Finite Group Databasebeta

← #13061773

Subgroups — #13061773

Order 1344.

520 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 131 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
1C3×C7×#124C211yesmaximal, characteristic, complemented
2#10621430C212yesmaximal, complemented
3#10621420C212yesmaximal, complemented
4#10621368C214yesmaximal, complemented
5#10621365C214yesmaximal, complemented
6#10621358C212yesmaximal, complemented
7#90374—31nomaximal, Hall
8C23×D21C2212yescomplemented
9C2×C3×C7×Q8C2211yescharacteristic, complemented
10C2×C3×C7×D4C2211yescharacteristic
11C2×#4268C2214yes—
12C2×D84C2212yes—
13C2×D84C2212yes—
14C2×Dic42C2212yescomplemented
15C3×C7×C22⋊C4C2214yescomplemented
16C3×C42×C7C2211yescharacteristic, complemented
17#66636C2212yescomplemented
18#66631C2214yescomplemented
19#66631C2214yescomplemented
20#66631C2214yescomplemented
21C4×Dic21C2212yescomplemented
22C7×#124D311yescharacteristic, complemented
23#6510—32no—
24#6500—32no—
25#6448—34no—
26#6445—34no—
27#6438—32no—
28#5683—71nomaximal, Hall
29C23×C3×C7C2312yes—
30C22×D21—42no—
31C22×D21—44no—
32C22×D21C2314yes—
33C3×C7×Q8—41no—
34C3×C7×D4—41no—
35C2×C3×C4×C7C2312yes—
36C2×C3×C4×C7C2311yescharacteristic
37C2×C3×C4×C7C2312yes—
38#4268—44no—
39C2×Dic21C2314yes—
40D84—42no—
41D84—42no—
42Dic42—42no—
43C23×D7—32no—
44C2×C7×Q8C2×D311yescharacteristic, complemented
45C2×C7×D4C2×D311yescharacteristic
46C2×C22⋊D7—34no—
47C2×D28—32no—
48C2×D28—32no—
49C2×Dic14—32no—
50C7×C22⋊C4C2×D314yescomplemented
51C42×C7C2×D311yescharacteristic, complemented
52C22⋊Dic7—32no—
53(C2×D7)⋊C4—34no—
54(C2×D7)⋊C4—34no—
55(C2×D7)⋊C4—34no—
56C4×Dic7—32no—
57C3×#124D711yescharacteristic, complemented
58#947—72no—
59#937—72no—
60#885—74no—
61#882—74no—
62#875—72no—
63C22×C3×C7—24no—
64C22×C3×C7—22no—
65C22×C3×C7C2411yescharacteristic
66C2×D21—24no—
67C2×D21—24no—
68C2×D21—24no—
69C2×D21—48no—
70C3×C4×C7—22no—
71C3×C4×C7—22no—
72C3×C4×C7—21no—
73Dic21—24no—
74H(3,𝔽4)—211noSylow
75C23×C7C22×D312yes—
76C22×D7—34no—
77C22×D7—122no—
78C22×D7—124no—
79C7×Q8—41no—
80C7×D4—41no—
81C2×C4×C7C22×D312yes—
82C2×C4×C7C22×D311yescharacteristic
83C2×C4×C7C22×D312yes—
84C22⋊D7—124no—
85C2×Dic7—34no—
86D28—122no—
87D28—122no—
88Dic14—122no—
89C23×D3—72no—
90C2×C3×Q8C2×D711yescharacteristic, complemented
91C2×C3×D4C2×D711yescharacteristic
92C2×#67—74no—
93C2×D12—72no—
94C2×D12—72no—
95C2×Dic6—72no—
96C3×C22⋊C4C2×D714yescomplemented
97C3×C42C2×D711yescharacteristic, complemented
98#217—72no—
99(C2×D3)⋊C4—74no—
100(C2×D3)⋊C4—74no—
101(C2×D3)⋊C4—74no—
102C4×Dic3—72no—
103C2×C3×C7—42no—
104C2×C3×C7H(5,𝔽2)11yescharacteristic
105C2×C3×C7H(5,𝔽2)12yes—
106D21—44no—
107#124—212no—
108#124—214no—
109#124—212no—
110#124D2111yescharacteristic, complemented
111#120—212no—
112#120—214no—
113C22×C7—22no—
114C22×C7—24no—
115C22×C7C23×D311yescharacteristic
116C2×D7—64no—
117C2×D7—64no—
118C2×D7—64no—
119C2×D7—128no—
120C4×C7—22no—
121C4×C7—21no—
122C4×C7—22no—
123Dic7—64no—
124C23×C3C22×D712yes—
125C22×D3—74no—
126C22×D3—282no—
127C22×D3—284no—
128C3×Q8—41no—
129C3×D4—41no—
130C2×C3×C4C22×D712yes—
131C2×C3×C4C22×D711yescharacteristic
132C2×C3×C4C22×D712yes—
133#67—284no—
134C2×Dic3—74no—
135D12—282no—
136D12—282no—
137Dic6—282no—
138C3×C7H(3,𝔽4)11yescharacteristic, complemented, Hall
139C24—212no—
140C2×Q8—212no—
141C2×Q8C2×D2111yescharacteristic, complemented
142C2×D4—214no—
143C2×D4—212no—
144C2×D4—212no—
145C2×D4C2×D2111yescharacteristic
146C22⋊C4—212no—
147C22⋊C4—214no—
148C22⋊C4—214no—
149C22⋊C4—214no—
150C22⋊C4C2×D2114yescomplemented
151C42—212no—
152C42C2×D2111yescharacteristic, complemented
153C2×C7—42no—
154C2×C7#100411yescharacteristic
155C2×C7#100912yes—
156D7—124no—
157C22×C3—22no—
158C22×C3—24no—
159C22×C3C23×D711yescharacteristic
160C2×D3—144no—
161C2×D3—144no—
162C2×D3—144no—
163C2×D3—288no—
164C3×C4—22no—
165C3×C4—21no—
166C3×C4—22no—
167Dic3—144no—
168C23—214no—
169C23—842no—
170C23—844no—
171C23C22×D2112yes—
172Q8—41no—
173Q8—842no—
174D4—41no—
175D4—842no—
176D4—842no—
177D4—844no—
178C2×C4—214no—
179C2×C4C22×D2112yes—
180C2×C4C22×D2111yescharacteristic
181C2×C4C22×D2112yes—
182C7#568311yescharacteristic, complemented, Sylow
183C2×C3—42no—
184C2×C3#654811yescharacteristic
185C2×C3#655312yes—
186D3—284no—
187C22—22no—
188C22—24no—
189C22—424no—
190C22—424no—
191C22—424no—
192C22—848no—
193C22C23×D2111yescharacteristic
194C4—21no—
195C4—22no—
196C4—22no—
197C4—424no—
198C3#9037411yescharacteristic, complemented, Sylow
199C2—42no—
200C2—844no—
201C2#1062165711yescharacteristic
202C2#1062166212yes—
subgroupG/H
center, FrattiniC22C23×D21
commutator, socle, cosocleC22×C3×C7C24
FittingC3×C7×#124C2
hypercenter#124D21
hypocenterC3×C7H(3,𝔽4)
LCSG ▷ C22×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ #124

