Finite Group Databasebeta

← #13054544

Subgroups — #13054544

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
2#10621419C211yesmaximal, characteristic, complemented
3#10620884C212yesmaximal, complemented
4C3×#6407C211yesmaximal, characteristic, complemented
5#10620513C212yesmaximal, complemented
6#90014—31nomaximal, Hall
7C2×C3×C7×D4C2211yescharacteristic
8C2×Dic42C2211yescharacteristic, complemented
9C2×C7×#67—22no—
10C7×#237—22no—
11D3×C7×D4—22no—
12C7×#235C2212yes—
13#66636—22no—
14C4×Dic21C2211yescharacteristic, complemented
15C3×#1136C2212yescomplemented
16#66571—22no—
17#66567—22no—
18C7×H(5,𝔽2)—31no—
19#6514—32no—
20#6499—31no—
21#6411—32no—
22#6407D311yescharacteristic, complemented
23#4746—71nomaximal, Hall
24C23×C3×C7D412yes—
25C22×D3×C7—22no—
26C3×C7×D4—22no—
27C2×C3×C4×C7C2311yescharacteristic
28C2×Dic21—21no—
29C2×Dic21—21no—
30Dic42—22no—
31C7×#67—24no—
32C7×#67—22no—
33C2×C7×Dic3—22no—
34C7×D12D412yes—
35C4×D3×C7—22no—
36C7×Dic6D412yes—
37C3×C7⋊C8—22no—
38C7×#41—32no—
39C7×#41—62no—
40C2×C7×D4—62no—
41C2×C7×D4—62no—
42C2×C7×D4C2×D311yescharacteristic
43C2×Dic14—31no—
44C22⋊Dic7—62no—
45#1144—62no—
46#1142—62no—
47C4×Dic7—31no—
48#1136C2×D312yescomplemented
49#1004D711yescharacteristic
50#936—71no—
51#909—72no—
52C3×#100—71no—
53#825—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—
60Dic21—41no—
61Dic21—41no—
62C7×Dic3—22no—
63#455—211noSylow
64C23×C7—62no—
65C23×C7D1212yes—
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—61no—
75C2×Dic7—61no—
76Dic14—62no—
77C7⋊C8—22no—
78C2×C3×D4C2×D711yescharacteristic
79C2×#67—22no—
80#237—22no—
81D3×D4—22no—
82#235C2×D712yes—
83C2×Dic6—71no—
84C3×M16—72no—
85#217—142no—
86C4×Dic3—71no—
87Dic12—142no—
88#204—142no—
89C2×C3×C7—21no—
90C2×C3×C7—42no—
91C2×C3×C7#12011yescharacteristic
92D3×C7—42no—
93H(5,𝔽2)—31no—
94#137—212no—
95#124—211no—
96#104—212no—
97#100—71no—
98C22×C7—22no—
99C22×C7—42no—
100C22×C7—62no—
101C22×C7—242no—
102C22×C7C2×D1211yescharacteristic
103C4×C7—62no—
104C4×C7D3×D412yes—
105Dic7—121no—
106Dic7—121no—
107C23×C3C22⋊D712yes—
108C22×D3—22no—
109C3×D4—22no—
110C2×C3×C4C22×D711yescharacteristic
111#67—24no—
112#67—22no—
113C2×Dic3—22no—
114C2×Dic3—141no—
115C2×Dic3—141no—
116D12C22⋊D712yes—
117C4×D3—22no—
118Dic6—142no—
119Dic6C22⋊D712yes—
120C3×C8—142no—
121C3×C7#45511yescharacteristic, complemented, Hall
122#41—32no—
123#41—62no—
124C2×Q8—211no—
125C2×D4—62no—
126C2×D4—62no—
127C2×D4D3×D711yescharacteristic
128Q16—422no—
129SD16—422no—
130M16—72no—
131C22⋊C4—422no—
132C42—211no—
133C2×C7—21no—
134C2×C7—42no—
135C2×C7—122no—
136C2×C7#88211yescharacteristic
137C22×C3—22no—
138C22×C3—42no—
139C22×C3C2×C22⋊D711yescharacteristic
140C2×D3—22no—
141C2×D3—82no—
142C3×C4C2×C22⋊D712yes—
143Dic3—22no—
144Dic3—281no—
145Dic3—281no—
146C23—62no—
147C23#424712yes—
148Q8—32no—
149Q8—422no—
150D4—22no—
151D4—32no—
152D4—64no—
153D4—62no—
154C2×C4—62no—
155C2×C4—62no—
156C2×C4—421no—
157C2×C4—421no—
158C2×C4C2×D3×D711yescharacteristic
159C8—142no—
160C7#474611yescharacteristic, complemented, Sylow
161C2×C3—21no—
162C2×C3—42no—
163C2×C3#651611yescharacteristic
164D3—42no—
165C22—22no—
166C22—42no—
167C22—62no—
168C22—242no—
169C22C2×#424711yescharacteristic
170C4—62no—
171C4—841no—
172C4—841no—
173C4D3×C22⋊D712yes—
174C3#9001411yescharacteristic, complemented, Sylow
175C2—21no—
176C2—42no—
177C2—122no—
178C2#1062113211yescharacteristic
subgroupG/H
centerC2#10621132
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. #10621419C2semidirect
3. #10620884C2semidirect
4. C3×#6407C2semidirect
5. #10620513C2semidirect
8. C2×Dic42C22semidirect
14. C4×Dic21C22semidirect
15. C3×#1136C22semidirect
22. #6407D3semidirect
48. #1136C2×D3semidirect
121. C3×C7#455semidirect
160. C7#4746semidirect
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
#106214192
#106208842
C3×#64072
#106205132
#900143
#47467