Finite Group Databasebeta

← #13054762

Subgroups — #13054762

Order 1344.

292 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 64 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
1C7×#1007C211yesmaximal, characteristic, complemented
2C3×#6510C211yesmaximal, characteristic, complemented
3#10620860C212yesmaximal, complemented
4#10620675C211yesmaximal, characteristic, complemented
5#10620514C212yesmaximal, complemented
6#90017—31nomaximal, Hall
7C2×C3×C7×Q8C2211yescharacteristic
8C7×#239—22no—
9D3×C7×Q8—22no—
10C7×#235—22no—
11C7×#235C2212yes—
12C2×C3×D28C2211yescharacteristic
13#66627C2212yes—
14C3×(C2×D7)⋊C4—22no—
15C3×C4×Dic7C2211yescharacteristic
16#66563—22no—
17#66560—22no—
18C7×2−1+4—31no—
19#6512—32no—
20#6510D311yescharacteristic, complemented
21#6411—32no—
22#6410—31no—
23#5062—71nomaximal, Hall
24C22×C3×D7—21no—
25C3×C7×Q8—22no—
26C2×C3×C4×C7D412yes—
27C2×C3×C4×C7C2311yescharacteristic
28C7×#67—22no—
29C7×D12—22no—
30C7×D12D412yes—
31C4×D3×C7—24no—
32C4×D3×C7—22no—
33C7×Dic6—22no—
34C7×Dic6D412yes—
35C2×C3×Dic7—21no—
36C3×D28—22no—
37C3⋊(C7⋊C8)—22no—
38C7×#41—32no—
39C7×#41—62no—
40C7×#41—62no—
41C2×C7×Q8—62no—
42C2×C7×Q8C2×D311yescharacteristic
43C2×D28C2×D311yescharacteristic
44#1143—62no—
45#1141—62no—
46(C2×D7)⋊C4—22no—
47C4×Dic7C2×D311yescharacteristic
48#1136—32no—
49#1007D711yescharacteristic, complemented
50C3×#124—71no—
51#932—72no—
52#836—71no—
53#805—72no—
54C22×C3×C7C2×D411yescharacteristic
55C2×D3×C7—22no—
56C2×C3×D7—42no—
57C2×C3×D7—41no—
58C3×C4×C7—22no—
59C3×C4×C7C2×D412yes—
60C3×Dic7—41no—
61C7×Dic3—22no—
62#454—211noSylow
63C22×D7—21no—
64C7×Q8—22no—
65C7×Q8—32no—
66C7×Q8—62no—
67C7×D4—32no—
68C7×D4—62no—
69C7×D4—62no—
70C2×C4×C7—64no—
71C2×C4×C7—62no—
72C2×C4×C7#6712yes—
73C2×C4×C7C22×D311yescharacteristic
74C2×Dic7—21no—
75D28—22no—
76C7⋊C8—62no—
77C2×C3×Q8C2×D711yescharacteristic
78C2×C3×D4—71no—
79#239—22no—
80D3×Q8—22no—
81#235—22no—
82#235C2×D712yes—
83C3×C22⋊C4—142no—
84C3×C42—71no—
85#214—142no—
86#213—142no—
87#208—72no—
88C2×C3×C7—21no—
89C2×C3×C7#12011yescharacteristic
90C3×D7—81no—
91D3×C7—42no—
922−1+4—31no—
93AGL(1,ℤ/8)—212no—
94#124—71no—
95#104—212no—
96#101—211no—
97C22×C7—62no—
98C22×C7C2×#6711yescharacteristic
99C2×D7—42no—
100C2×D7—41no—
101C4×C7—22no—
102C4×C7—62no—
103C4×C7D3×D412yes—
104Dic7—41no—
105C23×C3—141no—
106C3×Q8—22no—
107C3×D4—142no—
108C2×C3×C4—141no—
109C2×C3×C4C22⋊D712yes—
110C2×C3×C4C22×D711yescharacteristic
111#67—22no—
112D12—22no—
113D12C22⋊D712yes—
114C4×D3—24no—
115C4×D3—22no—
116Dic6—22no—
117Dic6C22⋊D712yes—
118C3⋊C8—142no—
119C3×C7#45411yescharacteristic, complemented, Hall
120#41—32no—
121#41—62no—
122#41—62no—
123C2×Q8—62no—
124C2×Q8D3×D711yescharacteristic
125C2×D4—71no—
126SD16—422no—
127D8—422no—
128M16—212no—
129C22⋊C4—142no—
130C42—71no—
131C2×C7—21no—
132C2×C7—122no—
133C2×C7#93711yescharacteristic
134D7—81no—
135C22×C3—282no—
136C22×C3—281no—
137C22×C3C2×C22⋊D711yescharacteristic
138C2×D3—22no—
139C3×C4—22no—
140C3×C4—281no—
141C3×C4C2×C22⋊D712yes—
142Dic3—22no—
143C23—141no—
144Q8—22no—
145Q8—32no—
146Q8—62no—
147D4—32no—
148D4—62no—
149D4—62no—
150D4—142no—
151C2×C4—64no—
152C2×C4—62no—
153C2×C4—141no—
154C2×C4#424512yes—
155C2×C4C2×D3×D711yescharacteristic
156C8—422no—
157C7#506211yescharacteristic, complemented, Sylow
158C2×C3—21no—
159C2×C3—561no—
160C2×C3#651611yescharacteristic
161D3—42no—
162C22—62no—
163C22—282no—
164C22—281no—
165C22C2×#424511yescharacteristic
166C4—22no—
167C4—62no—
168C4—281no—
169C4D3×C22⋊D712yes—
170C3#9001711yescharacteristic, complemented, Sylow
171C2—21no—
172C2—122no—
173C2—561no—
174C2#1062113011yescharacteristic
subgroupG/H
centerC2#10621130
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C2×D3×D7
FittingC2×C3×C7×Q8C22
socleC2×C3×C7#120
hypercenterC2×Q8D3×D7
hypocenterC3×C7#454
LCSG ▷ C2×C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
1. C7×#1007C2semidirect
2. C3×#6510C2semidirect
3. #10620860C2semidirect
4. #10620675C2semidirect
5. #10620514C2semidirect
20. #6510D3semidirect
49. #1007D7semidirect
119. C3×C7#454semidirect
157. C7#5062semidirect
170. C3#90017semidirect

Maximal subgroups

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

subgroupindex
C7×#10072
C3×#65102
#106208602
#106206752
#106205142
#900173
#50627