Finite Group Databasebeta

← #11739322

Subgroups — #11739322

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#90371C211yesmaximal, characteristic, complemented
3C7×#497C211yesmaximal, characteristic, complemented
4#89959C212yesmaximal, complemented
5#89957C212yesmaximal, complemented
6#89944C212yesmaximal, complemented
7#89937C212yesmaximal, complemented
8#89854C211yesmaximal, characteristic, complemented
9#89830C211yesmaximal, characteristic, complemented
10#89485C211yesmaximal, characteristic, complemented
11#89477C211yesmaximal, characteristic, complemented
12C2×#1161C2212yes—
13C2×C7×Q16C2212yes—
14C2×C7×SD16C2212yes—
15C7×C4⋊Q8C2211yescharacteristic
16C7×#124C2211yescharacteristic
17#6510C2212yes—
18#6509C2211yescharacteristic
19Q8×Dic7C2211yescharacteristic
20C2×#1143C2212yes—
21#6501C2211yescharacteristic
22D4×Dic7C2211yescharacteristic
23C2×#1141C2212yes—
24#6458—21no—
25#6458—21no—
26#6456—21no—
27#6456—21no—
28#6446—21no—
29(C22⋊D7)⋊C4—21no—
30#6442—21no—
31#6440—21no—
32#6437C2211yescharacteristic
33C42×D7C2211yescharacteristic
34#6433C2211yescharacteristic
35C7×#97C2211yescharacteristic
36#6409C2212yes—
37#6409C2212yes—
38#6395C2212yes—
39#6395C2212yes—
40(C2×D7)⋊C8C2212yes—
41Dic7⋊C8C2212yes—
42#6377C2211yescharacteristic
43#3219—71nomaximal, Sylow
44C2×C7×Q8C2312yes—
45C2×C7×Q8C2311yescharacteristic
46C2×C7×D4C2311yescharacteristic
47C2×C22⋊D7—21no—
48C22×Dic7—21no—
49#1161—42no—
50#1161—42no—
51C2×D28—22no—
52C2×D28—21no—
53C2×D28—21no—
54C2×D28C2312yes—
55C2×C4×D7—22no—
56C2×C4×D7D411yescharacteristic
57C2×C4×D7D411yescharacteristic
58C2×C4×D7C2311yescharacteristic
59C7×Q16—42no—
60C7×SD16—42no—
61C2×C7×C8C2312yes—
62C7×C4⋊C4—21no—
63C7×C4⋊C4—21no—
64C7×C22⋊C4—21no—
65C7×C22⋊C4—21no—
66C42×C7C2311yescharacteristic
67C22⋊Dic7—21no—
68C22⋊Dic7—21no—
69#1143—42no—
70#1141—42no—
71(C2×D7)⋊C4—21no—
72(C2×D7)⋊C4—22no—
73(C2×D7)⋊C4—22no—
74C4⋊Dic7—21no—
75C4⋊Dic7C2311yescharacteristic
76C4⋊Dic7C2311yescharacteristic
77Dic7⋊C4—21no—
78Dic7⋊C4—21no—
79Dic7⋊C4—21no—
80C4×Dic7—21no—
81C4×Dic7D411yescharacteristic
82C4×Dic7D411yescharacteristic
83C4×Dic7C2311yescharacteristic
84C2×C7⋊C8C2312yes—
85#532—71no—
86#529—71no—
87#497D711yescharacteristic, complemented
88#496—71no—
89#491—71no—
90#490—71no—
91#464—72no—
92#463—72no—
93#452—72no—
94#449—72no—
95#432—71no—
96C23×C7—21no—
97C22×D7—22no—
98C22×D7C2×D411yescharacteristic
99C7×Q8—22no—
100C7×Q8—22no—
101C7×Q8—22no—
102C7×D4—22no—
103C2×C4×C7—21no—
104C2×C4×C7—22no—
105C2×C4×C7C2×D411yescharacteristic
106C2×C4×C7C2×D411yescharacteristic
107C2×C4×C7C2411yescharacteristic
108C22⋊D7—42no—
109C2×Dic7—21no—
110C2×Dic7—21no—
111C2×Dic7—42no—
112C2×Dic7C2×D411yescharacteristic
113C2×Dic7C2×D411yescharacteristic
114C2×Dic7C2×D411yescharacteristic
115D28—22no—
116D28—22no—
117D28—42no—
118D28—42no—
119D28—44no—
120C4×D7—22no—
121C4×D7—41no—
122C4×D7—41no—
123C4×D7—44no—
124C7×C8—22no—
125C7⋊C8—22no—
126C2×#41—72no—
127C2×Q16C2×D712yes—
128C2×SD16—72no—
129C2×SD16C2×D712yes—
