Finite Group Databasebeta

← #11737931

Subgroups — #11737931

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×#462C211yesmaximal, characteristic, complemented
4#89960C212yesmaximal
5#89954C212yesmaximal, complemented
6#89862C211yesmaximal, characteristic, complemented
7#89793C211yesmaximal, characteristic, complemented
8#89622C211yesmaximal, characteristic, complemented
9D7×C4⋊C8C211yesmaximal, characteristic
10#89585C212yesmaximal
11#89566C212yesmaximal, complemented
12C2×C7×Q16C2212yes—
13C7×C4⋊Q8C2211yescharacteristic
14C4×C7×Q8C2211yescharacteristic
15#6509C2212yes—
16#6509C2211yescharacteristic
17Q8×Dic7C2212yes—
18#6507C2211yescharacteristic
19C2×#1144C2212yes—
20C2×C8×D7C2212yes—
21#6460—21no—
22#6460C2211yescharacteristic
23#6459—21no—
24#6455—21no—
25#6453—21no—
26#6452—21no—
27#6451C2211yescharacteristic
28#6450—21no—
29#6435—21no—
30C42×D7C2211yescharacteristic
31#6432C2211yescharacteristic
32C4×Dic14—21no—
33C4×Dic14C2211yescharacteristic
34C7×C4⋊C8C2211yescharacteristic
35C7×#103C2212yes—
36#6409C2212yes—
37#6409C2212yes—
38#6392C2212yes—
39#6390C2212yes—
40Dic7⋊C8C2212yes—
41#6381C2212yes—
42C4⋊(C7⋊C8)C2211yescharacteristic
43#3249—71nomaximal, Sylow
44C2×C7×Q8C2311yescharacteristic
45C2×C7×Q8C2312yes—
46C2×C4×D7D411yescharacteristic
47C2×C4×D7D411yescharacteristic
48C2×C4×D7C2311yescharacteristic
49C2×Dic14—21no—
50C2×Dic14—21no—
51C2×Dic14C2311yescharacteristic
52C7×Q16—42no—
53C2×C7×C8C2312yes—
54C7×C4⋊C4—21no—
55C7×C4⋊C4—21no—
56C7×C4⋊C4—21no—
57C7×C4⋊C4C2311yescharacteristic
58C42×C7—21no—
59C42×C7C2311yescharacteristic
60#1144—42no—
61(C2×D7)⋊C4—21no—
62(C2×D7)⋊C4—21no—
63(C2×D7)⋊C4—22no—
64C4⋊Dic7—21no—
65C4⋊Dic7—21no—
66C4⋊Dic7—22no—
67C4⋊Dic7C2311yescharacteristic
68C4⋊Dic7C2312yes—
69Dic7⋊C4—21no—
70Dic7⋊C4—21no—
71Dic7⋊C4—21no—
72Dic7⋊C4—21no—
73Dic7⋊C4—21no—
74Dic7⋊C4—22no—
75C4×Dic7—21no—
76C4×Dic7—22no—
77C4×Dic7D411yescharacteristic
78C4×Dic7D411yescharacteristic
79C4×Dic7C2311yescharacteristic
80C2×C7⋊C8C2312yes—
81C8×D7—42no—
82#533—71no—
83#533—71no—
84#477—71no—
85#477—71no—
86#477—72no—
87#467—72no—
88#467—72no—
89#462—72no—
90#462—71no—
91#462D711yescharacteristic, complemented
92C2×C4⋊C8—71no—
93C22×D7C2×D411yescharacteristic
94C7×Q8—22no—
95C7×Q8—21no—
96C7×Q8—21no—
97C7×Q8—22no—
98C2×C4×C7—22no—
99C2×C4×C7—21no—
100C2×C4×C7—21no—
101C2×C4×C7C2×D411yescharacteristic
102C2×C4×C7C2×D411yescharacteristic
103C2×C4×C7C2411yescharacteristic
104C2×Dic7—21no—
105C2×Dic7—21no—
106C2×Dic7—22no—
107C2×Dic7C2×D411yescharacteristic
108C2×Dic7C2×D411yescharacteristic
109C2×Dic7C2×D411yescharacteristic
110C4×D7—21no—
111C4×D7—21no—
112C4×D7—41no—
113C4×D7Q1614yescomplemented
114Dic14—21no—
115Dic14—21no—
116Dic14—42no—
117Dic14—42no—
118C7×C8—22no—
119C7⋊C8—22no—
120C2×Q16—72no—
121C2×Q16C2×D712yes—
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—72no—
131C22⋊Q8—71no—
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#107—72no—
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—42no—
157C4×C7—41no—
158C4×C7—41no—
159C4×C7C2×Q1612yes—
160C4×C7H(5,𝔽2)11yescharacteristic
161C4×C72−1+411yescharacteristic
162Dic7—21no—
163Dic7—21no—
164Dic7—41no—
165Dic7—41no—
166Dic7—42no—
167Dic7C2×Q1612yes—
168C2×Q8—71no—
169C2×Q8—141no—
170C2×Q8—141no—
171C2×Q8C22×D711yescharacteristic
172C2×Q8C22×D712yes—
173C22×C4—71no—
174C22×C4—71no—
175C22×C4—71no—
176Q16—42no—
177Q16—282no—
178C2×C8—72no—
179C2×C8—282no—
180C2×C8C22×D712yes—
181C4⋊C4—21no—
182C4⋊C4—21no—
183C4⋊C4—21no—
184C4⋊C4—71no—
185C4⋊C4—72no—
186C4⋊C4—141no—
187C4⋊C4—141no—
188C4⋊C4—141no—
189C4⋊C4—142no—
190C4⋊C4—141no—
191C4⋊C4—142no—
192C4⋊C4—141no—
193C4⋊C4—141no—
194C4⋊C4—141no—
195C4⋊C4C22×D711yescharacteristic
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—22no—
213Q8—21no—
214Q8—22no—
215Q8—141no—
216Q8—141no—
217Q8—282no—
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—142no—
229C2×C4—141no—
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—41no—
244C4—41no—
245C4—42no—
246C4—72no—
247C4—141no—
248C4—141no—
249C4—282no—
250C4—281no—
251C4—281no—
252C4D7×Q1612yes—
253C4#654811yescharacteristic
254C4#655411yescharacteristic
255C2—142no—
256C2#9029911yescharacteristic
257C2C2×D7×Q1611yescharacteristic
258C2#9046811yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#462C2
socleC22×C7C22×D4
prime radicalC22×D7C2×D4
hypercenter#462D7
hypocenterC7#3249
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ #462

Complements

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

NH ≅ G/Ntype
3. C7×#462C2semidirect
5. #89954C2semidirect
6. #89862C2semidirect
7. #89793C2semidirect
8. #89622C2semidirect
11. #89566C2semidirect
91. #462D7semidirect
113. 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×#4622
#899602
#899542
#898622
#897932
#896222
D7×C4⋊C82
#895852
#895662
#32497