Finite Group Databasebeta

← #11739331

Subgroups — #11739331

Order 896.

412 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 133 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, complemented
2#90416C211yesmaximal, characteristic
3C7×#498C211yesmaximal, characteristic, complemented
4#89865C211yesmaximal, characteristic, complemented
5#89860C211yesmaximal, characteristic, complemented
6#89636C214yesmaximal, complemented
7#89624C214yesmaximal, complemented
8#89465C211yesmaximal, characteristic, complemented
9C4×C8×D7C211yesmaximal, characteristic, complemented
10C2×#1161—21no—
11C7×C4⋊Q8C2211yescharacteristic
12C7×C4⋊Q8C2211yescharacteristic
13#6510—21no—
14#6509—21no—
15Q8×Dic7—21no—
16C2×#1132C2214yes—
17C2×C8×D7C2212yes—
18#6460C2212yes—
19#6458—21no—
20#6458C2212yes—
21#6456—21no—
22#6456C2212yes—
23#6455—21no—
24#6453—21no—
25#6450C2212yes—
26#6437C2211yescharacteristic
27C42×D7C2211yescharacteristic
28#6432C2211yescharacteristic
29C7×#106C2214yes—
30C4×C7×C8C2211yescharacteristic
31C8×Dic7C2212yes—
32#6384C2214yes—
33#6383C2214yes—
34C4×C7⋊C8C2211yescharacteristic
35#3167—71nomaximal, Sylow
36C2×C7×Q8—21no—
37C2×C7×Q8—21no—
38#1161—44no—
39C2×D28—22no—
40C2×D28—21no—
41C2×D28—21no—
42C2×D28C2312yes—
43C2×C4×D7—22no—
44C2×C4×D7D411yescharacteristic
45C2×C4×D7D411yescharacteristic
46C2×C4×D7C2311yescharacteristic
47C2×Dic14C2312yes—
48C2×C7×C8C2312yes—
49C7×C4⋊C4—21no—
50C7×C4⋊C4—21no—
51C7×C4⋊C4C2312yes—
52C7×C4⋊C4C2312yes—
53C42×C7C2311yescharacteristic
54(C2×D7)⋊C4—22no—
55(C2×D7)⋊C4—22no—
56(C2×D7)⋊C4—22no—
57C4⋊Dic7—21no—
58C4⋊Dic7—21no—
59C4⋊Dic7—22no—
60Dic7⋊C4—22no—
61C4×Dic7—22no—
62C4×Dic7D411yescharacteristic
63C4×Dic7D411yescharacteristic
64C4×Dic7C2311yescharacteristic
65C2×C7⋊C8C2312yes—
66#1132—28no—
67C8×D7—24no—
68#533—71no—
69#532—71no—
70#498D711yescharacteristic, complemented
71#492—71no—
72#487—71no—
73#486—71no—
74#465—74no—
75C4×SD16—74no—
76C2×C4×C8—71no—
77C22×D7—22no—
78C22×D7C2×D411yescharacteristic
79C7×Q8—42no—
80C7×Q8—42no—
81C2×C4×C7—22no—
82C2×C4×C7—22no—
83C2×C4×C7C2×D411yescharacteristic
84C2×C4×C7C2×D411yescharacteristic
85C2×C4×C7C2411yescharacteristic
86C2×Dic7—22no—
87C2×Dic7C2×D411yescharacteristic
88C2×Dic7C2×D411yescharacteristic
89C2×Dic7C2×D411yescharacteristic
90D28—24no—
91D28—44no—
92D28—42no—
93D28—42no—
94C4×D7—22no—
95C4×D7—22no—
96C4×D7—44no—
97C4×D7SD1614yescomplemented
98Dic14—24no—
99C7×C8#4114yescomplemented
100C7⋊C8—22no—
101C2×#41—141no—
102C2×SD16—74no—
103C22×C8—72no—
104C4⋊Q8—71no—
105C4⋊Q8C2×D711yescharacteristic
106C4⋊Q8C2×D711yescharacteristic
107#127—71no—
108#125—141no—
109#124—141no—
110C22⋊Q8—72no—
111C22⋊Q8—141no—
112#121—72no—
113#121—141no—
114C4×Q8—72no—
