Finite Group Databasebeta

← #11739040

Subgroups — #11739040

Order 896.

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

normal subgroups only

This group has 402 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, complemented
2#90416C211yesmaximal, characteristic
3C7×#487C211yesmaximal, characteristic, complemented
4#89857C211yesmaximal, characteristic, complemented
5#89850C211yesmaximal, characteristic, complemented
6#89589C214yesmaximal, complemented
7#89584C214yesmaximal, complemented
8#89466C211yesmaximal, characteristic, complemented
9C4×#1131C211yesmaximal, characteristic, complemented
10C2×#1161—21no—
11C7×C4⋊Q8C2211yescharacteristic
12C7×C4⋊Q8C2211yescharacteristic
13#6510—21no—
14#6509C2212yes—
15Q8×Dic7C2212yes—
16C2×#1143C2214yes—
17C2×#1131C2212yes—
18#6460—21no—
19#6458—21no—
20#6458C2212yes—
21#6456—21no—
22#6456C2212yes—
23#6455—21no—
24#6453—21no—
25#6450—21no—
26#6437C2211yescharacteristic
27C42×D7C2211yescharacteristic
28#6432C2211yescharacteristic
29C7×#103C2214yes—
30C4×C7×C8C2211yescharacteristic
31#6395C2214yes—
32#6389C2212yes—
33#6381C2214yes—
34C4×C7⋊C8C2211yescharacteristic
35#3182—71nomaximal, Sylow
36C2×C7×Q8—21no—
37C2×C7×Q8C2312yes—
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×Dic14—21no—
48C2×C7×C8C2312yes—
49C7×C4⋊C4—21no—
50C7×C4⋊C4—21no—
51C7×C4⋊C4—21no—
52C7×C4⋊C4C2312yes—
53C42×C7C2311yescharacteristic
54#1143—28no—
55(C2×D7)⋊C4—22no—
56(C2×D7)⋊C4—22no—
57(C2×D7)⋊C4—22no—
58C4⋊Dic7—21no—
59C4⋊Dic7—22no—
60C4⋊Dic7C2312yes—
61Dic7⋊C4—22no—
62C4×Dic7—22no—
63C4×Dic7D411yescharacteristic
64C4×Dic7D411yescharacteristic
65C4×Dic7C2311yescharacteristic
66C2×C7⋊C8C2312yes—
67#1131—24no—
68#533—71no—
69#532—71no—
70#500—71no—
71#492—71no—
72#487D711yescharacteristic, complemented
73#486—71no—
74#468—74no—
75#440—74no—
76C4×M16—71no—
77C22×D7—22no—
78C22×D7C2×D411yescharacteristic
79C7×Q8—24no—
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—22no—
97C4×D7—44no—
98Dic14—42no—
99C7×C8—22no—
100C7⋊C8#4114yescomplemented
101C2×#41—141no—
102C2×SD16—74no—
103C2×M16—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#107—74no—
121#103C2×D714yes—
122#102—74no—
123#97—72no—
124C4×C8—71no—
125C4×C8C2×D711yescharacteristic
126C22×C7C22×D411yescharacteristic
127C2×D7—21no—
128C2×D7—21no—
129C2×D7—21no—
130C2×D7—42no—
131C2×D7—42no—
132C2×D7—42no—
133C4×C7—42no—
134C4×C7—42no—
135C4×C7AGL(1,ℤ/8)12yes—
136C4×C7#13712yes—
137C4×C7C2×#4112yes—
138Dic7—21no—
139Dic7—21no—
140Dic7—21no—
141Dic7—42no—
142#41—284no—
143C2×Q8—21no—
144C2×Q8—141no—
145C2×Q8C22×D712yes—
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—
155M16—144no—
156C2×C8—72no—
157C2×C8C22×D712yes—
158C4⋊C4—21no—
159C4⋊C4—21no—
160C4⋊C4—21no—
161C4⋊C4—72no—
162C4⋊C4—142no—
163C4⋊C4—142no—
164C4⋊C4—141no—
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×C7C2×AGL(1,ℤ/8)11yescharacteristic
176C2×C7C2×#13711yescharacteristic
177D7—41no—
178D7—82no—
179C23—71no—
180C23—142no—
181Q8—24no—
182Q8—42no—
183Q8—282no—
184D4—144no—
185D4—284no—
186D4—282no—
187D4—282no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—71no—
191C2×C4—71no—
192C2×C4—71no—
193C2×C4—142no—
194C2×C4—142no—
195C2×C4—142no—
196C2×C4—142no—
197C2×C4—284no—
198C2×C4D4×D711yescharacteristic
199C2×C4D4×D711yescharacteristic
200C2×C4C23×D711yescharacteristic
201C8—22no—
202C8—74no—
203C7#318211yescharacteristic, complemented, Sylow
204C22—141no—
205C22—141no—
206C22—141no—
207C22—282no—
208C22—282no—
209C22—282no—
210C22C2×D4×D711yescharacteristic
211C4—42no—
212C4—42no—
213C4—141no—
214C4—141no—
215C4—141no—
216C4—282no—
217C4#647712yes—
218C4#648112yes—
219C4D7×#4112yes—
220C2—281no—
221C2—562no—
222C2#9035911yescharacteristic
223C2C2×#647711yescharacteristic
224C2C2×#648111yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#487C2
socleC22×C7C22×D4
prime radical#90416C2
hypercenter#487D7
hypocenterC7#3182
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ #487

Complements

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

NH ≅ G/Ntype
1. #90419C2semidirect
3. C7×#487C2semidirect
4. #89857C2semidirect
5. #89850C2semidirect
6. #89589C2semidirect
7. #89584C2semidirect
8. #89466C2semidirect
9. C4×#1131C2semidirect
72. #487D7semidirect
100. C7⋊C8#41semidirect
203. C7#3182semidirect

Maximal subgroups

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

subgroupindex
#904192
#904162
C7×#4872
#898572
#898502
#895892
#895842
#894662
C4×#11312
#31827