Finite Group Databasebeta

← #13054546

Subgroups — #13054546

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
2C3×#6499C211yesmaximal, characteristic, complemented
3#10620877C212yesmaximal, complemented
4C7×#833C211yesmaximal, characteristic, complemented
5#10620516C212yesmaximal, complemented
6#89518—31nomaximal, Hall
7C2×C3×C7×D4C2211yescharacteristic
8C2×#4268—22no—
9#66730—22no—
10D4×D21—22no—
11#66728C2212yes—
12C2×C3×Dic14C2211yescharacteristic, complemented
13C7×#208C2212yescomplemented
14C3×C22⋊Dic7—22no—
15C3×C4×Dic7C2211yescharacteristic, complemented
16#66570—22no—
17#66566—22no—
18#6548—31no—
19#6499D311yescharacteristic, complemented
20#6472—32no—
21C7×#100—31no—
22#6399—32no—
23#5061—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×C3×Dic7—21no—
35C2×C3×Dic7—21no—
36C3×Dic14—22no—
37C7×C3⋊C8—22no—
38C2×C7×D4C2×D311yescharacteristic
39C2×C22⋊D7—62no—
40#1159—62no—
41D4×D7—62no—
42#1157—32no—
43C2×Dic14C2×D311yescharacteristic, complemented
44C7×M16—32no—
45C22⋊Dic7—22no—
46C4×Dic7C2×D311yescharacteristic, complemented
47Dic28—62no—
48#1132—62no—
49#1004—71no—
50C3×#124—71no—
51#942—72no—
52#833D711yescharacteristic, complemented
53#805—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—
61C3×Dic7—41no—
62C3×Dic7—41no—
63#455—211noSylow
64C23×C7#6712yes—
65C22×D7—62no—
66C7×D4—22no—
67C2×C4×C7C22×D311yescharacteristic
68C22⋊D7—64no—
69C22⋊D7—62no—
70C2×Dic7—21no—
71C2×Dic7—21no—
72C2×Dic7—62no—
73D28—32no—
74C4×D7—62no—
75Dic14—22no—
76Dic14—32no—
77C7×C8—62no—
78C2×C3×Q8—71no—
79C2×C3×D4C2×D711yescharacteristic
80C2×#67—142no—
81#237—142no—
82D3×D4—142no—
83#235—72no—
84C3×C22⋊C4—142no—
85C3×C42—71no—
86#216—142no—
87#215—142no—
88#208C2×D712yescomplemented
89C2×C3×C7—21no—
90C2×C3×C7—42no—
91C2×C3×C7#12011yescharacteristic
92D21—42no—
93H(5,𝔽2)—211no—
94#137—212no—
95#124—71no—
96#104—212no—
97#100—31no—
98C22×C7—22no—
99C22×C7—42no—
100C22×C7C2×#6711yescharacteristic
101C2×D7—62no—
102C2×D7—242no—
103C4×C7D3×D412yes—
104Dic7—41no—
105Dic7—41no—
106Dic7—62no—
107C23×C3D2812yes—
108C22×D3—142no—
109C3×Q8—142no—
110C3×D4—22no—
111C2×C3×C4—141no—
112C2×C3×C4—141no—
113C2×C3×C4C22×D711yescharacteristic
114#67—144no—
115#67—142no—
116C2×Dic3—142no—
117D12—72no—
118C4×D3—142no—
119Dic6—72no—
120C3⋊C8—22no—
121C3×C7#45511yescharacteristic, complemented, Hall
122#41—212no—
123#41—422no—
124C2×Q8—71no—
125C2×D4—422no—
126C2×D4—422no—
127C2×D4D3×D711yescharacteristic
128Q16—422no—
129SD16—422no—
130M16—32no—
131C22⋊C4—142no—
132C42—71no—
133C2×C7—21no—
134C2×C7—42no—
135C2×C7#93711yescharacteristic
136D7—122no—
137C22×C3—22no—
138C22×C3—42no—
139C22×C3C2×D2811yescharacteristic
140C2×D3—142no—
141C2×D3—562no—
142C3×C4—281no—
143C3×C4—281no—
144C3×C4D4×D712yes—
145Dic3—142no—
146C23—422no—
147C23#424612yes—
148Q8—142no—
149Q8—212no—
150D4—22no—
151D4—212no—
152D4—424no—
153D4—422no—
154C2×C4—141no—
155C2×C4—141no—
156C2×C4—422no—
157C2×C4—422no—
158C2×C4C2×D3×D711yescharacteristic
159C8—62no—
160C7#506111yescharacteristic, complemented, Sylow
161C2×C3—21no—
162C2×C3—42no—
163C2×C3#644511yescharacteristic
164D3—282no—
165C22—22no—
166C22—42no—
167C22—422no—
168C22—1682no—
169C22C2×#424611yescharacteristic
170C4—281no—
171C4—281no—
172C4—422no—
173C4#6669412yes—
174C3#8951811yescharacteristic, complemented, Sylow
175C2—21no—
176C2—42no—
177C2—842no—
178C2#1062113411yescharacteristic
subgroupG/H
centerC2#10621134
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. C3×#6499C2semidirect
3. #10620877C2semidirect
4. C7×#833C2semidirect
5. #10620516C2semidirect
12. C2×C3×Dic14C22semidirect
13. C7×#208C22semidirect
15. C3×C4×Dic7C22semidirect
19. #6499D3semidirect
43. C2×Dic14C2×D3semidirect
46. C4×Dic7C2×D3semidirect
52. #833D7semidirect
88. #208C2×D7semidirect
121. C3×C7#455semidirect
160. C7#5061semidirect
174. C3#89518semidirect

Maximal subgroups

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

subgroupindex
#106216572
C3×#64992
#106208772
C7×#8332
#106205162
#895183
#50617