Complements

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

NH ≅ G/Ntype
1. C3×C7×#124C2semidirect
2. #10621430C2semidirect
3. #10621420C2semidirect
4. #10621368C2semidirect
5. #10621365C2semidirect
6. #10621358C2semidirect
8. C23×D21C22semidirect
9. C2×C3×C7×Q8C22semidirect
14. C2×Dic42C22semidirect
15. C3×C7×C22⋊C4C22semidirect
16. C3×C42×C7C22semidirect
17. #66636C22semidirect
18. #66631C22semidirect
19. #66631C22semidirect
20. #66631C22semidirect
21. C4×Dic21C22semidirect
22. C7×#124D3semidirect
44. C2×C7×Q8C2×D3semidirect
50. C7×C22⋊C4C2×D3semidirect
51. C42×C7C2×D3semidirect
57. C3×#124D7semidirect
90. C2×C3×Q8C2×D7semidirect
96. C3×C22⋊C4C2×D7semidirect
97. C3×C42C2×D7semidirect
110. #124D21semidirect
138. C3×C7H(3,𝔽4)semidirect
141. C2×Q8C2×D21semidirect
150. C22⋊C4C2×D21semidirect
152. C42C2×D21semidirect
182. C7#5683semidirect
198. C3#90374semidirect

Maximal subgroups

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

subgroupindex
C3×C7×#1242
#106214302
#106214202
#106213682
#106213652
#106213582
#903743
#56837