Finite Group Databasebeta

← #11739388

Subgroups — #11739388

Order 896.

360 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 125 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
2D7×C4⋊Q8C211yesmaximal, characteristic
3C7×#500C211yesmaximal, characteristic, complemented
4#89865C211yesmaximal, characteristic, complemented
5#89850C211yesmaximal, characteristic, complemented
6#89656C214yesmaximal
7#89647C214yesmaximal, complemented
8#89457C211yesmaximal, characteristic
9#89453C211yesmaximal, characteristic, complemented
10C2×Q8×D7—21no—
11C7×C4⋊Q8C2211yescharacteristic
12C7×C4⋊Q8C2211yescharacteristic
13#6509—21no—
14Q8×Dic7—21no—
15#6507—21no—
16#6460C2212yes—
17#6455—21no—
18D7×C4⋊C4—21no—
19D7×C4⋊C4C2212yes—
20#6453—21no—
21#6451—21no—
22#6451C2212yes—
23#6450C2212yes—
24C42×D7C2211yescharacteristic
25#6432C2211yescharacteristic
26#6432C2211yescharacteristic
27C7×#107C2214yes—
28C4×C7×C8C2211yescharacteristic
29(C2×D7)⋊C8C2212yes—
30#6390C2214yes—
31Dic7⋊C8C2212yes—
32#6384C2214yes—
33#6381C2214yes—
34C4×C7⋊C8C2211yescharacteristic
35#3230—71nomaximal, Sylow
36C2×C7×Q8—21no—
37C2×C7×Q8—21no—
38Q8×D7—44no—
39C2×C4×D7—22no—
40C2×C4×D7D411yescharacteristic
41C2×C4×D7D411yescharacteristic
42C2×C4×D7C2311yescharacteristic
43C2×Dic14—21no—
44C2×Dic14—22no—
45C2×Dic14C2312yes—
46C2×C7×C8C2312yes—
47C7×C4⋊C4—21no—
48C7×C4⋊C4—21no—
49C7×C4⋊C4C2312yes—
50C7×C4⋊C4C2312yes—
51C42×C7C2311yescharacteristic
52(C2×D7)⋊C4—22no—
53C4⋊Dic7—21no—
54C4⋊Dic7—21no—
55C4⋊Dic7—21no—
56C4⋊Dic7—22no—
57C4⋊Dic7C2312yes—
58Dic7⋊C4—22no—
59Dic7⋊C4—22no—
60Dic7⋊C4—22no—
61C4×Dic7—22no—
62C4×Dic7D411yescharacteristic
63C4×Dic7D411yescharacteristic
64C4×Dic7C2311yescharacteristic
65C2×C7⋊C8C2312yes—
66#533—71no—
67C2×C4⋊Q8—71no—
68#500—71no—
69#500D711yescharacteristic, complemented
70#487—71no—
71#487—71no—
72#484—74no—
73#477—74no—
74#431—71no—
75C22×D7C2×D411yescharacteristic
76C7×Q8—42no—
77C7×Q8—42no—
78C2×C4×C7—22no—
79C2×C4×C7—22no—
80C2×C4×C7C2×D411yescharacteristic
81C2×C4×C7C2×D411yescharacteristic
82C2×C4×C7C2411yescharacteristic
83C2×Dic7—22no—
84C2×Dic7—22no—
85C2×Dic7C2×D411yescharacteristic
86C2×Dic7C2×D411yescharacteristic
87C2×Dic7C2×D411yescharacteristic
88C4×D7—22no—
89C4×D7—22no—
90C4×D7—44no—
91C4×D7Q1614yes—
92Dic14—24no—
93Dic14—44no—
94Dic14—42no—
95C7×C8—22no—
96C7⋊C8—22no—
97C22×Q8—141no—
98C4⋊Q8—72no—
99C4⋊Q8—71no—
100C4⋊Q8—71no—
101C4⋊Q8—141no—
102C4⋊Q8—141no—
103C4⋊Q8C2×D711yescharacteristic
104C4⋊Q8C2×D711yescharacteristic
105#125—141no—
