Finite Group Databasebeta

← #13054766

Subgroups — #13054766

Order 1344.

292 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 60 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#10621660C211yesmaximal, characteristic, complemented
2C7×#947C211yesmaximal, characteristic, complemented
3#10620866C212yesmaximal, complemented
4C3×#6410C211yesmaximal, characteristic, complemented
5#10620515C212yesmaximal, complemented
6#89834—31nomaximal, Hall
7C2×C3×C7×Q8C2211yescharacteristic
8#66732—22no—
9Q8×D21—22no—
10#66728—22no—
11#66728C2212yes—
12C2×C7×D12C2211yescharacteristic
13C7×(C2×D3)⋊C4—22no—
14C4×C7×Dic3C2211yescharacteristic
15C3×#1136C2212yes—
16#66567—22no—
17#66562—22no—
18#6551—31no—
19C7×#124—31no—
20#6495—32no—
21#6410D311yescharacteristic, complemented
22#6379—32no—
23#4753—71nomaximal, Hall
24C22×D3×C7—21no—
25C3×C7×Q8—22no—
26C2×C3×C4×C7D412yes—
27C2×C3×C4×C7C2311yescharacteristic
28#4268—22no—
29D84—22no—
30D84D412yes—
31C4×D21—24no—
32C4×D21—22no—
33Dic42—22no—
34Dic42D412yes—
35C2×C7×Dic3—21no—
36C7×D12—22no—
37C3×C7⋊C8—22no—
38C2×C7×Q8C2×D311yescharacteristic
39C2×C7×D4—31no—
40#1161—62no—
41Q8×D7—62no—
42#1157—32no—
43#1157—62no—
44C7×C22⋊C4—62no—
45C42×C7—31no—
46#1142—62no—
47#1141—62no—
48#1136C2×D312yes—
49#1007—71no—
50#947D711yescharacteristic, complemented
51#908—72no—
52C3×#101—71no—
53#825—72no—
54C22×C3×C7C2×D411yescharacteristic
55C2×D21—22no—
56C2×D3×C7—42no—
57C2×D3×C7—41no—
58C3×C4×C7—22no—
59C3×C4×C7C2×D412yes—
60Dic21—22no—
61C7×Dic3—41no—
62#454—211noSylow
63C23×C7—61no—
64C7×Q8—22no—
65C7×D4—62no—
66C2×C4×C7—61no—
67C2×C4×C7D1212yes—
68C2×C4×C7C22×D311yescharacteristic
69C22⋊D7—62no—
70D28—32no—
71D28—62no—
72C4×D7—64no—
73C4×D7—62no—
74Dic14—32no—
75Dic14—62no—
76C7⋊C8—22no—
77C2×C3×Q8C2×D711yescharacteristic
78#239—142no—
79D3×Q8—142no—
80#235—72no—
81#235—142no—
82C2×D12C2×D711yescharacteristic
83C3×M16—72no—
84(C2×D3)⋊C4—22no—
85C4×Dic3C2×D711yescharacteristic
86D24—142no—
87#204—142no—
88C2×C3×C7—21no—
89C2×C3×C7#12011yescharacteristic
90D21—42no—
91D3×C7—81no—
922−1+4—211no—
93AGL(1,ℤ/8)—212no—
94#124—31no—
95#104—212no—
96#101—71no—
97C22×C7—122no—
98C22×C7—121no—
99C22×C7C2×D1211yescharacteristic
100C2×D7—62no—
101C4×C7—22no—
102C4×C7—121no—
103C4×C7D3×D412yes—
104Dic7—62no—
105C22×D3—21no—
106C3×Q8—22no—
107C2×C3×C4C22⋊D712yes—
108C2×C3×C4C22×D711yescharacteristic
109#67—142no—
110C2×Dic3—21no—
111D12—22no—
112D12—72no—
113D12—142no—
114C4×D3—144no—
115C4×D3—142no—
116Dic6—72no—
117Dic6—142no—
118C3×C8—142no—
119C3×C7#45411yescharacteristic, complemented, Hall
120#41—212no—
121#41—422no—
122#41—422no—
123C2×Q8—422no—
124C2×Q8D3×D711yescharacteristic
125C2×D4—31no—
126SD16—422no—
127D8—422no—
128M16—72no—
129C22⋊C4—62no—
130C42—31no—
131C2×C7—21no—
132C2×C7—241no—
133C2×C7#88211yescharacteristic
134D7—122no—
135C22×C3C2×C22⋊D711yescharacteristic
136C2×D3—42no—
137C2×D3—41no—
138C2×D3—142no—
139C3×C4—22no—
140C3×C4D4×D712yes—
141Dic3—41no—
142Dic3—142no—
143C23—61no—
144Q8—22no—
145Q8—212no—
146Q8—422no—
147D4—62no—
148D4—212no—
149D4—422no—
150D4—422no—
151C2×C4—61no—
152C2×C4—424no—
153C2×C4—422no—
154C2×C4#424712yes—
155C2×C4C2×D3×D711yescharacteristic
156C8—142no—
157C7#475311yescharacteristic, complemented, Sylow
158C2×C3—21no—
159C2×C3#650011yescharacteristic
160D3—81no—
161D3—282no—
162C22—122no—
163C22—121no—
164C22—422no—
165C22C2×#424711yescharacteristic
166C4—22no—
167C4—121no—
168C4—422no—
169C4#6669412yes—
170C3#8983411yescharacteristic, complemented, Sylow
171C2—21no—
172C2—241no—
173C2—842no—
174C2#1062113511yescharacteristic
subgroupG/H
centerC2#10621135
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. #10621660C2semidirect
2. C7×#947C2semidirect
3. #10620866C2semidirect
4. C3×#6410C2semidirect
5. #10620515C2semidirect
21. #6410D3semidirect
50. #947D7semidirect
119. C3×C7#454semidirect
157. C7#4753semidirect
170. C3#89834semidirect

Maximal subgroups

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

subgroupindex
#106216602
C7×#9472
#106208662
C3×#64102
#106205152
#898343
#47537