Finite Group Databasebeta

← C2×C3×#6503 #13061136

Subgroups — C2×C3×#6503 #13061136

Order 1344.

996 conjugacy classes of subgroups; only the 286 normal subgroups are shown.

normal subgroups only

This group has 996 conjugacy classes of subgroups — too many to draw. Only its 286 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
1C22×C3×C7×D4C211yesmaximal, characteristic, complemented
2C22×C3×C22⋊D7C214yesmaximal, complemented
3C22×C3×D28C211yesmaximal, characteristic, complemented
4C3×#6503C218yesmaximal, direct product factor
5C2×C3×C4×Dic7C211yesmaximal, characteristic, complemented
6C2×#6503C311yesmaximal, characteristic, direct product factor, Hall
7C24×C3×C7C2212yes—
8C23×C3×D7C2212yes—
9C2×C3×C7×D4—24no—
10C2×C3×C7×D4C2214yescomplemented
11C22×C3×C4×C7C2211yescharacteristic
12C2×C3×C22⋊D7—216no—
13C2×C3×C22⋊D7C22116yescomplemented
14C22×C3×Dic7C2212yes—
15C2×C3×D28—24no—
16C2×C3×D28C2214yescomplemented
17C3×C4×Dic7C2214yescomplemented
18C22×C7×D4C2×C311yescharacteristic, complemented
19C22×C22⋊D7C2×C314yescomplemented
20C22×D28C2×C311yescharacteristic, complemented
21#6503C2×C318yesdirect product factor
22C2×C4×Dic7C2×C311yescharacteristic, complemented
23C2×C3×#127—71nomaximal, Hall
24C23×C3×C7—28no—
25C23×C3×C7—24no—
26C23×C3×C7C2314yes—
27C23×C3×C7C2311yescharacteristic
28C22×C3×D7—24no—
29C22×C3×D7—28no—
30C22×C3×D7C2314yes—
31C3×C7×D4—28no—
32C2×C3×C4×C7D414yescomplemented
33C2×C3×C4×C7C2312yes—
34C3×C22⋊D7—232no—
35C2×C3×Dic7D418yescomplemented
36C2×C3×Dic7C2314yes—
37C3×D28—28no—
38C24×C7C22×C312yes—
39C23×D7C22×C312yes—
40C2×C7×D4—24no—
41C2×C7×D4C22×C314yescomplemented
42C22×C4×C7C22×C311yescharacteristic
43C2×C22⋊D7—216no—
44C2×C22⋊D7C22×C3116yescomplemented
45C22×Dic7C22×C312yes—
46C2×D28—24no—
47C2×D28C22×C314yescomplemented
48C4×Dic7C22×C314yescomplemented
49C22×C3×D4—74no—
50C22×C3×D4—71no—
51C22×C3×D4D711yescharacteristic, complemented
52C3×#127—78no—
53C2×C3×C42—71no—
54C22×C3×C7—28no—
55C22×C3×C7—24no—
56C22×C3×C7—48no—
57C22×C3×C7C2×D414yes—
58C22×C3×C7C2×D412yes—
59C22×C3×C7C2411yescharacteristic
60C2×C3×D7—24no—
61C2×C3×D7—28no—
62C2×C3×D7—48no—
63C3×C4×C7C2×D414yescomplemented
64C3×Dic7C2×D418yescomplemented
65C2×#127—71noSylow
66C23×C7—28no—
67C23×C7—24no—
68C23×C7C23×C314yes—
69C23×C7C23×C311yescharacteristic
70C22×D7—24no—
71C22×D7—28no—
72C22×D7C23×C314yes—
73C7×D4—28no—
74C2×C4×C7C3×D414yescomplemented
75C2×C4×C7C23×C312yes—
76C22⋊D7—232no—
77C2×Dic7C3×D418yescomplemented
78C2×Dic7C23×C314yes—
79D28—28no—
80C24×C3—72no—
81C24×C3C2×D712yes—
82C2×C3×D4—24no—
83C2×C3×D4—716no—
84C2×C3×D4—74no—
85C2×C3×D4—1416no—
86C2×C3×D4—144no—
87C2×C3×D4C2×D714yescomplemented
88C22×C3×C4—72no—
89C22×C3×C4C2×D711yescharacteristic
90C3×C42—74no—
91C2×C3×C7—44no—
92C2×C3×C7#12714yescomplemented
93C2×C3×C7C22×D412yes—
94C2×C3×C7C22×D411yescharacteristic
95C3×D7—44no—
96C22×D4—74no—
97C22×D4—71no—
98C22×D4C3×D711yescharacteristic, complemented
99#127—78no—
100C2×C42—71no—
101C22×C7—28no—
102C22×C7—24no—
103C22×C7—48no—
104C22×C7C2×C3×D414yes—
105C22×C7C2×C3×D412yes—
106C22×C7C24×C311yescharacteristic
107C2×D7—24no—
108C2×D7—28no—
109C2×D7—48no—
110C4×C7C2×C3×D414yescomplemented
111Dic7C2×C3×D418yescomplemented
112C23×C3—28no—
113C23×C3—24no—
114C23×C3—74no—
115C23×C3—148no—
116C23×C3—144no—
117C23×C3C22×D714yes—
118C23×C3C22×D711yescharacteristic
119C3×D4—28no—
120C3×D4—1432no—
121C3×D4—148no—
122C2×C3×C4—78no—
123C2×C3×C4—74no—
124C2×C3×C4C22⋊D714yescomplemented
125C2×C3×C4C22×D712yes—
126C3×C7C2×#12711yescharacteristic, complemented, Hall
127C24—72no—
128C24C2×C3×D712yes—
129C2×D4—24no—
130C2×D4—716no—
131C2×D4—74no—
132C2×D4—1416no—
133C2×D4—144no—
134C2×D4C2×C3×D714yescomplemented
135C22×C4—72no—
136C22×C4C2×C3×D711yescharacteristic
137C42—74no—
138C2×C7—44no—
139C2×C7C3×#12714yescomplemented
140C2×C7C22×C3×D412yes—
141C2×C7C22×C3×D411yescharacteristic
142D7—44no—
143C22×C3—28no—
144C22×C3—24no—
145C22×C3—48no—
146C22×C3—148no—
147C22×C3—144no—
148C22×C3—288no—
149C22×C3D4×D714yes—
150C22×C3C2×C22⋊D712yes—
151C22×C3C23×D711yescharacteristic
152C3×C4—78no—
153C3×C4C2×C22⋊D714yescomplemented
154C23—28no—
155C23—24no—
156C23—74no—
157C23—148no—
158C23—144no—
159C23C22×C3×D714yes—
160C23C22×C3×D711yescharacteristic
161D4—28no—
162D4—1432no—
163D4—148no—
164C2×C4—78no—
165C2×C4—74no—
166C2×C4C3×C22⋊D714yescomplemented
167C2×C4C22×C3×D712yes—
168C7C2×C3×#12711yescharacteristic, complemented, Sylow
169C2×C3—44no—
170C2×C3—284no—
171C2×C3#650314yesdirect product factor
172C2×C3C2×D4×D712yes—
173C2×C3C22×C22⋊D711yescharacteristic
174C22—28no—
175C22—24no—
176C22—48no—
177C22—148no—
178C22—144no—
179C22—288no—
180C22C3×D4×D714yes—
181C22C2×C3×C22⋊D712yes—
182C22C23×C3×D711yescharacteristic
183C4—78no—
184C4C2×C3×C22⋊D714yescomplemented
185C3C2×#650311yescharacteristic, direct product factor, Sylow
186C2—44no—
187C2—284no—
188C2C3×#650314yesdirect product factor
189C2C2×C3×D4×D712yes—
190C2C22×C3×C22⋊D711yescharacteristic
subgroupG/H
centerC23×C3C22×D7
commutator, cosocleC22×C7C24×C3
FrattiniC22C23×C3×D7
FittingC22×C3×C7×D4C2
socleC23×C3×C7C23
hypercenterC22×C3×D4D7
hypocenterC7C2×C3×#127
LCSG ▷ C22×C7 ▷ C7
UCSC1 ◁ C23×C3 ◁ C22×C3×D4

