Finite Group Databasebeta

← #11739166

Subgroups — #11739166

Order 896.

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

normal subgroups only

This group has 368 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#90407C211yesmaximal, characteristic
2D7×#125C211yesmaximal, characteristic, complemented
3C7×#490C211yesmaximal, characteristic, complemented
4#89843C211yesmaximal, characteristic, complemented
5#89837C211yesmaximal, characteristic, complemented
6#89549C212yesmaximal, complemented
7#89546C212yesmaximal, complemented
8#89538C212yesmaximal, complemented
9#89535C212yesmaximal, complemented
10#89477C211yesmaximal, characteristic, complemented
11#89474C211yesmaximal, characteristic, complemented
12C2×#1159—21no—
13C7×#127C2211yescharacteristic
14C7×#125C2211yescharacteristic
15#6501—21no—
16#6501C2212yes—
17#6499—21no—
18D4×Dic7—21no—
19D4×Dic7C2212yes—
20D7×C4⋊C4—21no—
21D7×C4⋊C4—21no—
22D7×C4⋊C4C2211yescharacteristic
23D7×C4⋊C4C2211yescharacteristic
24#6453C2211yescharacteristic
25#6453C2211yescharacteristic
26#6452—21no—
27#6452—21no—
28C42×D7C2211yescharacteristic
29#6433C2211yescharacteristic
30#6432C2211yescharacteristic
31C7×#102C2212yes—
32C7×#102C2212yes—
33C7×#97C2211yescharacteristic
34#6406C2212yes—
35#6406C2212yes—
36(C2×D7)⋊C8C2212yes—
37#6392C2212yes—
38#6391C2212yes—
39Dic7⋊C8C2212yes—
40#6382C2212yes—
41#6381C2212yes—
42#6377C2211yescharacteristic
43#3234—71nomaximal, Sylow
44C2×C7×D4—21no—
45C2×C7×D4—21no—
46C2×C7×D4C2312yes—
47C2×C22⋊D7—22no—
48C22×Dic7—22no—
49#1159—44no—
50C2×C4×D7—21no—
51C2×C4×D7—21no—
52C2×C4×D7D411yescharacteristic
53C2×C4×D7D411yescharacteristic
54C2×C4×D7C2311yescharacteristic
55C2×Dic14—21no—
56C2×C7×C8C2312yes—
57C7×C4⋊C4—21no—
58C7×C4⋊C4—21no—
59C7×C4⋊C4C2311yescharacteristic
60C7×C4⋊C4C2311yescharacteristic
61C42×C7C2311yescharacteristic
62C22⋊Dic7—22no—
63C22⋊Dic7—22no—
64C4⋊Dic7—21no—
65C4⋊Dic7—21no—
66C4⋊Dic7—21no—
67C4⋊Dic7C2311yescharacteristic
68C4⋊Dic7C2311yescharacteristic
69C4⋊Dic7C2312yes—
70Dic7⋊C4—21no—
71Dic7⋊C4—21no—
72Dic7⋊C4—21no—
73Dic7⋊C4—21no—
74Dic7⋊C4—21no—
75Dic7⋊C4—21no—
76C4×Dic7D411yescharacteristic
77C4×Dic7D411yescharacteristic
78C4×Dic7C2311yescharacteristic
79C2×C7⋊C8C2312yes—
80#532—71no—
81C2×#125—71no—
82#501—71no—
83#501—71no—
84#490—71no—
85#490D711yescharacteristic, complemented
86#481—72no—
87#480—72no—
88#478—72no—
89#478—72no—
90#432—71no—
91C23×C7—22no—
92C22×D7C2×D411yescharacteristic
93C7×D4—24no—
94C7×D4—42no—
95C7×D4—42no—
96C2×C4×C7—21no—
97C2×C4×C7—21no—
98C2×C4×C7C2×D411yescharacteristic
99C2×C4×C7C2×D411yescharacteristic
100C2×C4×C7C2411yescharacteristic
101C22⋊D7—44no—
102C2×Dic7—21no—
103C2×Dic7—21no—
104C2×Dic7—22no—
105C2×Dic7—44no—
106C2×Dic7C2×D411yescharacteristic
107C2×Dic7C2×D411yescharacteristic
108C2×Dic7C2×D411yescharacteristic
109C4×D7—22no—
110C4×D7—22no—
111C4×D7—41no—
112C4×D7—42no—
113C4×D7—42no—
114Dic14—42no—
115C7×C8—22no—
116C7⋊C8—22no—
117C2×#41—141no—
118C4⋊Q8—71no—
119#127C2×D711yescharacteristic
120#125—71no—
121#125—71no—
122#125—71no—
123#125—141no—
124#125—141no—
