Finite Group Databasebeta

← #13054547

Subgroups — #13054547

Order 1344.

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

normal subgroups only

This group has 304 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#10621657C211yesmaximal, characteristic
2C7×#936C211yesmaximal, characteristic, complemented
3#10620883C212yesmaximal, complemented
4C3×#6407C211yesmaximal, characteristic, complemented
5#10620515C212yesmaximal, complemented
6#89833—31nomaximal, Hall
7C2×C3×C7×D4C2211yescharacteristic
8C2×#4268—22no—
9#66730—22no—
10D4×D21—22no—
11#66728C2212yes—
12C2×C7×Dic6C2211yescharacteristic, complemented
13C7×#217—22no—
14C4×C7×Dic3C2211yescharacteristic, complemented
15C3×#1136C2212yescomplemented
16#66571—22no—
17#66568—22no—
18#6548—31no—
19C7×#124—31no—
20#6505—32no—
21#6407D311yescharacteristic, complemented
22#6379—32no—
23#4746—71nomaximal, Hall
24C23×C3×C7D412yes—
25C22×D21—22no—
26C3×C7×D4—22no—
27C2×C3×C4×C7C2311yescharacteristic
28#4268—24no—
29#4268—22no—
30C2×Dic21—22no—
31D84D412yes—
32C4×D21—22no—
33Dic42D412yes—
34C2×C7×Dic3—21no—
35C2×C7×Dic3—21no—
36C7×Dic6—22no—
37C3×C7⋊C8—22no—
38C2×C7×Q8—31no—
39C2×C7×D4C2×D311yescharacteristic
40C2×C22⋊D7—62no—
41#1159—62no—
42D4×D7—62no—
43#1157—32no—
44C7×C22⋊C4—62no—
45C42×C7—31no—
46#1144—62no—
47#1143—62no—
48#1136C2×D312yescomplemented
49#1004—71no—
50#936D711yescharacteristic, complemented
51#909—72no—
52C3×#100—71no—
53#825—72no—
54C22×C3×C7—22no—
55C22×C3×C7—42no—
56C22×C3×C7C2×D411yescharacteristic
57C2×D21—22no—
58C2×D21—82no—
59C3×C4×C7C2×D412yes—
60Dic21—22no—
61C7×Dic3—41no—
62C7×Dic3—41no—
63#455—211noSylow
64C23×C7D1212yes—
65C22×D7—62no—
66C7×Q8—62no—
67C7×D4—22no—
68C2×C4×C7—61no—
69C2×C4×C7—61no—
70C2×C4×C7C22×D311yescharacteristic
71C22⋊D7—64no—
72C22⋊D7—62no—
73C2×Dic7—62no—
74D28—32no—
75C4×D7—62no—
76Dic14—32no—
77C7⋊C8—22no—
78C2×C3×D4C2×D711yescharacteristic
79C2×#67—142no—
80#237—142no—
81D3×D4—142no—
82#235—72no—
83C2×Dic6C2×D711yescharacteristic, complemented
84C3×M16—72no—
85#217—22no—
86C4×Dic3C2×D711yescharacteristic, complemented
87Dic12—142no—
88#204—142no—
89C2×C3×C7—21no—
90C2×C3×C7—42no—
91C2×C3×C7#12011yescharacteristic
92D21—42no—
93H(5,𝔽2)—211no—
94#137—212no—
95#124—31no—
96#104—212no—
97#100—71no—
98C22×C7—22no—
99C22×C7—42no—
100C22×C7C2×D1211yescharacteristic
101C2×D7—62no—
102C2×D7—242no—
103C4×C7—121no—
104C4×C7—121no—
105C4×C7D3×D412yes—
106Dic7—62no—
107C23×C3C22⋊D712yes—
108C22×D3—142no—
109C3×D4—22no—
110C2×C3×C4C22×D711yescharacteristic
111#67—144no—
112#67—142no—
113C2×Dic3—21no—
114C2×Dic3—21no—
115C2×Dic3—142no—
116D12—72no—
117C4×D3—142no—
118Dic6—22no—
119Dic6—72no—
120C3×C8—142no—
121C3×C7#45511yescharacteristic, complemented, Hall
122#41—212no—
123#41—422no—
124C2×Q8—31no—
125C2×D4—422no—
126C2×D4—422no—
127C2×D4D3×D711yescharacteristic
128Q16—422no—
129SD16—422no—
130M16—72no—
131C22⋊C4—62no—
132C42—31no—
133C2×C7—21no—
134C2×C7—42no—
135C2×C7#88211yescharacteristic
136D7—122no—
137C22×C3—22no—
138C22×C3—42no—
139C22×C3C2×C22⋊D711yescharacteristic
140C2×D3—142no—
141C2×D3—562no—
142C3×C4D4×D712yes—
143Dic3—41no—
144Dic3—41no—
145Dic3—142no—
146C23—422no—
147C23#424712yes—
148Q8—62no—
149Q8—212no—
150D4—22no—
151D4—212no—
152D4—424no—
153D4—422no—
154C2×C4—61no—
155C2×C4—61no—
156C2×C4—422no—
157C2×C4—422no—
158C2×C4C2×D3×D711yescharacteristic
159C8—142no—
160C7#474611yescharacteristic, complemented, Sylow
161C2×C3—21no—
162C2×C3—42no—
163C2×C3#650011yescharacteristic
164D3—282no—
165C22—22no—
166C22—42no—
167C22—422no—
168C22—1682no—
169C22C2×#424711yescharacteristic
170C4—121no—
171C4—121no—
172C4—422no—
173C4#6669412yes—
174C3#8983311yescharacteristic, complemented, Sylow
175C2—21no—
176C2—42no—
177C2—842no—
178C2#1062113511yescharacteristic
subgroupG/H
centerC2#10621135
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C2×D3×D7
FittingC2×C3×C7×D4C22
socleC2×C3×C7#120
prime radical#10621657C2
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. C7×#936C2semidirect
3. #10620883C2semidirect
4. C3×#6407C2semidirect
5. #10620515C2semidirect
12. C2×C7×Dic6C22semidirect
14. C4×C7×Dic3C22semidirect
15. C3×#1136C22semidirect
21. #6407D3semidirect
48. #1136C2×D3semidirect
50. #936D7semidirect
83. C2×Dic6C2×D7semidirect
86. C4×Dic3C2×D7semidirect
121. C3×C7#455semidirect
160. C7#4746semidirect
174. C3#89833semidirect

Maximal subgroups

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

subgroupindex
#106216572
C7×#9362
#106208832
C3×#64072
#106205152
#898333
#47467