Finite Group Databasebeta

← #11738563

Subgroups — #11738563

Order 896.

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

normal subgroups only

This group has 348 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
1#90419C211yesmaximal, characteristic
2#90269C211yesmaximal, characteristic
3C7×#477C211yesmaximal, characteristic, complemented
4#89863C211yesmaximal, characteristic, complemented
5#89803C211yesmaximal, characteristic, complemented
6#89659C212yesmaximal
7#89647C212yesmaximal, complemented
8#89621C211yesmaximal, characteristic, complemented
9D7×C4⋊C8C211yesmaximal, characteristic
10#89585C212yesmaximal
11#89565C212yesmaximal, complemented
12C7×C4⋊Q8C2211yescharacteristic
13C4×C7×Q8C2211yescharacteristic
14#6509—21no—
15#6509C2211yescharacteristic
16Q8×Dic7—21no—
17#6507C2211yescharacteristic
18C2×#1144C2212yes—
19C2×Dic28C2212yes—
20C2×C8×D7C2212yes—
21#6460C2212yes—
22#6460C2211yescharacteristic
23#6459—21no—
24#6455—21no—
25#6453—21no—
26#6452—21no—
27#6451C2211yescharacteristic
28#6450C2212yes—
29#6435—21no—
30C42×D7C2211yescharacteristic
31#6432C2211yescharacteristic
32C4×Dic14—21no—
33C4×Dic14C2211yescharacteristic
34C7×#107C2212yes—
35C7×C4⋊C8C2211yescharacteristic
36C7×#103C2212yes—
37#6390C2212yes—
38Dic7⋊C8C2212yes—
39#6384C2212yes—
40#6384C2212yes—
41#6381C2212yes—
42C4⋊(C7⋊C8)C2211yescharacteristic
43#3249—71nomaximal, Sylow
44C2×C7×Q8—21no—
45C2×C7×Q8C2311yescharacteristic
46C2×C4×D7D411yescharacteristic
47C2×C4×D7D411yescharacteristic
48C2×C4×D7C2311yescharacteristic
49C2×Dic14—21no—
50C2×Dic14C2311yescharacteristic
51C2×Dic14C2312yes—
52C2×C7×C8C2312yes—
53C7×C4⋊C4—21no—
54C7×C4⋊C4—21no—
55C7×C4⋊C4C2312yes—
56C7×C4⋊C4C2311yescharacteristic
57C42×C7—21no—
58C42×C7C2311yescharacteristic
59#1144—42no—
60(C2×D7)⋊C4—21no—
61(C2×D7)⋊C4—21no—
62(C2×D7)⋊C4—22no—
63C4⋊Dic7—21no—
64C4⋊Dic7—21no—
65C4⋊Dic7—21no—
66C4⋊Dic7—22no—
67C4⋊Dic7C2311yescharacteristic
68Dic7⋊C4—21no—
69Dic7⋊C4—21no—
70Dic7⋊C4—21no—
71Dic7⋊C4—21no—
72Dic7⋊C4—21no—
73Dic7⋊C4—22no—
74C4×Dic7—21no—
75C4×Dic7—22no—
76C4×Dic7D411yescharacteristic
77C4×Dic7D411yescharacteristic
78C4×Dic7C2311yescharacteristic
79C2×C7⋊C8C2312yes—
80Dic28—42no—
81C8×D7—42no—
82#533—71no—
83#533—71no—
84#477—71no—
85#477—72no—
86#477D711yescharacteristic, complemented
87#467—72no—
88#467—72no—
89#462—71no—
90#462—71no—
91#462—72no—
92C2×C4⋊C8—71no—
93C22×D7C2×D411yescharacteristic
94C7×Q8—21no—
95C7×Q8—21no—
96C7×Q8—42no—
97C2×C4×C7—22no—
98C2×C4×C7—21no—
99C2×C4×C7—21no—
100C2×C4×C7C2×D411yescharacteristic
101C2×C4×C7C2×D411yescharacteristic
102C2×C4×C7C2411yescharacteristic
103C2×Dic7—21no—
104C2×Dic7—21no—
105C2×Dic7—22no—
106C2×Dic7C2×D411yescharacteristic
107C2×Dic7C2×D411yescharacteristic
108C2×Dic7C2×D411yescharacteristic
109C4×D7—21no—
110C4×D7—21no—
111C4×D7—41no—
112C4×D7Q1614yescomplemented
113Dic14—21no—
114Dic14—22no—
115Dic14—21no—
116Dic14—22no—
117Dic14—42no—
118C7×C8—22no—
119C7⋊C8—22no—
120C2×Q16—72no—
121C2×Q16—72no—
122C22×C8—72no—
123C4⋊Q8—71no—
124C4⋊Q8—71no—
125C4⋊Q8—71no—
126C4⋊Q8C2×D711yescharacteristic