125#125C2×D711yescharacteristic
126#124—141no—
127#121—72no—
128#121—141no—
129C4×D4—72no—
130C4×D4—141no—
131C2×C4⋊C4—71no—
132C2×C4⋊C4—71no—
133C2×C4⋊C4—141no—
134C2×C4⋊C4—141no—
135C2×C42—71no—
136#107—72no—
137#107—72no—
138#106—72no—
139#106—72no—
140C4⋊C8—72no—
141#102—72no—
142#102—72no—
143#102C2×D712yes—
144#102C2×D712yes—
145C22⋊C8—72no—
146#97—71no—
147#97C2×D711yescharacteristic
148C22×C7—42no—
149C22×C7—42no—
150C22×C7—42no—
151C22×C7C22×D411yescharacteristic
152C2×D7—21no—
153C2×D7—21no—
154C2×D7AGL(1,ℤ/8)12yes—
155C4×C7—21no—
156C4×C7—21no—
157C4×C7—41no—
158C4×C7—41no—
159C4×C72−1+412yes—
160Dic7—21no—
161Dic7—21no—
162Dic7—21no—
163Dic7—42no—
164Dic7—41no—
165Dic7—41no—
166#41—284no—
167C2×Q8—141no—
168C2×D4—21no—
169C2×D4—21no—
170C2×D4—142no—
171C2×D4C22×D712yes—
172C22×C4—71no—
173C22×C4—71no—
174C22×C4—71no—
175C22×C4—141no—
176C22×C4—141no—
177C22×C4—142no—
178C2×C8—72no—
179C2×C8C22×D712yes—
180C4⋊C4—21no—
181C4⋊C4—21no—
182C4⋊C4—71no—
183C4⋊C4—71no—
184C4⋊C4—72no—
185C4⋊C4—141no—
186C4⋊C4—141no—
187C4⋊C4—141no—
188C4⋊C4—141no—
189C4⋊C4—141no—
190C4⋊C4—141no—
191C4⋊C4—141no—
192C4⋊C4—141no—
193C4⋊C4—141no—
194C4⋊C4C22×D711yescharacteristic
195C4⋊C4C22×D711yescharacteristic
196C22⋊C4—142no—
197C22⋊C4—142no—
198C42—71no—
199C42—71no—
200C42—71no—
201C42C22×D711yescharacteristic
202C2×C7—82no—
203C2×C7#53611yescharacteristic
204C2×C7C2×AGL(1,ℤ/8)11yescharacteristic
205C2×C7#57511yescharacteristic
206D7—22no—
207C23—22no—
208C23—71no—
209Q8—282no—
210D4—24no—
211D4—42no—
212D4—42no—
213D4—284no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—71no—
217C2×C4—71no—
218C2×C4—71no—
219C2×C4—142no—
220C2×C4—142no—
221C2×C4—142no—
222C2×C4—141no—
223C2×C4—141no—
224C2×C4—281no—
225C2×C4—282no—
226C2×C4—282no—
227C2×C4—284no—
228C2×C4D4×D711yescharacteristic
229C2×C4D4×D711yescharacteristic
230C2×C4C23×D711yescharacteristic
231C8—22no—
232C8—142no—
233C7#323411yescharacteristic, complemented, Sylow
234C22—42no—
235C22—42no—
236C22—42no—
237C22—72no—
238C22—141no—
239C22—141no—
240C22C2×D4×D711yescharacteristic
241C4—21no—
242C4—21no—
243C4—41no—
244C4—41no—
245C4—141no—
246C4—141no—
247C4—141no—
248C4—282no—
249C4—281no—
250C4—281no—
251C4#655412yes—
252C2—82no—
253C2—142no—
254C2#9036611yescharacteristic
255C2D7×AGL(1,ℤ/8)11yescharacteristic
256C2#9046411yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#490C2
socleC22×C7C22×D4
prime radical#90407C2
hypercenter#490D7
hypocenterC7#3234
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ #490

Complements

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

NH ≅ G/Ntype
2. D7×#125C2semidirect
3. C7×#490C2semidirect
4. #89843C2semidirect
5. #89837C2semidirect
6. #89549C2semidirect
7. #89546C2semidirect
8. #89538C2semidirect
9. #89535C2semidirect
10. #89477C2semidirect
11. #89474C2semidirect
85. #490D7semidirect
233. C7#3234semidirect

Maximal subgroups

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

subgroupindex
#904072
D7×#1252
C7×#4902
#898432
#898372
#895492
#895462
#895382
#895352
#894772
#894742
#32347