130C2×D8—72no—
131C4⋊Q8C2×D711yescharacteristic
132#127—71no—
133#126—141no—
134#125—71no—
135#124—72no—
136#124C2×D711yescharacteristic
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#103—72no—
151#103—72no—
152#102—72no—
153#102—72no—
154C22⋊C8—72no—
155#97—71no—
156#97C2×D711yescharacteristic
157C22×C7—41no—
158C22×C7—41no—
159C22×C7—41no—
160C22×C7C22×D411yescharacteristic
161C2×D7—21no—
162C2×D7—21no—
163C2×D7—21no—
164C2×D7—42no—
165C2×D7—42no—
166C2×D7—42no—
167C4×C7—21no—
168C4×C7—21no—
169C4×C7—42no—
170C4×C7—41no—
171C4×C7H(5,𝔽2)12yes—
172Dic7—21no—
173Dic7—21no—
174Dic7—21no—
175Dic7—41no—
176Dic7—41no—
177#41—282no—
178#41—282no—
179C2×Q8C22×D712yes—
180C2×Q8C22×D711yescharacteristic
181C2×D4—72no—
182C2×D4—142no—
183C2×D4—141no—
184C2×D4—141no—
185C2×D4—141no—
186C2×D4C22×D711yescharacteristic
187C22×C4—71no—
188C22×C4—71no—
189C22×C4—71no—
190C22×C4—142no—
191C22×C4—141no—
192Q16—42no—
193SD16—42no—
194SD16—282no—
195D8—282no—
196C2×C8—72no—
197C2×C8C22×D712yes—
198C4⋊C4—21no—
199C4⋊C4—21no—
200C4⋊C4—71no—
201C4⋊C4—71no—
202C4⋊C4—141no—
203C4⋊C4—141no—
204C4⋊C4—141no—
205C4⋊C4—141no—
206C22⋊C4—21no—
207C22⋊C4—21no—
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—81no—
219C2×C7#53511yescharacteristic
220C2×C7#57511yescharacteristic
221C2×C7#57511yescharacteristic
222D7—41no—
223D7—82no—
224C23—21no—
225C23—71no—
226C23—142no—
227Q8—22no—
228Q8—22no—
229Q8—22no—
230D4—22no—
231D4—142no—
232D4—142no—
233D4—282no—
234D4—282no—
235D4—284no—
236D4—282no—
237C2×C4—21no—
238C2×C4—22no—
239C2×C4—71no—
240C2×C4—71no—
241C2×C4—71no—
242C2×C4—142no—
243C2×C4—141no—
244C2×C4—141no—
245C2×C4—281no—
246C2×C4—281no—
247C2×C4—284no—
248C2×C4—282no—
249C2×C4D4×D711yescharacteristic
250C2×C4D4×D711yescharacteristic
251C2×C4C23×D711yescharacteristic
252C8—22no—
253C8—142no—
254C7#321911yescharacteristic, complemented, Sylow
255C22—41no—
256C22—41no—
257C22—41no—
258C22—141no—
259C22—141no—
260C22—141no—
261C22—282no—
262C22—282no—
263C22—282no—
264C22C2×D4×D711yescharacteristic
265C4—21no—
266C4—21no—
267C4—42no—
268C4—41no—
269C4—141no—
270C4—141no—
271C4—141no—
272C4—281no—
273C4—281no—
274C4#654812yes—
275C2—81no—
276C2—281no—
277C2—562no—
278C2#9040611yescharacteristic
279C2#9046811yescharacteristic
280C2#9046811yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#497C2
socleC22×C7C22×D4
hypercenter#497D7
hypocenterC7#3219
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ #497

Complements

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

NH ≅ G/Ntype
1. #90416C2semidirect
2. #90371C2semidirect
3. C7×#497C2semidirect
4. #89959C2semidirect
5. #89957C2semidirect
6. #89944C2semidirect
7. #89937C2semidirect
8. #89854C2semidirect
9. #89830C2semidirect
10. #89485C2semidirect
11. #89477C2semidirect
87. #497D7semidirect
254. C7#3219semidirect

Maximal subgroups

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

subgroupindex
#904162
#903712
C7×#4972
#899592
#899572
#899442
#899372
#898542
#898302
#894852
#894772
#32197