127#125—141no—
128#125—141no—
129C22⋊Q8—71no—
130C22⋊Q8—71no—
131C22⋊Q8—72no—
132C22⋊Q8—141no—
133C22⋊Q8—141no—
134C4×Q8—72no—
135C4×Q8—71no—
136C4×Q8—141no—
137C4×Q8—141no—
138C4×Q8C2×D711yescharacteristic
139#117—141no—
140#117—141no—
141C2×C42—71no—
142#107—72no—
143#107C2×D712yes—
144C4⋊C8—72no—
145C4⋊C8—71no—
146C4⋊C8C2×D711yescharacteristic
147#103—72no—
148#103—72no—
149#103—72no—
150#103C2×D712yes—
151C22×C7C22×D411yescharacteristic
152C2×D7—21no—
153C2×D7—21no—
154C2×D7C2×Q1612yes—
155C4×C7—21no—
156C4×C7—41no—
157C4×C7—41no—
158C4×C7—42no—
159C4×C7C2×Q1612yes—
160C4×C7H(5,𝔽2)11yescharacteristic
161C4×C72−1+411yescharacteristic
162Dic7—21no—
163Dic7—21no—
164Dic7—41no—
165Dic7—42no—
166Dic7—41no—
167Dic7C2×Q1612yes—
168C2×Q8—21no—
169C2×Q8—72no—
170C2×Q8—71no—
171C2×Q8—141no—
172C2×Q8C22×D711yescharacteristic
173C22×C4—71no—
174C22×C4—71no—
175C22×C4—71no—
176Q16—282no—
177Q16—282no—
178C2×C8—72no—
179C2×C8—282no—
180C2×C8C22×D712yes—
181C4⋊C4—21no—
182C4⋊C4—21no—
183C4⋊C4—71no—
184C4⋊C4—141no—
185C4⋊C4—141no—
186C4⋊C4—142no—
187C4⋊C4—141no—
188C4⋊C4—141no—
189C4⋊C4—141no—
190C4⋊C4—142no—
191C4⋊C4—141no—
192C4⋊C4—141no—
193C4⋊C4—141no—
194C4⋊C4C22×D711yescharacteristic
195C4⋊C4C22×D712yes—
196C22⋊C4—141no—
197C22⋊C4—141no—
198C22⋊C4—142no—
199C42—21no—
200C42—71no—
201C42—71no—
202C42—71no—
203C42—142no—
204C42—141no—
205C42C22×D711yescharacteristic
206C2×C7#53711yescharacteristic
207C2×C7C22×Q1611yescharacteristic
208C2×C7#57511yescharacteristic
209D7—22no—
210C23—71no—
211Q8—21no—
212Q8—21no—
213Q8—42no—
214Q8—141no—
215Q8—142no—
216Q8—141no—
217Q8—142no—
218Q8—282no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—22no—
222C2×C4—71no—
223C2×C4—74no—
224C2×C4—71no—
225C2×C4—71no—
226C2×C4—141no—
227C2×C4—141no—
228C2×C4—141no—
229C2×C4—142no—
230C2×C4—141no—
231C2×C4—281no—
232C2×C4D4×D711yescharacteristic
233C2×C4D4×D711yescharacteristic
234C2×C4C23×D711yescharacteristic
235C8—22no—
236C8—142no—
237C7#324911yescharacteristic, complemented, Sylow
238C22—72no—
239C22—141no—
240C22—141no—
241C22C2×D4×D711yescharacteristic
242C4—21no—
243C4—42no—
244C4—41no—
245C4—41no—
246C4—72no—
247C4—141no—
248C4—141no—
249C4—281no—
250C4—282no—
251C4—281no—
252C4D7×Q1612yes—
253C4#655111yescharacteristic
254C4#655311yescharacteristic
255C2—142no—
256C2#9032011yescharacteristic
257C2C2×D7×Q1611yescharacteristic
258C2#9046411yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#477C2
socleC22×C7C22×D4
prime radicalC22×D7C2×D4
hypercenter#477D7
hypocenterC7#3249
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ #477

Complements

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

NH ≅ G/Ntype
3. C7×#477C2semidirect
4. #89863C2semidirect
5. #89803C2semidirect
7. #89647C2semidirect
8. #89621C2semidirect
11. #89565C2semidirect
86. #477D7semidirect
112. C4×D7Q16semidirect
237. C7#3249semidirect

Maximal subgroups

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

subgroupindex
#904192
#902692
C7×#4772
#898632
#898032
#896592
#896472
#896212
D7×C4⋊C82
#895852
#895652
#32497