Finite Group Databasebeta

← #13054767

Subgroups — #13054767

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
1C3×#6551C211yesmaximal, characteristic
2C7×#944C211yesmaximal, characteristic, complemented
3#10620869C212yesmaximal, complemented
4#10620675C211yesmaximal, characteristic, complemented
5#10620512C212yesmaximal, complemented
6#89867—31nomaximal, Hall
7C2×C3×C7×Q8C2211yescharacteristic
8C2×C7×Dic6C2211yescharacteristic
9C3×#1161—22no—
10C3×Q8×D7—22no—
11C3×#1157—22no—
12C3×#1157C2212yes—
13#66627C2212yes—
14C7×Dic3⋊C4—22no—
15C4×C7×Dic3C2211yescharacteristic
16#66569—22no—
17#66563—22no—
18#6551D311yescharacteristic
19C7×C4⋊Q8—31no—
20#6505—32no—
21#6410—31no—
22#6379—32no—
23#5246—71nomaximal, Hall
24C3×C7×Q8—22no—
25C2×C3×C4×C7D412yes—
26C2×C3×C4×C7C2311yescharacteristic
27C2×C7×Dic3—21no—
28C2×C7×Dic3—21no—
29C7×Dic6—22no—
30C3×C22⋊D7—22no—
31C3×D28—22no—
32C3×D28D412yes—
33C3×C4×D7—24no—
34C3×C4×D7—22no—
35C3×Dic14—22no—
36C3×Dic14D412yes—
37C3⋊(C7⋊C8)—22no—
38C2×C7×Q8—31no—
39C2×C7×Q8C2×D311yescharacteristic
40#1161—22no—
41Q8×D7—22no—
42#1157—22no—
43#1157C2×D312yes—
44C7×C4⋊C4—62no—
45C42×C7—31no—
46#1144—62no—
47#1143—62no—
48#1136—32no—
49C3×2−1+4—71no—
50#951—72no—
51#944D711yescharacteristic, complemented
52#837—72no—
53#836—71no—
54C22×C3×C7C2×D411yescharacteristic
55C2×C3×D7—22no—
56C3×C4×C7—22no—
57C3×C4×C7C2×D412yes—
58C3×Dic7—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×C7#6712yes—
67C2×C4×C7C22×D311yescharacteristic
68C22⋊D7—22no—
69D28—22no—
70D28#6712yes—
71C4×D7—24no—
72C4×D7—22no—
73Dic14—22no—
74Dic14#6712yes—
75C7⋊C8—62no—
76C3×#41—72no—
77C3×#41—142no—
78C3×#41—142no—
79C2×C3×Q8—142no—
80C2×C3×Q8C2×D711yescharacteristic
81C2×Dic6C2×D711yescharacteristic
82#216—142no—
83#214—142no—
84Dic3⋊C4—22no—
85C4×Dic3C2×D711yescharacteristic
86#208—72no—
87C2×C3×C7—21no—
88C2×C3×C7#12011yescharacteristic
89C3×D7—42no—
902−1+4—71no—
91#137—212no—
92C4⋊Q8—31no—
93#104—212no—
94#101—211no—
95C22×C7C2×#6711yescharacteristic
96C2×D7—22no—
97C4×C7—22no—
98C4×C7—62no—
99C4×C7—121no—
100C4×C7C2×#6712yes—
101Dic7—22no—
102C3×Q8—22no—
103C3×Q8—72no—
104C3×Q8—142no—
105C3×D4—72no—
106C3×D4—142no—
107C3×D4—142no—
108C2×C3×C4—144no—
109C2×C3×C4—142no—
110C2×C3×C4C22⋊D712yes—
111C2×C3×C4C22×D711yescharacteristic
112C2×Dic3—21no—
113C2×Dic3—21no—
114Dic6—22no—
115C3⋊C8—142no—
116C3×C7#45611yescharacteristic, complemented, Hall
117#41—72no—
118#41—142no—
119#41—142no—
120C2×Q8—31no—
121C2×Q8—142no—
122C2×Q8D3×D711yescharacteristic
123Q16—422no—
124SD16—422no—
125M16—212no—
126C4⋊C4—62no—
127C42—31no—
128C2×C7—21no—
129C2×C7#95311yescharacteristic
130D7—42no—
131C22×C3—142no—
132C22×C3C2×C22⋊D711yescharacteristic
133C3×C4—22no—
134C3×C4—142no—
135C3×C4D4×D712yes—
136Dic3—22no—
137Dic3—41no—
138Q8—22no—
139Q8—62no—
140Q8—72no—
141Q8—142no—
142D4—72no—
143D4—142no—
144D4—142no—
145C2×C4—61no—
146C2×C4—61no—
147C2×C4—144no—
148C2×C4—142no—
149C2×C4#424512yes—
150C2×C4C2×D3×D711yescharacteristic
151C8—422no—
152C7#524611yescharacteristic, complemented, Sylow
153C2×C3—21no—
154C2×C3—282no—
155C2×C3#650011yescharacteristic
156C22—142no—
157C22C2×#424511yescharacteristic
158C4—22no—
159C4—62no—
160C4—121no—
161C4—142no—
162C4D7×#6712yes—
163C3#8986711yescharacteristic, complemented, Sylow
164C2—21no—
165C2—282no—
166C2#1062112511yescharacteristic
subgroupG/H
centerC2#10621125
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C2×D3×D7
FittingC2×C3×C7×Q8C22
socleC2×C3×C7#120
prime radicalC3×#6551C2
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. #10620869C2semidirect
4. #10620675C2semidirect
5. #10620512C2semidirect
51. #944D7semidirect
116. C3×C7#456semidirect
152. C7#5246semidirect
163. C3#89867semidirect

Maximal subgroups

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

subgroupindex
C3×#65512
C7×#9442
#106208692
#106206752
#106205122
#898673
#52467