Finite Group Databasebeta

← #11740589

Subgroups — #11740589

Order 896.

410 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 121 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
1C7×#529C211yesmaximal, characteristic, complemented
2C2×#6438C211yesmaximal, characteristic, complemented
3#89840C212yesmaximal, complemented
4#89831C212yesmaximal, complemented
5#89822C211yesmaximal, characteristic, complemented
6#89818C211yesmaximal, characteristic, complemented
7#89812C211yesmaximal, characteristic, complemented
8#89808C211yesmaximal, characteristic
9#89800C212yesmaximal, complemented
10#89788C212yesmaximal, complemented
11#89697C211yesmaximal, characteristic, complemented
12C22×D28C2211yescharacteristic
13C22×Dic14C2211yescharacteristic
14C7×#126—21no—
15C7×#125C2211yescharacteristic, complemented
16C7×#124C2211yescharacteristic, complemented
17C7×#123—21no—
18C7×C22⋊Q8C2211yescharacteristic
19C7×#121C2211yescharacteristic
20C4×C7×Q8C2211yescharacteristic, complemented
21C4×C7×D4—21no—
22C4×C7×D4C2211yescharacteristic, complemented
23C7×#117—21no—
24C2×C42×C7C2211yescharacteristic
25C2×#1144C2212yescomplemented
26C2×#1143C2212yes—
27C2×#1142C2212yescomplemented
28C2×#1141C2212yes—
29C2×(C2×D7)⋊C4—21no—
30C2×(C2×D7)⋊C4—21no—
31#6438—21no—
32#6438—21no—
33#6438C2212yes—
34#6438C2212yes—
35C22⋊(C7⋊C8)C2211yescharacteristic
36C22⋊(C7⋊C8)C2211yescharacteristic
37#6384C2212yescomplemented
38#6384C2212yescomplemented
39#6383C2212yes—
40#6383C2212yes—
41C4⋊(C7⋊C8)C2212yescomplemented
42C4×C7⋊C8C2212yescomplemented
43#3211—71nomaximal, Sylow
44C23×D7—21no—
45C2×C7×Q8C2311yescharacteristic
46C2×C7×D4—21no—
47C2×C7×D4C2311yescharacteristic
48C22×C4×C7—21no—
49C22×C4×C7D411yescharacteristic
50C22×C4×C7D411yescharacteristic
51C22×C4×C7C2311yescharacteristic
52C22×Dic7—21no—
53C2×D28—21no—
54C2×D28—42no—
55C2×D28C2312yes—
56C2×Dic14—21no—
57C2×Dic14—42no—
58C2×Dic14C2312yes—
59C7×C4⋊C4—21no—
60C7×C4⋊C4—21no—
61C7×C4⋊C4—21no—
62C7×C4⋊C4—21no—
63C7×C4⋊C4C2311yescharacteristic
64C7×C4⋊C4C2311yescharacteristic
65C7×C22⋊C4—21no—
66C7×C22⋊C4—21no—
67C7×C22⋊C4—21no—
68C7×C22⋊C4—21no—
69C7×C22⋊C4—21no—
70C42×C7—21no—
71C42×C7D411yescharacteristic
72C42×C7D411yescharacteristic
73C42×C7C2311yescharacteristic
74C42×C7C2311yescharacteristic
75#1144—42no—
76#1143—42no—
77#1142—42no—
78#1141—42no—
79(C2×D7)⋊C4—22no—
80(C2×D7)⋊C4—22no—
81(C2×D7)⋊C4—22no—
82(C2×D7)⋊C4—22no—
83C2×C7⋊C8C2312yes—
84C2×C7⋊C8C2312yes—
85#529D711yescharacteristic, complemented
86C2×#124—71no—
87#495—72no—
88#488—72no—
89#464—72no—
90#463—72no—
91#451—71no—
92#450—71no—
93#448—71no—
94#447—71no—
95#431—71no—
96C23×C7—21no—
97C23×C7C2×D411yescharacteristic
98C22×D7—22no—
99C22×D7—42no—
100C22×D7—42no—
101C22×D7—42no—
102C7×Q8—22no—
103C7×D4—22no—
104C7×D4—42no—
105C2×C4×C7—22no—
106C2×C4×C7—22no—
107C2×C4×C7—21no—
108C2×C4×C7—21no—
109C2×C4×C7—21no—
110C2×C4×C7—41no—
111C2×C4×C7—42no—
112C2×C4×C7C2×D411yescharacteristic
113C2×C4×C7C2×D411yescharacteristic
114C2×C4×C7C2×D411yescharacteristic
115C2×C4×C7C2×D411yescharacteristic
116C2×C4×C7C2×D411yescharacteristic
117C2×C4×C7C2411yescharacteristic
118C2×Dic7—22no—
119C2×Dic7—42no—
120D28—22no—
121D28—22no—
122D28—42no—
123Dic14—22no—
124Dic14—22no—
125Dic14—42no—
126C7⋊C8—22no—
127C7⋊C8—22no—
128C22×Q8—71no—
129C22×D4—71no—
130C2×Q16—72no—
131C2×SD16—72no—
132C2×SD16—72no—
133C2×D8—72no—
134#126—21no—
135#125C2×D711yescharacteristic, complemented
136#124—72no—
137#124—72no—
138#124—141no—
139#124—141no—
140#124C2×D711yescharacteristic, complemented
141#123—21no—
142C22⋊Q8C2×D711yescharacteristic
143#121C2×D711yescharacteristic
144C4×Q8C2×D711yescharacteristic, complemented
145C4×D4—21no—
146C4×D4C2×D711yescharacteristic, complemented
