Finite Group Databasebeta

← #11738460

Subgroups — #11738460

Order 896.

360 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 123 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
1D7×C4⋊Q8C211yesmaximal, characteristic
2#90269C211yesmaximal, characteristic
3C7×#475C211yesmaximal, characteristic, complemented
4#89866C211yesmaximal, characteristic, complemented
5#89792C211yesmaximal, characteristic, complemented
6#89634C212yesmaximal
7#89627C212yesmaximal, complemented
8#89623C211yesmaximal, characteristic, complemented
9#89605C211yesmaximal, characteristic
10#89577C212yesmaximal
11#89564C212yesmaximal, complemented
12C2×Q8×D7—21no—
13C7×C4⋊Q8C2211yescharacteristic
14C4×C7×Q8C2211yescharacteristic
15#6509C2211yescharacteristic
16#6507—21no—
17#6507C2211yescharacteristic
18#6460C2211yescharacteristic
19#6459—21no—
20D7×C4⋊C4—21no—
21D7×C4⋊C4C2212yes—
22#6452—21no—
23#6451—21no—
24#6451C2211yescharacteristic
25#6451C2212yes—
26#6435—21no—
27C42×D7C2211yescharacteristic
28#6432C2211yescharacteristic
29C4×Dic14—21no—
30C4×Dic14C2211yescharacteristic
31C7×#106C2212yes—
32C7×C4⋊C8C2211yescharacteristic
33C7×#103C2212yes—
34#6409C2212yes—
35(C2×D7)⋊C8C2212yes—
36#6391C2212yes—
37#6390C2212yes—
38C8×Dic7C2212yes—
39#6384C2212yes—
40#6382C2212yes—
41#6382C2212yes—
42C4⋊(C7⋊C8)C2211yescharacteristic
43#3229—71nomaximal, Sylow
44C2×C7×Q8—21no—
45C2×C7×Q8C2311yescharacteristic
46Q8×D7—44no—
47C2×C4×D7—22no—
48C2×C4×D7D411yescharacteristic
49C2×C4×D7D411yescharacteristic
50C2×C4×D7C2311yescharacteristic
51C2×Dic14—21no—
52C2×Dic14—21no—
53C2×Dic14—22no—
54C2×Dic14C2311yescharacteristic
55C2×C7×C8C2312yes—
56C7×C4⋊C4—21no—
57C7×C4⋊C4—21no—
58C7×C4⋊C4C2312yes—
59C7×C4⋊C4C2311yescharacteristic
60C42×C7—21no—
61C42×C7C2311yescharacteristic
62(C2×D7)⋊C4—21no—
63(C2×D7)⋊C4—21no—
64C4⋊Dic7—21no—
65C4⋊Dic7—21no—
66C4⋊Dic7C2312yes—
67C4⋊Dic7C2311yescharacteristic
68Dic7⋊C4—21no—
69Dic7⋊C4—21no—
70Dic7⋊C4—21no—
71Dic7⋊C4—21no—
72Dic7⋊C4—21no—
73Dic7⋊C4—22no—
74Dic7⋊C4—22no—
75C4×Dic7—21no—
76C4×Dic7D411yescharacteristic
77C4×Dic7D411yescharacteristic
78C4×Dic7C2311yescharacteristic
79C2×C7⋊C8C2312yes—
80#533—71no—
81C2×C4⋊Q8—71no—
82#498—72no—
83#487—72no—
84#483—72no—
85#483—72no—
86#475—71no—
87#475—71no—
88#475—71no—
89#475D711yescharacteristic, complemented
90#431—71no—
91C22×D7C2×D411yescharacteristic
92C7×Q8—21no—
93C7×Q8—21no—
94C7×Q8—42no—
95C2×C4×C7—21no—
96C2×C4×C7—22no—
97C2×C4×C7—21no—
98C2×C4×C7C2×D411yescharacteristic
99C2×C4×C7C2×D411yescharacteristic
100C2×C4×C7C2411yescharacteristic
101C2×Dic7—21no—
102C2×Dic7—21no—
103C2×Dic7—22no—
104C2×Dic7C2×D411yescharacteristic
105C2×Dic7C2×D411yescharacteristic
106C2×Dic7C2×D411yescharacteristic
107C4×D7—21no—
108C4×D7—21no—
109C4×D7—22no—
110C4×D7—44no—
111C4×D7SD1614yes—
112Dic14—21no—
113Dic14—21no—
114Dic14—42no—
115Dic14—44no—
116Dic14—42no—
117C7×C8—22no—
118C7⋊C8—22no—
119C22×Q8—141no—
120C4⋊Q8—72no—
121C4⋊Q8—71no—
122C4⋊Q8—71no—
123C4⋊Q8—71no—
124C4⋊Q8—141no—
125C4⋊Q8—141no—
126C4⋊Q8C2×D711yescharacteristic
