Finite Group Databasebeta

← #11739132

Subgroups — #11739132

Order 896.

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

normal subgroups only

This group has 390 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#90419C211yesmaximal, characteristic, complemented
2#90359C211yesmaximal, characteristic
3C7×#489C211yesmaximal, characteristic, complemented
4#89861C211yesmaximal, characteristic, complemented
5#89832C211yesmaximal, characteristic, complemented
6#89582C212yesmaximal, complemented
7#89565C212yesmaximal, complemented
8#89555C212yesmaximal, complemented
9#89534C212yesmaximal, complemented
10#89487C211yesmaximal, characteristic, complemented
11#89477C211yesmaximal, characteristic, complemented
12C2×#1161C2211yescharacteristic
13C2×#1159C2211yescharacteristic
14C7×C4⋊Q8C2211yescharacteristic
15C7×#124C2211yescharacteristic
16#6509—21no—
17Q8×Dic7—21no—
18#6507C2211yescharacteristic
19C2×#1144C2212yes—
20#6503C2211yescharacteristic
21C2×#1142C2212yes—
22C2×Dic28C2212yes—
23C2×#1132C2212yes—
24#6460C2212yes—
25#6455—21no—
26#6453—21no—
27#6450C2212yes—
28#6447—21no—
29#6447—21no—
30(C22⋊D7)⋊C4—21no—
31(C22⋊D7)⋊C4—21no—
32#6438C2211yescharacteristic
33C42×D7C2211yescharacteristic
34#6432C2211yescharacteristic
35C7×#103C2212yes—
36C7×#102C2212yes—
37C7×#97C2211yescharacteristic
38(C2×D7)⋊C8C2212yes—
39Dic7⋊C8C2212yes—
40#6384C2212yes—
41#6383C2212yes—
42#6377C2211yescharacteristic
43#3216—71nomaximal, Sylow
44C2×C7×Q8—21no—
45C2×C7×Q8C2311yescharacteristic
46C2×C7×D4C2311yescharacteristic
47C2×C22⋊D7—21no—
48C2×C22⋊D7—21no—
49C2×C22⋊D7—21no—
50C22×Dic7—21no—
51#1161—42no—
52#1159—42no—
53C2×D28—21no—
54C2×D28C2311yescharacteristic
55C2×C4×D7—21no—
56C2×C4×D7D411yescharacteristic
57C2×C4×D7D411yescharacteristic
58C2×C4×D7C2311yescharacteristic
59C2×Dic14C2312yes—
60C2×Dic14C2311yescharacteristic
61C2×C7×C8C2312yes—
62C7×C4⋊C4—21no—
63C7×C4⋊C4C2312yes—
64C7×C22⋊C4—21no—
65C7×C22⋊C4—21no—
66C42×C7C2311yescharacteristic
67#1144—42no—
68#1142—42no—
69(C2×D7)⋊C4—22no—
70(C2×D7)⋊C4—21no—
71(C2×D7)⋊C4—21no—
72C4⋊Dic7—21no—
73C4⋊Dic7—21no—
74C4⋊Dic7—22no—
75Dic7⋊C4—21no—
76Dic7⋊C4—21no—
77Dic7⋊C4—22no—
78C4×Dic7—22no—
79C4×Dic7D411yescharacteristic
80C4×Dic7D411yescharacteristic
81C4×Dic7C2311yescharacteristic
82C2×C7⋊C8C2312yes—
83Dic28—42no—
84#1132—42no—
85#533—71no—
86#532—71no—
87#497—71no—
88#497—71no—
89#489—71no—
90#489D711yescharacteristic, complemented
91#462—72no—
92#460—72no—
93#452—72no—
94#452—72no—
95#432—71no—
96C23×C7—21no—
97C22×D7—21no—
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—
108C22⋊D7—42no—
109C22⋊D7—42no—
110C2×Dic7—22no—
111C2×Dic7—21no—
112C2×Dic7—42no—
113C2×Dic7C2×D411yescharacteristic
114C2×Dic7C2×D411yescharacteristic
115C2×Dic7C2×D411yescharacteristic
116D28—22no—
117D28—42no—
118C4×D7—22no—
119C4×D7—22no—
120C4×D7—41no—
121C4×D7—42no—
122Dic14—22no—
123Dic14—22no—
124Dic14—22no—
125C7×C8—22no—
126C7⋊C8—22no—
127C2×#41—71no—
128C2×#41—71no—
129C2×Q16—72no—
130C2×Q16—72no—
131C2×SD16—72no—
132C2×SD16—72no—
133C4⋊Q8—71no—
134C4⋊Q8—71no—
135C4⋊Q8C2×D711yescharacteristic