106C22⋊Q8—72no—
107C22⋊Q8—141no—
108C4×Q8—72no—
109C4×Q8—141no—
110#117—141no—
111C2×C4⋊C4—72no—
112C2×C4⋊C4—141no—
113C2×C42—71no—
114#107—74no—
115#107C2×D714yes—
116C4⋊C8—72no—
117#103—74no—
118#103—74no—
119C22⋊C8—72no—
120C4×C8—71no—
121C4×C8C2×D711yescharacteristic
122C22×C7C22×D411yescharacteristic
123C2×D7—21no—
124C2×D7—21no—
125C2×D7C2×Q1612yes—
126C4×C7—42no—
127C4×C7—42no—
128C4×C7C2×Q1612yes—
129C4×C7AGL(1,ℤ/8)12yes—
130C4×C72−1+412yes—
131Dic7—21no—
132Dic7—21no—
133Dic7—42no—
134Dic7—42no—
135Dic7C2×Q1612yes—
136C2×Q8—21no—
137C2×Q8—21no—
138C2×Q8—72no—
139C2×Q8—142no—
140C2×Q8—141no—
141C2×Q8—284no—
142C22×C4—71no—
143C22×C4—71no—
144C22×C4—71no—
145C22×C4—142no—
146C2×C8—72no—
147C2×C8C22×D712yes—
148C4⋊C4—21no—
149C4⋊C4—21no—
150C4⋊C4—72no—
151C4⋊C4—142no—
152C4⋊C4—142no—
153C4⋊C4—142no—
154C4⋊C4—142no—
155C4⋊C4—141no—
156C4⋊C4—141no—
157C4⋊C4—141no—
158C4⋊C4C22×D712yes—
159C4⋊C4C22×D712yes—
160C22⋊C4—142no—
161C42—71no—
162C42—71no—
163C42—71no—
164C42—142no—
165C42C22×D711yescharacteristic
166C2×C7#53611yescharacteristic
167C2×C7C22×Q1611yescharacteristic
168C2×C7C2×AGL(1,ℤ/8)11yescharacteristic
169D7—22no—
170C23—71no—
171Q8—42no—
172Q8—42no—
173Q8—144no—
174Q8—284no—
175Q8—282no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—71no—
179C2×C4—74no—
180C2×C4—71no—
181C2×C4—71no—
182C2×C4—142no—
183C2×C4—142no—
184C2×C4—142no—
185C2×C4—142no—
186C2×C4—284no—
187C2×C4D4×D711yescharacteristic
188C2×C4D4×D711yescharacteristic
189C2×C4C23×D711yescharacteristic
190C8—22no—
191C8—142no—
192C7#323011yescharacteristic, complemented, Sylow
193C22—72no—
194C22—141no—
195C22—141no—
196C22C2×D4×D711yescharacteristic
197C4—42no—
198C4—42no—
199C4—72no—
200C4—141no—
201C4—141no—
202C4—282no—
203C4—282no—
204C4#647412yes—
205C4D7×Q1612yes—
206C4#655112yes—
207C2—142no—
208C2#9041511yescharacteristic
209C2C2×#647411yescharacteristic
210C2C2×D7×Q1611yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#500C2
socleC22×C7C22×D4
prime radicalC22×D7C2×D4
hypercenter#500D7
hypocenterC7#3230
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ #500

Complements

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

NH ≅ G/Ntype
3. C7×#500C2semidirect
4. #89865C2semidirect
5. #89850C2semidirect
7. #89647C2semidirect
9. #89453C2semidirect
69. #500D7semidirect
192. C7#3230semidirect

Maximal subgroups

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

subgroupindex
#904192
D7×C4⋊Q82
C7×#5002
#898652
#898502
#896562
#896472
#894572
#894532
#32307