Finite Group Databasebeta

← #11739127

Subgroups — #11739127

Order 896.

400 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 117 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#90419C211yesmaximal, characteristic, complemented
2D7×#124C211yesmaximal, characteristic
3C7×#489C211yesmaximal, characteristic, complemented
4#89861C211yesmaximal, characteristic, complemented
5#89827C211yesmaximal, characteristic, complemented
6#89574C212yesmaximal, complemented
7#89572C212yesmaximal, complemented
8#89542C212yesmaximal
9#89540C212yesmaximal, complemented
10#89486C211yesmaximal, characteristic, complemented
11#89479C211yesmaximal, characteristic, complemented
12C2×Q8×D7C2211yescharacteristic
13C2×D4×D7C2211yescharacteristic
14C7×C4⋊Q8C2211yescharacteristic, complemented
15C7×#124C2211yescharacteristic
16#6510C2211yescharacteristic
17#6509—21no—
18Q8×Dic7—21no—
19#6499C2211yescharacteristic
20C2×#1131C2212yes—
21#6460—21no—
22#6455C2212yes—
23#6453C2212yescomplemented
24#6450—21no—
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
39C8×Dic7C2212yescomplemented
40#6384C2212yescomplemented
41#6383C2212yes—
42#6377C2211yescharacteristic, complemented
43#3140—71nomaximal, Sylow
44C23×D7—21no—
45C2×C7×Q8—21no—
46C2×C7×Q8C2311yescharacteristic
47C2×C7×D4C2311yescharacteristic
48C2×C22⋊D7—21no—
49Q8×D7—24no—
50D4×D7—24no—
51C2×D28C2311yescharacteristic
52C2×C4×D7—21no—
53C2×C4×D7D411yescharacteristic
54C2×C4×D7D411yescharacteristic
55C2×C4×D7C2311yescharacteristic
56C2×Dic14—21no—
57C2×Dic14—21no—
58C2×Dic14C2311yescharacteristic
59C2×C7×C8C2312yes—
60C7×C4⋊C4—21no—
61C7×C4⋊C4C2312yes—
62C7×C22⋊C4—21no—
63C7×C22⋊C4—21no—
64C42×C7C2311yescharacteristic
65C22⋊Dic7—21no—
66C22⋊Dic7—21no—
67(C2×D7)⋊C4—22no—
68(C2×D7)⋊C4—21no—
69(C2×D7)⋊C4—21no—
70(C2×D7)⋊C4—21no—
71(C2×D7)⋊C4—21no—
72C4⋊Dic7—21no—
73C4⋊Dic7—22no—
74C4⋊Dic7C2312yes—
75Dic7⋊C4—22no—
76C4×Dic7—22no—
77C4×Dic7D411yescharacteristic
78C4×Dic7D411yescharacteristic
79C4×Dic7C2311yescharacteristic
80C2×C7⋊C8C2312yes—
81#1131—24no—
82#533—71no—
83C2×#124—71no—
84#489—71no—
85#489—71no—
86#489—71no—
87#489D711yescharacteristic, complemented
88#488—72no—
89#488—72no—
90#419—72no—
91#418—72no—
92C4×M16—71no—
93C23×C7—21no—
94C22×D7—21no—
95C22×D7—42no—
96C22×D7—42no—
97C22×D7—42no—
98C22×D7C2×D411yescharacteristic
99C7×Q8—22no—
100C7×Q8—42no—
101C7×D4—22no—
102C2×C4×C7—22no—
103C2×C4×C7—21no—
104C2×C4×C7C2×D411yescharacteristic
105C2×C4×C7C2×D411yescharacteristic
106C2×C4×C7C2411yescharacteristic
107C22⋊D7—42no—
108C2×Dic7—22no—
109C2×Dic7—21no—
110C2×Dic7C2×D411yescharacteristic
111C2×Dic7C2×D411yescharacteristic
112C2×Dic7C2×D411yescharacteristic
113D28—22no—
114C4×D7—41no—
115C4×D7—41no—
116C4×D7—42no—
117C4×D7#4114yes—
118Dic14—22no—
119Dic14—42no—
120Dic14—42no—
121C7×C8—22no—
122C7⋊C8—22no—
123C22×Q8—71no—
124C22×D4—71no—
125C2×M16—72no—
126C4⋊Q8—71no—
127C4⋊Q8C2×D711yescharacteristic, complemented
128#125—72no—
129#124—71no—
130#124—71no—
131#124—71no—
132#124—141no—
133#124—141no—
134#124C2×D711yescharacteristic
135C22⋊Q8—141no—
136C22⋊Q8—141no—
137C4×Q8—141no—
138C4×Q8—141no—
139#117—72no—
140C2×C22⋊C4—141no—
141C2×C22⋊C4—141no—
142C2×C42—71no—
143#103—72no—
144#103—72no—
