—
244Q8—142no—
245Q8—282no—
246D4—22no—
247D4—142no—
248D4—142no—
249D4—142no—
250D4—282no—
251D4—282no—
252D4—284no—
253D4—282no—
254C2×C4—21no—
255C2×C4—22no—
256C2×C4—71no—
257C2×C4—71no—
258C2×C4—71no—
259C2×C4—142no—
260C2×C4—142no—
261C2×C4—141no—
262C2×C4—281no—
263C2×C4—284no—
264C2×C4—282no—
265C2×C4D4×D711yescharacteristic
266C2×C4D4×D711yescharacteristic
267C2×C4C23×D711yescharacteristic
268C8—22no—
269C8—142no—
270C7#321311yescharacteristic, complemented, Sylow
271C22—41no—
272C22—41no—
273C22—41no—
274C22—72no—
275C22—141no—
276C22—141no—
277C22—282no—
278C22—281no—
279C22—282no—
280C22—281no—
281C22—281no—
282C22—282no—
283C22—564no—
284C22C2×D4×D711yescharacteristic
285C4—21no—
286C4—21no—
287C4—42no—
288C4—41no—
289C4—141no—
290C4—141no—
291C4—141no—
292C4—281no—
293C4#655312yes—
294C2—81no—
295C2—142no—
296C2—561no—
297C2—562no—
298C2#9036511yescharacteristic
299C2D7×AGL(1,ℤ/8)11yescharacteristic
300C2#9046811yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#489C2
socleC22×C7C22×D4
hypercenter#489D7
hypocenterC7#3213
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. #90416C2semidirect
2. D7×#124C2semidirect
3. C7×#489C2semidirect
4. #89861C2semidirect
5. #89830C2semidirect
6. #89591C2semidirect
7. #89578C2semidirect
8. #89559C2semidirect
9. #89545C2semidirect
10. #89484C2semidirect
11. #89477C2semidirect
15. C7×C4⋊Q8C22semidirect
19. C2×#1143C22semidirect
23. C2×#1132C22semidirect
27. #6456C22semidirect
33. #6437C22semidirect
35. C7×#103C22semidirect
37. C7×#97C22semidirect
39. Dic7⋊C8C22semidirect
40. #6384C22semidirect
42. #6377C22semidirect
91. #489D7semidirect
139. C4⋊Q8C2×D7semidirect
157. #103C2×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
#904162
D7×#1242
C7×#4892
#898612
#898302
#895912
#895782
#895592
#895452
#894842
#894772
#32137