Complements

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

NH ≅ G/Ntype
1. C22×C3×C7×D4C2semidirect
2. C22×C3×C22⋊D7C2semidirect
3. C22×C3×D28C2semidirect
4. C3×#6503C2direct (N × H)
5. C2×C3×C4×Dic7C2semidirect
6. C2×#6503C3direct (N × H)
10. C2×C3×C7×D4C22semidirect
13. C2×C3×C22⋊D7C22semidirect
16. C2×C3×D28C22semidirect
17. C3×C4×Dic7C22semidirect
18. C22×C7×D4C2×C3semidirect
19. C22×C22⋊D7C2×C3semidirect
20. C22×D28C2×C3semidirect
21. #6503C2×C3direct (N × H)
22. C2×C4×Dic7C2×C3semidirect
32. C2×C3×C4×C7D4semidirect
35. C2×C3×Dic7D4semidirect
41. C2×C7×D4C22×C3semidirect
44. C2×C22⋊D7C22×C3semidirect
47. C2×D28C22×C3semidirect
48. C4×Dic7C22×C3semidirect
51. C22×C3×D4D7semidirect
63. C3×C4×C7C2×D4semidirect
64. C3×Dic7C2×D4semidirect
74. C2×C4×C7C3×D4semidirect
77. C2×Dic7C3×D4semidirect
87. C2×C3×D4C2×D7semidirect
92. C2×C3×C7#127semidirect
98. C22×D4C3×D7semidirect
110. C4×C7C2×C3×D4semidirect
111. Dic7C2×C3×D4semidirect
124. C2×C3×C4C22⋊D7semidirect
126. C3×C7C2×#127semidirect
134. C2×D4C2×C3×D7semidirect
139. C2×C7C3×#127semidirect
153. C3×C4C2×C22⋊D7semidirect
166. C2×C4C3×C22⋊D7semidirect
168. C7C2×C3×#127semidirect
171. C2×C3#6503direct (N × H)
184. C4C2×C3×C22⋊D7semidirect
185. C3C2×#6503direct (N × H)
188. C2C3×#6503direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×C7×D42
C22×C3×C22⋊D72
C22×C3×D282
C3×#65032
C2×C3×C4×Dic72
C2×#65033
C2×C3×#1277