147#117—21no—
148C2×C22⋊C4—141no—
149C2×C22⋊C4—141no—
150C2×C42C2×D711yescharacteristic
151C4⋊C8—72no—
152#103—72no—
153#103—72no—
154#102—72no—
155#102—72no—
156C22⋊C8—71no—
157C22⋊C8—71no—
158C4×C8—72no—
159C22×C7—21no—
160C22×C7—21no—
161C22×C7—41no—
162C22×C7—41no—
163C22×C7—41no—
164C22×C7#13512yes—
165C22×C7C22×D411yescharacteristic
166C2×D7—42no—
167C2×D7—42no—
168C2×D7—42no—
169C2×D7—84no—
170C4×C7—21no—
171C4×C7—21no—
172C4×C7—21no—
173C4×C7—21no—
174C4×C7—21no—
175C4×C7—41no—
176C4×C7—41no—
177C4×C7—41no—
178C4×C7H(5,𝔽2)12yes—
179Dic7—42no—
180C24—141no—
181C2×Q8—72no—
182C2×Q8—141no—
183C2×Q8—282no—
184C2×Q8C22×D711yescharacteristic
185C2×D4—21no—
186C2×D4—72no—
187C2×D4—141no—
188C2×D4—282no—
189C2×D4C22×D711yescharacteristic
190C22×C4—21no—
191C22×C4—141no—
192C22×C4C22⋊D711yescharacteristic
193C22×C4C22⋊D711yescharacteristic
194C22×C4C22×D711yescharacteristic
195Q16—282no—
196SD16—282no—
197SD16—282no—
198D8—282no—
199C2×C8—72no—
200C2×C8—72no—
201C4⋊C4—21no—
202C4⋊C4—21no—
203C4⋊C4—21no—
204C4⋊C4—21no—
205C4⋊C4C22×D711yescharacteristic
206C4⋊C4C22×D711yescharacteristic
207C22⋊C4—21no—
208C22⋊C4—21no—
209C22⋊C4—21no—
210C22⋊C4—21no—
211C22⋊C4—21no—
212C22⋊C4—142no—
213C22⋊C4—142no—
214C22⋊C4—142no—
215C22⋊C4—142no—
216C42—21no—
217C42C22⋊D711yescharacteristic
218C42C22⋊D711yescharacteristic
219C42C22×D711yescharacteristic
220C42C22×D711yescharacteristic
221C2×C7—22no—
222C2×C7—81no—
223C2×C7#53511yescharacteristic
224C2×C7C2×#13511yescharacteristic
225C2×C7#57511yescharacteristic
226D7—82no—
227C23—21no—
228C23—142no—
229C23—282no—
230C23—282no—
231C23—282no—
232C23C2×C22⋊D711yescharacteristic
233Q8—22no—
234Q8—142no—
235Q8—142no—
236Q8—282no—
237D4—22no—
238D4—42no—
239D4—142no—
240D4—142no—
241D4—282no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—41no—
248C2×C4—42no—
249C2×C4—142no—
250C2×C4—282no—
251C2×C4C2×C22⋊D711yescharacteristic
252C2×C4C2×C22⋊D711yescharacteristic
253C2×C4C2×C22⋊D711yescharacteristic
254C2×C4C2×C22⋊D711yescharacteristic
255C2×C4C2×C22⋊D711yescharacteristic
256C2×C4C23×D711yescharacteristic
257C8—142no—
258C8—142no—
259C7#321111yescharacteristic, complemented, Sylow
260C22—21no—
261C22—21no—
262C22—41no—
263C22—41no—
264C22—41no—
265C22—282no—
266C22—282no—
267C22—282no—
268C22—564no—
269C22#651312yes—
270C22C22×C22⋊D711yescharacteristic
271C4—21no—
272C4—21no—
273C4—21no—
274C4—21no—
275C4—21no—
276C4—41no—
277C4—41no—
278C4—41no—
279C4—282no—
280C4#655312yes—
281C2—22no—
282C2—81no—
283C2—562no—
284C2C2×#651311yescharacteristic
285C2#9051311yescharacteristic
286C2#9052111yescharacteristic
subgroupG/H
centerC22C22×C22⋊D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#529C2
socleC22×C7C22×D4
prime radical#89808C2
hypercenter#529D7
hypocenterC7#3211
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #529

Complements

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

NH ≅ G/Ntype
1. C7×#529C2semidirect
2. C2×#6438C2semidirect
3. #89840C2semidirect
4. #89831C2semidirect
5. #89822C2semidirect
6. #89818C2semidirect
7. #89812C2semidirect
9. #89800C2semidirect
10. #89788C2semidirect
11. #89697C2semidirect
15. C7×#125C22semidirect
16. C7×#124C22semidirect
20. C4×C7×Q8C22semidirect
22. C4×C7×D4C22semidirect
25. C2×#1144C22semidirect
27. C2×#1142C22semidirect
37. #6384C22semidirect
38. #6384C22semidirect
41. C4⋊(C7⋊C8)C22semidirect
42. C4×C7⋊C8C22semidirect
85. #529D7semidirect
135. #125C2×D7semidirect
140. #124C2×D7semidirect
144. C4×Q8C2×D7semidirect
146. C4×D4C2×D7semidirect
259. C7#3211semidirect

Maximal subgroups

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

subgroupindex
C7×#5292
C2×#64382
#898402
#898312
#898222
#898182
#898122
#898082
#898002
#897882
#896972
#32117