136#127—71no—
137#125—141no—
138#124—71no—
139#124C2×D711yescharacteristic
140C22⋊Q8—72no—
141C22⋊Q8—141no—
142#121—141no—
143#121—141no—
144C4×Q8—72no—
145C4×Q8—141no—
146C4×D4—141no—
147C4×D4—141no—
148#117—141no—
149C2×C42—71no—
150C4⋊C8—72no—
151#103—72no—
152#103C2×D712yes—
153#102—72no—
154#102C2×D712yes—
155C22⋊C8—72no—
156#97—71no—
157#97C2×D711yescharacteristic
158C22×C7—41no—
159C22×C7—41no—
160C22×C7—41no—
161C22×C7C22×D411yescharacteristic
162C2×D7—21no—
163C2×D7—21no—
164C2×D7—21no—
165C2×D7—41no—
166C2×D7—41no—
167C2×D7—41no—
168C4×C7—21no—
169C4×C7—21no—
170C4×C7—41no—
171C4×C7—42no—
172C4×C7H(5,𝔽2)12yes—
173Dic7—21no—
174Dic7—21no—
175Dic7—42no—
176Dic7—41no—
177Dic7#13712yes—
178#41—282no—
179#41—282no—
180C2×Q8—21no—
181C2×Q8—71no—
182C2×Q8—72no—
183C2×Q8C22×D711yescharacteristic
184C2×D4—71no—
185C2×D4—141no—
186C2×D4—141no—
187C2×D4—141no—
188C2×D4—141no—
189C2×D4C22×D711yescharacteristic
190C22×C4—71no—
191C22×C4—71no—
192C22×C4—71no—
193C22×C4—141no—
194C22×C4—141no—
195Q16—282no—
196Q16—282no—
197SD16—282no—
198SD16—282no—
199C2×C8—72no—
200C2×C8C22×D712yes—
201C4⋊C4—21no—
202C4⋊C4—141no—
203C4⋊C4—141no—
204C4⋊C4—142no—
205C4⋊C4—142no—
206C4⋊C4—141no—
207C4⋊C4—141no—
208C4⋊C4C22×D712yes—
209C22⋊C4—21no—
210C22⋊C4—21no—
211C22⋊C4—142no—
212C22⋊C4—141no—
213C22⋊C4—141no—
214C42—71no—
215C42—71no—
216C42—71no—
217C42—142no—
218C42C22×D711yescharacteristic
219C2×C7—81no—
220C2×C7#53511yescharacteristic
221C2×C7C2×#13711yescharacteristic
222C2×C7#57511yescharacteristic
223D7—41no—
224D7—81no—
225C23—21no—
226C23—71no—
227C23—141no—
228Q8—22no—
229Q8—42no—
230Q8—142no—
231Q8—142no—
232Q8—142no—
233D4—22no—
234D4—142no—
235D4—282no—
236D4—282no—
237D4—282no—
238D4—282no—
239C2×C4—22no—
240C2×C4—21no—
241C2×C4—71no—
242C2×C4—71no—
243C2×C4—71no—
244C2×C4—142no—
245C2×C4—142no—
246C2×C4—142no—
247C2×C4—141no—
248C2×C4—281no—
249C2×C4—282no—
250C2×C4—282no—
251C2×C4D4×D711yescharacteristic
252C2×C4D4×D711yescharacteristic
253C2×C4C23×D711yescharacteristic
254C8—22no—
255C8—142no—
256C7#321611yescharacteristic, complemented, Sylow
257C22—41no—
258C22—41no—
259C22—41no—
260C22—141no—
261C22—141no—
262C22—141no—
263C22—281no—
264C22—281no—
265C22—281no—
266C22C2×D4×D711yescharacteristic
267C4—21no—
268C4—21no—
269C4—42no—
270C4—41no—
271C4—72no—
272C4—141no—
273C4—141no—
274C4—281no—
275C4—282no—
276C4#655312yes—
277C2—81no—
278C2—281no—
279C2—561no—
280C2#9036511yescharacteristic
281C2#9046411yescharacteristic
282C2D7×#13711yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#489C2
socleC22×C7C22×D4
prime radical#90359C2
hypercenter#489D7
hypocenterC7#3216
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. #89832C2semidirect
6. #89582C2semidirect
7. #89565C2semidirect
8. #89555C2semidirect
9. #89534C2semidirect
10. #89487C2semidirect
11. #89477C2semidirect
90. #489D7semidirect
256. C7#3216semidirect

Maximal subgroups

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

subgroupindex
#904192
#903592
C7×#4892
#898612
#898322
#895822
#895652
#895552
#895342
#894872
#894772
#32167