145#103—72no—
146#103C2×D712yescomplemented
147#102—72no—
148#102—72no—
149#102—72no—
150#102C2×D712yes—
151#97—71no—
152#97C2×D711yescharacteristic, complemented
153C4×C8—72no—
154C22×C7—41no—
155C22×C7—41no—
156C22×C7—41no—
157C22×C7C22×D411yescharacteristic
158C2×D7—21no—
159C2×D7—21no—
160C2×D7—41no—
161C2×D7—41no—
162C2×D7—41no—
163C2×D7—84no—
164C2×D7#12412yes—
165C4×C7—21no—
166C4×C7—21no—
167C4×C7—41no—
168C4×C7—42no—
169C4×C7C2×#4112yes—
170Dic7—21no—
171Dic7—21no—
172Dic7—41no—
173Dic7—42no—
174Dic7#12412yes—
175C24—141no—
176C2×Q8—21no—
177C2×Q8—71no—
178C2×Q8—141no—
179C2×Q8—144no—
180C2×Q8—141no—
181C2×Q8C22×D711yescharacteristic
182C2×D4—71no—
183C2×D4—144no—
184C2×D4—141no—
185C2×D4C22×D711yescharacteristic
186C22×C4—71no—
187C22×C4—71no—
188C22×C4—71no—
189C22×C4—141no—
190M16—144no—
191C2×C8—72no—
192C2×C8C22×D712yes—
193C4⋊C4—21no—
194C4⋊C4—72no—
195C4⋊C4—142no—
196C4⋊C4—142no—
197C4⋊C4—141no—
198C4⋊C4C22×D712yes—
199C22⋊C4—21no—
200C22⋊C4—21no—
201C22⋊C4—141no—
202C22⋊C4—141no—
203C22⋊C4—142no—
204C22⋊C4—141no—
205C22⋊C4—141no—
206C22⋊C4—141no—
207C22⋊C4—141no—
208C42—71no—
209C42—71no—
210C42—71no—
211C42—142no—
212C42C22×D711yescharacteristic
213C2×C7—81no—
214C2×C7C2×#12411yescharacteristic
215C2×C7#57511yescharacteristic
216C2×C7#57511yescharacteristic
217D7—22no—
218D7—81no—
219C23—21no—
220C23—71no—
221C23—141no—
222C23—282no—
223C23—282no—
224C23—282no—
225Q8—22no—
226Q8—42no—
227Q8—142no—
228Q8—282no—
229Q8—282no—
230D4—22no—
231D4—142no—
232D4—282no—
233C2×C4—22no—
234C2×C4—21no—
235C2×C4—71no—
236C2×C4—74no—
237C2×C4—71no—
238C2×C4—71no—
239C2×C4—142no—
240C2×C4—141no—
241C2×C4—281no—
242C2×C4—281no—
243C2×C4—282no—
244C2×C4D4×D711yescharacteristic
245C2×C4D4×D711yescharacteristic
246C2×C4C23×D711yescharacteristic
247C8—22no—
248C8—142no—
249C7#314011yescharacteristic, complemented, Sylow
250C22—41no—
251C22—41no—
252C22—41no—
253C22—72no—
254C22—141no—
255C22—141no—
256C22—281no—
257C22—281no—
258C22—281no—
259C22—564no—
260C22C2×D4×D711yescharacteristic
261C4—21no—
262C4—21no—
263C4—42no—
264C4—41no—
265C4—72no—
266C4—141no—
267C4—141no—
268C4—282no—
269C4—281no—
270C4D7×#4112yes—
271C2—81no—
272C2—142no—
273C2—561no—
274C2D7×#12411yescharacteristic
275C2#9046411yescharacteristic
276C2#9046811yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#489C2
socleC22×C7C22×D4
prime radicalC2×D4×D7C22
hypercenter#489D7
hypocenterC7#3140
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. #90419C2semidirect
3. C7×#489C2semidirect
4. #89861C2semidirect
5. #89827C2semidirect
6. #89574C2semidirect
7. #89572C2semidirect
9. #89540C2semidirect
10. #89486C2semidirect
11. #89479C2semidirect
14. C7×C4⋊Q8C22semidirect
23. #6453C22semidirect
31. #6432C22semidirect
32. C7×#103C22semidirect
34. C7×#97C22semidirect
35. #6409C22semidirect
38. #6390C22semidirect
39. C8×Dic7C22semidirect
40. #6384C22semidirect
42. #6377C22semidirect
87. #489D7semidirect
127. C4⋊Q8C2×D7semidirect
146. #103C2×D7semidirect
152. #97C2×D7semidirect
249. C7#3140semidirect

Maximal subgroups

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

subgroupindex
#904192
D7×#1242
C7×#4892
#898612
#898272
#895742
#895722
#895422
#895402
#894862
#894792
#31407