Finite Group Databasebeta

← #13054775

Subgroups — #13054775

Order 1344.

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

normal subgroups only

This group has 284 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
2C7×#944C211yesmaximal, characteristic, complemented
3#10620883C212yesmaximal, complemented
4C3×#6410C211yesmaximal, characteristic, complemented
5#10620515C212yesmaximal, complemented
6#89867—31nomaximal, Hall
7C2×C3×C7×Q8C2211yescharacteristic
8#66732—22no—
9Q8×D21—22no—
10#66728—22no—
11#66728C2212yes—
12C2×C7×Dic6C2211yescharacteristic
13C7×Dic3⋊C4—22no—
14C4×C7×Dic3C2211yescharacteristic
15C3×#1136C2212yes—
16#66571—22no—
17#66568—22no—
18#6551—31no—
19C7×C4⋊Q8—31no—
20#6505—32no—
21#6410D311yescharacteristic, complemented
22#6379—32no—
23#4752—71nomaximal, Hall
24C3×C7×Q8—22no—
25C2×C3×C4×C7D412yes—
26C2×C3×C4×C7C2311yescharacteristic
27#4268—22no—
28D84—22no—
29D84D412yes—
30C4×D21—24no—
31C4×D21—22no—
32Dic42—22no—
33Dic42D412yes—
34C2×C7×Dic3—21no—
35C2×C7×Dic3—21no—
36C7×Dic6—22no—
37C3×C7⋊C8—22no—
38C2×C7×Q8—31no—
39C2×C7×Q8C2×D311yescharacteristic
40#1161—62no—
41Q8×D7—62no—
42#1157—32no—
43#1157—62no—
44C7×C4⋊C4—62no—
45C42×C7—31no—
46#1144—62no—
47#1143—62no—
48#1136C2×D312yes—
49#1007—71no—
50#944D711yescharacteristic, complemented
51#909—72no—
52C3×#101—71no—
53#825—72no—
54C22×C3×C7C2×D411yescharacteristic
55C2×D21—22no—
56C3×C4×C7—22no—
57C3×C4×C7C2×D412yes—
58Dic21—22no—
59C7×Dic3—22no—
60C7×Dic3—41no—
61#456—211noSylow
62C7×Q8—22no—
63C7×Q8—62no—
64C2×C4×C7—61no—
65C2×C4×C7—61no—
66C2×C4×C7D1212yes—
67C2×C4×C7C22×D311yescharacteristic
68C22⋊D7—62no—
69D28—32no—
70D28—62no—
71C4×D7—64no—
72C4×D7—62no—
73Dic14—32no—
74Dic14—62no—
75C7⋊C8—22no—
76C2×C3×Q8C2×D711yescharacteristic
77#239—142no—
78D3×Q8—142no—
79#235—72no—
80#235—142no—
81C2×Dic6C2×D711yescharacteristic
82C3×M16—72no—
83Dic3⋊C4—22no—
84C4×Dic3C2×D711yescharacteristic
85Dic12—142no—
86#204—142no—
87C2×C3×C7—21no—
88C2×C3×C7#12011yescharacteristic
89D21—42no—
902−1+4—211no—
91#137—212no—
92C4⋊Q8—31no—
93#104—212no—
94#101—71no—
95C22×C7C2×D1211yescharacteristic
96C2×D7—62no—
97C4×C7—22no—
98C4×C7—62no—
99C4×C7—121no—
100C4×C7D3×D412yes—
101Dic7—62no—
102C3×Q8—22no—
103C2×C3×C4C22⋊D712yes—
104C2×C3×C4C22×D711yescharacteristic
105#67—142no—
106C2×Dic3—21no—
107C2×Dic3—21no—
108D12—72no—
109D12—142no—
110C4×D3—144no—
111C4×D3—142no—
112Dic6—22no—
113Dic6—72no—
114Dic6—142no—
115C3×C8—142no—
116C3×C7#45611yescharacteristic, complemented, Hall
117#41—212no—
118#41—422no—
119#41—422no—
120C2×Q8—31no—
121C2×Q8—422no—
122C2×Q8D3×D711yescharacteristic
123Q16—422no—
124SD16—422no—
125M16—72no—
126C4⋊C4—62no—
127C42—31no—
128C2×C7—21no—
129C2×C7#88211yescharacteristic
130D7—122no—
131C22×C3C2×C22⋊D711yescharacteristic
132C2×D3—142no—
133C3×C4—22no—
134C3×C4D4×D712yes—
135Dic3—22no—
136Dic3—41no—
137Dic3—142no—
138Q8—22no—
139Q8—62no—
140Q8—212no—
141Q8—422no—
142D4—212no—
143D4—422no—
144D4—422no—
145C2×C4—61no—
146C2×C4—61no—
147C2×C4—424no—
148C2×C4—422no—
149C2×C4#424712yes—
150C2×C4C2×D3×D711yescharacteristic
151C8—142no—
152C7#475211yescharacteristic, complemented, Sylow
153C2×C3—21no—
154C2×C3#650011yescharacteristic
155D3—282no—
156C22—422no—
157C22C2×#424711yescharacteristic
158C4—22no—
159C4—62no—
160C4—121no—
161C4—422no—
162C4#6669412yes—
163C3#8986711yescharacteristic, complemented, Sylow
164C2—21no—
165C2—842no—
166C2#1062113511yescharacteristic
subgroupG/H
centerC2#10621135
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C2×D3×D7
FittingC2×C3×C7×Q8C22
socleC2×C3×C7#120
prime radical#10621660C2
hypercenterC2×Q8D3×D7
hypocenterC3×C7#456
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
2. C7×#944C2semidirect
3. #10620883C2semidirect
4. C3×#6410C2semidirect
5. #10620515C2semidirect
21. #6410D3semidirect
50. #944D7semidirect
116. C3×C7#456semidirect
152. C7#4752semidirect
163. C3#89867semidirect

Maximal subgroups

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

subgroupindex
#106216602
C7×#9442
#106208832
C3×#64102
#106205152
#898673
#47527