Finite Group Databasebeta

← #11739187

Subgroups — #11739187

Order 896.

396 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 396 conjugacy classes of subgroups — too many to draw. Only its 109 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
2#90382C211yesmaximal, characteristic, complemented
3C7×#491C211yesmaximal, characteristic, complemented
4#89858C211yesmaximal, characteristic, complemented
5#89839C211yesmaximal, characteristic, complemented
6#89591C212yesmaximal, complemented
7#89583C212yesmaximal, complemented
8#89582C212yesmaximal, complemented
9#89570C212yesmaximal, complemented
10#89484C211yesmaximal, characteristic, complemented
11#89477C211yesmaximal, characteristic, complemented
12C2×#1161C2212yes—
13C7×C4⋊Q8C2211yescharacteristic
14C7×#125C2211yescharacteristic
15#6510C2212yes—
16C2×#1144C2212yes—
17C2×#1143C2212yes—
18C2×D56C2212yes—
19C2×#1132C2212yes—
20#6461—21no—
21#6460C2211yescharacteristic
22#6458—21no—
23#6458—21no—
24#6458C2211yescharacteristic
25#6457—21no—
26#6456—21no—
27#6456—21no—
28#6456—21no—
29#6456C2211yescharacteristic
30#6455—21no—
31#6450C2211yescharacteristic
32#6438C2211yescharacteristic
33#6437C2211yescharacteristic
34C42×D7C2211yescharacteristic
35C7×#103C2212yes—
36C7×#103C2212yes—
37C7×#97C2211yescharacteristic
38(C2×D7)⋊C8C2212yes—
39Dic7⋊C8C2212yes—
40#6383C2212yes—
41#6383C2212yes—
42#6377C2211yescharacteristic
43#3219—71nomaximal, Sylow
44C2×C7×Q8C2312yes—
45#1161—42no—
46#1161—42no—
47C2×D28—21no—
48C2×D28—22no—
49C2×D28—21no—
50C2×D28—21no—
51C2×D28C2312yes—
52C2×D28C2311yescharacteristic
53C2×C4×D7—21no—
54C2×C4×D7—22no—
55C2×C4×D7D411yescharacteristic
56C2×C4×D7D411yescharacteristic
57C2×C4×D7C2311yescharacteristic
58C2×Dic14C2311yescharacteristic
59C2×C7×C8C2312yes—
60C7×C4⋊C4—21no—
61C7×C4⋊C4—21no—
62C7×C4⋊C4—21no—
63C7×C4⋊C4—21no—
64C7×C4⋊C4C2311yescharacteristic
65C7×C4⋊C4C2311yescharacteristic
66C42×C7C2311yescharacteristic
67#1144—42no—
68#1143—42no—
69(C2×D7)⋊C4—21no—
70(C2×D7)⋊C4—21no—
71(C2×D7)⋊C4—21no—
72(C2×D7)⋊C4—21no—
73(C2×D7)⋊C4—22no—
74(C2×D7)⋊C4—22no—
75(C2×D7)⋊C4—21no—
76C4⋊Dic7—21no—
77Dic7⋊C4—21no—
78C4×Dic7—21no—
79C4×Dic7D411yescharacteristic
80C4×Dic7D411yescharacteristic
81C4×Dic7C2311yescharacteristic
82C2×C7⋊C8C2312yes—
83D56—42no—
84#1132—42no—
85#532—71no—
86#529—71no—
87#497—71no—
88#496—71no—
89#491D711yescharacteristic, complemented
90#490—71no—
91#464—72no—
92#463—72no—
93#452—72no—
94#449—72no—
95#432—71no—
96C22×D7—21no—
97C22×D7—22no—
98C22×D7C2×D411yescharacteristic
99C7×Q8—22no—
100C7×Q8—22no—
101C2×C4×C7—21no—
102C2×C4×C7—21no—
103C2×C4×C7—22no—
104C2×C4×C7C2×D411yescharacteristic
105C2×C4×C7C2×D411yescharacteristic
106C2×C4×C7C2411yescharacteristic
107C2×Dic7—21no—
108C2×Dic7C2×D411yescharacteristic
109C2×Dic7C2×D411yescharacteristic
110C2×Dic7C2×D411yescharacteristic
111D28—22no—
112D28—22no—
113D28—22no—
114D28—42no—
115D28—42no—
116D28—42no—
117D28—44no—
118C4×D7—22no—
119C4×D7—41no—
120C4×D7—41no—
121C4×D7—42no—
122C4×D7—44no—
123Dic14—22no—
124C7×C8—22no—
125C7⋊C8—22no—
126C2×#41—72no—
127C2×Q16—72no—
128C2×SD16—72no—
129C2×SD16—72no—
130C2×D8—72no—
131C4⋊Q8C2×D711yescharacteristic
132#127—71no—
133#126—141no—
134#125C2×D711yescharacteristic
