Finite Group Databasebeta

← C67×#1462 #22555893

Subgroups — C67×#1462 #22555893

Order 8576.

440 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 104 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×C22⋊C4×C67C211yesmaximal, characteristic
2#407×C67C212yesmaximal, complemented
3#323×C67C214yesmaximal, complemented
4C25×C67C411yescharacteristic
5C23×C4×C67C411yescharacteristic
6C23×C4×C67C2211yescharacteristic
7C2×M16×C67—22no—
8C2×C22⋊C4×C67—22no—
9C2×C22⋊C4×C67—22no—
10C2×C22⋊C4×C67C412yes—
11C2×C22⋊C4×C67C2212yes—
12C22⋊C8×C67—22no—
13C22⋊C8×C67C2214yescomplemented
14C24×C67—21no—
15C24×C67—21no—
16C24×C67—24no—
17C24×C67—42no—
18C24×C67—42no—
19C24×C67C2×C411yescharacteristic
20C24×C67C2×C412yes—
21C22×C4×C67—21no—
22C22×C4×C67—21no—
23C22×C4×C67—42no—
24C22×C4×C67—42no—
25C22×C4×C67C2×C412yes—
26C22×C4×C67C2×C411yescharacteristic
27C22×C4×C67D412yes—
28C22×C4×C67D412yes—
29C22×C4×C67C2311yescharacteristic
30M16×C67—44no—
31C2×C8×C67—24no—
32C22⋊C4×C67—24no—
33C22⋊C4×C67—24no—
34C23×C67—22no—
35C23×C67—22no—
36C23×C67—22no—
37C23×C67—24no—
38C23×C67—24no—
39C23×C67—22no—
40C23×C67—22no—
41C23×C67—42no—
42C23×C67—42no—
43C23×C67—416no—
44C23×C67—42no—
45C23×C67—42no—
46C23×C67—42no—
47C23×C67—42no—
48C23×C67C22⋊C411yescharacteristic
49C23×C67C22⋊C411yescharacteristic
50C23×C67M1614yescomplemented
51C23×C67C22×C411yescharacteristic
52C2×C4×C67—24no—
53C2×C4×C67—22no—
54C2×C4×C67—41no—
55C2×C4×C67—42no—
56C2×C4×C67—41no—
57C2×C4×C67—42no—
58C2×C4×C67—42no—
59C2×C4×C67—42no—
60C2×C4×C67C22⋊C412yes—
61C2×C4×C67C2×D412yes—
62C8×C67—44no—
63C22×C67—28no—
64C22×C67—21no—
65C22×C67—21no—
66C22×C67—21no—
67C22×C67—21no—
68C22×C67—21no—
69C22×C67—21no—
70C22×C67—24no—
71C22×C67—44no—
72C22×C67—48no—
73C22×C67—48no—
74C22×C67—44no—
75C22×C67—42no—
76C22×C67—42no—
77C22×C67#9912yes—
78C22×C67#10012yes—
79C22×C67C2×C22⋊C411yescharacteristic
80C22×C67C2×M1612yes—
81C4×C67—22no—
82C4×C67—22no—
83C4×C67—42no—
84C2×C67—22no—
85C2×C67—22no—
86C2×C67—22no—
87C2×C67—44no—
88C2×C67#40711yescharacteristic
89C2×C67C2×#9911yescharacteristic
90C2×C67C2×#10011yescharacteristic
91#1462C6711yesmaximal, characteristic, direct product factor, Sylow
92C67#146211yescharacteristic, direct product factor, Sylow
93C22×C22⋊C4C2×C6711yescharacteristic
94#407C2×C6712yescomplemented
95#323C2×C6714yescomplemented
96C25C4×C6711yescharacteristic
97C23×C4C4×C6711yescharacteristic
98C23×C4C22×C6711yescharacteristic
99C2×M16—22no—
100C2×C22⋊C4—22no—
101C2×C22⋊C4—22no—
102C2×C22⋊C4C4×C6712yes—
103C2×C22⋊C4C22×C6712yes—
104C22⋊C8—22no—
105C22⋊C8C22×C6714yescomplemented
106C24—21no—
107C24—21no—
108C24—24no—
109C24—42no—
110C24—42no—
111C24C2×C4×C6711yescharacteristic
112C24C2×C4×C6712yes—
113C22×C4—21no—
114C22×C4—21no—
115C22×C4—42no—
116C22×C4—42no—
117C22×C4C2×C4×C6712yes—
118C22×C4C2×C4×C6711yescharacteristic
119C22×C4D4×C6712yes—
120C22×C4D4×C6712yes—
121C22×C4C23×C6711yescharacteristic
122M16—44no—
123C2×C8—24no—
124C22⋊C4—24no—
125C22⋊C4—24no—
126C23—22no—
127C23—22no—
128C23—22no—
129C23—24no—
130C23—24no—
131C23—22no—
132C23—22no—
133C23—42no—
134C23—42no—
135C23—416no—
136C23—42no—
137C23—42no—
138C23—42no—
139C23—42no—
140C23C22⋊C4×C6711yescharacteristic
141C23C22⋊C4×C6711yescharacteristic
142C23M16×C6714yescomplemented
143C23C22×C4×C6711yescharacteristic
144C2×C4—24no—
145C2×C4—22no—
146C2×C4—41no—
147C2×C4—42no—
148C2×C4—41no—
149C2×C4—42no—
150C2×C4—42no—
151C2×C4—42no—
152C2×C4C22⋊C4×C6712yes—
153C2×C4C2×D4×C6712yes—
154C8—44no—
155C22—28no—
156C22—21no—
157C22—21no—
158C22—21no—
159C22—21no—
160C22—21no—
161C22—21no—
162C22—24no—
163C22—44no—
164C22—48no—
165C22—44no—
166C22—48no—
167C22—42no—
168C22—42no—
169C22#99×C6712yes—
170C22#100×C6712yes—
171C22C2×C22⋊C4×C6711yescharacteristic
172C22C2×M16×C6712yes—
173C4—22no—
174C4—22no—
175C4—42no—
176C2—22no—
177C2—22no—
178C2—22no—
179C2—44no—
180C2#407×C6711yescharacteristic
181C2C2×#99×C6711yescharacteristic
182C2C2×#100×C6711yescharacteristic
subgroupG/H
center, socleC22×C67C2×C22⋊C4
commutatorC23C22×C4×C67
Frattini, cosocleC22×C4C23×C67
prime radicalC25×C67C4
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C67 ◁ C23×C4×C67 ◁ G

Complements

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

NH ≅ G/Ntype
2. #407×C67C2semidirect
3. #323×C67C2semidirect
13. C22⋊C8×C67C22semidirect
50. C23×C67M16semidirect
91. #1462C67direct (N × H)
92. C67#1462direct (N × H)
94. #407C2×C67semidirect
95. #323C2×C67semidirect
105. C22⋊C8C22×C67semidirect
142. C23M16×C67semidirect

Maximal subgroups

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

subgroupindex
C22×C22⋊C4×C672
#407×C672
#323×C672
#146267