115C4×Q8—141no—
116C4×D4—72no—
117C4×D4—141no—
118#117—141no—
119C2×C42—71no—
120#106C2×D714yes—
121#103—74no—
122#102—74no—
123C4×C8—72no—
124C4×C8—71no—
125C4×C8C2×D711yescharacteristic
126C22×C7C22×D411yescharacteristic
127C2×D7—21no—
128C2×D7—21no—
129C2×D7—42no—
130C2×D7—42no—
131C2×D7—42no—
132C2×D7C2×SD1612yes—
133C4×C7—42no—
134C4×C7—42no—
135C4×C7C2×SD1612yes—
136C4×C7#13512yes—
137C4×C7C2×#4112yes—
138Dic7—21no—
139Dic7—21no—
140Dic7—42no—
141Dic7C2×SD1612yes—
142#41—284no—
143C2×Q8—21no—
144C2×Q8—21no—
145C2×Q8—72no—
146C2×D4—72no—
147C2×D4—142no—
148C2×D4—141no—
149C2×D4—141no—
150C22×C4—71no—
151C22×C4—71no—
152C22×C4—71no—
153C22×C4—142no—
154SD16—148no—
155C2×C8—72no—
156C2×C8—144no—
157C2×C8C22×D712yes—
158C4⋊C4—21no—
159C4⋊C4—21no—
160C4⋊C4—142no—
161C4⋊C4—142no—
162C4⋊C4—141no—
163C4⋊C4—141no—
164C4⋊C4C22×D712yes—
165C4⋊C4C22×D712yes—
166C22⋊C4—142no—
167C22⋊C4—142no—
168C22⋊C4—142no—
169C42—71no—
170C42—71no—
171C42—71no—
172C42—142no—
173C42C22×D711yescharacteristic
174C2×C7#53211yescharacteristic
175C2×C7C22×SD1611yescharacteristic
176C2×C7C2×#13511yescharacteristic
177D7—22no—
178D7—82no—
179C23—71no—
180C23—142no—
181Q8—42no—
182Q8—42no—
183Q8—144no—
184D4—144no—
185D4—284no—
186D4—282no—
187D4—282no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—71no—
191C2×C4—74no—
192C2×C4—71no—
193C2×C4—71no—
194C2×C4—142no—
195C2×C4—142no—
196C2×C4—142no—
197C2×C4—284no—
198C2×C4D4×D711yescharacteristic
199C2×C4D4×D711yescharacteristic
200C2×C4C23×D711yescharacteristic
201C8—142no—
202C8#116114yescomplemented
203C7#316711yescharacteristic, complemented, Sylow
204C22—72no—
205C22—141no—
206C22—141no—
207C22—282no—
208C22—282no—
209C22—282no—
210C22C2×D4×D711yescharacteristic
211C4—42no—
212C4—42no—
213C4—72no—
214C4—141no—
215C4—141no—
216C4—282no—
217C4D7×SD1612yes—
218C4#647912yes—
219C4C2×#116112yes—
220C2—142no—
221C2—562no—
222C2#9041611yescharacteristic
223C2C2×D7×SD1611yescharacteristic
224C2C2×#647911yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#498C2
socleC22×C7C22×D4
prime radical#90416C2
hypercenter#498D7
hypocenterC7#3167
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ #498

Complements

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

NH ≅ G/Ntype
1. #90419C2semidirect
3. C7×#498C2semidirect
4. #89865C2semidirect
5. #89860C2semidirect
6. #89636C2semidirect
7. #89624C2semidirect
8. #89465C2semidirect
9. C4×C8×D7C2semidirect
70. #498D7semidirect
97. C4×D7SD16semidirect
99. C7×C8#41semidirect
202. C8#1161semidirect
203. C7#3167semidirect

Maximal subgroups

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

subgroupindex
#904192
#904162
C7×#4982
#898652
#898602
#896362
#896242
#894652
C4×C8×D72
#31677