Finite Group Databasebeta

← #13054763

Subgroups — #13054763

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
2#10621430C211yesmaximal, characteristic, complemented
3#10620867C212yesmaximal, complemented
4C3×#6410C211yesmaximal, characteristic, complemented
5#10620513C212yesmaximal, complemented
6#90017—31nomaximal, Hall
7C2×C3×C7×Q8C2211yescharacteristic
8C2×D84C2211yescharacteristic
9C7×#239—22no—
10D3×C7×Q8—22no—
11C7×#235—22no—
12C7×#235C2212yes—
13#66631—22no—
14C4×Dic21C2211yescharacteristic
15C3×#1136C2212yes—
16#66568—22no—
17#66562—22no—
18C7×2−1+4—31no—
19#6512—32no—
20#6510—31no—
21#6411—32no—
22#6410D311yescharacteristic, complemented
23#4753—71nomaximal, Hall
24C22×D21—21no—
25C3×C7×Q8—22no—
26C2×C3×C4×C7D412yes—
27C2×C3×C4×C7C2311yescharacteristic
28C2×Dic21—21no—
29D84—22no—
30C7×#67—22no—
31C7×D12—22no—
32C7×D12D412yes—
33C4×D3×C7—24no—
34C4×D3×C7—22no—
35C7×Dic6—22no—
36C7×Dic6D412yes—
37C3×C7⋊C8—22no—
38C7×#41—32no—
39C7×#41—62no—
40C7×#41—62no—
41C2×C7×Q8—62no—
42C2×C7×Q8C2×D311yescharacteristic
43C2×D28—31no—
44#1143—62no—
45#1141—62no—
46(C2×D7)⋊C4—62no—
47C4×Dic7—31no—
48#1136C2×D312yes—
49#1007D711yescharacteristic, complemented
50#947—71no—
51#908—72no—
52C3×#101—71no—
53#825—72no—
54C22×C3×C7C2×D411yescharacteristic
55C2×D21—42no—
56C2×D21—41no—
57C2×D3×C7—22no—
58C3×C4×C7—22no—
59C3×C4×C7C2×D412yes—
60Dic21—41no—
61C7×Dic3—22no—
62#454—211noSylow
63C22×D7—61no—
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×C7D1212yes—
73C2×C4×C7C22×D311yescharacteristic
74C2×Dic7—61no—
75D28—62no—
76C7⋊C8—22no—
77C2×C3×Q8C2×D711yescharacteristic
78#239—22no—
79D3×Q8—22no—
80#235—22no—
81#235C2×D712yes—
82C2×D12—71no—
83C3×M16—72no—
84(C2×D3)⋊C4—142no—
85C4×Dic3—71no—
86D24—142no—
87#204—142no—
88C2×C3×C7—21no—
89C2×C3×C7#12011yescharacteristic
90D21—81no—
91D3×C7—42no—
922−1+4—31no—
93AGL(1,ℤ/8)—212no—
94#124—211no—
95#104—212no—
96#101—71no—
97C22×C7—62no—
98C22×C7C2×D1211yescharacteristic
99C2×D7—122no—
100C2×D7—121no—
101C4×C7—22no—
102C4×C7—62no—
103C4×C7D3×D412yes—
104Dic7—121no—
105C22×D3—141no—
106C3×Q8—22no—
107C2×C3×C4C22⋊D712yes—
108C2×C3×C4C22×D711yescharacteristic
109#67—22no—
110C2×Dic3—141no—
111D12—22no—
112D12—142no—
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—211no—
126SD16—422no—
127D8—422no—
128M16—72no—
129C22⋊C4—422no—
130C42—211no—
131C2×C7—21no—
132C2×C7—122no—
133C2×C7#88211yescharacteristic
134D7—241no—
135C22×C3C2×C22⋊D711yescharacteristic
136C2×D3—22no—
137C2×D3—282no—
138C2×D3—281no—
139C3×C4—22no—
140C3×C4C2×C22⋊D712yes—
141Dic3—22no—
142Dic3—281no—
143C23—421no—
144Q8—22no—
145Q8—32no—
146Q8—62no—
147D4—32no—
148D4—62no—
149D4—62no—
150D4—422no—
151C2×C4—64no—
152C2×C4—62no—
153C2×C4—421no—
154C2×C4#424712yes—
155C2×C4C2×D3×D711yescharacteristic
156C8—142no—
157C7#475311yescharacteristic, complemented, Sylow
158C2×C3—21no—
159C2×C3#651611yescharacteristic
160D3—42no—
161D3—561no—
162C22—62no—
163C22—842no—
164C22—841no—
165C22C2×#424711yescharacteristic
166C4—22no—
167C4—62no—
168C4—841no—
169C4D3×C22⋊D712yes—
170C3#9001711yescharacteristic, complemented, Sylow
171C2—21no—
172C2—122no—
173C2—1681no—
174C2#1062113211yescharacteristic
subgroupG/H
centerC2#10621132
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. #10621430C2semidirect
3. #10620867C2semidirect
4. C3×#6410C2semidirect
5. #10620513C2semidirect
22. #6410D3semidirect
49. #1007D7semidirect
119. C3×C7#454semidirect
157. C7#4753semidirect
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
#106214302
#106208672
C3×#64102
#106205132
#900173
#47537