Finite Group Databasebeta

← #13054772

Subgroups — #13054772

Order 1344.

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

normal subgroups only

This group has 284 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
2#10621427C211yesmaximal, characteristic, complemented
3#10620884C212yesmaximal, complemented
4C3×#6410C211yesmaximal, characteristic, complemented
5#10620513C212yesmaximal, complemented
6#90018—31nomaximal, Hall
7C2×C3×C7×Q8C2211yescharacteristic
8C2×Dic42C2211yescharacteristic
9C7×#239—22no—
10D3×C7×Q8—22no—
11C7×#235—22no—
12C7×#235C2212yes—
13#66629—22no—
14C4×Dic21C2211yescharacteristic
15C3×#1136C2212yes—
16#66571—22no—
17#66567—22no—
18C7×2−1+4—31no—
19#6514—32no—
20#6507—31no—
21#6411—32no—
22#6410D311yescharacteristic, complemented
23#4752—71nomaximal, Hall
24C3×C7×Q8—22no—
25C2×C3×C4×C7D412yes—
26C2×C3×C4×C7C2311yescharacteristic
27C2×Dic21—21no—
28C2×Dic21—21no—
29Dic42—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×Dic14—31no—
44#1144—62no—
45#1142—62no—
46Dic7⋊C4—62no—
47C4×Dic7—31no—
48#1136C2×D312yes—
49#1007D711yescharacteristic
50#944—71no—
51#909—72no—
52C3×#101—71no—
53#825—72no—
54C22×C3×C7C2×D411yescharacteristic
55C2×D3×C7—22no—
56C3×C4×C7—22no—
57C3×C4×C7C2×D412yes—
58Dic21—22no—
59Dic21—41no—
60C7×Dic3—22no—
61#456—211noSylow
62C7×Q8—22no—
63C7×Q8—32no—
64C7×Q8—62no—
65C7×D4—32no—
66C7×D4—62no—
67C7×D4—62no—
68C2×C4×C7—64no—
69C2×C4×C7—62no—
70C2×C4×C7D1212yes—
71C2×C4×C7C22×D311yescharacteristic
72C2×Dic7—61no—
73C2×Dic7—61no—
74Dic14—62no—
75C7⋊C8—22no—
76C2×C3×Q8C2×D711yescharacteristic
77#239—22no—
78D3×Q8—22no—
79#235—22no—
80#235C2×D712yes—
81C2×Dic6—71no—
82C3×M16—72no—
83Dic3⋊C4—142no—
84C4×Dic3—71no—
85Dic12—142no—
86#204—142no—
87C2×C3×C7—21no—
88C2×C3×C7#12011yescharacteristic
89D3×C7—42no—
902−1+4—31no—
91#137—212no—
92C4⋊Q8—211no—
93#104—212no—
94#101—71no—
95C22×C7—62no—
96C22×C7C2×D1211yescharacteristic
97C4×C7—22no—
98C4×C7—62no—
99C4×C7D3×D412yes—
100Dic7—62no—
101Dic7—121no—
102C3×Q8—22no—
103C2×C3×C4C22⋊D712yes—
104C2×C3×C4C22×D711yescharacteristic
105#67—22no—
106C2×Dic3—141no—
107C2×Dic3—141no—
108D12—22no—
109D12C22⋊D712yes—
110C4×D3—24no—
111C4×D3—22no—
112Dic6—22no—
113Dic6—142no—
114Dic6C22⋊D712yes—
115C3×C8—142no—
116C3×C7#45611yescharacteristic, complemented, Hall
117#41—32no—
118#41—62no—
119#41—62no—
120C2×Q8—62no—
121C2×Q8—211no—
122C2×Q8D3×D711yescharacteristic
123Q16—422no—
124SD16—422no—
125M16—72no—
126C4⋊C4—422no—
127C42—211no—
128C2×C7—21no—
129C2×C7—122no—
130C2×C7#88211yescharacteristic
131C22×C3C2×C22⋊D711yescharacteristic
132C2×D3—22no—
133C3×C4—22no—
134C3×C4C2×C22⋊D712yes—
135Dic3—22no—
136Dic3—142no—
137Dic3—281no—
138Q8—22no—
139Q8—32no—
140Q8—62no—
141Q8—422no—
142D4—32no—
143D4—62no—
144D4—62no—
145C2×C4—64no—
146C2×C4—62no—
147C2×C4—421no—
148C2×C4—421no—
149C2×C4#424712yes—
150C2×C4C2×D3×D711yescharacteristic
151C8—142no—
152C7#475211yescharacteristic, complemented, Sylow
153C2×C3—21no—
154C2×C3#651611yescharacteristic
155D3—42no—
156C22—62no—
157C22C2×#424711yescharacteristic
158C4—22no—
159C4—62no—
160C4—422no—
161C4—841no—
162C4D3×C22⋊D712yes—
163C3#9001811yescharacteristic, complemented, Sylow
164C2—21no—
165C2—122no—
166C2#1062113211yescharacteristic
subgroupG/H
centerC2#10621132
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C2×D3×D7
FittingC2×C3×C7×Q8C22
socleC2×C3×C7#120
prime radicalC7×#1007C2
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. #10621427C2semidirect
3. #10620884C2semidirect
4. C3×#6410C2semidirect
5. #10620513C2semidirect
22. #6410D3semidirect
116. C3×C7#456semidirect
152. C7#4752semidirect
163. C3#90018semidirect

Maximal subgroups

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

subgroupindex
C7×#10072
#106214272
#106208842
C3×#64102
#106205132
#900183
#47527