127#125—141no—
128C22⋊Q8—71no—
129C22⋊Q8—71no—
130C22⋊Q8—141no—
131C4×Q8—71no—
132C4×Q8—141no—
133C4×Q8C2×D711yescharacteristic
134#117—141no—
135C2×C4⋊C4—72no—
136C2×C4⋊C4—141no—
137C2×C42—71no—
138#106—72no—
139#106—72no—
140#106—72no—
141#106C2×D712yes—
142C4⋊C8—71no—
143C4⋊C8C2×D711yescharacteristic
144#103—72no—
145#103—72no—
146#103—72no—
147#103C2×D712yes—
148C22⋊C8—72no—
149C4×C8—72no—
150C22×C7C22×D411yescharacteristic
151C2×D7—21no—
152C2×D7—21no—
153C2×D7C2×SD1612yes—
154C4×C7—21no—
155C4×C7—41no—
156C4×C7—42no—
157C4×C7—41no—
158C4×C7C2×SD1612yes—
159C4×C72−1+411yescharacteristic
160C4×C72−1+411yescharacteristic
161Dic7—21no—
162Dic7—41no—
163Dic7—42no—
164Dic7—41no—
165Dic7C2×SD1612yes—
166Dic7#13712yes—
167C2×Q8—21no—
168C2×Q8—71no—
169C2×Q8—141no—
170C2×Q8—142no—
171C2×Q8—141no—
172C2×Q8—284no—
173C2×Q8C22×D711yescharacteristic
174C22×C4—71no—
175C22×C4—71no—
176C22×C4—71no—
177C22×C4—142no—
178C2×C8—72no—
179C2×C8C22×D712yes—
180C4⋊C4—21no—
181C4⋊C4—21no—
182C4⋊C4—72no—
183C4⋊C4—71no—
184C4⋊C4—142no—
185C4⋊C4—142no—
186C4⋊C4—141no—
187C4⋊C4—141no—
188C4⋊C4—141no—
189C4⋊C4—141no—
190C4⋊C4—141no—
191C4⋊C4—141no—
192C4⋊C4—141no—
193C4⋊C4C22×D712yes—
194C4⋊C4C22×D711yescharacteristic
195C22⋊C4—141no—
196C22⋊C4—141no—
197C42—21no—
198C42—71no—
199C42—71no—
200C42—71no—
201C42—141no—
202C42C22×D711yescharacteristic
203C2×C7#53611yescharacteristic
204C2×C7C22×SD1611yescharacteristic
205C2×C7C2×#13711yescharacteristic
206D7—22no—
207C23—71no—
208Q8—21no—
209Q8—21no—
210Q8—42no—
211Q8—141no—
212Q8—141no—
213Q8—282no—
214Q8—284no—
215Q8—282no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—22no—
219C2×C4—71no—
220C2×C4—71no—
221C2×C4—74no—
222C2×C4—71no—
223C2×C4—142no—
224C2×C4—141no—
225C2×C4—141no—
226C2×C4—141no—
227C2×C4—141no—
228C2×C4—142no—
229C2×C4—284no—
230C2×C4D4×D711yescharacteristic
231C2×C4D4×D711yescharacteristic
232C2×C4C23×D711yescharacteristic
233C8—22no—
234C8—142no—
235C7#322911yescharacteristic, complemented, Sylow
236C22—72no—
237C22—141no—
238C22—141no—
239C22C2×D4×D711yescharacteristic
240C4—21no—
241C4—42no—
242C4—41no—
243C4—41no—
244C4—72no—
245C4—72no—
246C4—141no—
247C4—282no—
248C4—281no—
249C4—281no—
250C4D7×SD1612yes—
251C4#655111yescharacteristic
252C4#655411yescharacteristic
253C2—142no—
254C2#9031911yescharacteristic
255C2C2×D7×SD1611yescharacteristic
256C2D7×#13711yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#475C2
socleC22×C7C22×D4
prime radicalC22×D7C2×D4
hypercenter#475D7
hypocenterC7#3229
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ #475

Complements

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

NH ≅ G/Ntype
3. C7×#475C2semidirect
4. #89866C2semidirect
5. #89792C2semidirect
7. #89627C2semidirect
8. #89623C2semidirect
11. #89564C2semidirect
89. #475D7semidirect
235. C7#3229semidirect

Maximal subgroups

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

subgroupindex
D7×C4⋊Q82
#902692
C7×#4752
#898662
#897922
#896342
#896272
#896232
#896052
#895772
#895642
#32297