Finite Group Databasebeta

← #13056698

Subgroups — #13056698

Order 1344.

304 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 304 conjugacy classes of subgroups — too many to draw. Only its 63 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×C7×#124C211yesmaximal, characteristic, complemented
3#10621425C212yesmaximal, complemented
4#10620672C211yesmaximal, characteristic, complemented
5#10620644C212yesmaximal, complemented
6#89833—31nomaximal, Hall
7C2×C3×C7×Q8C2211yescharacteristic, complemented
8C2×C3×C7×D4C2211yescharacteristic
9C2×#4268—22no—
10#66730—22no—
11D4×D21—22no—
12#66728C2212yes—
13C3×C7×C22⋊C4—22no—
14C3×C42×C7C2211yescharacteristic, complemented
15#66635—22no—
16#66634—22no—
17#66627C2212yescomplemented
18#6548—31no—
19C7×#124D311yescharacteristic, complemented
20#6505—32no—
21#6407—31no—
22#6379—32no—
23#5061—71nomaximal, Hall
24C23×C3×C7D412yes—
25C22×D21—22no—
26C3×C7×Q8—22no—
27C3×C7×D4—22no—
28C2×C3×C4×C7—21no—
29C2×C3×C4×C7—21no—
30C2×C3×C4×C7C2311yescharacteristic
31#4268—24no—
32#4268—22no—
33C2×Dic21—22no—
34D84D412yes—
35C4×D21—22no—
36Dic42D412yes—
37C3⋊(C7⋊C8)—22no—
38C2×C7×Q8C2×D311yescharacteristic, complemented
39C2×C7×D4C2×D311yescharacteristic
40C2×C22⋊D7—62no—
41#1159—62no—
42D4×D7—62no—
43#1157—32no—
44C7×C22⋊C4—22no—
45C42×C7C2×D311yescharacteristic, complemented
46#1144—62no—
47#1143—62no—
48#1136—32no—
49#1004—71no—
50C3×#124D711yescharacteristic, complemented
51#942—72no—
52#833—71no—
53#805—72no—
54C22×C3×C7—22no—
55C22×C3×C7—42no—
56C22×C3×C7C2×D411yescharacteristic
57C2×D21—22no—
58C2×D21—82no—
59C3×C4×C7—41no—
60C3×C4×C7—41no—
61C3×C4×C7C2×D412yes—
62Dic21—22no—
63#455—211noSylow
64C23×C7#6712yes—
65C22×D7—62no—
66C7×Q8—22no—
67C7×D4—22no—
68C2×C4×C7—21no—
69C2×C4×C7—21no—
70C2×C4×C7C22×D311yescharacteristic
71C22⋊D7—64no—
72C22⋊D7—62no—
73C2×Dic7—62no—
74D28—32no—
75C4×D7—62no—
76Dic14—32no—
77C7⋊C8—62no—
78C2×C3×Q8C2×D711yescharacteristic, complemented
79C2×C3×D4C2×D711yescharacteristic
80C2×#67—142no—
81#237—142no—
82D3×D4—142no—
83#235—72no—
84C3×C22⋊C4—22no—
85C3×C42C2×D711yescharacteristic, complemented
86#216—142no—
87#215—142no—
88#208—72no—
89C2×C3×C7—21no—
90C2×C3×C7—42no—
91C2×C3×C7#12011yescharacteristic
92D21—42no—
93H(5,𝔽2)—211no—
94#137—212no—
95#124D2111yescharacteristic, complemented
96#104—212no—
97#100—211no—
98C22×C7—22no—
99C22×C7—42no—
100C22×C7C2×#6711yescharacteristic
101C2×D7—62no—
102C2×D7—242no—
103C4×C7—41no—
104C4×C7—41no—
105C4×C7D3×D412yes—
106Dic7—62no—
107C23×C3C22⋊D712yes—
108C22×D3—142no—
109C3×Q8—22no—
110C3×D4—22no—
111C2×C3×C4—21no—
112C2×C3×C4—21no—
113C2×C3×C4C22×D711yescharacteristic
114#67—144no—
115#67—142no—
116C2×Dic3—142no—
117D12—72no—
118C4×D3—142no—
119Dic6—72no—
120C3⋊C8—142no—
121C3×C7#45511yescharacteristic, complemented, Hall
122#41—212no—
123#41—422no—
124C2×Q8C2×D2111yescharacteristic, complemented
125C2×D4—422no—
126C2×D4—422no—
127C2×D4C2×D2111yescharacteristic
128Q16—422no—
129SD16—422no—
130M16—212no—
131C22⋊C4—22no—
132C42C2×D2111yescharacteristic, complemented
133C2×C7—21no—
134C2×C7—42no—
135C2×C7#93711yescharacteristic
136D7—122no—
137C22×C3—22no—
138C22×C3—42no—
139C22×C3C2×C22⋊D711yescharacteristic
140C2×D3—142no—
141C2×D3—562no—
142C3×C4—41no—
143C3×C4—41no—
144C3×C4D4×D712yes—
145Dic3—142no—
146C23—422no—
147C23#426812yes—
148Q8—22no—
149Q8—212no—
150D4—22no—
151D4—212no—
152D4—424no—
153D4—422no—
154C2×C4—21no—
155C2×C4—21no—
156C2×C4—422no—
157C2×C4—422no—
158C2×C4C22×D2111yescharacteristic
159C8—422no—
160C7#506111yescharacteristic, complemented, Sylow
161C2×C3—21no—
162C2×C3—42no—
163C2×C3#650011yescharacteristic
164D3—282no—
165C22—22no—
166C22—42no—
167C22—422no—
168C22—1682no—
169C22C2×#426811yescharacteristic
170C4—41no—
171C4—41no—
172C4—422no—
173C4D4×D2112yes—
174C3#8983311yescharacteristic, complemented, Sylow
175C2—21no—
176C2—42no—
177C2—842no—
178C2#1062142011yescharacteristic
subgroupG/H
centerC2#10621420
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C22×D21
FittingC3×C7×#124C2
socleC2×C3×C7#120
prime radical#10621657C2
hypercenter#124D21
hypocenterC3×C7#455
LCSG ▷ C2×C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C2 ◁ C2×C4 ◁ #124

Complements

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

NH ≅ G/Ntype
2. C3×C7×#124C2semidirect
3. #10621425C2semidirect
4. #10620672C2semidirect
5. #10620644C2semidirect
7. C2×C3×C7×Q8C22semidirect
14. C3×C42×C7C22semidirect
17. #66627C22semidirect
19. C7×#124D3semidirect
38. C2×C7×Q8C2×D3semidirect
45. C42×C7C2×D3semidirect
50. C3×#124D7semidirect
78. C2×C3×Q8C2×D7semidirect
85. C3×C42C2×D7semidirect
95. #124D21semidirect
121. C3×C7#455semidirect
124. C2×Q8C2×D21semidirect
132. C42C2×D21semidirect
160. C7#5061semidirect
174. C3#89833semidirect

Maximal subgroups

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

subgroupindex
#106216572
C3×C7×#1242
#106214252
#106206722
#106206442
#898333
#50617