Finite Group Databasebeta

← #13054543

Subgroups — #13054543

Order 1344.

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

normal subgroups only

This group has 304 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×#1004C211yesmaximal, characteristic
2C3×#6499C211yesmaximal, characteristic, complemented
3#10620871C212yesmaximal, complemented
4#10620672C211yesmaximal, characteristic, complemented
5#10620514C212yesmaximal, complemented
6#90014—31nomaximal, Hall
7C2×C3×C7×D4C2211yescharacteristic
8C2×C7×#67—22no—
9C7×#237—22no—
10D3×C7×D4—22no—
11C7×#235C2212yes—
12C2×C3×Dic14C2211yescharacteristic, complemented
13#66627C2212yescomplemented
14C3×C22⋊Dic7—22no—
15C3×C4×Dic7C2211yescharacteristic, complemented
16#66569—22no—
17#66564—22no—
18C7×H(5,𝔽2)—31no—
19#6514—32no—
20#6499D311yescharacteristic, complemented
21#6411—32no—
22#6407—31no—
23#5061—71nomaximal, Hall
24C23×C3×C7D412yes—
25C22×D3×C7—22no—
26C3×C7×D4—22no—
27C2×C3×C4×C7C2311yescharacteristic
28C7×#67—24no—
29C7×#67—22no—
30C2×C7×Dic3—22no—
31C7×D12D412yes—
32C4×D3×C7—22no—
33C7×Dic6D412yes—
34C2×C3×Dic7—21no—
35C2×C3×Dic7—21no—
36C3×Dic14—22no—
37C3⋊(C7⋊C8)—22no—
38C7×#41—32no—
39C7×#41—62no—
40C2×C7×D4—62no—
41C2×C7×D4—62no—
42C2×C7×D4C2×D311yescharacteristic
43C2×Dic14C2×D311yescharacteristic, complemented
44C22⋊Dic7—22no—
45#1144—62no—
46#1142—62no—
47C4×Dic7C2×D311yescharacteristic, complemented
48#1136—32no—
49#1004D711yescharacteristic
50C3×#124—71no—
51#942—72no—
52#833—71no—
53#805—72no—
54C22×C3×C7—22no—
55C22×C3×C7—42no—
56C22×C3×C7C2×D411yescharacteristic
57C2×D3×C7—22no—
58C2×D3×C7—82no—
59C3×C4×C7C2×D412yes—
60C3×Dic7—41no—
61C3×Dic7—41no—
62C7×Dic3—22no—
63#455—211noSylow
64C23×C7—62no—
65C23×C7#6712yes—
66C7×Q8—32no—
67C7×D4—22no—
68C7×D4—32no—
69C7×D4—64no—
70C7×D4—62no—
71C2×C4×C7—62no—
72C2×C4×C7—62no—
73C2×C4×C7C22×D311yescharacteristic
74C2×Dic7—21no—
75C2×Dic7—21no—
76Dic14—22no—
77C7⋊C8—62no—
78C2×C3×Q8—71no—
79C2×C3×D4C2×D711yescharacteristic
80C2×#67—22no—
81#237—22no—
82D3×D4—22no—
83#235C2×D712yes—
84C3×C22⋊C4—142no—
85C3×C42—71no—
86#216—142no—
87#215—142no—
88#208—72no—
89C2×C3×C7—21no—
90C2×C3×C7—42no—
91C2×C3×C7#12011yescharacteristic
92D3×C7—42no—
93H(5,𝔽2)—31no—
94#137—212no—
95#124—71no—
96#104—212no—
97#100—211no—
98C22×C7—22no—
99C22×C7—42no—
100C22×C7—62no—
101C22×C7—242no—
102C22×C7C2×#6711yescharacteristic
103C4×C7—62no—
104C4×C7D3×D412yes—
105Dic7—41no—
106Dic7—41no—
107C23×C3C22⋊D712yes—
108C22×D3—22no—
109C3×Q8—142no—
110C3×D4—22no—
111C2×C3×C4—141no—
112C2×C3×C4—141no—
113C2×C3×C4C22×D711yescharacteristic
114#67—24no—
115#67—22no—
116C2×Dic3—22no—
117D12C22⋊D712yes—
118C4×D3—22no—
119Dic6C22⋊D712yes—
120C3⋊C8—142no—
121C3×C7#45511yescharacteristic, complemented, Hall
122#41—32no—
123#41—62no—
124C2×Q8—71no—
125C2×D4—62no—
126C2×D4—62no—
127C2×D4D3×D711yescharacteristic
128Q16—422no—
129SD16—422no—
130M16—212no—
131C22⋊C4—142no—
132C42—71no—
133C2×C7—21no—
134C2×C7—42no—
135C2×C7—122no—
136C2×C7#93711yescharacteristic
137C22×C3—22no—
138C22×C3—42no—
139C22×C3C2×C22⋊D711yescharacteristic
140C2×D3—22no—
141C2×D3—82no—
142C3×C4—281no—
143C3×C4—281no—
144C3×C4C2×C22⋊D712yes—
145Dic3—22no—
146C23—62no—
147C23#424512yes—
148Q8—32no—
149Q8—142no—
150D4—22no—
151D4—32no—
152D4—64no—
153D4—62no—
154C2×C4—62no—
155C2×C4—62no—
156C2×C4—141no—
157C2×C4—141no—
158C2×C4C2×D3×D711yescharacteristic
159C8—422no—
160C7#506111yescharacteristic, complemented, Sylow
161C2×C3—21no—
162C2×C3—42no—
163C2×C3#651611yescharacteristic
164D3—42no—
165C22—22no—
166C22—42no—
167C22—62no—
168C22—242no—
169C22C2×#424511yescharacteristic
170C4—62no—
171C4—281no—
172C4—281no—
173C4D3×C22⋊D712yes—
174C3#9001411yescharacteristic, complemented, Sylow
175C2—21no—
176C2—42no—
177C2—122no—
178C2#1062113011yescharacteristic
subgroupG/H
centerC2#10621130
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C2×D3×D7
FittingC2×C3×C7×D4C22
socleC2×C3×C7#120
prime radicalC7×#1004C2
hypercenterC2×D4D3×D7
hypocenterC3×C7#455
LCSG ▷ C2×C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×D4

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C3×#6499C2semidirect
3. #10620871C2semidirect
4. #10620672C2semidirect
5. #10620514C2semidirect
12. C2×C3×Dic14C22semidirect
13. #66627C22semidirect
15. C3×C4×Dic7C22semidirect
20. #6499D3semidirect
43. C2×Dic14C2×D3semidirect
47. C4×Dic7C2×D3semidirect
121. C3×C7#455semidirect
160. C7#5061semidirect
174. C3#90014semidirect

Maximal subgroups

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

subgroupindex
C7×#10042
C3×#64992
#106208712
#106206722
#106205142
#900143
#50617