135#124—72no—
136#124—71no—
137#123—141no—
138C22⋊Q8—71no—
139#121—71no—
140#121—141no—
141#121—141no—
142C4×Q8—71no—
143C4×D4—71no—
144C4×D4—141no—
145C4×D4—141no—
146C4×D4—141no—
147#117—141no—
148C2×C42—71no—
149C4⋊C8—72no—
150#103C2×D712yes—
151#103C2×D712yes—
152#102—72no—
153#102—72no—
154C22⋊C8—72no—
155#97—71no—
156#97C2×D711yescharacteristic
157C22×C7C22×D411yescharacteristic
158C2×D7—21no—
159C2×D7—21no—
160C2×D7—21no—
161C2×D7—42no—
162C2×D7—41no—
163C2×D7—41no—
164C2×D7—42no—
165C2×D7—42no—
166C2×D7—41no—
167C4×C7—21no—
168C4×C7—21no—
169C4×C7—42no—
170C4×C7—41no—
171C4×C7—41no—
172C4×C7H(5,𝔽2)12yes—
173Dic7—21no—
174Dic7—21no—
175Dic7—21no—
176Dic7—41no—
177#41—282no—
178#41—282no—
179C2×Q8—71no—
180C2×Q8C22×D712yes—
181C2×D4—72no—
182C2×D4—71no—
183C2×D4—141no—
184C2×D4—142no—
185C2×D4—141no—
186C2×D4—141no—
187C22×C4—71no—
188C22×C4—71no—
189C22×C4—71no—
190C22×C4—141no—
191C22×C4—142no—
192Q16—282no—
193SD16—282no—
194SD16—282no—
195D8—282no—
196C2×C8—72no—
197C2×C8C22×D712yes—
198C4⋊C4—21no—
199C4⋊C4—21no—
200C4⋊C4—21no—
201C4⋊C4—21no—
202C4⋊C4—141no—
203C4⋊C4—141no—
204C4⋊C4C22×D711yescharacteristic
205C4⋊C4C22×D711yescharacteristic
206C22⋊C4—141no—
207C22⋊C4—141no—
208C22⋊C4—142no—
209C22⋊C4—142no—
210C22⋊C4—141no—
211C22⋊C4—141no—
212C22⋊C4—141no—
213C42—71no—
214C42—71no—
215C42—71no—
216C42—141no—
217C42C22×D711yescharacteristic
218C2×C7#53511yescharacteristic
219C2×C7#57511yescharacteristic
220C2×C7#57511yescharacteristic
221D7—41no—
222D7—82no—
223D7—81no—
224C23—71no—
225C23—141no—
226C23—142no—
227Q8—22no—
228Q8—22no—
229Q8—142no—
230D4—142no—
231D4—142no—
232D4—142no—
233D4—282no—
234D4—282no—
235D4—282no—
236D4—284no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—22no—
240C2×C4—71no—
241C2×C4—71no—
242C2×C4—71no—
243C2×C4—142no—
244C2×C4—141no—
245C2×C4—281no—
246C2×C4—281no—
247C2×C4—282no—
248C2×C4—284no—
249C2×C4D4×D711yescharacteristic
250C2×C4D4×D711yescharacteristic
251C2×C4C23×D711yescharacteristic
252C8—22no—
253C8—142no—
254C7#321911yescharacteristic, complemented, Sylow
255C22—141no—
256C22—141no—
257C22—141no—
258C22—281no—
259C22—282no—
260C22—282no—
261C22—281no—
262C22—282no—
263C22—281no—
264C22C2×D4×D711yescharacteristic
265C4—21no—
266C4—21no—
267C4—41no—
268C4—41no—
269C4—42no—
270C4—141no—
271C4—141no—
272C4—141no—
273C4—281no—
274C4#655312yes—
275C2—281no—
276C2—561no—
277C2—562no—
278C2#9036511yescharacteristic
279C2#9046811yescharacteristic
280C2#9046811yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#491C2
socleC22×C7C22×D4
hypercenter#491D7
hypocenterC7#3219
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ #491

Complements

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

NH ≅ G/Ntype
1. #90416C2semidirect
2. #90382C2semidirect
3. C7×#491C2semidirect
4. #89858C2semidirect
5. #89839C2semidirect
6. #89591C2semidirect
7. #89583C2semidirect
8. #89582C2semidirect
9. #89570C2semidirect
10. #89484C2semidirect
11. #89477C2semidirect
89. #491D7semidirect
254. C7#3219semidirect

Maximal subgroups

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

subgroupindex
#904162
#903822
C7×#4912
#898582
#898392
#895912
#895832
#895822
#895702
#894842
#894772
#32197